FLAP: Unveiling (ββ) 0ν -Decay

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1 FLAP: Unveiling (ββ) 0ν -Decay A. Meroni SISSA, Trieste October 18, (ββ) 0ν -decay- Introduction Neutrinos are the rst evidence of physics beyond the standard model. oscillation data of the last ten years, since SK discoveries in 1998, have reveal that they have non zero mass which implies that they mix. At present we know that we have compelling evidence of three light active states with masses not exceeding 1 ev. In the simplest framework the three avour states (ν e, ν µ, ν τ ) are coherent quantum superposition of 3 light mass states (ν 1, ν 2, ν 3 ) and their mixing is described in good approximation with a unitary mixing matrix parametrized in terms of three angles (θ 12, θ 13, θ 23 ) and one dirac phase δ plus two eventual Majorana phases. The standard parametrization of the PMNS mixing matrix U P MNS is given by: U P MNS = R 23 c 13 0 s 13 e iδ s 13 e iδ 0 c 13 R 12 diag(1, e iα/2, e iβ/2 ) (1) The amplitude and the frequencies of avour oscillation phenomena are governed by the three angles θ ij and by two swuared mass dierences. The most recent values of these parameters is summarized in Table 1. However there are still many open questions in neutrino physics: we do Parameter Best t 3σ range m 2 [10 5 ev 2 ] (NH or IH) m 2 A [10 3 ev 2 ] (NH) (IH) sin 2 θ 12 /10 1 (NH or IH) sin 2 θ 13 /10 2 (NH) (IH) sin 2 θ 23 /10 1 (NH) (IH) Table 1: Global t analysis including the Daya Bay and RENO results Fogli et al. ArXiV: not know neither the absolute scale of the neutrino masses nor the kind of hierarchy (normal or inverted), the values of the CP violating phases. More interestingly we do not know yet the very nature of massive neutrinos, wheter they are Dirac particle or Majorana particles. This issue is essential to determine the underlying symmetries of the electroweak interaction. If neutrinos has masses under 100 ev, the most sensitive experiments to the existence of Majorana neutrinos are those investigating (ββ) 0ν -Decay. If double beta beta is (A, Z) (A, Z + 2) + 2e + 2 ν e (2) the process we would like to consider is without the emission of antineutrinos i.e. the following one: 1

2 (A, Z) (A, Z + 2) + 2e. (3) This process in the SM is forbidden. This process is generated at second order in perturbation expansion of SM weak interaction. Indeed, in this process the total lepton number is not conserved (is violated by two units i.e. L = ±2) and then is possible only if neutrinos are neutral Majorana particle that correspond to their own antiparticle 1. Is common to study (ββ) 0ν -decay amplitude as function of the eective Majorana mass parameter which is dened like this: light <m> = (U ej ) 2 m j (4) Before discussing the general framework under wich one can derive this formula, let us dicuss the genric features of a Majorana eld. 2 Properties of Majorana fermion elds The Majorana eld χ(x) is dened by the following pure algebraic condition: j C χ t (x) = ξχ(x) (5) where ξ = ±1 and C is the charge conjugation matrix. We would like to show now that no conserved lepton charges can allow us to distinguish a neutrino described by a Majorana eld from its anti-neutrino. 2.1 Majorana mass term The Majorana mass term is constructed as follow: L M = 1 2 [(ν αl) c M αβ ν βl + ν αl M αβ (ν αl) c ] (6) where M is in general a n n complex matrix. First we want to notice the simmetricity of the mass matrix. A complex symmetric matrix can always expressed in the form M = (U ) t mu where U is a unitary matrix 2 If we insert this into the mass term we nd L M = 1 2 χ jmχ j χ j = U jβ ν βl + U t jβ (ν αl) c = C χ t j (7) Evidently from equation (6) one can see that there is no global gauge transformation under which the mass term in its most general form could be invariant. This implies that no global gauge lepton charges could be conserved. For instance, the electric charge is not conserved, hence it is zero. (In other words Vector currents are zero for Majorana elds). So, neutrino are truly neutral particles with spin 1/2. In addition also Lepton number is violated (by two units). Hence no possible lepton charge could allow us to distinguish neutrinos from anti-neutrinos.. (See further) 1 Experimentally (ββ) 0ν -decay is dierent from normal double beta decay because looking to the electron spectrum one can recognize the nature of the neutrino: if the spectrum is continuous then ν e ν e while if it is a line, ν would be a Majorana particle. 2 Briey one can say that MM has positive eigenstates m 2 and it is hermitian. So one can always diagonalize it with a unitary matrix V s.t. MM = V m 2 V and further one can write M = V mu with U = m 1 V M s. t. UU = m 1 V MM V m 1 = 1. Details in note 8 [?], For a complex matrix is true that M = V mu with U and V unitary. It is valid that MM = V m 2 V, but since M = M t = (U ) t mv t hence (U ) t mu t = V m 2 V This implies that [U t V, m] = 0 so U t V is diagonal and unitary. Details in [?] 2

