# Phenomenology of neutrinoless double beta decay

###### Abstract

Neutrinoless double beta decay () violates lepton number by two units, a positive observation therefore necessarily implies physics beyond the standard model. Here, three possible contributions to decay are briefly reviewed: (a) The mass mechanism and its connection to neutrino oscillations; (b) Left-right symmetric models and the lower limit on the right-handed boson mass; and (c) R-parity violating supersymmetry. In addition, the recently published “extended black box” theorem is briefly discussed. Combined with data from oscillation experiments this theorem provides proof that the decay amplitude must receive a non-zero contribution from the mass mechanism, if neutrinos are indeed Majorana particles.

IFIC/06-26

## 1 Introduction

Since the discovery of neutrino oscillations [1] most papers on neutrinoless double beta decay () have exclusively concentrated on its implications for Majorana neutrino masses. However, as is well-known, any model beyond the standard model of particle physics, which allows for lepton number violation, potentially contributes to decay. Thus, the basic physics of decay can be summarized as:

(1) |

The factor contains some (unknown, but lepton number violating) particle physics parameters. To determine the numerical value of input from both, experiment and theoretical nuclear physics, is needed. Experiments limit (or measure) , for a discussion of various different experiments see, for example [2]. in eq. (1) stands for a nuclear structure matrix element. Different particle physics contributions to decay depend on different matrix elements. No definite consensus about the value and, most importantly, the error of nuclear matrix elements exist up to now. For a thorough discussion see [3]. Finally, is a leptonic phase space integral, its value can be calculated quite precisely [4].

This talk concentrates exclusively on particle physics aspects of decay. The classic “black box” [5] theorem and its recently published “extended” version [6] are briefly discussed, before reviewing constraints on left-right symmetric models and supersymmetry with R-parity violation derived from a lower limit on the decay half-live. Last but not least, expectations for the mass mechanism of decay in light of neutrino oscillation data are discussed. It is curious to note, that combining the “extended black box” with oscillation data [6] already today demonstrates that there must be a non-zero contribution from the mass mechanism to the decay amplitude, if neutrinos are indeed Majorana particles.

## 2 decay and the Black Box

From the experimental point of view lepton number violation in decay is observed through the appearance of two electrons in the final state with no missing energy. Many different, possible mechanisms have been discussed in the literature. Interestingly, however, one can show [5] that independent of which contribution to decay is the dominant one, neutrinos are guaranteed to have a non-zero Majorana mass, if decay is observed. The proof of this “black box” theorem [5] essentially follows from the observation that any effective low-energy operator inducing decay will contribute also - possibly at the some order in perturbation theory, for sure in some higher order - to the () entry of the Majorana neutrino mass matrix (). A perfect cancellation of all different contributions to would then require a special symmetry and the proof of the black box theorem is completed by showing that no such symmetry can exist [7] in any gauge model containing the standard model charged current interaction.

This well-known theorem has recently been extended to the case of three generations of neutrinos and arbitrary lepton number and lepton flavour violating processes [6]. Combined with data from oscillation experiments this “extended” black box theorem can be used to show that . The proof involves two steps. In the first step it is shown that any effective operator generating lepton number violating processes of the form , where and stand symbolically for any set of SM particles with , necessarily generates a non-zero entry in the Majorana neutrino mass matrix in higher order of perturbation theory. As for the original black box, one can show that there is no possible symmetry allowing for a perfect cancellation of different contributions to this entry. In the second step, then all allowed neutrino mass matrices with are constructed. It is then easy to show that none of the possible five structures is consistent with oscillation data. One can thus conclude that is guaranteed for Majorana neutrinos [6] already today.

The above theorem(s) do not state which mechanism of decay is the dominant one. Two instructive examples, in which the mass mechanism might indeed not be the dominant contribution to decay, are therefore discussed next.

### 2.1 Left-right symmetry

For decay, with its typical low energy scale of a few MeV, all calculations can be done with the effective Hamiltonian [4]

(2) |

Here, and are the hadronic and leptonic charged currents, stands for . is the Fermi constant, , i.e. the mixing angle between the and bosons, and .

The Hamiltonian of eq. (2) gives rise to the diagrams in fig. (1). The graphs on the left and the middle represent so-called “long-range” contributions. The graph to the left is due to a product of two and corresponds to the mass mechanism of decay, proportional to (see discussion in the next section). The graph in the middle is proportional to and . The graph to the right is proportional to [8]. Here, .

Formally, the long-range contribution in LR models are suppressed only by one power of /, compared to the short-range contribution, which is quadratic in /. Many calculations therefore have taken into account only the long-range LR contributions. However, as first pointed out by Mohapatra [9] and confirmed by a detailed calculation of the relevant nuclear matrix elements [8], the short-range contribution can be much more important then the long-range one. This at first sight contradictive statement can be easily understood. In left-right symmetric models the mixing between the active, left (and light) neutrinos with the heavy, sterile ones can be estimated “á la seesaw” to be very roughly of the order one gets . Then, with a limit of

(3) |

can then be derived. Note that the limit disappears as goes to infinity, as it should. Note also that the uncertainty in this limit due to the uncertainty in the nuclear matrix element calculation scales only as and thus is quite insensitive to the details of the nuclear model.

