How to interpret a discovery or null result of the 0ν2β decay
HHow to interpret a discovery or null result of the ν β decay Zhi-zhong Xing ∗ Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, ChinaCenter for High Energy Physics, Peking University, Beijing 100080, China
Zhen-hua Zhao † and Ye-Ling Zhou ‡ Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
Abstract
The Majorana nature of massive neutrinos will be crucially probed in the next-generationexperiments of the neutrinoless double-beta (0 ν β ) decay. The effective mass term of thisprocess, (cid:104) m (cid:105) ee , may be contaminated by new physics. So how to interpret a discovery ornull result of the 0 ν β decay in the foreseeable future is highly nontrivial. In this paper weintroduce a novel three-dimensional description of |(cid:104) m (cid:105) ee | , which allows us to see its sensitivityto the lightest neutrino mass and two Majorana phases in a transparent way. We take a lookat to what extent the free parameters of |(cid:104) m (cid:105) ee | can be well constrained provided a signal ofthe 0 ν β decay is observed someday. To fully explore lepton number violation, all the sixeffective Majorana mass terms (cid:104) m (cid:105) αβ (for α, β = e, µ, τ ) are calculated and their lower boundsare illustrated with the two-dimensional contour figures. The effect of possible new physics onthe 0 ν β decay is also discussed in a model-independent way. We find that the result of |(cid:104) m (cid:105) ee | in the normal (or inverted) neutrino mass ordering case modified by the new physics effect maysomewhat mimic that in the inverted (or normal) mass ordering case in the standard three-flavor scheme. Hence a proper interpretation of a discovery or null result of the 0 ν β decay maydemand extra information from some other measurements. PACS number(s): 14.60.Pq, 13.15.+g, 12.15.FfKeywords: Majorana neutrino, 0 ν β decay, CP violation, new physics ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] a r X i v : . [ h e p - ph ] A p r Introduction
One of the burning questions in nuclear and particle physics is whether massive neutrinos arethe Majorana fermions [1]. The latter must be associated with the phenomena of lepton numberviolation (LNV), such as the neutrinoless double-beta (0 ν β ) decays of some even-even nuclei inthe form of ( A, Z ) → ( A, Z + 2) + 2 e − [2]. On the other hand, the Majorana zero modes may haveprofound consequences or applications in solid-state physics [3]. That is why it is fundamentallyimportant to verify the existence of elementary Majorana fermions in Nature. The most suitablecandidate of this kind is expected to be the massive neutrinos [4].However, the tiny masses of three known neutrinos make it extremely difficult to identify theirMajorana nature. The most promising experimental way is to search for the 0 ν β decays. Thanksto the Schechter-Valle theorem [5], a discovery of the 0 ν β decay mode will definitely pin down theMajorana nature of massive neutrinos no matter whether this LNV process is mediated by othernew physics (NP) particles or not. The rate of such a decay mode can be expressed asΓ ν = G ν ( Q, Z ) (cid:12)(cid:12) M ν (cid:12)(cid:12) |(cid:104) m (cid:105) ee | , (1)where G ν is the phase-space factor, M ν denotes the relevant nuclear matrix element (NME), and (cid:104) m (cid:105) ee stands for the effective Majorana neutrino mass term. In the standard three-flavor scheme, (cid:104) m (cid:105) ee = m U e + m U e + m U e (2)with m i (for i = 1 , ,
3) being the neutrino masses and U ei being the matrix elements of thePontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix [6]. Given current neutrinooscillation data [7], the three neutrinos may have a normal mass ordering (NMO) m < m < m or an inverted mass ordering (IMO) m < m < m . In the presence of NP, (cid:104) m (cid:105) ee is likely tobe contaminated by extra contributions which can be either constructive or destructive. While anobservation of the 0 ν β decay must point to an appreciable value of |(cid:104) m (cid:105) ee | , a null experimentalresult does not necessarily mean that massive neutrinos are the Dirac fermions because (cid:104) m (cid:105) ee ∼ ν β decay in the foreseeable future ishighly nontrivial and deserves special attention [10, 11, 12]. In this work we focus on the sensitivityof |(cid:104) m (cid:105) ee | to the unknown parameters in the neutrino sector, which include the absolute neutrinomass scale, the Majorana CP-violating phases, and even possible NP contributions. Beyond