3 The mass matrix is symmetric. One can show this using the fact that a minus sign appears when two fermion elds operators interchange and that C = C t : ψ αl M αβ C ψ t βl = ( ψ αl M αβ C ψ t βl )t = ψ βl C t M t αβ ( ψ αl ) t = ψ βl M t αβ C( ψ αl ) t = ψ αl M t βα C ψ t βl that veries M αβ = M t αβ No global charges are conserved Using the language of LH and RH eld we can clarify why the majorana mass term in equation 7 is violating U(1) global gauge transformation. Since we need that the weak charged currents and the charge lepton mass term would not change let us say that if the phase is set equal to 1: If the elds χ kl and l kl were transformed as: χ R (x) = (χ(x) c ) R = Cχ(x) L t = (χ(x)l ) c (8) then the eld χ kr is transformed as: χ kl = eiλ χ kl l kl = eiλ l kl (9) χ R (x) = (e iλ χ(x) L) c χ R (x) = e iλ (χ(x) L) c χ R (x) = e iλ χ R (x) (10) Hence the neutrino mass term in 7 is no more invariant. This means that the majorana condition could not absorb phases. Neutrino is a neutral particle: Dirac equation We can show that this eld describes a true neutral particle because applaying charge conjugation on the Dirac equation and implementing it with the Majorana condition we get: [iγ µ ( µ ie χ A µ ) m]ξχ(x) [iγ µ ( µ +ie χ A µ ) m]c χ t (x) = [iγ µ ( µ +ie χ A µ ) m]ξχ(x) = 0 (11) so χ(x) satises both Dirac equations with negative and positive charge hence e χ = 0. Majorana condition implies zero charge hence U(1) global charges are not conserved. In this description we discover that a Majorana particle is its own antiparticle. Vector and Tensor currents are zero Moreover we know that any vector current is zero as well as tensor currents. Writing (7) as χ = ξχ t C (remember C = C t ) we have: χγ µ χ = ξχ t C γ µ (ξ C χ t (x)) = χ t α(γ µ ) t αβ χt β = χγµ χ = 0 χσ µν χ = ξξ χ t (C σ µν C) χ t = χ t ( σ µν ) t χ t = χσ µν χ = 0 (12) These results are important because the former represents the charge vector current while the latter represents the magnetic dipole moment current together to the electric dipole moment (with γ 5 ). 3 3 These current are 5 dim operators in the Lagrangian L dim=5 χσ µν χf µν + χσ µν γ 5 χf µν. The former is classically the magnetic dipole moment, U = µ B, while the latter is the electric dipole moment D E. 3

4 3 Multiple mechanisms Neutrinoless double beta decay is allowed by a wide range of models, from the standard mechanism of light Majorana neutrino exchange [?] to those such as Left-Right Symmetry [?] or R-parity violating Supersymmetry (SUSY) [?]. These mechanisms might trigger (ββ) 0ν -decay individually or together. In Left-Right Symmetric models, for example, there is an additional contribution from the exchange of heavy right-handed neutrinos. In R-parity violating SUSY, on the other hand, (ββ) 0ν -decay can be mediated by heavy particles such as neutralino or gluino. These dierent mechanisms can interfere only if the lepton current structure coincides and hence it can be factorized. If, on the contrary, these are not-interfering mechanisms, i.e. the leptonic current structure is dierent, then the interference term is suppressed by a factor depending on the considered nucleus. [?]. The (ββ) 0ν -decay half-life of a certain nucleus can therefore be written as function of some lepton number violating (LNV) parameters, each of them connected with a dierent mechanism i: [T 0ν 1/2 ] 1 = G 0ν (E, Z) i η i M 0ν i 2 (13) where η i, G 0ν (E, Z) and Mi 0ν are respectively the LNV parameter, the phase space factor and the nuclear matrix element (NME) related to a single mechanism for the given nucleus 4. We outline below the LNV mechanisms we focus on. 3.1 Light Majorana Neutrino exchange mechanism The standard scenario to allow (ββ) 0ν -decay is the exchange of a light Majorana left-handed neutrino, χ L, via V A weak interactions. The basic diagram corresponding to this mechanism is drawn in Fig. 1. V A W e χ L W e V A Figure 1: Feynman diagrams for the (ββ) 0ν -decay, generated by the light Majorana neutrino exchange. The following term in the S-matrix (or scattering matrix) gives the contribution to the matrix element of the process in second order of perturbation theory in G F : ( S (2) = T = ( i)2 2! e i HI w.i. = ( i)2 4 2! (x)dx e i H S.I. dx 4 1dx 4 2N [ L w.i. I ( GF 2 ) 2 I (x)dx ) (14) (x 1 )L w.i. I (x 2 ) ] ( T jα(x h 1 )jβ h (x 2)e i ) HI S.I. (x)dx (15) dx 4 1dx 4 2N [(ē L γ α ν el )(ē L γ β ν el )] T 4 Notice that in this case the formula is valid for interfering mechanisms only. ( j h α(x 1 )j h β (x 2)e i H S.I. I (x)dx ) (16) 4

