Constraints on Absolute Neutrino Majorana Mass from Current Data

Transcription

1 THE UNIVERSE Vol. 2, No. 3 July-Septeber 2014 Invited Regular Article Constraints on Absolute Neutrino Majorana Mass fro Current Data Yanqi Huang 1 and Bo-Qiang Ma1, 2, 3, 1 School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing , China 2 Collaborative Innovation Center of Quantu Matter, Beijing, China 3 Center for High Energy Physics, Peking University, Beijing , China We present new constraints on the neutrino Majorana asses fro the current data of neutrinoless double beta decay and neutrino flavour ixing. With the latest results of 0ν progresses fro various isotopes, including the recent calculations of the nuclear atrix eleents, we find that the strongest constraint of the effective Majorana neutrino ass is fro the 136 Xe data of the EXO-200 and KaLAND-Zen collaborations. Further ore, by cobining the 0ν experiental data with the neutrino ixing paraeters fro new analyses, we get the ass upper liits of neutrino ass eigenstates and flavour eigenstates and suggest several relations aong these neutrino asses. PACS nubers: Pq, Er, Ff, L 1. Introduction The experients with solar, atospheric, reactor and accelerator neutrinos in recent decades [1 9] have provided copelling evidences for the neutrino oscillation. In the threegeneration neutrino fraework, the neutrino oscillation is caused by the nonzero neutrino asses due to the ixing aong the ass eigenstates in the flavour eigenstates. There have been a nuber of new easureents on the neutrino ass splitting recently [10 13]. However, the absolute ass scale of neutrinos is still unclear yet. It is also not clarified whether neutrinos are Dirac or Majorana ferions. Fortunately, the neutrinoless double beta decay (0ν) process provides us soe extra inforation on neutrino properties. The double beta decay of eleven nuclei, whose Q-values are larger than 2 MeV such as 76 Ge, 82 Se, 100 Mo, 130 Te and 136 Xe, have been studied up to date [14 22]. For alost all the eleven isotopes, the twoneutrino double beta decay (2ν) processes (A, Z) (A, Z + 2) + 2e + 2 ν e, (1) Eail: abq@pku.edu.cn are observed. The 0ν process (A, Z) (A, Z + 2) + 2e, (2) if occurs, can serve as signals of new physics such as the total lepton nuber violation and the Majorana nature of neutrinos. The only observation of decay events was reported in the year of 2001 by the Heidelberg-Moscow experient [14, 15], which claied the half-life tie T1/2 0ν = yr on 76 Ge at 68% CL. But the data fro the GERaniu Detector Array (GERDA) experient [21] in 2013 indicate that the half-life tie of 76 Ge should be no lower than yr (90% CL), with the Heidelberg-Moscow result unconfired. Recently, the Enriched Xenon Observatory (EXO- 200) collaboration [22] and the Kaioka Liquid Scintillator Anti-Neutrino Detector-Zero Neutrino Double Beta (KaLAND-Zen) collabaration [19] released their new results fro which we can obtain the strongest constraint on neutrino Majorana ass. The lower liits on 0ν half tie of several isotopes reported by recent experients are listed in Table I. We can only obtain the effective Majorana neutrino ass fro the half-life tie of 0ν process. In order to get constraints of each ass 65

