Lecture 2: Is the total lepton number conserved? Are neutrinos Dirac or Majorana fermions?

Here is then what we know today about the masses and mixing of the elementary fermions. The neutrino sector is really strange, i.e. very different compared to the charged fermions. CKM matrix is nearly diagonal PMNS matrix is `democratic’, i.e. matrix elements are of similar magnitude, except Ue3 which is evaluated here near the middle of its allowed range Neutrinos are many (at least 6) orders of magnitude lighter than the charged fermions.

To solve the dilemma of `unnaturally’ small neutrino mass we can give up on renormalizability and add operators of dimension d > 4 that are suppressed by inverse powers of some scale L but are consistent with the SM symmetries. Weinberg already in 1979 (PLR 43, 1566) showed that there is only one dimension d=5 gauge-invariant operator given the particle content of the standard model: L(5) = C(5)/L (LceH)(HTeL) +h.c. Here Lc = LTC, where C is charge conjugation and e = -it2. This operator clearly violates the lepton number by two units and represents neutrino Majorana mass L(M) = C(5)/L v2/2 (nLc nL) + h.c. If L is larger than v, the Higgs vacuum expectation value, the neutrinos will be `naturally’ lighter than the charged fermions.

The energy scale L is more or less the energy above which the effective operator expressions above are no longer valid. In order to estimate the magnitude of L suppose that C(5) ~ O(1) and neutrino mass ~ 0.1 eV. Then ~ v2/mn ~ 1015 GeV It is remarkable, but perhaps a coincidence, that this scale L is quite near the scale at which the running gauge coupling constants meet MGUT ~ 1015-16 GeV.

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A summary of various ways the mass terms can be constructed is shown here following the lectures by S. Willenbrock, hep-ph/0410370

{ } See-Saw Mechanism NR The most popular theory of why neutrinos are so light is the — See-Saw Mechanism (Gell-Mann, Ramond, Slansky (1979), Yanagida(1979), Mohapatra, Senjanovic(1980)) { Familiar light neutrino  Very heavy neutrino } NR Here are some formulae using the formalism just shown trying to explain what it is all about.

How can we tell whether the total lepton number is conserved? A partial list of processes where the lepton number would be violated: Neutrinoless bb decay: (Z,A) -> (Z+-2,A) + 2e(-+), T1/2 > ~1025 y Muon conversion: m- + (Z,A) -> e+ + (Z-2,A), BR < 10-12 Anomalous kaon decays: K+ -> p-m+m+ , BR < 10-9 Flux of ne from the Sun: BR < 10-4 Flux of ne from a nuclear reactor: BR < ? Observing any of these processes would mean that the lepton number is not conserved, and that neutrinos are massive Majorana particles. It turns out that the study of the 0nbb decay is by far the most sensitive test of the total lepton number conservation, so we restrict further discussion to this process.

APS Joint Study on the Future of Neutrino Physics (2004) We recommend, as a high priority, a phased program of sensitive searches for neutrinoless double beta decay (first on the list of recommendations) The answer to the question whether neutrinos are their own antiparticles is of central importance, not only to our understanding of neutrinos, but also to our understanding of the origin of mass.

Whatever processes cause 0, its observation would imply the existence of a Majorana mass term: Schechter and Valle,82 0 e– u d ()R L W By adding only Standard model interactions we obtain (n)R  (n)L Majorana mass term Hence observing the 0nbb decay guaranties that n are massive Majorana particles.

136Xe and 136Ce are stable against b decay, but unstable Double bb decay is observable because even-even nuclei are more bound than the odd-odd ones ( due to the pairing interaction) Atomic masses of A=136 nuclei 136Xe and 136Ce are stable against b decay, but unstable against bb decay (b-b- for 136Xe and b+b+ for 136Ce)

In double beta decay two neutrons imbedded in the ground state of an even-even nucleus are simultaneously transformed into two protons plus two electrons (plus possibly something else). The phenomenon exists thanks to the pairing interaction between nucleons that makes even-even nuclei more bound than the odd-odd ones. The 2nbb decay, in which two electron plus two antineutrinos are emitted, is allowed by the usual rules, while the neutrinoless decay 0nbb violates lepton number conservation and is forbidden in the Standard Model.

Symbolic representation of the two bb decay modes virtual state of the intermediate nucleus virtual state of the intermediate nucleus From G. Gratta

Candidate Nuclei for Double Beta Decay Q (MeV) Abund.(%) 48Ca48Ti 4.271 0.187 76Ge76Se 2.040 7.8 82Se82Kr 2.995 9.2 96Zr96Mo 3.350 2.8 100Mo100Ru 3.034 9.6 110Pd110Cd 2.013 11.8 116Cd116Sn 2.802 7.5 124Sn124Te 2.228 5.64 130Te130Xe 2.533 34.5 136Xe136Ba 2.479 8.9 150Nd150Sm 3.367 5.6 All candidate nuclei on this list have Q > 2MeV. The ones with an arrow are used in the present or planned large mass experiments.