5 We can recast the weak lepton product part using ν T k C = ν c k = ξ kν k and C 1 γ α C = γ T α. Since neutrino mixing take place one can write: ν el = light j U ej ν jl (17) where U ej, j = 1, 2, 3..., (dierent from the matrix U ek ) are the elements of the rst row of the Pontecorvo, Maki, Nakagawa and Sakata (PMNS) neutrino mixing matrix and m j is the light neutrino mass, m j < 1 ev, j = 1, 2, 3... The latter satisfy the unitarity condition: We have j=1 U ej 2 = 1. (18) U ej = U ej e iα j/2, (19) where α j, with j = 1, 2, 3, are three phases. Only the phase dierences (α j α k ) can play a physical role.the PMNS matrix U is not assumed to be CP conserving and at least two of the elements U ej contain physical CP violating phases [21, 22] (see also, e.g., [5]). In the case of 3 light neutrinos and the standard parametrization of U [5], the elements U e2 and U e3 contain the two physical CP violating Majorana phases [21] and U e3 contains the Dirac phase as well. k (ē L γ α ν el )(ē L γ β ν el ) = k ξ k (U ek ) 2 ēγ α P L ν k ν k }{{} P Lγ β Cē T = k (U ek ) 2 ēγ α P L ν k ν T k C CP L γ T β C Cē T (20) ξ k (U ek ) 2 ēγ α P L S(x 1 x 2 )P L γ β Cē T (21) where S(x 1 x 2 ) is the propagator: e iq(x 1 x 2 ) (/q + m k )dq e iq(x 1 x 2 ) P L S(x 1 x 2 )P L = P L ( i) q 2 m 2 P L = P L ( i)(m k ) k q 2 m 2 P L (22) k One can dene an eective Majorana mass <m> corresponding to the contribution from standard (V A) charged current (CC) weak interaction as follows (all m j 0): light <m> = m 1 Ue1 2 + m 2 Ue2 2 + m 3 Ue3 2 = (U ej ) 2 m j = = m 1 U e1 2 + m 2 U e2 2 e iα + m 3 U e3 2 e i β = m (1) ee + m (2) ee e iα + m (3) ee e i β j (23) where β = β 2δ. A pictorial representaion is depicted in Fig. 2. In the case of the light Majorana neutrino exchange mechanism of (ββ) 0ν -decay, the LNV parameter is given by: η ν = <m> m e. (24) One can show (appendix of [?]) that taking some approximations, that usually are made in calculating (ββ) 0ν -decay amplitudes, we can express the hadron current operator as 5

6 Figure 2: Pictorial representation A αβ J α (x 1 )J β (x 2 ) = J β (x 2 )J α (x 1 ) (25) hence the hadron part is a symmetric operator. Then in the matrix element the hadron part can be written as: ēγ α P L γ β Cē T A αβ = ēγ α γ β P R Cē T A αβ = ē(g αβ (γ αγ β γ β γ α ))P R Cē T A αβ (26) Hence is clear that the contribution to (ββ) 0ν -decay is given by: <m>[ē(1 + γ 5 )e c ]A αβ (27) where < m >is the eective Majorana neutrino mass. The leptonic structure is fundamental in roder to understand possible interfering mechanisms that can contribute in the decay. Let us make some further remark. First of all is this process is observed the value of <m>should be universal for all the nuclei if neutrinos have masses smaller than few MeV and this is the case. Further the parameter < m > is substantially dierent from the masses of the neutrinos that could be determined from other experiments such as Tritium Beta decay experiment. In fact < m > may be suppressed by destructive interference between the terms in the expression of <m> that is cancellations can occur from the complexities of the mixing matrix elements in the case of CP nonconservation. In case large cancellation appear of course one would be lead to conclude that 1 loop contributions must be considered like contributions coming from the exchange of the Higgs boson. The observation of (ββ) 0ν decay would mean that the amplitude transforming two d quarks into two u quarks and producing two electrons is non vanishing. This fact alone would prove that neutrinos has Majorana masses and then they are Majorana particles i.e. they are their own antiparticles. This was pointed out by Schechter and Valle in [46] in 1982 is the so called blackbox theorem that is depicted in Fig. 3. The black box may contain any mechanism whatsoever for generating (ββ) 0ν decay. Using four vertices of the the non abelian gauge symmetry on which the SM is based and connecting the lines together as in Fig. 3 shows that we develop an amplitude which gives a nonzero Majorana mass for the electron neutrino. This result more importantely is connected with the possibility that other mechanisms could be dominant in the decay amplitude. For example (ββ) 0ν decay could be triggered by heavy neutrino exchange, where the heavy neutrinos are responsible for light neutrino masses via the seesaw mechanism. Another example could be would be R-parity violating SUSY particles generating (ββ) 0ν decay, where via loops the same particles generate light neutrino masses. If instead turns out that (ββ) 0ν decay is a process triggered by the exchange of some supersymmetric Majorana particles, then one can use the Schechter-Valle theorem to argue that a nonzero Majorana mass for the 6