2 Invited Regular Article THE UNIVERSE Vol. 2, No. 3 July-Septeber 2014 eigenstates and flavour eigenstates, it is necessary to cobine the data of double beta decay experients and oscillation experients, since the oscillation provides us two ass square differences, as well as the ixing angles θ 12, θ 23, θ 13 and the CP phase δ in the Pontecorvo-Maki- Nakawaga-Sakata (PMNS) ixing atrix [23, 24]. In this paper we provide new constraints on absolute neutrino Majorana ass fro analysis of the latest data of 0ν processes, cobined with also new results of neutrino ixing paraeters. In Sec. 2, we obtain the upper liits of the effective Majorana neutrino ass. Then we calculate the ass constraints of different eigenstates by using the CP phase as a variable in Sec. 3 and Sec. 4. In Sec. 5, we present the conclusion and soe discussions on soe possible relations aong neutrino asses. 2. Neutrino Effective Majorana Mass The half-life tie of oν process is deterined by three kinds of contributions, i.e., the particle ass factor, the phase space integral factor G 0ν and the nuclear atrix eleent (NME) M 0ν. Both the latter two vary with different isotopes. If only considering the contribution of the light Majorana neutrino, the half-life tie of a given isotope (A, Z) is written as [25, 26] [T 0ν ( )] 1 0ν 2 = G 01(Q, Z) M 0ν(A) 2, (3) e where e is the rest ass of electron. Here we introduce the effective Majorana neutrino ass 0ν = j U 2 ej j, (4) to represent the contribution of the Majorana neutrino during the decay process. U lj (l = e, µ, τ, j = 1, 2, 3) are the eleents of product atrix of the PMNS atrix V and a diagonal atrix which contains two additional Majorana CP phases α 2 and α 3, c 12 c 13 s 12 c 13 s 13 e iδ V (θ 12, θ 23, θ 13, δ) = c 12 s 23 s 13 e iδ s 12 c 23 s 12 s 23 s 13 e iδ + c 12 c 13 s 23 c 13, (5) c 12 c 23 s 13 e iδ + s 12 s 23 s 12 c 23 s 13 e iδ c 12 s 23 c 23 c U = V 0 e ı α e ı α 31 2, (6) where s ij and c ij denote sin θ ij and cos θ ij. The flavour ixing of the three neutrino generations is written as ν l (x) = j U lj ν j (x), (7) where the subscripts l = e, µ, τ denote the flavour eigenstates and j = 1, 2, 3 represent the ass eigenstates. For each of the five isotopes entioned above in Table I, the phase space integral factor is deterined and has been calculated [26 28]. However, the NMEs are ore coplicated and we can only obtain the by approxiation ethods. There are several coon approaches, e.g., the interacting shell odel (ISM) [29], energy density functional (EDF) ethod [30], interacting boson odel (IBM) [31], quasiparticle rando phase approxiation (QRPA) [32], self-consistent renoralized quasiparticle rando phase approxiation (SRQRPA) [33] and 66

3 THE UNIVERSE Vol. 2, No. 3 July-Septeber 2014 Invited Regular Article Table I. The lower liits of half-life tie T 0ν 1/2 of several isotopes observed by recent experients [15 22]. isotope experient T 0ν 1/2 [1024 yr] 76 Ge Heidelberg-Moscow 9.6 GERDA Se NEMO Mo NEMO Te CUORICINO Xe KaLAND-Zen KaLAND-Zen KaLAND-Zen EXO Table II. The phase space integral factors [26 28] and the nuclear atrix eleents fro soe approxiation ethods [29 34] for these five isotopes. isotope 76 Ge 82 Se 100 Mo 130 Te 136 Xe G 01 [10 14 yr 1 ] ISM(U) EDF(U) IBM QRPA-A SPQRP-A SPQRPA-B SkM-HFB-QRPA the Skyre Hartree-Fock-Bogoliubov quasiparticle rando phase approxiation (SkM-HFB- QRPA) [34]. The NMEs fro soe of these ethods together with the phase space factors are listed in Table II. According to Eq. (3), we can obtain the expression of the effective Majorana neutrino ass with the half-life tie as known quantity, 0ν = e M 0ν G 01 T 0ν 1/2. (8) Table III. The upper liits of effective Majorana neutrino ass fro recent experients, with,1 0ν and 0ν,2, which are related to the NMEs, denoting the inial and axial values of the ass upper liits. isotope experient 0ν,1 [ev] 0ν,2 [ev] 76 Ge Heidelberg-Moscow GERDA Se NEMO Mo NEMO Te CUORICINO Xe KaLAND-Zen KaLAND-Zen KaLAND-Zen EXO Table IV. The global fit of neutrino ixing paraeters [37], with NH and IH denoting the noral and inverted hierarchies of the ass eigenstates. The second square difference of ass is defined as 2 = 2 3 ( )/2 in noral hierarchy, and 2 in inverted hierarchy. paraeter best fit±1σ 3σ range 2 21(10 5 ev 2 ) (10 3 ev 2 )(NH) (10 3 ev 2 )(IH) sin 2 θ 12 (NH or IH) sin 2 θ 23 (NH) sin 2 θ 23 (IH) sin 2 θ 13 (NH) sin 2 θ 13 (IH) The results of the constrains on 0ν fro different experients are showed in Table III. Coparing with the half-life tie data in Table I, we find that the EXO-200 and KaLAND-Zen experients provide the strongest constraint on 67

