Searching for the Dirac Nature of Neutrinos: Combining Neutrinoless Double Beta Decay and Neutrino Mass Measurements

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Nuclear Physics B (Proc. Suppl.) 246–247 (2014) 150–160 www.elsevier.com/locate/npbps

Searching for the Dirac Nature of Neutrinos: Combining Neutrinoless Double Beta Decay and Neutrino Mass Measurements S. Pascolia , C. F. Wongb a IPPP,

Department of Physics, Durham University, Durham, DH1 3LE, United Kingdom of Physics, The Chinese University of Hong Kong, N.T., Hong Kong

b Department

Abstract We studied the neutrinoless double beta decay process to tackle the issue about the nature of neutrino. Establishing the nature of neutrinos, whether they are Dirac or Majorana particles is one of the fundamental questions we need to answer in particle physics, and is related to the conservation of lepton number. Neutrinoless double beta decay ((ββ)0ν ) is the tool of choice for testing the Majorana nature of neutrinos. However, up to now, this process has not been observed, but a wide experimental effort is taking place worldwide and soon new results will become available. Different mechanisms can induce (ββ)0ν -decay and might interfere with each other, potentially leading to suppressed contributions to the decay rate. This possibility would become of great interest if upcoming neutrino mass measurements from KATRIN and cosmological observations found that mν > 0.2 eV but no positive signal was observed in (ββ)0ν -decay experiments. We focus on the possible interference between light Majorana neutrino exchange with other mechanisms, such as heavy sterile neutrinos and R-parity violating supersymmetric models. We show that in some cases the use of different nuclei would allow to disentangle the different contributions and allow to test the hypothesis of destructive interference. Finally, we present a model in which such interference can emerge and we discuss the range of parameters which would lead to a significant suppression of the decay rate. Keywords: Majorana Neutrino, Double Beta Decay, Neutrino Mass

1. Introduction After the discovery of neutrino oscillation, the search of nature of neutrino has gained new momentum. The most sensitive search is provided by (ββ)0ν -decay which plays an important role in neutrino physics. This process also implies the Lepton Number Violation (LNV) and put Seesaw Mechanism in favor. Thus the study of (ββ)0ν -decay plays an important role in neutrino physics. However, up to now, this process has not been seen experimentally yet. The underlying mechanism of the process cannot be identified either. In the past 10 years, efforts have been put to determination of the dominant mechanism for (ββ)0ν -decay. To Email addresses: [email protected] (S. Pascoli), [email protected] (C. F. Wong)

0920-5632/$ – see front matter © 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.nuclphysbps.2013.10.079

determine the leading mechanism, various techniques have been discussed, e.g. analysis of the angular distribution [1], and comparison of the nuclear matrix elements between different nuclei ([2], [3]). However, it is also possible that the contributions from different mechanisms are of the same order. For instance, the light neutrino and heavy neutrino exchange mechanism may destructively interfere each other, reducing the decay rate of the process, as first considered in Ref. [4]. This may explain why the detection of neutrinoless double beta decay is still absent, and give a hint that even neutrino is Majorana, we may still fail to detect the signal of (ββ)0ν -decay in some nuclei. We are interested in studying the case in which a future neutrino mass measurements from KATRIN [5, 6] and cosmological observations [7, 8] turn out to be incompatible with null results in neutrinoless double beta

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decay experiments, but might be reconciled if destructive interference is allowed between different mechanisms. Therefore, we study if it is possible to test the presence of destructive interference by combining the searches in different nuclei. 2. Different Mechanisms for (ββ)0ν -decay Neutrinoless double decay can be induced by various lepton-number violating mechanisms, including: (a) light Majorana neutrinos exchange [9, 10, 11]; (b) heavy Majorana neutrinos exchange [12, 13]; (c) R-parity violation (RPV) with short-range exchange [14, 15] and long-range exchange [16]; (d) right-handed leptonic and hadronic currents coupling [17]; (e) Kaluza-Klein neutrino exchange via extra dimension [18] and others. From a theoretical perspective, the contributions from (b) - (e), the “new-physics” mechanisms, are generically expected to be subdominant, as the couplings of the “new-physics” particles will be small. Recently, the possibility of large contributions from these mechanisms have been explored with interesting implications for (ββ)0ν -decay [19, 20, 21, 22]. Thus it is worthy to study these “new-physics” mechanisms and estimate the possible ranges of their magnitudes. Here we focus on the first three mechanisms in order to highlight the impact of destructive interference between multiple mechanisms [22]. Similar considerations also apply in other cases. The inverse half-life of (ββ)0ν -decay is Γi

