Rodejohann, W.

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Vol. 43 (2012) ACTA PHYSICA POLONICA B No 1NEUTRINO-LESS DOUBLE BETA DECAY:NEUTRINO PHYSICS∗Werner RodejohannMax-Planck-Institut für KernphysikPostfach 103980, 69029 Heidelberg, Germany(Received December 13, 2011)Some neutrino physics aspects of neutrino-less double beta decay arediscussed: this includes the possibility to test or rule out the inverted neu-trino mass ordering, distinguishing neutrino mass models, or the effects oflight sterile neutrinos.DOI:10.5506/APhysPolB.43.71PACS numbers: 14.60.Pq, 11.30.Fs, 14.60.St1. IntroductionNeutrino-less double beta decay (0νββ) is a process of fundamental im-portance for particle physics. It is defined as the transition of a nucleus intoa nucleus with proton number larger by two units, and the emission of twoelectrons(A,Z)→ (A,Z + 2) + 2 e− (0νββ) . (1)There are no leptons in the initial state, but two in the final state. Observa-tion of 0νββ would, therefore, show that lepton number, an accidental andclassical symmetry of the Standard Model of particle physics, is violated byNature. The process, therefore, stands on equal footing with baryon numberviolation, i.e. proton decay. For this reason a huge amount of experimentaland theoretical activity is pursued in order to detect and predict the process(see e.g. [1, 2, 3, 4] for recent reviews).In order to extract more specific physics from a signal or an improvedsignal, one has to assume something about the underlying physics. Here onehas basically two choices:∗ Presented at the XXXV International Conference of Theoretical Physics “Matter tothe Deepest”, Ustroń, Poland, September 12–18, 2011.(71)72 W. Rodejohann1. Standard interpretation:neutrino-less double beta decay is mediated by light and massive Ma-jorana neutrinos (the ones which oscillate) and all other mechanismspotentially leading to 0νββ give negligible or no contribution;2. Non-Standard interpretations:neutrino-less double beta decay is mediated by some other lepton num-ber violating physics, and light massive Majorana neutrinos (the oneswhich oscillate) potentially leading to 0νββ give negligible or no con-tribution.Here we will focus on the standard interpretation, which is presumably thebest motivated one. However, one always has to keep in mind that particlesother than light Majorana neutrinos can mediate the decay, such as parti-cles associated with R-parity violating SUSY, left–right symmetric theories,Higgs triplets, heavy neutrinos, leptoquarks, etc. A recent review on allpossibilities including methods to distinguish them from each other, with acomplete list of references, can be found in [3].Experimentally, the life-time reach depends strongly on whether the ex-periment is background-dominated, or background-free(T 0ν1/2)−1∝aM ε t without backgrounda ε√M tB∆Ewith background, (2)with B background index in counts/(keV kg yr), ∆E is the energy reso-lution, ε the efficiency, a the abundance of the isotope, M the mass andt the measurement time. The “master formula” for 0νββ mediated by amechanism x is (T 0ν1/2)−1= Gx(Q,Z) |Mx(A,Z) ηx|2 , (3)with Gx(Q,Z) the (calculable) phase space factor,Mx(A,Z) the (problem-atic) nuclear physics matrix element and ηx the (interesting) particle physicsparameter. Therefore, an improvement of a factor 2 of the particle physicsparameter implies a total factor of 16 improvement of background index,energy resolution, mass and time. Given that upcoming next generationexperiments are at the very end of what is currently technologically feasible,this factor is far from trivial.Neutrino-less Double Beta Decay: Neutrino Physics 732. Neutrino physics and 0νββ2.1. General aspectsThe quark level Feynman diagram for 0νββ is shown in Fig. 1. Neutrinoshave to be massive and Majorana particles for the process to occur. Theamplitude is proportional to the “effective mass”〈mee〉 =∣∣∣∑U2eimi∣∣∣ = ∣∣∣∣∣∣m(1)ee ∣∣∣+ ∣∣∣m(2)ee ∣∣∣ e2iα + ∣∣∣m(3)ee ∣∣∣ e2iβ∣∣∣ , (4)and is a function of the 3 neutrino masses mi, the lepton mixing matrixelements from the first row (or θ12 and θ13) and the two Majorana phases.All in all, it depends on 7 out of the 9 parameters of neutrino physics(only δ and θ23 do not appear) and therefore contains a lot of information.A typical plot of 〈mee〉 as a function of the smallest mass can be seen inFig. 