7 electron neutrino is generated via loops by the same particles and the light active states are then Majorana particles. From another poit of view it is also important to stress that if any lepton avour violating process, besides (ββ) 0ν decay, is observed this also would imply the existence of a Majorana mass for the neutrinos. Figure 3: Black Box diagram <m> in the case of normal or inverted ordering Depending on the sign of m 2 A and the value of the lightest neutrino mass i.e. the absolute neutrino mass scale, m min, the neutrino mass spectrum is dened in the following. It is worth expressing as well the three neutrino masses in terms of the m 2 and m 2 A, measured in neutrino oscillation experiments and the absolute neutrino mass scale determined by m min. For the element of the PMNS matrix U ej 2, j = 1, 2, 3, the following relation hold: U e1 2 = cos 2 θ (1 sin 2 θ 13 ), U e2 2 = sin 2 θ (1 sin 2 θ 13 ), U e3 2 = sin 2 θ 13, Normal Hierarchical(NH): m 1 m 2 < m 3, and then m 2 = (m m2 ) 1/2, m 3 = (m m2 A )1/2, being m 1 m min. The <m> results: <m> = m min cos 2 θ (1 sin 2 θ 13 ) + m 2 min + m2 sin2 θ (1 sin 2 θ 13 )e iα m 2 min + m2 A sin2 θ 13 e iα 31 (28) m 2 sin2 θ cos 2 θ 13 e iα 21 + m 2 A sin2 θ 13 e iα 31 (29) Inverted Hierarchical(IH): m 3 m 1 < m 2, m 1 = (m m2 A m2 ) 1/2, m 2 = m 2 21, being m 3 m min.the <m> results: <m> = m 2 min + m2 A m2 cos2 θ (1 sin 2 θ 13 ) + m 2 min + m2 A sin2 θ (1 sin 2 θ 13 )e iα m min sin 2 θ 13 e iα 31 m 2 min + m2 A cos 2 θ + sin 2 θ e iα 21 (1 sin 2 θ 13 ) (30) m 2 A cos 2 θ + sin 2 θ e iα 21 cos 2 θ 13 (31) In this equation we have neglected the solar square mass dierence because, from the existing data, since m 2 / m 2 A = ± ( at 3 σ) then one can say that: 7

8 1 QD <m> [ev] IH NH 1e m MIN [ev] Figure 4: arxiv: m 2 m 2 min m min sin 2 θ 13 m 2 m2 min + m2 A cos 2θ (32) A Actually the terms like m min sin 2 θ 13 can always be neglected if sin 2 θ 13 cos 2θ. Quasi-Degenerate (QD): m 1 m 2 m 3 m 0 and m 2 j m2 A, m ev <m> = m 0 (cos 2 θ + sin 2 θ e iα 21 )(1 sin 2 θ 13 ) + sin 2 θ 13 e iα 31 m 0 (cos 2 θ + sin 2 θ e iα 21 )(1 sin 2 θ 13 ) (33) Thus the <m> depends on the lightest neutrino mass, the two Majorana phases present in the PMNS mixing and on the type of neutrino mass spectrum. Using the data on the neutrino oscillation parameters it is possible to show (see, e.g., [5]) that: in the case of normal hierarchical spectrum one has <m> < ev, if the spectrum is with inverted hierarchy, 0.01 ev < <m> < 0.05 ev, a larger value of <m> is possible if the light neutrino mass spectrum is with partial hierarchy or of quasi-degenerate type. In the latter case <m> can be close to the existing upper limits. The fact that max <m> in the case of NH spectrum is considerably smaller than min <m> for the IH and QD spectrum opens the possibility to obtain information about the neutrino mass pattern from a measurement of <m> 0. More specically a positive result in the future generation of (ββ) 0ν -decay experiments with <m> > 0.01 ev would imply that the NH spectrum is strongly disfavored (if not excluded). If the future (ββ) 0ν -decay experiments show that <m> < 0.01 ev both the IH and QD spectrum will be ruled out for massive Majorana neutrinos, If in addition, it is establish from oscillation experiment that m 2 A < 0 then one would be let to conclude that either the massive neutrino are Dirac fermions, or the neutrinos are Majorana particles but there are additional contributions to (ββ) 0ν -decay amplitude which interfere destructively. Let us resume now what we discovered up to now. The studies of (ββ) 0ν -decay and a measurement of a nonzero value of <m> few 10 2 ev : 8