One can distinguish the two modes by measuring the sum electron energy One can distinguish the two modes by measuring the sum electron energy. Ultimately, though, the 2n decay is an unavoidable background to the 0nbb. With 2% resolution: ratio 1:106 10 minutes ratio 1:100 from S. Elliott

If (or when) the 0nbb decay is observed two problems must be resolved: What is the mechanism of the decay, i.e., what kind of virtual particle is exchanged between the affected nucleons (or quarks)? b)How to relate the observed decay rate to the fundamental parameters, i.e., what is the value of the corresponding nuclear matrix element?

What is the nature of the `black box’? In other words, what is the mechanism of the 0nbb decay? Light or heavy Majorana neutrino. Model extended to include right-handed WR. Mixing extended between the left and right-handed neutrinos. Light Majorana neutrino, only Standard Model weak interactions Supersymmetry with R-parity violation. Many new particles invoked. Light Majorana neutrinos exist also. Heavy Majorana neutrino interacting with WR. Model extended to include right-handed current interactions.

If the first graph (exchange of light Majorana neutrinos, i.e. the assumption that the `standard’ neutrinos are Majorana) is responsible, we will be able to determine the neutrino mass from the 0nbb rate. If any of the other graphs dominate, we cannot do that.

The relative size of heavy (AH) vs. light particle (AL) exchange to the decay amplitude is (a crude estimate): AL ~ GF2 mbb/<k2>, AH ~ GF2 MW4/L5 , where L is the heavy scale and k ~ 50 MeV is the virtual neutrino momentum. For L ~ 1 TeV and mbb ~ 0.1 – 0.5 eV AL/AH ~ 1, hence both mechanism contribute equally. If L >> 1 TeV, the heavy particle exchange results in unobservably small 0nbb rate. A possible key in deciding which mechanism dominates is in linking LNV to LFV violation.

In the standard model lepton flavor conservation (LFV) is a consequence of vanishing neutrino masses. However, the observation of neutrino oscillations shows that neutrinos are massive and that the flavor is not conserved. Hence a more general theory must contain LFV of charged leptons generated probably at some high scale. There is a long history of searches for LFV with charged leptons, like m -> e + g, muon conversion m- + (Z,A) -> e- + (Z,A), or m+ -> e+ + e+ + e- . Impressive limits for the branching ratios have been established: < 1.2x10-11 < 8x10-13

There are ambitious new proposals with much better sensitivities: MECO (unfortunately dead now): Bm ->e < 5x10-17 on Al MEG: Bm -> e+g < 5x 10-14 i.e. improvement by a factor of ~ 1000 - 10000. The direct effect of neutrino mass is “GIM suppressed” by a factor of (Dm2n/MW2)2 ~ 10-50 hence unobservable. g W m n e

Linking LNV to LFV Summary: SM extensions with low ( TeV) scale LNV ** Left-right symmetric model, R-parity violating SUSY, etc. possibly G0nbb unrelated to mbb2 SM extensions with high (GUT) scale LNV [ ] ** In absence of fine-tuning or hierarchies in flavor couplings. Important caveat! See: V. Cirigliano et al., PRL93,231802(2004)

We can relate <mbb> to other observables related to the What is the relation of the deduced fundamental parameters and the neutrino mixing matrix? Or, in other words, what is the relation between the 0nbb decay rate and the absolute neutrino mass? As long as the mass eigenstates ni are Majorana neutrinos, the 0nbb decay will occur, with the rate 1/T1/2= G(Etot,Z) (M0n)2 <mbb>2, where G(Etot,Z) is easily calculable phase space factor, M0n is the nuclear matrix element, calculable with difficulties (and discussed later), and <mbb> = Si |Uei|2 exp(iai) mi, where ai are unknown Majorana phases (only two of them are relevant). We can relate <mbb> to other observables related to the absolute neutrino mass.

<mbb> vs. the absolute mass scale minimum mass, not observable from observational cosmology, M = m1+m2+m3 from b decay blue shading: normal hierarchy, Dm231 > 0. red shading: inverted hierarchy Dm231 < 0 shading:best fit parameters, lines 95% CL errors. Thanks to A. Piepke

Moore’s law of 0nbb decay: There is a steady progress in the sensitivity of the searches for 0nbb decay. Several experiments that are funded and almost ready to go will reach sensitivity to ~0.1 eV. There is one (so far unconfirmed) claim that the 0nbb decay of 76Ge was actually observed. The deduced mass <mbb> would be then 0.3-0.7 eV.

