Neutrinoless : 2015 review

Stefano Dell’Oro,1, ∗ Simone Marcocci,1, † Matteo Viel,2, 3, ‡ and Francesco Vissani4, 1, § 1INFN, Gran Sasso Science Institute, Viale F. Crispi 7, 67100 L’Aquila, 2INAF, Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34131 Trieste, Italy 3INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy 4INFN, Laboratori Nazionali del Gran Sasso, Via G. Acitelli 22, 67100 Assergi (AQ), Italy (Dated: April 20, 2016) The discovery of masses through the observation of oscillations boosted the importance of neutrinoless double beta decay (0νββ). In this paper, we review the main features of this process, underlining its key role both from the experimental and theoretical point of view. In particular, we contextualize the 0νββ in the panorama of lepton-number violating processes, also assessing some possible particle mechanisms mediating the process. Since the 0νββ existence is correlated with neutrino masses, we also review the state-of-art of the theoretical understanding of neutrino masses. In the final part, the status of current 0νββ experiments is presented and the prospects for the future hunt for 0νββ are discussed. Also, experimental data coming from cosmological surveys are considered and their impact on 0νββ expectations is examined.

PACS numbers: 14.60.Pq,23.40.-s DOI: 10.1155/2016/2162659

u u I. INTRODUCTION d d d u W- In 1937, almost ten years after Paul Dirac’s “The quan- - tum theory of ” [1,2], Ettore Majorana proposed e ν a new way to represent in a relativistic quan- M e- tum field theory [3], and remarked that this could be - especially useful for neutral particles. A single Majorana W d u quantum field characterizes the situation in which par- d d ticles and antiparticles coincide, as it happens for the u u photon. Giulio Racah stressed that such a field could FIG. 1. Diagram of the 0νββ process due to the exchange of fully describe massive , noting that the theory massive Majorana neutrinos, here denoted generically by νM. by E. Majorana leads to physical predictions essentially different from those coming from Dirac theory, [4]. Two years later, Wendell Furry [5] studied within this scenario In the attempt to investigate the nature of the a new process similar to the “double beta disintegration”, 0νββ process, various other theoretical possibilities were introduced by Maria Goeppert-Mayer in 1935 [6]. It is considered, beginning by postulating new super-weak in- the double beta decay without neutrino emission, or neu- teractions [7,8]. However, the general interest has always trinoless double beta decay (0νββ). This process assumes remained focused on the neutrino mass mechanism. In a simple form, namely fact, this scenario is supported by two important facts: (A, Z) → (A, Z + 2) + 2e−. (1) 1. On the theoretical side, the triumph of the Stan- dard Model (SM) of electroweak interactions in the The of the 0νββ process, written in 1970s [9–11] led to formulate the discussion of new terms of the particles we know today and of massive Ma- physics signals using the language of effective op- jorana neutrinos, is given Fig.1. arXiv:1601.07512v2 [hep-ph] 19 Apr 2016 erators, suppressed by powers of the new physics The main and evident feature of the 0νββ transi- mass scale. There is only one operator that is sup- tion is the explicit violation of the number of leptons pressed only by one power of the new mass scale and, more precisely, the creation of a pair of . and violates the global symmetries of the SM or, The discovery of 0νββ would therefore demonstrate that more precisely, the lepton number: it is the one lepton number is not a symmetry of nature. This, in that gives rise to Majorana neutrino masses [12] turn, would support the exciting theoretical picture that (see also Refs. [13–16]). leptons played a part in the creation of the matter- antimatter asymmetry in the universe. 2. On the experimental side, some anomalies in neu- trino physics, which emerged in throughout 30 years, found their natural explanation in terms of ∗ [email protected] oscillations of massive neutrinos [17]. This expla- † [email protected] nation was confirmed by several experiments (see ‡ [email protected] Refs. [18, 19] for reviews). Thus, although oscilla- § [email protected] tion phenomena are not sensitive to the Majorana 2