4 Invited Regular Article THE UNIVERSE Vol. 2, No. 3 July-Septeber 2014 (a) Constrains on Mass and Flavour Eigenstates (ev) (ev) (b) Now we already obtain an upper liit of 0ν. According to Eq. (4), if considering the ixing paraeters we get to the constraints of three ass eigenstates of Majorana neutrino. Here we use the global fitting results [36, 37] of ixing paraeters listed in Table IV as the inputs. The latest T2K result [38] suggests the negative axial violation of CP phase δ = 90 or 270, which agrees with the prediction of the axial CP violation (δ = ±90 ) fro the µ-τ interchange syetry [39]. In our calculation, we take the CP phase δ as a paraeter too. Since there is little inforation about the Majorana CP phases, we hypothesize that α 21 = α 31 = 0. Thus, we have only three ass eigenvalues as unknown quantities in the three independent equations below Fig. 1. The constraints on neutrino Majorana ass eigenvalues, ass upper liits i (i=1,2,3) as functions of the CP phase δ. The upper figure (a) corresponds to the noral hierarchy, and the figure (b) for inverted hierarchy. The error bars denote the ±1σ range influences of the ixing paraeters. Since 1 and 2 are very close to each other, the two lines appear to overlap. the effective Majorana neutrino ass 0ν < 0ν,in = ev, (9) based on the 136 Xe data and EDF approxiation approach. It is necessary to notice that though the GERDA group reports a strong lower liit of the 0ν decay half-life tie, it does not provide the strongest constraint on Majorana neutrino ass [35], since the phase space integral factor of 136 Xe is uch larger than that of 76 Ge. 0ν = (c 12 c 13 ) (s 12 c 13 ) s 2 13e 2iδ 3, (10) = 2 21, (11) = (NH), (12) = (IH). (13) Then we draw the curves that show how the upper liits of neutrino ass eigenvalues 1, 2, 3 change with the CP phase δ, as seen fro Fig. 1. In the figure we can find that these three functions i (δ) (i = 1, 2, 3) are all trigonoetric functions with the sae phase. The period of the i (δ) is not 2π but π. The π period is resulted fro the fact that the δ appears as 2δ for in Eq. (9). If δ satisfies the axial CP violation assuption, naely δ = ±90, the sign of δ has no influence for constraining the neutrino Majorana ass in this way, for the range of ±90 is just the right period of one π. The first two ass eigenvalues 1 and 2 are very close to each other. The third one has a larger difference with respect to the, but the distinction is still within 10 ev. The error bars 68

5 THE UNIVERSE Vol. 2, No. 3 July-Septeber 2014 Invited Regular Article in the figure reflect the ±1σ range influences of the ixing paraeters. This error range alost covers the difference between 1 and 2. This indicates that the three ass eigenstates ν i (i = 1, 2, 3) are degenerate: ν 1 and ν 2 are strongly degenerate, and ν 3 is a little bit weaker. That helps us to understand why the ixing angle between ν 1 and ν 2 is large. On the other hand, since the level of error is related to 0ν, it is necessary to iprove the observation on 0ν process, for getting stronger ass upper liits or even exactly events to enrich the knowledge of the neutrino ass eigenvalues. Flavour eigenstates In the threegeneration neutrino fraework, there are three ass eigenstates that evolute with the tie and three flavour eigenstates that take part in the weak interaction. Equipped with the upper liits of each ass eigenvalue fro the effective Majorana neutrino ass, any calculations becoe feasible. It is the flavour eigenstate ass constraints that interest us ost since they have the direct correlation to the dynaic processes. We introduce the ass operator ˆ that satisfies the relation j = ν j ˆ ν j, (14) where j = 1, 2, 3 for the ass eigenstates. Fro Eq. (7), the asses of flavour eigenstates, i.e., l = ν l ˆ ν l (l = e, µ, τ), are written as e = c 12 c s 12 c s 13 e iδ 2 3, µ = c 12 s 23 s 13 e iδ s 12 c (15) + s 12 s 23 s 13 e iδ + c 12 c s 23 c , (16) (ev) (ev) (a) e (b) e Fig. 2. The constraints on asses of Majorana neutrino flavour eigenstates. We draw the curves by regarding the CP violate angle δ as an independent variable, with l (l = e, µ, τ) denoting the ass upper liits. The upper figure (a) corresponds to the noral hierarchy, and the figure (b) for inverted hierarchy. 1. The l -δ relations are also trigonoetriclike functions. Its frequency is the sae as the ass eigenstate. It is easy to understand since li i (δ) (i = 1, 2, 3) are trigonoetric functions of δ, and the coefficients of cos δ ters in Eqs. (16) and (17) cancel each other due to the facts that θ and that li 1 = li 2. τ = c 12 c 23 s 13 e iδ + s 12 s s 12 c 23 s 13 e iδ c 12 s c 23 c We use δ as a paraeter to draw the curves of the functions l (δ) (l = e, µ, τ), as shown in Fig. 2. Fro the figure we find that: (17) 2. In the NH condition, the ass upper liits of flavour eigenstate asses have the hierarchy li e < li µ < li τ. (18) But in the IH condition, the hierarchy changes into li τ < li µ < li e. (19) 69