thepopular Vissani graph [13] which gives a two-dimensional description of the dependence of |(cid:104) m (cid:105) ee | on the smallest neutrino mass, we introduce a novel three-dimensional description of the sensitivityof |(cid:104) m (cid:105) ee | to both the smallest neutrino mass and the Majorana phases in the standard three-flavorscheme. We single out the Majorana phase which may make |(cid:104) m (cid:105) ee | sink into a decline in theNMO case, and show that a constructive NP contribution is possible to compensate that declineand enhance |(cid:104) m (cid:105) ee | to the level which more or less mimics the case of the IMO. On the otherhand, the destructive NP contribution is not impossible to suppress |(cid:104) m (cid:105) ee | to the level which isindiscoverable, even though the neutrino mass ordering is inverted or nearly degenerate. Given adiscovery of the 0 ν β decay, the possibility of constraining the unknown parameters is discussedin several cases. We also examine the dependence of |(cid:104) m (cid:105) αβ | (for α, β = e, µ, τ ) on the absoluteneutrino mass scale and three CP-violating phases of the PMNS matrix U , and conclude that some2ther possible LNV processes have to be measured in order to fully understand an experimentaloutcome of the 0 ν β decay and even determine the Majorana phases. |(cid:104) m (cid:105) ee | In the standard three-flavor scheme the unitary PMNS matrix U can be parameterized in terms ofthree rotation angles ( θ , θ , θ ) and three phase angles ( δ , ρ , σ ) in the following way [7]: U e = c c e i ρ/ , U e = s c ,U e = s e i σ/ , U µ = c s e i( δ + ρ/ , (3)where c ij ≡ cos θ ij and s ij ≡ sin θ ij (for ij = 12 , , δ is referred to as the Dirac phase sinceit measures the strength of CP violation in neutrino oscillations, ρ and σ are referred to as theMajorana phases and have nothing to do with neutrino oscillations. The phase convention takenin Eq. (3) is intended to forbid δ to appear in the effective Majorana mass term of the 0 ν β decay: |(cid:104) m (cid:105) ee | = (cid:12)(cid:12) m c c e i ρ + m s c + m s e i σ (cid:12)(cid:12) . (4)The merit of this phase convention is obvious. In the extreme case of the NMO or IMO (i.e., m = 0 or m = 0), which is allowed by current experimental data, one of the two Majorana phasesautomatically disappears from |(cid:104) m (cid:105) ee | . Note, however, that δ is intrinsically of the Majorana naturebecause it can enter other effective Majorana mass terms (e.g., (cid:104) m (cid:105) eµ and (cid:104) m (cid:105) µτ [14]).A measurement of the 0 ν β decay allows us to determine or constrain |(cid:104) m (cid:105) ee | . So far the mostpopular way of presenting |(cid:104) m (cid:105) ee | has been the Vissani graph [13]. It illustrates the allowed rangeof |(cid:104) m (cid:105) ee | against m or m by inputting the experimental values of θ and θ and allowing ρ and σ to vary in the interval [0 ◦ , ◦ ). In the NMO case |(cid:104) m (cid:105) ee | may sink into a decline when m lies in the range 0 . . |(cid:104) m (cid:105) ee | . In comparison, there is a lower bound |(cid:104) m (cid:105) ee | (cid:38) .
02 eVin the IMO case, and it is always larger than the upper bound of |(cid:104) m (cid:105) ee | in the NMO case whenthe lightest neutrino mass is smaller than about 0.01 eV [15]. This salient feature enables us toconfirm or rule out the IMO, if the future 0 ν β -decay experiments can reach a sensitivity below0.02 eV. Nevertheless, the Vissani graph is unable to tell the dependence of |(cid:104) m (cid:105) ee | on ρ and σ . Forexample, which Majorana phase is dominantly responsible for the significant decline of |(cid:104) m (cid:105) ee | inthe NMO case? To answer such questions and explore the whole parameter space, let us generalizethe two-dimensional Vissani graph by introducing a novel three-dimensional description of |(cid:104) m (cid:105) ee | .Fig. 1 is a three-dimensional illustration of the lower and upper bounds of |(cid:104) m (cid:105) ee | in the NMOand IMO cases. In our numerical calculations we have input the best-fit values of ∆ m , ∆ m , θ and θ obtained from a recent global analysis of current neutrino oscillation data [16]. Forsimplicity, the uncertainties of these four parameters are not taken into account because they donot change the main features of |(cid:104) m (cid:105) ee | . The unknown Majorana phases ρ and σ are allowed tovary in the range [0 ◦ , ◦ ), and the neutrino mass m or m is constrained via the Planck data(i.e., m + m + m < .