nature of neutrinos [20], the concept of neutrino Dirac massive particle Majorana massive particle mass has changed its status in physics, from the one of hypothesis to the one of fact. This, of course, strengthened the case for light massive neutrinos to -- play a major role for the 0νββ transition. For these reasons, besides being an interesting nuclear 0 0 process, 0νββ is a also a key tool for studying neutri- nos, probing whether their nature is the one of Majorana particles and providing us with precious information on + + the neutrino mass scale and ordering. Even though the predictions of the 0νββ lifetime still suffer from numer- ous uncertainties, great progresses in assessing the expec- FIG. 2. Massive fields in their rest frames. The arrows show tations for this process have been and are being made. the possible directions of the spin. (Left) The 4 states of These will be discussed later in this review. Dirac massive field. The signs indicate the charge that distin- guishes particles and antiparticles, e. g. the electric charge of About the present review In recent years, several an electron. (Right) The 2 states of Majorana massive field. review papers concerning neutrinoless double beta decay The symbol “zero” indicates the absence of any U(1) charge: particles and antiparticles coincide. have been written. They certainly witness the vivid inter- est of the scientific community in this topic. Each work emphasizes one or more relevant aspects such as the ex- We would like to warn the Reader that other attitudes perimental part [21–25], the nuclear physics [26, 27], the in the discussion are surely possible, and it is indeed the connection with neutrino masses [28, 29], other particle case for some of the mentioned review works. Using less physics mechanisms [30–33], . . . The present work is not stringent limits from cosmology and disregarding the un- an exception. We mostly focus on the first three aspects. certainties from nuclear physics is equivalent to assume This choice is motivated by our intention to follow the the most favorable situation for the experiments. This theoretical ideas that describe the most plausible expec- could be considered beneficial for the people involved in tations for the experiments. In particular, after a general experimental search for the neutrinoless double beta de- theoretical introduction (Secs.II andIII), we examine the cay. However, we prefer to adhere to a more problematic present knowledge on neutrino masses (Sec.IV) and the view in the present work, simply because we think that status of expectations from nuclear physics in (Sec.V). it more closely reflects the present status of facts. Con- Then we review the experimental situation (Sec.VI) and sidering the numerous experiments involved in the field, emphasize the link between neutrinoless double beta de- we deem that an updated discussion on these two issues cay and cosmology (Sec.VII). has now become quite urgent. This will help us to as- A more peculiar aspect of this review is the effort to fol- sess and appreciate better the progresses expected in the low the historical arguments, without worrying too much close future, concerning the cosmological measurements about covering once more well-known material or about of neutrino masses and perhaps also the theoretical cal- presenting an exhaustive coverage of the huge recent lit- culations of the relevant nuclear matrix elements. erature on the subject. Another specific characteristic is the way the information on the neutrino Majorana mass is dealt with. In order to pass from this quantity to the II. THE TOTAL LEPTON NUMBER (potentially measurable) decay rate, we have to dispose of quantitative information on the neutrino masses and No elementary process where the number of leptons or on the matrix elements of the transition, which in turn re- the number of hadrons varies has been observed yet. This quires the description of the nuclear wave functions and suggests the hypothesis that the lepton number L and the of the operators that are implied. Therefore, our ap- B are subject to conservations laws. However, we proach is to consider the entire available information on do not have any deep justification for which these laws neutrino masses and, in particular, the one coming from should be exact. In fact, it is possible to suspect that cosmology. We argue that the recent progresses (espe- their validity is just approximate or circumstantial, since cially those coming from the Planck satellite data [34]) it is related to the range of energies that we can explore play a very central role for the present discussion. On in laboratories.1 the other side, the matrix elements have to be calculated In this section, we discuss the status of the investi- (rather than measured) and are thus subject to uncer- gations on the total lepton number in the SM and in tainties which are difficult to assess reliably. Moreover, the adopted methods of calculation do not precisely re- produce other measurable quantities (single beta decay, two-neutrino double beta decay, . . . ). We thus prefer to 1 Notice also that the fact that neutral leptons (i. e. neutrinos or adopt a cautious/conservative assessment of the theoret- antineutrinos) are very difficult to observe restricts the experi- ical ranges of these matrix elements. mental possibilities to test the total lepton number. 