≡

0ν −1 [T 1/2 ]i

= Gi | ην Mν,i + ηN MN,i + ηλ Mλ,i +ηq Mq,i |2 ,

(1)

where Gi is the common phase space factor. Here, i indicates the nuclear species. Mν,i , MN,i , Mλ,i , Mq,i are the corresponding nuclear matrix elements (NME), which describe the nuclear effects in (ββ)0ν -decay. The subscripts ν, N, λ, q refer to light neutrino, heavy neutrino, short-range and long-range RPV mechanisms respectively. ην , ηN and ηλ , ηq are the lepton number violating (LNV) parameters respectively, defined as, light 1  < mν > mββ ην = (Uek )2 mk ≡ ≡ , me k me me

ηN = mp

heavy 

(VeLj )2

j

1 , Mj

151

md˜ πα s λ2 111 mp · [1 + ( R )2 ]2 , 2 4 6 G F m ˜ mg˜ mu˜ L

(4)

 λ λ 1 1 11k 1k1 d [sin2θ(k) ( 2 − 2 )]. √ md˜ (k) md˜ (k) k 2 2G F 1 2

(5)

ηλ 

dR

ηq =

All the LNV parameters in the equations above are independent of nuclear structure. L In Eq. (2), me is the electron mass, mk , Uek are the light ν masses and the elements of mixing matrix between light ν mass and flavor state, respectively. Sterile neutrinos with masses M j much higher than the average propagating momentum, M j  100 MeV, give the contribution in Eq. (3). We indicate with VeLj the mixing between the heavy mass states and electron neutrinos. In Eq. (4 - 5), G F is the Fermi constant, α s is the SU(3) gauge coupling constant, λi jk is the trilinear coupling constant in the R-parity violation superpotential. mu˜ L , md˜R and mg˜ are the masses of u-squark, d-squark and gluino, respectively. The definitions of md˜1 (k) , md˜2 (k) d and sin2θ(k) can be referred to Ref. [16]. The physical meaning or more detailed explanations of these SUSY parameters can be referred to Ref. [23]. Given the evidence for light ν masses, the corresponding mechanism for neutrinoless double beta decay is the most discussed one. Its Feynman diagram is given in Fig. 1. As shown in Eq. (2), the predictions for the decay rate of (ββ)0ν -decay depends critically on |mν |, which is related to the neutrino masses and the CP-violating phases. Future information on the value of ν mass, from e.g. tritium β decay experiments and cosmology, on the type of ν mass ordering and on the mixing angles will allow to obtain new predictions for |mν | and conse0ν . quently for T 1/2 For the exchange neutrino mass larger than 100 MeV, the process becomes ’short-range’ and the inter-nucleon

(2)

(3)

Figure 1: Feynman diagram giving rise to the standard light ν mechanism contribution to (ββ)0ν -decay.

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Figure 2: The short-range and long-range R-parity violating contribution to (ββ)0ν -decay.

distance is considerably small. In this case the interacting nucleons are not point-like [24]. The nuclear repulsive force has to be considered in the calculation of nuclear current, and the approximation of the propagator in light ν exchange does not apply here. These lead to the second term in Eq. (1), whose contribution is proportional to the average of the inverse of heavy ν masses. The heavy ν is related to the neutrino mass generation, thus VeLj and also the contribution from heavy ν mechanism are constrained by Seesaw, which will be discussed in detail in Section 3. Supersymmetry with R-parity violation can also induce (ββ)0ν -decay. There are two possible trilinear RPV contributions. The first case is the short-range mechanism, exchanging the heavy super-particles [14] to induce the LNV process, see Fig. 2 (left), thus the process is ’short-range’. The corresponding contribution to the decay rate can be parameterized by ηλ . The other possibility is the squark-neutrino mechanism [16], which suggests that SUSY trigger (ββ)0ν -decay through long range exchange of virtual light neutrino and heavy squark, see Fig. 2 (right). In this case, the light neutrino in the propagator is not necessarily massive. Both short-range and long-range RPV mechanisms are dominated and enhanced by pion-exchange [15, 16]. The R-parity violating interactions can lead to generation of Majorana neutrino masses. This can constraint the trilinear coupling constant. Different from the mass mechanisms, the trilinear RPV coupling is not related to right-handed ν nor Seesaw, it generates the light ν mass through the squark-quark loop (Fig. 4). Thus the trilinear coupling constants λi jk , the LNV parameters ηλ and ηq , will not suffer from the small active-sterile mixing as the case in heavy ν mechanism, but they all are sensitive to the masses of squarks and gluinos. The contributions from both short-range and long-range RPV mechanisms are suppressed by the mass of heavy s-particles, which will be discussed in detail in section 4.