23 eV at the 95% confidence level [17]). Some comments on Fig. 1 arein order. (1) The upper bound of |(cid:104) m (cid:105) ee | is trivial, because it can be obtained by simply taking ρ = σ = 0 ◦ . (2) The lower bound of |(cid:104) m (cid:105) ee | is nontrivial, because it is a result of the maximalcancellation among the three components of |(cid:104) m (cid:105) ee | for given values of ρ , σ and m or m . (3) In3 og | h m i ee | e V l og | h m i ee | e V ρ [ ◦ ] ρ [ ◦ ] σ [ ◦ ] ρ [ ◦ ] ρ [ ◦ ] σ [ ◦ ] σ [ ◦ ] l og m e V l og m e V σ [ ◦ ] l og m e V l og m e V NMOIMO
Figure 1: Three-dimensional illustration of the lower (blue) and upper (light orange) bounds of |(cid:104) m (cid:105) ee | as functions of the lightest neutrino mass and two Majorana phases in the NMO or IMOcase.the NMO case it is the phase ρ that may lead the lower bound of |(cid:104) m (cid:105) ee | to a significant decline(even down to zero). In comparison, |(cid:104) m (cid:105) ee | is essentially insensitive to σ in both the NMO andIMO cases. (4) The allowed range of |(cid:104) m (cid:105) ee | in the IMO case exhibits a “steady flow” profile,which is consistent with the two-dimensional Vissani graph. Its lower bound ( ∼ .
02 eV) appearsat ρ = 180 ◦ for a specific value of m and arbitrary values of σ , but a deadly cancellation amongthe three components of |(cid:104) m (cid:105) ee | has no way to happen. (5) When the neutrino mass spectrum isnearly degenerate (i.e., m (cid:39) m (cid:39) m (cid:38) .
05 eV), the results of |(cid:104) m (cid:105) ee | in the NMO and IMOcases are almost indistinguishable.The parameter space for the vanishing of |(cid:104) m (cid:105) ee | in the NMO case is of particular interest,because it points to a null result of the 0 ν β decay although massive neutrinos are the Majo-rana particles. However, the “dark well” of |(cid:104) m (cid:105) ee | versus the ρ - m plane in Fig. 1 has a sharpchampagne-bottle profile at the ground. This characteristic can be understood by figuring out thecorrelation between m and ρ from |(cid:104) m (cid:105) ee | = 0. Namely, m c c + 2 m m c s c cos ρ + m s c = m s . (5)Given the best-fit values of ∆ m , ∆ m , θ and θ [16], Fig. 2 shows the ρ - m correlation whichcorresponds to the contour of the champagne-bottle profile of |(cid:104) m (cid:105) ee | in Fig. 1. One can see thatthe “dark well” appears when ρ lies in the range 160 ◦ — 200 ◦ and m varies from 0 . . σ . Such a fine structure of cancellation has been missed before.As a matter of fact, a three-dimensional description of |(cid:104) m (cid:105) ee | against two free parameters isequivalent to a set of two-dimensional contour figures which project the values of |(cid:104) m (cid:105) ee | onto the4 .000 0.002 0.004 0.006 0.008120140160180200220240 ρ [ ◦ ] m [eV] Figure 2: A correlation between m and ρ as constrained by the vanishing of |(cid:104) m (cid:105) ee | in the NMOcase, corresponding to the contour of the champagne-bottle profile of |(cid:104) m (cid:105) ee | in Fig. 1.parameter-space planes, if only its upper or lower bound is considered. In order to clearly presentthe correspondence between the numerical result of |(cid:104) m (cid:105) ee | and that of a given parameter whichis difficult to be identified in a three-dimensional graph, we show the contour figures for the lowerbound of |(cid:104) m (cid:105) ee | on the ρ - σ , m - ρ (or m - ρ ) and m - σ (or m - σ ) planes in the NMO (or IMO) casein Fig. 3 (or Fig. 4). For the sake of completeness, we calculate