9 can establish the Majorana nature of massive neutrinos. the (ββ) 0ν -decay experiments are presently the only feasible experiments capable of doing that; can give informations on the type of neutrino mass spectrum. More specically, a measured value of <m> few 10 2 ev can provide, in particular, unique constraints on, or even can allow one to determine, the type of neutrino mass spectrum if neutrinos ν i are Majorana particles. The neutrino mass pattern can be with normal ordering: m 1 < m 2 < m 3 that correspond to m 2 A m2 31 > 0 being the neutrino mass square dierence responsible for the the (dominant) atmospheric neutrino oscillations inverted ordering: m 3 < m 1 < m 2 that correspond to m 2 A m2 32 < 0. can provide also unique information on the absolute scale of neutrino masses or on the lightest neutrino mass; with additional information from other sources (Tritium decay experiments or cosmological and astrophysical data considerations) on the absolute neutrino mass scale, the (ββ) 0ν - decay experiments can provide unique information on the Majorana CP-violation phases α 1 and α Right Heavy Neutrino exchange mechanism We assume that the neutrino mass spectrum includes, in addition to the three light Majorana neutrinos, heavy Majorana states N k with masses M k much larger than the typical energy scale of the (ββ) 0ν -decay, M k 100 MeV; we will consider the case of M k > 10 GeV. Such a possibility arises if the weak interaction Lagrangian includes right-handed (RH) sterile neutrino elds which couple to the LH avour neutrino elds via the neutrino Yukawa coupling and possess a Majorana mass term. The heavy Majorana neutrinos N k can mediate the (ββ) 0ν -decay similar to the light Majorana neutrinos via the V A charged current weak interaction. The dierence between the two mechanisms is that, unlike in the light Majorana neutrino exchange which leads to a long range inter-nucleon interactions, in the case of M k > 10 GeV of interest the momentum dependence of the heavy Majorana neutrino propagators can be neglected (i.e., the N k propagators can be contracted to points) and, as a consequence, the corresponding eective nucleon transition operators are local. The LNV parameter in the case when the (ββ) 0ν -decay is generated by the (V A) CC weak interaction due to the exchange of N k can be written as: η L N = heavy k Uek 2 m p M k, (34) where m p is the proton mass and U ek is the element of the neutrino mixing matrix through which N k couples to the electron in the weak charged lepton current. If the weak interaction Lagrangian contains also (V + A) (i.e., right-handed (RH)) charged currents coupled to a RH charged weak boson W R, as, L L+R = g 2 2 [(ēγ α(1 γ 5 )ν el )W µ + (ēγ α (1 + γ 5 )ν er )W µr ] (35) where ν er = k V ekn kr, C N k T = ξn k. Here V ek are the elements of a mixing matrix by which N k couple to the electron in the (V +A) charged lepton current, M W is the mass of the Standard Model charged weak boson, M W = 82 GeV, and MW R is the mass of W R. It follows from the existing data that [23] W R > 2.5 TeV. For instance, in the SU(2) L SU(2) R U(1) theories we can have also a contribution to the (ββ) 0ν -decay amplitude generated by the exchange of virtual N k coupled to the electron in the 9