It is important to establish the leading mechanism in (ββ)0ν -decay and extract the values of the corresponding LNV parameter. However, it is also possible that the contributions from different mechanisms are of the same order. For instance, the light neutrino and heavy neutrino exchange mechanism may destructively interfere with each other, reducing the decay rate of the process, as first considered in Ref. [4]. In fact, the parameters ην , ηN , ηλ and ηq could be complex, in presence of CP-violation (or have opposite signs, for CP conservation), and cancellations in the decay rate can take place, making the half-life time much longer than naively expected. In the next 2 sections, we will investigate how do heavy ν and R-parity violation mechanisms relate to the light ν mass generation. These relations will constrain their contributions to (ββ)0ν -decay and reveal under which circumstances, the “new-physics” mechanisms would be as large as standard light ν mechanism and thus the cancellation becomes possible. 3. Heavy Neutrino Contribution 3.1. Heavy ν Mechanism We first consider the contribution from “heavy ν exchange”. It scales as M −2 j and thus the corresponding amplitude would be strongly suppressed and this mechanism subdominant [25]. For the sake of simplicity, here we assume that (ββ)0ν decay is only induced by the exchange of Majorana light neutrinos and heavy neutrinos. In this case, the decay rate simplifies to Γi = Gi | ην Mν,i + ηN MN,i |2 .

(6)

The mixing with the heavy neutrinos will be constrained from the Seesaw Mechanism, which shows that light heavy   2 (Uek ) mk = − (Ve j )2 M j . k

(7)

j

Consequently, the decay rate can be rewritten as  2 me mp MN,i 2 j (Ve j ) M j Γi = Gi | Mν,i · (−1 + )| . me M 2j Mν,i (8) The second term corresponds to the contribution from heavy neutrino exchange. The equation above shows that if M j is very large, the light neutrino exchange dominates (ββ)0ν -decay [25].

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This conclusion can be avoided if neutrino masses are not generated at tree level. An example is given by inverse seesaw or extended seesaw models in which another set of heavy sterile neutrinos is introduced. Their mixing with the active ones cancels exactly the contribution to ν masses which are 0 at tree level. Specifically, in the basis of active neutrinos, νL , and heavy neutrinos N1 , N2 , the mass matrix after electroweak symmetry breaking will read [26, 27, 20] ⎛ ⎜⎜⎜ 0 ⎜ M = ⎜⎜⎜⎜ mD ⎝ mD

mD μ Λ

mD Λ μ

⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎠ .

(9)

By setting μ and = 0, the light neutrino mass is zero at tree level. In this case the loop contribution would dominate in the neutrino mass [26] and the relation in Eq. (7) will not apply, thus the heavy neutrino mechanism is allowed to dominate in (ββ)0ν -decay, or significantly cancel the contribution of the light ν mechanism. A more complete discussion about the dominance of heavy neutrino mechanism can be referred to Ref. [26]. It gives an example that in (ββ)0ν -decay, the contribution from heavy neutrino mechanism can be as large as the standard light neutrino mechanism, but the parameters Λ, μ will be very constrained. Fig. 3 shows that when Yukawa coupling = 0.01, 0.001 10−4 , 10−5 and 10−6 , the decay-rate in Eq. (6) will be smaller than future experimental sensitivity in some particular regions of Λ–μ space1 . This figure further shows the possibility that heavy ν mechanism and standard light ν mechanism effectively cancel each other. The intersections between the colored areas and the black line represent the situations that heavy ν contribution = light ν contribution and at the same time the parameters satisfy the combining constraints from different neutrino experiments. In this case, the destructive interference or even exact cancellation is possible and leads to the absence of observation in the future experiments. A more detailed analysis of the cancellation for different nuclei will be shown in Section 6. 3.2. The Other Similar Mechanism (ββ)0ν -decay can also be induced by other mechanisms like KK neutrino (extra-dimension), bilinear Rparity violation. These two cases are similar to heavy 1 According to Ref. [28], the sensitivities of next-to-next generation experiments would be < mββ >  27 - 41 meV. Here Fig. 3 conservatively requires that < mββ >≤ 50 meV.

Figure 3: The allowed regions where combined contributions from heavy ν and standard light ν mechanisms are smaller than the future sensitivity, and the corresponding light ν mass and active-sterile ν mixing are compatible with the neutrino experiments. The red, green, blue, orange and purple plots correspond to Yukawa coupling = 0.01, 0.001, 10−4 , 10−5 and 3 × 10−6 . The solid black line represents when heavy ν contribution = light ν contribution, and the intersections between the black line and the colored areas are the regions that are phenomenologically interesting.