the contour figures for the lowerbounds of all the six effective Majorana mass terms defined as (cid:104) m (cid:105) αβ = m U α U β + m U α U β + m U α U β , (6)where the subscripts α and β run over e , µ and τ . There are at least two good reasons for considering |(cid:104) m (cid:105) αβ | : (a) only the 0 ν β decay itself cannot offer sufficient information to fix the three unknownparameters of |(cid:104) m (cid:105) ee | ; (b) if a null result of the 0 ν β decay is observed, one will have to search forsome other LNV processes so as to identify the Majorana nature of massive neutrinos. The typicalLNV processes which are associated with (cid:104) m (cid:105) αβ include the µ − → e + conversion in the nuclearbackground, neutrino-antineutrino oscillations, rare LNV decays of B and D mesons, and so on[15]. In Figs. 3 and 4 the contours for the lower bounds of |(cid:104) m (cid:105) αβ | are presented by gradient colorsand their corresponding magnitudes are indicated by the legends. In particular, the purple areasstand for the parameter space where significant cancellations (i.e., |(cid:104) m (cid:105) αβ | < − eV) can takeplace. When the m -associated term of |(cid:104) m (cid:105) αβ | is not suppressed by s ∼ σ . Hence a combined analysis of the 0 ν β decay andsome other LNV processes will be greatly helpful to determine or constrain both ρ and σ . m , and ρ from a signal of the ν β decay In the standard three-flavor scheme we have studied the possible profile (especially the lower bound)of |(cid:104) m (cid:105) ee | against the unknown mass and phase parameters. Inversely, the unknown parameterscan be constrained if the 0 ν β decay is discovered and the magnitude of |(cid:104) m (cid:105) ee | is determined. Agood example of this kind is the strong constraint on the parameter space of m and ρ in Eq. (5)5 h m i ee | |h m i µµ | |h m i ττ ||h m i eµ | |h m i eτ | |h m i µτ | ρ [ ◦ ] ρ [ ◦ ] σ [ ◦ ] σ [ ◦ ] σ [ ◦ ] [eV] |h m i ee | |h m i µµ | |h m i ττ ||h m i eµ | |h m i eτ | |h m i µτ | ρ [ ◦ ] ρ [ ◦ ] log ( m / eV) log ( m / eV) log ( m / eV) [eV] |h m i ee | |h m i µµ | |h m i ττ ||h m i eµ | |h m i eτ | |h m i µτ | σ [ ◦ ] σ [ ◦ ] log ( m / eV) log ( m / eV) log ( m / eV) [eV] Figure 3: The lower bounds of |(cid:104) m (cid:105) αβ | changing with m , ρ and σ in the NMO case.6 h m i ee | |h m i µµ | |h m i ττ ||h m i eµ | |h m i eτ | |h m i µτ | ρ [ ◦ ] ρ [ ◦ ] σ [ ◦ ] σ [ ◦ ] σ [ ◦ ] [eV] |h m i ee | |h m i µµ | |h m i ττ ||h m i eµ | |h m i eτ | |h m i µτ | ρ [ ◦ ] ρ [ ◦ ] log ( m / eV) log ( m / eV) log ( m / eV) [eV] |h m i ee | |h m i µµ | |h m i ττ ||h m i eµ | |h m i eτ | |h m i µτ | σ [ ◦ ] σ [ ◦ ] log ( m / eV) log ( m / eV) log ( m / eV) [eV] Figure 4: The lower bounds of |(cid:104) m (cid:105) αβ | changing with m , ρ and σ in the IMO case.7r Fig. 2 based on the assumption |(cid:104) m (cid:105) ee | = 0, which is more or less equivalent to a null resultof the 0 ν β decay provided the experimental sensitivity has been good enough. So it makes senseto ask the following question: to what extent the unknown parameters can be constrained from asignal of the 0 ν β decay?Let us try to answer this question in an ideal situation with no concern about the experimentalerror bars. The first issue is to derive the correlation between m (or m ) and ρ like that given inEq. (5) by eliminating σ . Since Eq. (4) can be viewed as an implicit function ρ = f ( m i , σ ) forgiven values of θ , θ and |(cid:104) m (cid:105) ee | , one may eliminate σ by substituting