3 a number of minimal extensions, focusing on theoretical considerations. In particular, we introduce the possibil- direction of motion direction of motion ity that neutrinos are endowed with Majorana mass and consider a few possible manifestations of lepton number violating phenomena. The case of the 0νββ will instead 0 0 be addressed in the rest of this work. A fast lepton with A fast lepton with negative helicity yelds µ- positive helicity yelds µ+ A. B and L symmetries in the SM → it must be called νµ → it must be called νµ The SM in its minimal formulation has various global symmetries, including B and L, which are called “ac- cidental”. This is due to the specific particle content FIG. 3. The chiral nature of weak interactions allows us to of the model and to the hypothesis of renormalizability. define what is a neutrino and what it an antineutrino in the ultra-relativistic limit, when chirality coincides with helicity Some combinations of these symmetries, like for example and the value of the mass plays only a minor role. “B − L”, are conserved also non-perturbatively. This is sufficient to forbid the 0νββ transition completely in the SM. In other words, a hypothetical evidence for such a δpp0 δλλ0 . For any value of the momentum, there are 2 transition would directly point out to physics beyond the spin (or helicity) states: SM. At the same time, the minimal formulation of the SM implies that neutrinos are massless, and this contra- a∗(p +)|vac.i = |p ↑ i and a∗(p −)|vac.i = |p ↓ i. dicts the experimental findings. Therefore, the question (5) of how to modify the SM arises, and this in turn poses Fig.2 illustrates the comparison between the particle the related burning question concerning the nature of content both of a Dirac and a Majorana field in the case neutrino masses. p = 0 (rest frame). Evidently, a Majorana neutrino is incompatible with any U(1) transformation, e. g. L or the weak hypercharge B. Majorana neutrinos (that is however broken in the vacuum). In general, L will be violated by the presence of Majorana mass. In 1937, Majorana proposed a theory of massive and In the SM, the neutrino field appears only in the com- “real” fermions, [3]. This theory contains less fields than bination the one used by Dirac for the description of the elec- ψ = P ψ (6) tron [1,2] and, in this sense, it is simpler. Following the L L formalism introduced in 1933 by Fermi when describing where PL ≡ (1 − γ5)/2 is the so-called chiral projector the β decay [35], the condition of reality for a quantized (TableI). It is then possible to implement the hypothesis fermionic field can be written as: of Majorana in the most direct way by defining the real field χ = Cχ¯t (2) ¯t χ ≡ ψL + CψL. (7) where C is the charge conjugation matrix, while † In fact, we can conversely obtain the SM field by a pro- χ¯ ≡ χ γ0 is the Dirac conjugate of the field. In partic- ular, Majorana advocated a specific choice of the Dirac jection: γ-matrices, such that Cγt = 1, which simplifies various 0 ψ ≡ P χ. (8) equations. The free particle Lagrangian density formally L L coincides with the usual one: 1 C. Ultrarelativistic limit and massive neutrinos L = χ¯(i∂/ − m)χ. (3) Majorana 2 The discovery that parity is a violated symmetry in Following Majorana’s notations, the decomposition of weak interactions [36, 37] was soon followed by the un- the quantized fields into oscillators is: derstanding that the charged current (which contains X the neutrino field) always includes the left chiral pro- χ(x) = [a(pλ) ψ(x; pλ) + a∗(pλ) ψ∗(x; pλ)] (4) jector [38–40] (see Secs.IIB andIIIA). p,λ It is interesting to note the following implication. Within the hypothesis that neutrino are massless, the where λ = ±1 is the relative orientation between the becomes equivalent to two Weyl equa- spin and the momentum (helicity). We adopt the nor- tions [41] corresponding to the Hamiltonian functions malization for the wave functions: R dx|ψ(t, x)|2 = 1, ∗ 0 0 ∗ 0 0 and for the oscillators: a(pλ) a (p λ )+a (p λ ) a(pλ) = Hν/ν¯ = ∓cp σ (9) 4 where σ are the three Pauli matrices and the two signs ipate in SM interactions and can therefore be endowed apply to the neutral leptons that thanks to the interac- with a Majorana mass M, still respecting the SM gauge tion produce charged leptons of charge ∓1, respectively. symmetries. However, they do participate in the new in- In other words, we can define these states as neutrinos teractions, and more importantly for the discussion, they and antineutrinos, respectively. Moreover, by looking at can mix with the ordinary neutrinos via the Dirac mass Eq. (9), one can see that the energy eigenstates are also terms, mDirac. Therefore, in presence of RH neutrinos, helicity eigenstates. More precisely, the spin of the neu- the SM Lagrangian (after spontaneous symmetry break- trino (antineutrino) is antiparallel (parallel) to its mo- ing) will include