ν exchange since they are also constrained by treelevel neutrino mass generation. Similar to the traditional Type I Seesaw Mechanism, the singlet ν in extradimension [18, 29] and bilinear R-parity violation [30] can also generate neutrino mass through tree-level diagrams. However the exchange particles of these mechanisms are not the heavy sterile neutrinos, but the KK neutrino in extra dimensions, or neutralinos which relate to the bilinear R-parity violation. In a word, extra-dimension and bilinear RPV mechanisms will suffer from the similar constraint as Eq. (7) and thus their contributions to (ββ)0ν -decay will be negligible, unless the exchange particles are around O(100 MeV). However, such light KK neutrinos or SUSY particles are not in favor. So in the following, extradimension will not be discussed, since it is just similar to the heavy ν mechanism. We will just focus on the models which generate ν masses differently with the typical Seesaw. Within R-parity violation, besides the bilinear terms, there also exist trilinear terms, which are related to the loop-level mass generation of neutrino and different to the traditional Type I Seesaw. This mechanism will be discussed in detail in Section 4.

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in Fig. 4.2 The relation between neutrino mass matrix and trilinear coupling constants is as following [33], 3   λ λ  [sin(2θk )mq j 16π2 i jk i jk log(x jk ) (x jk − 1)log(x1jk ) × ( jk 2 + jk2 ) + ( j ↔ k)], (11) x2 − 1 x1 − 1(x2jk − 1)

Miiq =

Figure 4: Squark-quark loop contribution to ν mass generation.

where 4. R-Parity Violation Contribution

sin(2θk ) = 2mqk (Ak + μtanβ)×

R-parity is defined as R = (-1)3B+L+2S , with B, L being the baryon and lepton numbers, and S the spin. The R-parity violation may trigger LNV process and allows additional terms in super-potential.

[(m2q˜ k − m2q˜ k − 0.34MZ2 cos(2β))2 − L

R

4(mkq (Ak + μtanβ))2 ]−1/2 , x1jk = m2q j /m2q˜ k , 1

WRPV = λi jk Li L j Ekc + λi jk Li Q j Dck + μi Li H2 .

(10)

The super-potential WRPV can lead to lepton number violation and Majorana neutrino masses in the absence of right-handed neutrino and see-saw mechanism. The last term in Eq. (10) is the bilinear term, it can also induce the ν mass and (ββ)0ν -decay [14, 30]. However, as stated in the previous section, the bilinear contribution to ν mass and (ββ)0ν -decay is similar to the heavy neutrino mechanism. Following the similar derivation in ref [25], we can prove that it is subdominant. For simplicity, we focus on the trilinear λ coupling and its contribution to (ββ)0ν -decay. By using the relation between λ and the neutrino mass matrix, we can get the upper limit of λ s and check the possibility that the R-parity violation mechanisms are in the same order with light neutrino exchange. 4.1. Generating ν Mass Matrix Through Trilinear RPV Eqs. (4) and (5) show that both ηλ and ηq depend on the trilinear coupling λ . To compare the contributions from light ν and RPV mechanism, the range of different λi jk has to be estimated. This will be achievable through the relation between trilinear coupling and neutrino mass matrix. The R-parity violation can generate neutrino mass through tree-level and loop self-energy diagrams. The tree-level contributions come from the bilinear term in Eq. (10), which is the part we are not concerned here (as we discussed in previous section). The loop diagram, as shown in Figure 4, involves the quark-squark mixing and the trilinear coupling λ . For the sake of simplicity, in the following estimation, we ignored the other contributions and assumed that ν mass generation mainly comes from the loop-diagram

(12)

x2jk = m2q j /m2q˜ k .

(13)

2

The definitions of m2q˜ k and m2q˜ k can be referred to Ref. 1

2

[33], while m2q˜ k and m2q˜ k can be referred to Ref. [23]. L L All of them are functions of m0 (universal scalar mass) and m1/2 (universal gaugino mass) [23]. We followed the assumptions in Ref [33] and set up the values for different SUSY parameters. Since the values of ηλ and ηq depend on the values of m0 and m1/2 , the universal scalar and gaugino mass are the key factors in our estimation and they are supposed to be of same order. In the following, we will simply assume that m0 = m1/2 and estimate the contributions of “short-range trilinear RPV” and “long-range trilinear RPV” mechanisms and compare them with the standard light ν mechanism. 4.2. The Short-Range R-Parity Violation The so-called “short-range RPV” mechanism, means that at the quark level, the R-parity violating process arises from exchanging the heavy SUSY particles like gluinos, neutralinos, squarks (Fig. 2 (left)), etc. All these processes involve only the λ111 coupling constant. And among all these terms, the gluino exchange is supposed to dominate [34]. On the other hand, at the hadron level, the onepion and two-pion exchange are supposed to dominate [15, 35]. This further enhances the values of the nuclear 2 Similar as the heavy sterile ν case, it is always expected that if R-parity violation generates the light ν mass, the corresponding treelevel masses always dominate over the loop induced masses [31]. However, reference [32] showed that it is possible that the loop contribution exceeds the tree-level. In this paper, we just simply adopt the conventional hypothesis that different approaches to ν mass generation do not compensate each other and thus we can estimate the limit of squark-quark loop contribution without knowing the others.