it with the solution of ∂ρ/∂σ | σ ∗ = 0. In this way we obtain the maximum and minimum of ρ as functions of m i :cos ρ max,min = − m c c + m s c − (cid:0) m s ± |(cid:104) m (cid:105) ee | (cid:1) m m c s c . (7)If |(cid:104) m (cid:105) ee | vanishes, then it is straightforward for Eq. (7) to reproduce Eq. (5). The maximum andminimum of σ as functions of m i can similarly be obtained:cos σ max,min = − m s + m s c − (cid:0) m c c ± |(cid:104) m (cid:105) ee | (cid:1) m m s c s . (8)However, σ is actually insensitive to |(cid:104) m (cid:105) ee | as shown in Fig. 1. Hence the constraint on σ mustbe rather loose even if the 0 ν β decay is observed. For this reason we simply focus on the possibleconstraints on ρ and m (or m ) in the following.Of course, the value of |(cid:104) m (cid:105) ee | extracted from a measurement of the 0 ν β decay via Eq. (1)must involve a large uncertainty originating from the NME M ν , while the phase-space factor G ν ( Q, Z ) can be precisely calculated. Following Ref. [18], we introduce a dimensionless factor F to parameterize the uncertainty of |(cid:104) m (cid:105) ee | inheriting from that of the NME: F = M ν max /M ν min ,where M ν max and M ν min stand respectively for the maximal and minimal values of the NME whichare consistently calculated in a given framework. It is apparent that F (cid:38) F = 1 cannotbe reached until the NME is accurately determined. Given a value of F , the “true” value of |(cid:104) m (cid:105) ee | may lie in the range (cid:2) |(cid:104) m (cid:105) ee | / √ F , |(cid:104) m (cid:105) ee |√ F (cid:3) [18]. In our numerical calculation we take F = 1 and F = 2 for illustration. Fig. 5 shows the allowed regions of m (or m ) and ρ for a few typical valuesof |(cid:104) m (cid:105) ee | . The effect of F can be seen when comparing between the cases of F = 1 and F = 2.Two comments are in order. (1) If |(cid:104) m (cid:105) ee | is vanishingly small (e.g., |(cid:104) m (cid:105) ee | = 0 . ρ can beconstrained in the range [140 ◦ , ◦ ] in the NMO case. If a larger value of |(cid:104) m (cid:105) ee | is measured (e.g.,0 .
005 eV or 0 .
05 eV), the allowed range of ρ will saturate the full interval [0 , ◦ ). To fix the valueof ρ needs the input of m . Hence some additional information about m from the cosmologicalobservation or from the direct beta-decay experiment will be greatly helpful. (2) The situation inthe IMO case is quite similar: ρ can be constrained in a narrow range if |(cid:104) m (cid:105) ee | approaches itsminimal value (i.e., 0 .
02 eV), but it is allowed to take any value in the range [0 , ◦ ) if |(cid:104) m (cid:105) ee | ismuch larger (e.g., 0 .
05 eV). Here again is some additional information about m required to pindown the value of ρ . |(cid:104) m (cid:105) ee | When a NP contribution to the 0 ν β decay is concerned, the situation can be quite complicatedbecause it may compete with the standard effect (i.e., the one from the three light Majorana8 [ ◦ ] ρ [ ◦ ] m [eV] m [eV] m [eV] m [eV] m [eV] m [eV] F = 1 F = 2 F = 1 F = 2 NMO, |h m i ee | = 0 . |h m i ee | = 0 .
005 eV NMO, |h m i ee | = 0 .
05 eVIMO, |h m i ee | = 0 .
02 eV IMO, |h m i ee | = 0 .
05 eV IMO, |h m i ee | = 0 . Figure 5: The regions of the smallest neutrino mass ( m or m ) and the Majorana phase ρ asconstrained by an “observed” value of |(cid:104) m (cid:105) ee | . In the NMO case |(cid:104) m (cid:105) ee | = 0 . .
005 eVand 0 .
05 eV are taken, and in the IMO case |(cid:104) m (cid:105) ee | = 0 .
02 eV, 0 .