### 2.2 R-parity violation

In the standard model lepton number is conserved, because there is (a) no right-handed neutrino and (b) only one Higgs doublet with . In supersymmetric models, on the other hand, if one does not assume lepton number conservation a priori, one can write down the following (trilinear) lepton number violating terms

Here, the tilde indicates the scalar superpartners of the usual quarks and leptons. A product of two of the terms in eq. (2.2), together with an MSSM neutralino and/or gluino interaction lead to decay diagrams without any virtual neutrinos being exchanged, as first pointed out in [10, 11]. A dedicated calculation of all diagrams [12], together with a limit of y for Ge leads to

(5) |

It is interesting to note, that such a small value of generates at 1-loop level an entry in the Majorana neutrino mass matrix of eV only.

## 3 Neutrino oscillations and decay

If the mass mechanism is dominant, the decay half-live is proportional to the (square of the) () element of the Majorana neutrino mass matrix. For three generations of light neutrinos, this so-called “effective Majorana” mass can be expressed as:

(6) |

Eq.(6) contains a priori seven unknowns: Three mass eigenstates, two angles and two phases. With the help of data from neutrino oscillation experiments, one can trade two mass eigenstates for the observed and and relate the two angles to the solar () and reactor angle (). For the case of normal hierarchy, , eq.(6) can then be written as

(7) |

while for the case of inverse hierarchy, , it is given by

(8) |

Fig. (2) shows the resulting allowed range of for both, normal and inverse hierarchy, taking into account the latest results from a global fit to all neutrino oscillation data [13]. The lower limit on , which appears in the case of inverse hierarchy, can be understood trivially. For and eq. (8) reads approximately

(9) |

Thus, as soon as data tells us that , exact cancellation is no longer a possibility. This statement remains true for any finite , simply because is guaranteed by data nowadays. Fig. (3) shows how this lower limit evolves with future data from neutrino oscillation experiments. A possible future smaller upper bound on would make it easier for decay experiments to rule out inverse hierarchy.

There is no such simple quantitative lower limit for the case of normal hierarchy. Fig. (2), to the left, aims at demonstrating this point. If , a lower limit appears if

(10) |

However, from this superficial look at the data at the point exact cancellation yielding seems possible. However, this is equivalent to saying and it is exactly this possibility which is ruled out by the “extended black box” theorem [6].

In fig. (4) finally a summary of the current status of various experimental attempts on measuring/limiting the absolute scale of neutrino masses is given. The light and darker blue areas are allowed for the decay half live of Ge for normal and inverse hierarchy, calculated with matrix elements from [14]. Note, that matrix elements from [15] lead to slightly larger half-lives, see also the discussion in [3]. The green area labeled “Mainz & Troitsk” shows the latest upper limits derived from endpoint measurements in H decay [16, 17]. The bar labeled “KATRIN” represents the expected sensitiviy of the next generation H experiment KATRIN [18]. Note, that KATRIN claims a final sensitivity of eV ( 90 % c.l.) or a 5 discovery threshold of eV. Various limits on the absolute neutrino mass scale from cosmology have been published recently, derived from CMB data combined with information from large scale structure surveys. For three generations of neutrinos numbers ranging from eV, depending on input and bias, have been published. For a detailed discussion see, for example, the review [19]. The horizontal gray band indicates the range of the finite claimed by some members of the Heidelberg-Moscow experiment [20]. Note that this result is highly controversial, see for example the discussion by Barabash in [2]. The vertical red lines indicate the sensitivity of two future Ge experiments. GERDA [21] is currently in phase I, phase II is funded. In the future Majorana [22] and/or GERDA phase III can test the range allowed by inverse hierarchy.

## 4 Conclusions

Lower limits on the decay half live can be used to constrain various particle physics parameters. However, from the point of view of particle physics it would be interesting to determine the dominant contribution to decay. Very little work has been done in this direction. Angular correlations between the eletrons [4] or a comparative study of and decay [23] might be able to disentangle left-left and left-right-handed combinations of currents (of the long range type). However, other contributions to decay possibly exist and ultimately it might be that only a combination of various different pieces of experimental data will provide the correct and final answer.

Acknowledgments

I would like to thank S.G. Kovalenko and J.W.F. Valle for various discussions on the subject. Financial support by Spanish grant FPA2005-01269, by European Commission Human Potential Program RTN network MRTN-CT-2004-503369 and the EU Network of Astroparticle Physics (ENTApP) WP1, as well as the spanish MCyT Ramon y Cajal program is acknowledged.

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