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matrix elements, which will be shown and discussed in the following sections. Since ηλ involves only λ111 , we used Eq. (11) to extract its limit and ignore the other coupling constants. Eq. (11) can be rewritten as q = mββ  M11 3   λ λ × ≥ 16π2 111 111 log(x11 ) (x211 + 1)log(x111 ) ) × 2] [sin(2θ1 )mq1 × ( 11 2 + 11 x2 − 1 (x1 − 1)(x211 − 1)

(14) = λ2 111 C, where C is a constant and depends on the value of m0 C=

log(x211 ) (x211 + 1)log(x111 ) 3 1 1( [sin(2θ )m + 11 )]. q 8π2 x211 − 1 (x1 − 1)(x211 − 1)

Eq. (14) shows that λ2 111 ≤ mν /C. Substitute it into Eq. (4), we get the upper limit of ηλ , ηλ ≤

md˜ πα s (mν /C) mp · [1 + ( R )2 ]2 . 2 4 6 G F m ˜ mg˜ mu˜ L

(15)

dR

Similar as the study of heavy neutrino mechanism, in (ββ)0ν -decay, there may exist sizable contribution from short-range R-parity violation, but the RPV generated light ν mass has to be compatible with the neutrino experiments, i.e. mββ ≤ cosmology bound. This will constrain the parameters λ111 3 and m0 , and thus the maximum of ηλ is constrained as well (Eq. (15)). Similar as the discussion in Section 3, here we assume that 0ν −1 Γi ≡ [T 1/2 ]i = Gi | ην Mν,i + ηλ Mλ,i |2 ,

(16)

and analyze in which ranges of λ111 and m0 , the combined contribution from short-range RPV and light ν mechanism6 will be smaller than the future experimental sensitivities. The plots are shown in Fig. 5 (left side), which correspond to different values of μ (the relation between μ and λ111 can be referred to Eq. (11)). Fig. 5 indicates that short-range RPV and standard light ν mechanism can effectively cancel each other only in very fine-tuned areas. The tiny intersections between the colored areas and the black line represent the situations that short-range RPV contribution exactly cancels 3 Eq. (11) shows that λ k 111 is related to sin(2θ ), which depends on μtanβ whose value is not confirmed yet. In Ref. [33], tanβ is assumed to be 20. On the other hand, μ should be of the same order of sparticle masses [23]. In this paper, we will follow these assumptions and analyze the value of λ111 in different cases: μ = m0 , 5 m0 and 10 m0 .

155

the light ν contribution. Again, this will lead to the absence of future measurement. The phenomenological consequences will be discussed in detail in Section 5. The discussion above just reveals the upper limit of ηλ , it is still possible that ηλ ην , as Eq. (15) just shows the possible maximum of ηλ . It shows the possibility that the short-range R-parity violation mechanism could be in the same order with the standard light ν mechanism, or even dominate in the (ββ)0ν -decay. 4.3. The Long-Range R-Parity Violation Besides the previous contribution, the trilinear Rparity violation can also induce the (ββ)0ν -decay by squark-neutrino mechanism [16], which suggests that the neutrino mediated (ββ)0ν -decay originating from Rparity violating interactions. This mechanism is comparatively long range. And the LNV parameter for this “long-range (squark-neutrino) RPV” mechanism is ηq (Eq.(5)). By using the similar trick as before, the upper limit of ηq can be estimated and is related to the value of mβ β. Here we use the same assumption in the estimation of “short-range” contribution, λ111 dominates among the trilinear coupling constants 4 . Which means that Eq. (5) changes to ηq =

λ2 1 1 111 d [sin2θ(1) ( 2 − 2 )]. √ md˜ (1) md˜ (1) 2 2G F 1 2

(17)

As showed in the previous subsection, λ2 111 = mββ /C, where 3 [sin(2θ1 )mq1 × 16π2 log(x11 ) (x211 + 1)log(x111 ) ( 11 2 + 11 ). × 2] x2 − 1 (x1 − 1)(x211 − 1)

C=

Thus the maximum of ηq is ηq ≤

mβ β/C 1 1 d − 2 )]. [sin2θ(1) ( 2 √ m m 2 2G F d˜1 (1) d˜2 (1)

(18)

Similar to previous subsection, the constraint of mββ  should be satisfied and thus the parameters λ111 and m0 are constrained. The allowed regions of combined contributions from long-range RPV and light ν mechanisms are plotted in Fig. 5 (right side). 4 According to ref [32], The maximum of λ λ   112 121 and λ113 λ131  ), thus we would just get the similar approximation if we Max(λ2 111 assume λ112 λ121 or λ113 λ131 dominates in Eq. (5).