05 eV and 0 . F .neutrinos as discussed above) either constructively or destructively. If the NP effect is significantenough, the simple relation between Γ ν and |(cid:104) m (cid:105) ee | in Eq. (1) has to be modified. This will makethe interpretation of a discovery or null result of the 0 ν β decay more uncertain. Here we aim tostudy the issue in a model-independent way. Namely, we parameterize the possible NP contributionto |(cid:104) m (cid:105) ee | in terms of its modulus and phase relative to the standard contribution, without goinginto details of any specific NP model [19, 20].An interesting and very likely case is that different contributions can add in a coherent wayso that their constructive or destructive interference may happen [19, 21]. If the helicities of twoelectrons emitted in the NP-induced 0 ν β channel are identical to those in the standard channel,then the overall rate of the 0 ν β decay in Eq. (1) can be modified in the following way:Γ ν = G ν ( Q, Z ) (cid:12)(cid:12) M ν (cid:104) m (cid:105) ee + M ν NP m (cid:12)(cid:12) ≡ G ν ( Q, Z ) (cid:12)(cid:12) M ν (cid:12)(cid:12) (cid:12)(cid:12) (cid:104) m (cid:105) (cid:48) ee (cid:12)(cid:12) , (9)where M ν NP denotes the NME subject to the NP process, m is a particle-physics parameterdescribing the NP contribution, and (cid:104) m (cid:105) (cid:48) ee represents the effective Majorana mass term defined as (cid:104) m (cid:105) (cid:48) ee = m U e + m U e + m U e + m NP (10)with m NP ≡ m M ν NP /M ν . Unless M ν NP is identical with M ν like the case of NP coming fromthe light sterile neutrinos [22], m NP generally differs from one isotope to another. Hence using9 og | h m i ee | e V l og | m N P | e V l og | m N P | e V l og m e V l og m e V NMO IMO
Figure 6: The lower (blue) and upper (light orange) bounds of |(cid:104) m (cid:105) (cid:48) ee | as functions of m (or m )and | m NP | in the NMO (or IMO) case.different isotopes to detect the 0 ν β decays is helpful for us to learn whether there is NP beyondthe standard scenario, but their different NMEs may involve different uncertainties.To see the interference between the NP term m NP = | m NP | e i φ NP and the standard one (cid:104) m (cid:105) ee in |(cid:104) m (cid:105) (cid:48) ee | , we plot the lower and upper bounds of |(cid:104) m (cid:105) (cid:48) ee | vs m (or m ) and | m NP | in the NMO (orIMO) case in Fig. 6. For given values of m (or m ) and | m NP | , the lower and upper bounds of |(cid:104) m (cid:105) (cid:48) ee | can be expressed as (cid:12)(cid:12) (cid:104) m (cid:105) (cid:48) ee (cid:12)(cid:12) upper = m | U e | + m | U e | + m | U e | + | m NP | , (cid:12)(cid:12) (cid:104) m (cid:105) (cid:48) ee (cid:12)(cid:12) lower = max (cid:110) , m i | U ei | − (cid:12)(cid:12) (cid:104) m (cid:105) (cid:48) ee (cid:12)(cid:12) upper , | m NP | − (cid:12)(cid:12) (cid:104) m (cid:105) (cid:48) ee (cid:12)(cid:12) upper (cid:111) (11)for i = 1 , ,
3. These results can be directly derived with the help of the “coupling-rod” diagramof the 0 ν β decay in the presence of the NP [9]. By setting m NP →
0, we simply arrive at theresults of |(cid:104) m (cid:105) ee | obtained before in the standard three-flavor scheme [13]. Some comments on ournumerical results are in order.(1) The parameter space in the NMO case can be divided into three regions according to theprofile of the lower bound of |(cid:104) m (cid:105) (cid:48) ee | : (a) the region with m < .
001 eV and | m NP | < .
001 eV,where the NP contribution is negligibly small and thus |(cid:104) m (cid:105) (cid:48) ee | approximates to (cid:12)(cid:12) (cid:104) m (cid:105) (cid:48) ee (cid:12)(cid:12) (cid:39) |(cid:104) m (cid:105) ee | (cid:38) (cid:12)(cid:12)(cid:12)(cid:12)(cid:113) ∆ m s c − (cid:113) ∆ m s (cid:12)(cid:12)(cid:12)(cid:12) ; (12)(b) the region with m > .