10 hypothetical (V +A) CC part of the weak interaction Lagrangian. In this case the corresponding LNV parameter can be written as: η R N = ( MW M W R ) 4 heavy k Vek 2 m p M k. (36) Here V ek are the elements of a mixing matrix by which N k couple to the electron in the (V + A) charged lepton current, M W is the mass of the Standard Model charged weak boson, M W = 82 GeV, and M W R is the mass of W R. It follows from the existing data that [23] W R > 2.5 TeV. If CP invariance does not hold, which we will assume to be the case in what follows, U ek and V ek will contain physical CP violating phases at least for some k and thus the parameters η L and ηr will not be real. N N V A V +A W L e W R e χ jl,n kl N kr W L e W R e V A V +A Figure 5: Feynman diagrams for the (ββ) 0ν -decay, generated by the light or heavy Majorana neutrino exchange (left panel) and the heavy (RH) Majorana neutrino exchange (right panel). As can be shown, the nuclear matrix elements corresponding to the two mechanisms of (ββ) 0ν -decay with exchange of heavy Majorana neutrinos N k, described in the present subsection, are the same and are given in [18]. We will denote them by M 0ν. N Finally, it is important to note that the current factor in the (ββ) 0ν -decay amplitude describing the two nal state electrons, has dierent forms in the cases of (ββ) 0ν -decay mediated by (V A) and (V + A) CC weak interactions 5, namely, ē(1 + γ 5 )e c 2e L (e c ) R and ē(1 γ 5 )e c 2e R (e c ) L, respectively, where e c = C(ē) T, C being the charge conjugation matrix. The dierence in the chiral structure of the two currents leads to a relatively strong suppression of the interference between the terms in the (ββ) 0ν -decay amplitude involving the two dierent electron current factors (see further). 3.3 SUSY: R-parity breaking ( /R p ) The SUSY models with R-parity non-conservation include LNV couplings which can trigger the (ββ) 0ν -decay. Let us recall that the R-parity is a multiplicative quantum number dened by R = ( 1) 2S+3B+L, where S, B and L are the spin, the baryon and lepton numbers of a given particle. The ordinary (Standard Model) particles have R = +1, while their superpartners carry R = 1. The LNV couplings emerge in this class of SUSY models from the R-parity breaking part of the superpotential W /Rp = λ ijk L i L j E c k + λ ijk L iq j D c k + µ il i H 2, (37) 5 The procedure is the same dened in the light neutrino exchange section. One has in this case ēγ α P R γ β Cē T A αβ = ēp L e c A αβ 10

11 d R λ 111 e L λ 111 u L d R d λ R 111 e L ũ L g u L g d R e L ũ L u L d R ũ L λ 111 u L e L d R d R λ 111 u L e L d R g d R λ 111 u L e L Figure 6: Feynamn diagrams for (ββ) 0ν -decay due to the gluino exchange mechanism. hepph/ where L, Q stand for lepton and quark SU(2) L doublet left-handed superelds, while E c, D c for lepton and down quark singlet superelds. Here, we concentrate only on the trilinear λ - couplings 6. The (ββ) 0ν -decay can probe only the rst generation lepton number violating coupling λ 111 because only the rst generation fermions u, d, e are involved in the process. The λ -couplings of the rst family of particles and sparticles relevant for (ββ) 0ν -decay are given in terms of the elds of the LH electron, electron neutrino ν el, LH selectron ẽ L and sneutrino ν el, u L,R - and d L,R -quarks and u- and d-squarks, ũ L,R, d L,R, by: L /Rp = λ 111 [ ) ( e c ũ (ū L dl )( R d νer c R + (ē L ν el )d L R d L ) + (ū L dl )d R ( ẽ L ν el ) ] + h.c.. (38) At the quark-level there are basically two types of R/ p SUSY mechanisms of (ββ) 0ν -decay: a short-range one with exchange of heavy Majorana and scalar SUSY particles (gluinos and squarks, and/or neutralinos and selectrons) [24, 25, 26, 27, 28, 29], and a long-range mechanism involving the exchange of both heavy squarks and light Majorana neutrinos [30, 31, 32, 33, 34]. We will call the latter the squark-neutrino mechanism. In this model (ββ) 0ν -decay can be triggered essentially by the exchange of a supersymmetric particle such as a neutralino of the model or a gluino. The complete set of diagrams can be found in [?]. In this paper we concentrate on the gluino contribution to (ββ) 0ν -decay due to the λ coupling (see diagrams in Fig. 6).The Lagrangian terms corresponding to gluino interactions, L g, with fermions and their superpartners are: L g = λ αβ 2g 3 2 ( qα L g a q β L qα R g a q β R ) + h.c. (39) After some Fiertz rearrangement, one can obtain the gluino contribution to the the (ββ) 0ν -decay matrix element: [ M g = G2 F η g (J P S J P S 1 2m p 4 J µν T J T µν) + η g(j ] P S J P S ) [ē(1 + γ 5 )e c ] (40) where we dene the corresponding LNV parameter as: [ η g = 2Λ ( ) ] m 4 dr 1 +, η g = 4Λ ( ) m 2 dr, Λ = πα s(λ 111 )2 9 mũl 9 mũl G 2 F m4 dr m P m g, (41) 6 By the way the R-parity potential can also be enlarged by a term which violated baryon number but proton stability forbids the simultaneous presence of lepton and baryon number violating terms in the superpotential. 11