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It is noteworthy that in Fig. 5, the intersections on the left side are smaller than on the right side, which means that the cancellation between “short-range RPV and light ν” is more fine-tuned than the cancellation between “long-range RPV and light ν”. This is reasonable, since the short-range mechanism is expected to be smaller than the long-range mechanism (the exchange

0.4

0.3

0.3

Λ' 111

Λ' 111

μ = m0 , 0.4

0.2

0.1

0.0

0.2

0.1

200

400

600

800 1000 1200 1400

200

400

m 0  GeV

600

800 1000 1200 1400

m 0  GeV

0.4

0.3

0.3

Λ' 111

Λ111

μ = 5m0 , 0.4

0.2

0.1

0.0

0.2

0.1

200

400

600

0.0

800 1000 1200 1400

200

400

m 0  GeV

600

800 1000 1200 1400

m 0  GeV

0.4

0.3

0.3

Λ111

Λ' 111

μ = 10m0 , 0.4

0.2

0.1

0.0

0.2

0.1

200

400

600

800 1000 1200 1400

m 0  GeV

0.0

200

400

600

800 1000 1200 1400

m 0  GeV

Figure 5: The allowed regions where combined contributions from RPV (left, short-range RPV mechanism; right, long-range RPV mechanism) and standard light ν mechanisms are smaller than the future sensitivity (mββ  ≤ 50 meV) of (ββ)0ν -decay experiments, and the corresponding mββ  is compatible with the experiment results. The red, green and blue plots correspond to μ = m0 , 5 m0 , 10 m0 . The solid black line represents when short-range (or long-range) RPV contribution = light ν contribution, and the intersections between the black line and the colored areas are the regions which are phenomenologically interesting.

particle is heavier in short-range RPV); the short-range RPV is more unlikely to be at the same order of standard light ν mechanism. 5. The Nuclear Matrix Elements Judging from the sections above, it would be possible to argue that the “new-physics” mechanisms are in the same order of the standard “light ν exchange” mechanism. In order to quantitatively analyze the cancellation effects, the precise evaluation of the nuclear matrix elements (NME) are important. The evaluation of nuclear matrix elements suffers large uncertainties which in turn affects the predictions for the half-life time of the decay and the extraction of information on neutrino masses and mixing parameters from future measurements. They depend on the nuclear structure of the reaction, the adjustment of parameters like strength parameter g ph , g pp , the choice of the coupling constant gA , the measurements of Two-neutrino double beta decay and the exchange mechanism considered for the process [2, 36, 37]. In recent years, a wide effort has been devoted to the computation of the NME, using two main approaches: QRPA (quasiparticle random phase approximation)[37, 38] and NSM (nuclear shell model) [39, 40]. Encouragingly, the computed values tend to vary in a smaller range, despite still large uncertainties. In this paper, the values of NMEs for all four mechanisms are taken from Ref.[22], where 8 variants of NMEs are listed, and each variants corresponds to different gA values, NN potential and model space size. Since the ratio of the NMEs between the nuclei are similar for different variants (only the ratios of NMEs matter in our analysis, as shown in the following contents), and the discussion of ‘different nuclei model and parameters’ is beyond the scope of this paper, here we just present the largest variants of NMEs in Ref.[22], as shown in Table 1. All the nuclear matrix elements were calculated under QRPA since this approach keeps the uncertainties of NMEs largely under control [41]. The values of NME vary according to the species of nucleus. Table 1 shows the NMEs of four nuclei, which are the promising candidates in the future (ββ)0ν -decay experiments. 6. Destructive Interference in Γi Now we study if it is possible to test the presence of destructive interference by combining the searches in different nuclei. The amplitude of (ββ)0ν -decay for 76 Ge

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G01 × 10 yr [2] Variant n.8 in [22] Mν MN Mλ Mq 15

76

Ge 7.93

82

Se 35.2

5.82 412 596 728

5.66 408 594 720

100

Mo 57.3

130

5.15 404 589 691

4.70 384 540 641

Te 55.4

Table 1: The phase space factor and nuclear matrix elements for light, heavy neutrino exchange, short-range and long-range RPV SUSY model.