01 eV and |(cid:104) m (cid:105) ee | being still dominant over | m NP | , where |(cid:104) m (cid:105) (cid:48) ee | has alower bound (cid:12)(cid:12) (cid:104) m (cid:105) (cid:48) ee (cid:12)(cid:12) (cid:39) |(cid:104) m (cid:105) ee | (cid:38) (cid:12)(cid:12)(cid:12)(cid:12) m c c − (cid:113) m + ∆ m s c − (cid:113) m + ∆ m s (cid:12)(cid:12)(cid:12)(cid:12) ; (13)and (c) the region with | m NP | being dominant over |(cid:104) m (cid:105) ee | , where the lower bound of |(cid:104) m (cid:105) (cid:48) ee | issimply the value of | m NP | . If | m NP | is comparable in magnitude with |(cid:104) m (cid:105) ee | of the IMO casein the standard three-flavor scheme, it will be impossible to distinguish the NMO case with NP10rom the IMO case without NP by only measuring the 0 ν β decay. This observation would makesense in the following situation: a signal of the 0 ν β decay looking like the IMO case in thestandard scenario were measured someday, but the IMO itself were in conflict with the “available”cosmological constraint on the sum of three neutrino masses. Note also that at the junctions of theaforementioned three regions, |(cid:104) m (cid:105) (cid:48) ee | can be vanishingly small either because |(cid:104) m (cid:105) ee | and | m NP | are both very small or because they undergo a deadly cancellation.(2) The profile of the lower bound of |(cid:104) m (cid:105) (cid:48) ee | in the IMO case is structurally simpler, as shownin Fig. 6. In the region dominated by |(cid:104) m (cid:105) ee | , |(cid:104) m (cid:105) (cid:48) ee | just behaves like |(cid:104) m (cid:105) ee | in the standardscenario and has a lower bound: (cid:12)(cid:12) (cid:104) m (cid:105) (cid:48) ee (cid:12)(cid:12) (cid:39) |(cid:104) m (cid:105) ee | (cid:38) (cid:12)(cid:12)(cid:12)(cid:12) m c c − (cid:113) m + ∆ m s c (cid:12)(cid:12)(cid:12)(cid:12) . (14)On the other hand, |(cid:104) m (cid:105) (cid:48) ee | will be saturated by | m NP | when the latter is dominant over |(cid:104) m (cid:105) ee | . Atthe junction of these two regions, (cid:104) m (cid:105) ee and m NP are comparable in magnitude and have a chanceto cancel each other. This unfortunate possibility would deserve special attention if the IMO wereverified by the cosmological data but a signal of the 0 ν β decay were not observed in an experimentsensitive to the |(cid:104) m (cid:105) ee | interval in the IMO case of the standard scenario. While most of the particle theorists believe that massive neutrinos must be the Majorana fermions,an experimental test of this belief is mandatory. Today a number of 0 ν β -decay experiments areunderway for this purpose. It is therefore imperative to consider how to interpret a discovery ornull result of the 0 ν β decay beforehand, before this will finally turn into reality.In this work we have tried to do so by presenting some new ideas and results which are essentiallydifferent from those obtained before. First, we have introduced a three-dimensional description ofthe effective Majorana mass term |(cid:104) m (cid:105) ee | by going beyond the conventional Vissani graph. Thisnew description allows us to look into the sensitivity of |(cid:104) m (cid:105) ee | (especially its lower bound) to thelightest neutrino mass and two Majorana phases in a more transparent way. For example, we haveshown that it is the Majorana phase ρ ∼ π that may make |(cid:104) m (cid:105) ee | sink into a decline in the NMOcase. Second, we have extended our discussion to all the six effective Majorana masses |(cid:104) m (cid:105) αβ | (for α, β = e, µ, τ ) which are associated with a number of different LNV processes, and presented a setof two-dimensional contour figures for their lower bounds. We stress that such a study makes sensebecause a measurement of the 0 ν β decay itself does not allow us to pin down the two Majoranaphases. Third, we have studied to what extent m (or m ) and ρ can be well constrained provideda discovery of the 0 ν β decay (i.e., a definite value of |(cid:104) m (cid:105) ee | ) is made someday. It is found that thesmaller |(cid:104) m (cid:105) ee | is, the stronger the constraint will be. Finally, the effect of possible NP contributingto the 0 ν β decay has been discussed in a model-independent way. 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