1. The dark gray/blue (light gray/yellow) areas are for the normal(inverted) ordering, while the darker areas inside the lighter ones are forthe best-fit values of the oscillation parameters (the range is induced by theunknown Majorana phases).WνiνiWdLdLuLe−Le−LuLUeiqUeiFig. 1. Left: quark level Feynman diagram for the standard interpretation ofneutrino-less double beta decay. Right: effective mass as function of the small-est neutrino mass.As can be seen from Eq. (3), to go from a life-time measurement or limitto the effective mass, precise knowledge of the nuclear matrix elements isnecessary. Unfortunately, a large uncertainty is present here, stemming fromdifferent calculational approaches (such as QRPA, NSM, IBM, etc.), anduncertainties within each individual approach. The procedure is to interpretthe different results for the matrix elements as “theory error”. A typicalcompilation of results, carefully checked for convention issues, is shown inFig. 2, taken from Ref. [5], where the references and details can be found.74 W. Rodejohann 0 2 4 6 8 1048Ca76Ge82Se96Zr100Mo110Pd116Cd124Sn130Te136Xe150NdNMEIsotopeNSMQRPA (Tue)QRPA (Jy)IBMIBMGCMPHFBPseudo-SU(3) 0.1 1 1048Ca76Ge82Se96Zr100Mo110Pd116Cd124Sn130Te136Xe150NdT1/2 [1027 y]Isotope0.27 < sin2(θ12) < 0.38Fig. 2. Top: Compilation of nuclear matrix element results. Bottom: necessaryhalf-life to rule out the inverted ordering. The horizontal (yellow) lines indicatefuture experimental limits.2.2. The inverted hierarchyAs can be seen from Fig. 1, the effective mass in the case of the invertedmass ordering cannot vanish. Hence, nature provides us two importantscales, namely the minimal and maximal effective mass for m3 = 0〈mee〉IHmax ' cos2 θ13√∆m2A and 〈mee〉IHmin ' cos2 θ13√∆m2A cos 2θ12 . (5)If we knew that the mass ordering is inverted, then going below 〈mee〉IHminwould rule out the Majorana nature of neutrinos. Alternatively, going below〈mee〉IHmin would rule out the inverted ordering under the assumptions thatneutrinos are Majorana particles. Though θ12 is the best-known neutrinoparameter, its current 3σ range implies a factor of 2 range for 〈mee〉IHmin.Recall from above that this factor 2 implies a total factor 16 in backgroundindex, energy resolution, mass and time. Hence, a better knowledge of thisNeutrino-less Double Beta Decay: Neutrino Physics 75angle would be of crucial importance to evaluate the physics potential ofneutrino-less double beta decay experiments. Fig. 2, taken from [5], showsthe necessary half-life to rule out the inverted ordering. The uncertaintyfrom the nuclear matrix elements and θ12 is taken into account. Though theinverted ordering cannot be fully covered with upcoming next generationexperiments, we will surely enter its regime [5].2.3. Ruling out flavor symmetry modelsThe peculiar mixing scheme that leptons display forces one to introduceflavor symmetries [6]. Ref. [7] lists about 70 different models which all givetri-bimaximal mixing. The natural question on how to distinguish them fromeach other could be answered by neutrino mass experiments such as neutrino-less double beta decay (in the standard interpretation). The reason is thatvery often “sum-rules” between the neutrino masses are induced within themodels. While masses cannot be predicted by flavor symmetries, relationsbetween them can very well be predicted. Examples are m1 + m2 = m3,or 2/m2 + 1/m3 = 1/m1. Here the masses are understood to be complex,i.e. the absolute values and the Majorana phases are constrained by thesesum-rules. Consequently, the effective mass will only take specific valuesin its general parameter space, which can be used to rule out some of themodels. Fig. 3 shows the results of such an analysis, taken from Ref. [8].10-310-210-1<mee> (eV)0.01 0.1mβ (eV)10-310-210-10.01 0.13σ 30% error3σ exactTBM exactNormal Inverted2/m2  + 1/m3  = 1/m11/m1  + 1/m2  = 1/m3Fig. 3. Direct neutrino parametermβ as measurable in KATRIN versus the effectivemass