12 and the hadronic currents have the form: J P S = ū α (1 + γ 5 )d α J µν T = ūα σ µν (1 + γ 5 )d α. (42) Assuming the dominance of the gluino exchange in the short-range mechanism, one obtains the following expressionfor the corresponding LNV parameter: η λ = πα s 6 λ G 2 F m4 dr m p m g [ ( ) ] m 2 2 dr 1 +. (43) Here, G F is the Fermi constant, α s = g3 2/(4π), g 3 being the SU(3) c gauge coupling constant. mũl, m dr and m g are masses of the LH u-squark, RH d-squark and gluino, respectively. At the hadron level we assume dominance of the pion-exchange mode. The enhancement of the pion exchange mode with respect to the conventional two-nucleon mechanism is due to the long-range character of nuclear interaction and the details of the bosonization of the π π + + e + e vertex. The nuclear matrix element associated with the gluino exchange mechanism, M 0ν λ, was calculated in [35, 36]. The electron current factor in the term of the (ββ) 0ν -decay amplitude corresponding to the gluino exchange mechanism under discussion has the form ē(1 + γ 5 )e c 2e L (e c ) R, i.e., it coincides with that of the light (or heavy LH) Majorana neutrino exchange. Thus, when calculating the (ββ) 0ν -decay half-life, the interference between the two terms in the (ββ) 0ν -decay amplitude, corresponding to the indicated two mechanisms, will not be suppressed. Therefore these two mechanisms can interfere. The (ββ) 0ν -decay half-life of a certain nucleus i due to gluino exchange is given by: [T1/2,i 0ν ] 1 = G 0ν i (E, Z) η λ Mi,λ 0ν 2 (44) where here in afterwards we use η λ as the LNV parameter related to gluino exchange. Typical values of the NME, Mi,λ 0ν 7, can be found in [?]. 4 Analysis We illustrate the possibility to get information about the dierent LNV parameters when two or more mechanisms are operative in (ββ) 0ν -decay, analysing the following two cases. First we consider two competitive not-interfering mechanisms of (ββ) 0ν -decay: light left-handed Majorana neutrino exchange and heavy right-handed Majorana neutrino exchange. In this case the interference term arising in the (ββ) 0ν -decay half-life from the product of the contributions due to the two mechanisms in the (ββ) 0ν -decay amplitude, is strongly suppressed [40] as a consequence of the dierent chiral structure of the nal state electron current in the two amplitudes. The latter leads to a dierent phase-space factor for the interference term, which is typically by a factor of 10 smaller than the standard one (corresponding to the contribution to the (ββ) 0ν -decay half-life of each of the two mechanisms). More specically, the suppression factors for 76 Ge, 82 Se, 100 Mo and 130 T e read, respectively [40]: 0.13; 0.08; and It is particularly small for 48 Ca: In the analysis which follows we will neglect the contribution of the interference term in the (ββ) 0ν -decay half-life. The eect of taking into account the interference term on the results thus obtained, as our numerical calculations have shown, does not exceed approximately 10%. In the case of negligible interference term, the inverse value of the (ββ) 0ν -decay half-life for a given isotope (A,Z) is given by: mũl 1 T1/2,i 0ν G0ν i (E, Z) = η ν 2 (M 0ν i,ν) 2 + η R 2 (M 0ν i,n) 2, (45) 7 It has been shown [?] that dominant contribution in /R p SUSY to (ββ) 0ν -decay is realized via the pion mode of hadronization and this must be taken into account computing the NMEs. 12