Determination of <mbb> can be only as accurate as the accuracy of the nuclear matrix elements M0n. There are two basic methods: 1) Nuclear shell model (SM) treats complicated configurations, but only few single-particle states (one shell or less). 2) Quasiparticle Random Phase Approximation (QRPA) can treat many single-particle states, but only simple particle-hole configurations. Most existing calculations are QRPA, only very few are SM. The spread of calculated values is often used as a measure of uncertainty. Often, however, it merely reflects a spread in choice of adjustable parameters in QRPA or other issues not directly related to nuclear structure (e.g. the treatment of the short range nucleon-nucleon repulsion)

nuclear matrix element for 76Ge A provocative question: Do we know at all how large the matrix elements for the 0nbb decay really are? Or, in other words, why there is so much variation among the published calculated matrix elements? from Bahcall et al hep-ph/0403167 , spread of published values of squared nuclear matrix element for 76Ge This suggests an uncertainty of as much as a factor of 5. Is it really so bad?

QRPA-RQRPA results of Rodin et al, erratum in Nucl. Phys QRPA-RQRPA results of Rodin et al, erratum in Nucl. Phys. A793, 213 (2007). Shell model results from Caurier et al, arXiv:0709.0277. In contrast, a reasonably narrow range. ISM ISM

Important source of uncertainty: Cancellation between the J = 0 (pairing) contribution and the J 0 (broken pairs) contribution:

The same effects are observed in shell model and in QRPA. Pairing part Broken pairs parts

Due to the cancellation between the pairing J =0 and broken pairs J 0 contributions the 0n m.e. gets most of its contributions from r < 2fm. Hence the treatment of short range effects is important and reduces the m.e. This is for 76Ge, other nuclei look similar. This is a general behavior. (slide by F. Simkovic)

0nbb half-lives for <mbb> = 50 meV based on the matrix elements of Rodin et al. 76Ge (0.8 - 1.2) x 1027 y 82Se (2.2 - 4.0) x 1026 y 100Mo (2.0 - 3.8) x 1026 y 130Te (1.7 - 5.2) x 1026 y 136Xe (0.4 - 1.3) x 1027 y (no 2n observed yet) Note: Calculated matrix elements decrease with increasing A, but the phase-space factors usually increase, particularly the Coulomb factor, hence relatively little variation of T1/2 with A. Note: The sensitivity to <mbb> scales as 1/(T1/2)1/2

The most sensitive double beta decay experiments to date are based on 76-Germanium. Heidelberg-Moscow (76Ge) energy spectrum Q value Half-life limit: 1.9 x 1025 years (H-M and IGEX) Majorana neutrinos ruled out for masses greater than ~0.35-1.0 eV

0 discovery claim HV. Klapdor-Kleingrothaus, et. al, Nuclear Instruments and Methods A 553 (2004) 371-406

Halflife deduced: 1.50+7.55-0.71 x1025 y at 95%C.L.

What should happen next? In a number of new experiments ( CUORE, EXO, Majorana, MOON, SuperNEMO, etc) the amount of source will be increased from the present ~10 kg to ~100 kg, and the sensitivity from the ~1025 years to ~1027 y, covering the `degenerate’ mass region. This should open the door for ~ton 0nbb decay experiments. 3) Next generation of experiments on LFV will extend the sensitivity considerably. In parallel, running of LHC will shed light on the existence of particles with ~TeV masses. 4) Hopefully, progress in the nuclear structure calculation will remove some or most of the uncertainty in the 0nbb nuclear matrix elements.

Spare slides

0nbb candidate isotopes: Q value and natural abundance High Q value reduces radioactive backgrounds, large abundance makes the experiment cheaper.

1/T1/2 = G(E,Z) (MGT2n)2 Nuclear matrix elements for the 2n decay deduced from measured halflives. Note the pronounced shell dependence. 1/T1/2 = G(E,Z) (MGT2n)2 easily calculable phase space factor

From Rodin et al. arXiv:0706.4304. Assessment of the uncertainty in M0n with QRPA and its variant RQRPA.

Comparison between QRPA and SM: Note that the SM results are typically smaller than QRPA. it is not yet clear which is better. QRPA, RQRPA: V.Rodin, A. Faessler, F. Š., P. Vogel, Erratum, nucl-th/07006.4304 shell model: E. Caurier at al. Rev. Mod. Phys. 77, 427 (2005). Slide by F. Simkovic, ignore the bottom line on the graph

The two methods, QRPA and SM are very different (slide by Poves (SM)):

The recent results of two leading QRPA groups agree quite well. The results differ for 82Se, 96Zr Is it a problem of choosing of single particle energies?