is assumed to be negligible i.e., ΓGe ∼ 0, due to interference between two mechanisms and consequently no signal for 0νββ-decay can be observed in this nucleus. Base on this assumption, we calculate the amplitudes for other nuclei and test if they are significantly different from zero. The analysis will be performed in three different cases: i) The cancellation between light and heavy neutrino exchange mechanisms; ii) The cancellation between light neutrino and short range RPV mechanisms; iii) The cancellation between light neutrino and long range RPV mechanisms. 6.1. Redefinition of mββ First consider the interference between light and heavy neutrino mechanisms. Assume νN = GGe | ην Mν,Ge + ηN MN,Ge |2  0, ΓGe Mν,Ge implying that ηN = −ην × . MN,Ge

(19) (20)

Substituting Eq. (20) into the calculation of the amplitudes of other nuclei, then the effective neutrino mass, mββ , which will be measured by the future experiments and was previously expected to be simply light 2 k (U ek ) mk of light neutrino, should be corrected as Mν,Ge me (ην Mν,i − ην × MN,i ) |, Mν,i MN,Ge Mν,Ge · MN,i )|. (21) =| mν  × (1 − MN,Ge · Mν,i

| mββ |i =|

The | mββ |i in L.H.S. of Eq. (21) is the value of “effective mass” expected to be observed in future experlight iments5 , while in R.H.S. mν  ≡ k (Uek )2 mk is the effective light ν mass from light ν mechanism. 5 Here we compare the measured “effective mass” rather than decay rates of different nuclei because the phase space factor Gi is not involved in Eq. (21). The cancellation effect of different nuclei would be obvious by comparing | mββ |i .

M

M

ν,i N,i It follows immediately that if Mν,Ge  MN,Ge , only partial cancellation will take place and | mββ |i might be significantly different from 0. Eq. (21) also manifests that the future measurement depends on the value of mν . The sensitivities of | mββ |i of next-to-next generation experiments are expected to be around 27 - 41 meV [28], thus the value of mν  cannot be too small. If the ν mass spectrum is normal hierarchy, then the corresponding | mββ |i would be too small for the future measurement (even without cancellation effect [42]). Thus in the following, only inverted hierarchy and quasi-degenerate will be discussed. The range of mν  is assumed to be 0.01 eV (minimum of inverted hierarchy) - 1 eV (maximum of quasi-degenerate). Similarly, we can evaluate | mββ |i in the case of destructive interference between light ν and RPV SUSY mechanisms. If cancellation effect takes place between light ν and short-range RPV, then Eq. (20) and (21) will change to

Mν,Ge . Mλ,Ge Mν,Ge · Mλ,i |i =| mν  × (1 − )|. Mλ,Ge · Mν,i

ηλ = −ην × | mββ

(22) (23)

If cancellation effect takes place between light ν and long-range R-parity violation, then Eq. (20) and (21) will change to Mν,Ge . Mq,Ge Mν,Ge · Mq,i |i =| mν  × (1 − )|. Mq,Ge · Mν,i

ηq = −ην × | mββ

(24) (25)

With Eqs. (21), (23) and (25), we can estimate the cancellation effect on the future measurements. 6.2. The Future Measurements It is expected that the next-to-next generation experiments will observe (ββ)0ν -decay if the ν mass spectrum is inverted hierarchy or quasi-degenerate. Therefore, we will focus on the inverted hierarchy and quasidegenerate. In Table 2, the expected values of | mββ |i , in the case of destructive interference between different mechanisms, are reported. | mββ |(max) corresponds to the maximum of mν  and | mββ |(min) corresponds to the minimum of mν  in inverted hierarchy. In Table 2, the results correspond to the NMEs in Table 1 [22]. Table 2 reveals that if “exact cancellation” takes place in the decay of 76 Ge, then it is unlikely to observe any signals from the future experiments of 82 Se, since 76 Ge

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(eV) | mββ |(max) | mββ |(min) | mββ |(max) | mββ |(min) | mββ |(max) | mββ |(min)

76

82 100 130 Ge Se Mo Te cancellation between heavy ν and light ν 0 0.0183 0.108 0.154 0 1.83×10−4 1.08×10−3 1.54×10−3 cancellation between short RPV and light ν 0 0.0248 0.117 0.122 0 2.48×10−4 1.17×10−3 1.22×10−3 cancellation between long RPV and light ν 0 0.0170 0.0727 0.0903 0 1.70×10−4 7.27×10−4 9.03×10−4

Table 2: Expected values of | mββ |i for different destructive interferences.