13 where the index i denotes the isotope. The values of the phase space factor G 0ν i (E, Z) and of the NMEs M 0ν i,ν and M 0ν i,n for 76 Ge, 82 Se, 100 Mo and 130 Te are listed in Table 2. The parameters η ν and η R are dened in eqs. (24) and (36). In the second illustrative case we consider (ββ) 0ν -decay triggered by two active and interfering mechanisms: the light Majorana neutrino exchange and the gluino exchange. In this case, for a given nucleus, the inverse of the (ββ) 0ν -decay half-life is given by: 1 T1/2,i 0ν G0ν i (E, Z) = η ν 2 (M 0ν i,ν) 2 + η λ 2 (M 0ν i,λ) cos αm 0ν i,λm 0ν i,ν η ν η λ. (46) Here η λ is the basic parameter of the gluino exchange mechanism dened in eq. (43) and α is the relative phase of η λ and η ν. The values of NMEs of the mechanisms considered are listed in Table 2. In the illustrative examples of how one can extract information about η ν, η R, etc. we use as input hypothetical values of the (ββ) 0ν -decay half-life of 76 Ge satisfying the existing lower limits and the value claimed by the Heidelberg-Moscow collaboration [41]. The observation of (ββ) 0ν decay for dierent isotopes is crucial to extract information about the dierent mechanisms that can induce the decay. In the analysis we are going to perform we will employ the recent bound from the EXO collaboration that reports a lower limit on the half-life of the neutrinoless double-beta decay [?] T 0ν 1/2 (136 Xe) > y (90 % CL). (47) We consider as well the strong limits on the (ββ) 0ν decay half-life achieved in experiments such as NEMO3 [?] and CUORICINO [13]. In order to obtain information of the dierent LNV parameters involved in the (ββ) 0ν decay, we consider the following lower bounds: T 0ν 1/2 (76 Ge) y[?], T 0ν 1/2 (76 Ge) = y[?] T1/2 0ν Se) y[?], T1/2 0ν Mo) y[?], T 0ν 1/2 (130 Te) y[13] (48) In the analysis which follow we will present numerical results rst for g A = 1.25 and using the NMEs calculated with the large size single particle basis (large basis) and the Charge Dependent Bonn (CD-Bonn) potential. Later results for g A = 1.0 and NMEs calculated with the Argonne potential will also be reported. 5 Experiment bounds and phenomenology 5.1 (ββ) 0ν -decay Experiments Neutrino oscillations are not sensitive to the absolute scale of neutrino masses. Information on the absolute neutrino mass scale can be derived in 3 H β-decay experiments and from cosmological and astrophysical data. The most stringent upper bounds on the ν e mass were obtained in the Troitzk and Mainz experiments: <m> < 2.3 ev at 95% C.L. (49) The most stringent upper limits on <m> were set by the IGEX [9], CUORICINO [13] and NEMO3 [12] experiments with 76 Ge, 130 Te and 100 Mo, respectively 8. As well we have to cite the Hidelberg-Moscow experiment. 8 The NEMO3 collaboration has searched for (ββ) 0ν -decay of 82 Se and other isotopes as well. 13

14 The IGEX collaboration has obtained for the half-life of 76 Ge, T 0ν 1/2 > yr (90% C.L.), from which the limit <m> < ( ) ev was derived [9]. The NEMO3 and CUORICINO experiments, designed to reach a sensitivity to < m > ( ) ev, set the limits: <m> < ( ) ev [12] and <m> < ( ) ev [13] (90% C.L.), where estimated uncertainties in the NME are accounted for. The two upper limits were derived from the experimental lower limits on the half-lives of 100 Mo and 130 Te, T1/2 0ν > yr (90%C.L.) [12] and T1/2 0ν > yr (90%C.L.) [13]. The best lower limit on the half-life of 76 Ge, T1/2 0ν > yr (90% C.L.), was found in the Heidelberg-Moscow 76 Ge experiment [8]. A positive (ββ) 0ν -decay signal at > 3σ, corresponding to T1/2 0ν = ( ) 1025 yr (99.73% C.L.) and implying <m> = ( ) ev, is claimed to have been observed in [10], while a later analysis reports evidence for (ββ) 0ν -decay at 6σ with <m> = 0.32 ± 0.03 ev [11]. The KATRIN experiment is planned to reach a sensitivity of <m> 0.20 ev, i.e. it will probe the region of the QD spectrum. Most importantly, a large number of projects aim at a sensitivity to <m> ( ) ev [14]: CUORE ( 130 Te), GERDA ( 76 Ge), SuperNEMO, EXO ( 136 Xe), MAJORANA ( 76 Ge), MOON ( 100 Mo), COBRA ( 116 Cd), XMASS ( 136 Xe), CANDLES ( 48 Ca), KamLAND- Zen ( 136 Xe), SNO+ ( 150 Nd), etc. These experiments, in particular, will test the positive result claimed in [10]. 14

15 Table 2: The phase-space factor G 0ν (E 0, Z) and the nuclear matrix elements M 0ν ν (light Majorana neutrino exchange mechanism), M 0ν N (heavy Majorana neutrino exchange mechanism), M 0ν λ (mechanism of gluino exchange dominance in SUSY with trilinear R-parity breaking term) and M 0ν q (squark-neutrino mechanism) for the (ββ) 0ν -decays of 76 Ge, 100 Se, 100 Mo and 130 T e. The nuclear matrix elements were obtained within the Selfconsistent Renormalized Quasiparticle Random Phase Approximation (SRQRPA). See text for details. Nuclear G 0ν (E 0, Z) M 0ν ν M 0ν N M 0ν λ 0ν M q transition [y 1 ] g A = g A = g A = g A = NN pot. m.s Ge 76 Se Argonne intm large CD-Bonn intm large Se 82 Kr Argonne intm large CD-Bonn intm large Mo 100 Ru Argonne intm large CD-Bonn intm large T e 130 Xe Argonne intm large CD-Bonn intm large

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