for two specific neutrino mass sum-rules, 2/m2 + 1/m3 = 1/m1 and 1/m1 +1/m2 = 1/m3.76 W. Rodejohann2.4. Sterile neutrinosMore and more evidence accumulates which hints to the existence ofsterile neutrinos with mass at the eV-scale. The most recent one is the “re-actor anomaly” [9], which denotes the outcome of a re-evaluation of neutrinospectra from nuclear reactors. A systematic upward shift of about 3% wasfound, and the results of past very short baseline reactor experiments, in-terpreted back then as negative, were in fact observing a deficit. This isin agreement with effects of sterile neutrinos with ∆m2st ' 1 eV2, and mix-ing of the order of 0.15 with electron neutrinos [9]. What is the effect forneutrino-less double beta decay?For one sterile neutrino, the effective mass is now a sum of four terms,the additional term quantifying the sterile contribution〈mee〉 = | |Ue1|2m1 + |Ue2|2m2 e2iα +∣∣U2e3∣∣m3 e2iβ︸ ︷︷ ︸〈mee〉act+ |Ue4|2m4 e2iΦ1︸ ︷︷ ︸〈mee〉st| . (6)Here |〈mee〉act| is the contribution discussed so far, and 〈mee〉st the sterileone. Suppose the easiest case that there is only one sterile neutrino, and thatit is heavier than the 3 active ones (the complete list of possibilities includingthe case of two sterile neutrinos can be found in Refs. [10,11]). Recall fromabove that |〈mee〉actNH| can vanish and that |〈mee〉actIH | cannot vanish, having atypical value of 0.02 eV. With the typical sterile neutrino parameters givenabove one has that∣∣〈mee〉st∣∣ '√∆m2st |Ue4|2 ' 0.02 eV{∣∣〈mee〉actNH∣∣'∣∣〈mee〉actIH ∣∣ . (7)Therefore, if the active neutrinos are normally ordered, the effective masscannot vanish anymore, whereas it can vanish when they are inversely or-dered [11]. Hence, the usual standard phenomenology has been completelyturned around. Given that the addition of light sterile neutrinos is pre-sumably the simplest modification of the standard picture, it shows thatwhen discussing the physics potential of 0νββ one should carefully list one’sassumptions.3. Final commentsThe field of neutrino-less double beta decay is currently entering a veryexciting stage. The strongest constraint on the process (half-life larger than1.9×1025 yrs for 76Ge [12]) is the leading limit since 10 years, and will finallybe improved in 2012. A variety of experiments with different experimentaltechniques and different isotopes is upcoming. Multi-isotope and -techniqueNeutrino-less Double Beta Decay: Neutrino Physics 77determination will be crucial to increase the credibility of a signal, helpful totest calculations of nuclear matrix elements, and distinguish the mechanismof 0νββ.I would like to thank Marek Biesiada for his kind invitation and warmhospitality in Ustroń. This work was supported by the ERC under theStarting Grant MANITOP and by the DFG in the Transregio 27 “Neutrinosand Beyond”.REFERENCES[1] F.T. Avignone, S.R. Elliott, J. Engel, Rev. Mod. Phys. 80, 481 (2008)[arXiv:0708.1033 [nucl-ex]].[2] A. Giuliani, Acta Phys. Pol. B 41, 1447 (2010).[3] W. Rodejohann, Int. J. Mod. Phys. E20, 1833 (2011) [arXiv:1106.1334[hep-ph]].[4] J.J. Gomez-Cadenas et al., arXiv:1109.5515 [hep-ex].[5] A. Dueck, W. Rodejohann, K. Zuber, Phys. Rev. D83, 113010 (2011)[arXiv:1103.4152 [hep-ph]].[6] For reviews, see G. Altarelli, F. Feruglio, Rev. Mod. Phys. 82, 2701 (2010)[arXiv:1002.0211 [hep-ph]]; H. Ishimori et al., Prog. Theor. Phys. Suppl.183, 1 (2010) [arXiv:1003.3552 [hep-th]].[7] J. Barry, W. Rodejohann, Phys. Rev. D81, 093002 (2010)[arXiv:1003.2385 [hep-ph]].[8] J. Barry, W. Rodejohann, Nucl. Phys. B842, 33 (2011) [arXiv:1007.5217[hep-ph]].[9] G. Mention et al., Phys. Rev. D83, 073006 (2011) [arXiv:1101.2755[hep-ex]].[10] S. Goswami, W. Rodejohann, J. High Energy Phys. 0710, 073 (2007)[arXiv:0706.1462 [hep-ph]].[11] J. Barry, W. Rodejohann, H. Zhang, J. High Energy Phys. 1107, 091 (2011)[arXiv:1105.3911 [hep-ph]].[12] H.V. Klapdor-Kleingrothaus et al., Eur. Phys. J. A12, 147 (2001)[arXiv:hep-ph/0103062v1].

Neutrino-less Double Beta Decay: Neutrino Physics

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Neutrino-less Double Beta Decay: Neutrino Physics

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