and 82 Se have similar values of NMEs. For 100 Mo and 130 Te, the | mββ | could still be as large as the future experimental sensitivity, which is around O(0.01 - 0.1 eV). These results reflect that even if significant destructive interference really exists between various mechanisms in (ββ)0ν -decay, the observation is still possible depending on the ν mass spectrum and the nuclei. To show the dependence of ν mass spectrum and the cancellation effect more clearly, we reproduced the ν mass spectrum in Fig. 6 (corresponding to 82 Se) and Fig. 7 (corresponding to 130 Te). The cancellation effects between light ν and heavy ν mechanisms are shown (the cancellation effects between light ν and RPV mechanisms are similar). In Figs. 6 and 7, the | mββ |i corresponding to traditional assumptions (light ν mechanism dominating the (ββ)0ν -decay), is also presented as a comparison 6 . The double beta decay of 82 Se is used as the first example, since under our assumption, the cancellation effect in this nuclei is most obvious. Fig. 6 shows that due to the strong cancellation effect, the | mββ |82 S e is almost two orders smaller than previously expected. Moreover, it is important to note that the intersections between the next-to-next generation sensitivity (red horizontal line) and the | mββ |82 S e of destructive interference scenario (green solid lines) are inside the colored area, which is ruled out by the cosmology result. This means that under our assumption, it would be impossible to observe the neutrinoless double beta decay of 82 Se in the next-to-next generation experiments. However, we should keep in mind that the cancellation effect depends on the ratio of the NMEs and thus it should be different for different nuclei. The failure in 6 The plots in Figs. 6 and 7 correspond to the up-to-date data for the oscillation parameters [43], including the most-updated measurement of large θ13 [44].

Figure 6: The expected values of | mββ |82 S e corresponding to different ν mass spectra. The green solid curves correspond to the destructive interference between light ν and heavy ν mechanisms, while the blue dashed lines correspond to the traditional assumption (no interference, light ν dominating). The red horizontal line represents the general sensitivity of next-to-next generation of SuperNEMO experiment (63 meV [45, 28]). The colored area (m0 > 0.58 eV) is the region that is ruled out by the cosmology result [8].

Figure 7: The expected values of | mββ |130 T e corresponding to different ν mass spectra. The black solid curves correspond to the destructive interference between light ν and heavy ν mechanisms, while the blue dashed lines correspond to the traditional assumption (no interference, light ν dominating). The red horizontal line represents the general sensitivity of next-to-next generation of CUORE experiment (30 meV [46, 28]). The colored area (m0 > 0.58 eV) is the region that is ruled out by the cosmology result [8].

one experiment does not lead to the absence of observations in all future experiments. For example, in the decay of 130 Te, the cancellation effect only reduces the | mββ |130 T e a few times smaller than previously expected, as shown in Fig. 7. Fig. 7 shows that the cancellation effect in the decay of 130 Te is not as strong as the decay of 82 Se. This time, the intersections between the horizontal red line and the black solid lines do not lie in the the ruled out area. If the ν mass ordering is Quasi-Degenerate, or more specifically, m0 is at the order of O(1 eV), the neutrinoless double beta decay is still detectable in the future experiment even though destructive interference exists. Therefore, under our assumption, 130 Te is a good

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candidate for future measurements. Furthermore, it is important to note that in both Fig. 6 and 7, even in the “no interference” circumstance, if the neutrino mass spectrum is with normal hierarchy, (ββ)0ν -decay still cannot be observed in the next-to-next generation experiments. Thus, it is hard to distinguish if the absence of future observation comes from the cancellation effect or the insignificance of mν . However, in principle, the cancellation effect is different for different nuclei; it depends on the ratio of the nuclear matrix elements. If destructive interference really exists, (ββ)0ν -decay is still expected to be observable in some certain nuclei. Thus, more experiments must be performed before making any conclusions. 7. Conclusion In this paper, we proved that the heavy ν mechanism (or RPV mechanisms) could effectively interfere with the contribution from light ν exchange. Due to the absence of observation, we suggested that the potential interference is destructive. In summary, we discussed the existence and the effect of destructive interferences among different mechanisms of neutrinoless double beta decay. For simplicity, we only discussed the cancellation between two mechanisms, but it is straightforward to generalize the analysis to the interference between three or more mechanisms. This paper shows that destructive interference reduces the decay rate significantly and may lead to the failure in observing the signal of (ββ)0ν decay in the experiments and we have to choose the nuclei carefully for future measurements. For example, if exact cancellation takes place in the decay of 76 Ge, then future experiments corresponding to 82 Se may not observe any signal due to the similar ratios of NMEs. Meanwhile, 130 Te would be a good candidate for testing the LNV process. Finally, besides the improvement of experimental sensitivity of (ββ)0ν -decay, intense efforts should also be made on the theoretical study. We must determine the uncertainties of the NMEs under different mechanisms and reduce these uncertainties, otherwise even when we can measure the (ββ)0ν -decay rates in experiments, we cannot extract much information about the effective ν mass, neutrino mass hierarchy, the details of heavy sterile neutrino (or the RPV parameters), etc. References [1] A. V. Borisov, A. Ali, and D. V. Zhuridov, arXiv:hepph/0606072.

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