6 Invited Regular Article THE UNIVERSE Vol. 2, No. 3 July-Septeber 2014 Since 3 is the largest (or sallest) one in the NH (or IH) case, Eqs. (18) and (19) indicate that the dependence on 3 becoes lager when the generation nuber increases. 3. We define the differences between ass upper liits of flavour eigenstates as µe = µ e, (20) τµ = τ µ. (21) The absolute values of these two differences are the sae in the noral hierarchy case and inverted hierarchy case, i.e., µe (NH) µe (IH) > 0, (22) τµ (NH) τµ (IH) > 0. (23) And e is alost the sae in the two cases. Fro 0ν processes we obtain the upper liit for the sued neutrino ass l = j 0.37 ev (NH) or 0.35 ev (IH), l j (24) which is coparable with the bound 0.23 ev obtained fro cosological observations [40]. 4. Conclusions In conclusion, by analysing the latest results of 0ν processes fro various isotopes, we give a new constraint of the effective Majorana neutrino ass in 136 Xe with the EXO-200 and KaLAND-Zen data [19, 22]. The strongest ass upper liit of 0ν ranges fro to ev depending on the different approxiation ethods when calculating nuclear atrix eleents. Further ore, cobining with the global fitting results of neutrino ixing paraeters, we calculate the upper ass liits of three ass eigenstates ν 1, ν 2, ν 3 and three flavour eigenstates ν e, ν µ and ν τ. The ass eigenvalue upper liits are very close to each other, and are periodic with period π. There are soe hierarchal and invariant relations which ight indicate inner relations aong the flavour eigenstates. The studies on the nature of neutrinos are beneficial to explore new physics. The 0ν decay process is iportant to explore whether the neutrinos are Majorana ferions or Dirac ferions, and it is also very useful to provide the liits of their absolute ass scales, as reflected fro our analysis. Such kind of experients can provide ore strong constraints on the effective Majorana neutrino ass in the future, and they can also clarify possible relations aong ass eigenstates and flavour eigenstates of neutrinos. Acknowledgents This work is supported by National Natural Science Foundation of China (Grants No and No ) and the National Fund for Fostering Talents of Basic Science (Grant Nos. J and J ). It is also supported by the Principal Fund for Undergraduate Research of Peking University. References [1] Q. R. Ahad et al. [SNO Collaboration], Phys. Rev. Lett. 89, (2002), nucl-ex/ [2] M. Altann et al. [GNO Collaboration], Phys. Lett. B616, 174 (2005), hep-ex/ [3] S. Fukuda et al. [Super-Kaiokande Collaboration], Phys. Lett. B539, 179 (2002), hepex/ [4] T. Araki et al. [KaLAND Collaboration], Phys. Rev. Lett. 94, (2005), hepex/ [5] M. H. Ahn et al. [K2K Collaboration], Phys. Rev. D74, (2006), hep-ex/ [6] P. Adason et al. [MINOS Collaboration], Phys. Rev. Lett. 101, (2008), arxiv: [7] P. Adason et al. [MINOS Collaboration], Phys. Rev. Lett. 107, (2011), arxiv:

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