the terms, mentum. See Fig.3 for illustration. The one-to-one connection between chirality and he- licity holds only in the ultrarelativistic limit, when the 1 L = −ν¯ mDirac ν + ν M C ν¯t + h. c. (10) mass of the neutrinos is negligible. This is typically the mass Ri `i L` 2 Ri i Ri case that applies for detectable neutrinos, since the cross sections are bigger at larger energies. However, these remarks do not imply in any way that where ` = e, µ, τ and i = 1, 2, 3. It is easy to understand neutrinos are massless. On the contrary, we know that that, at least generically, this framework implies that the neutrinos are massive. lepton number is broken. A consequence of the chiral nature of weak interac- tions is that, if we assume that neutrinos have the type Let us assume the existence of RH neutrinos, either of mass introduced by Dirac, we have a couple of states embedded in an unified group or not, and let us suppose that are sterile under weak interactions in the ultrarel- that they are heavy (this happens, for example, if the ativistic limit. Conversely, the fact that the left chiral scale of the new gauge is large and the couplings state exists can be considered a motivation in favor of of the RH neutrinos to the scalar bosons implementing the hypothesis of Majorana. In fact, this does not re- spontaneous symmetry breaking of the new gauge group quire the introduction of the right chiral state, as instead are not small). In this case, upon integrating away the required by the Dirac hypothesis. Most importantly, it heavy neutrinos from the theory, the light neutrinos will should be noticed that in the case of Majorana mass it receive Majorana mass, with size inversely proportional is not possible to define the difference between a neutrino to the mass of the RH ones, [13–16]. This is the cele- and an antineutrino in a Lorentz invariant way. brated Type I Seesaw Model. In other words, the hy- pothesis of heavy RH neutrinos allows us to account for the observed small mass of the neutrinos. Unfortunately, D. Right-handed neutrinos and unified groups we cannot predict the size of the light neutrino mass pre- cisely, unless we know both M and mDirac. The similarity between L and B is perceivable already In principle, RH neutrinos could also be quite light. within the SM. The connection is even deeper within the An extreme possibility is that some of them have masses so called Grand Unified Theories (GUT), i. e. gauge the- of the order of eV or less and give rise to new flavor ories with a single gauge coupling at a certain high en- oscillations observable in terrestrial laboratories [45–47]. ergy scale. The standard prototypes are SU(5) [42] and This could help to address some experimental anoma- SO(10) [43, 44]. GUTs undergo a series of symmetry- lies [48, 49]. However, as it has been known since breaking stages at lower energies, eventually reproducing long [50, 51], that the presence of eV neutrinos would the SM. They lead to predictions on the couplings of the also imply large effects in cosmology, both in the num- model and suggest the existence of new particles, even if ber of relativistic species and in the value of the neutrino theoretical uncertainties make it difficult to obtain reli- mass. These effects are not in agreement with the ex- able predictions. The possibilities to test these theories isting information from cosmology (see Sec.IVC) and, are limited, and major manifestations could be violations for this reason, we will not investigate this hypothesis of L and B. further (we refer an interested Reader to the various dis- The matter content of GUT theories is particularly rel- cussion on the impact of eV neutrinos on the 0νββ, see evant for the discussion. In fact, the organization of each e. g. Refs. [52–54]). family of the SM suggests the question whether right- handed neutrinos (RH) exist along with the other 7 RH In view of the evidences of neutrinos masses, theories particles (Fig.4). This question is answered affirmatively like SO(10) are particularly appealing, since they offer a in some extensions of the SM. For example, this is true natural explanation of light Majorana neutrino masses. for gauge groups that also include a SU(2)R factor, on However, a complete theory able to link in a convincing top of the usual SU(2)L factor. In the SO(10) gauge way masses (including those of neutrinos) and to group, which belongs to this class of models, each family provide us reliable predictions of new phenomena, such of matter includes the 15 SM particles plus 1 RH neu- as 0νββ, does not exist yet. Despite many attempts were trino. made in the past, it seems that this enterprise is still in It should be noted that RH neutrinos do not partic- its initial stages. 5

15 particle per matter family sibility of explaining the baryon number excess through Leptogenesis theories. However, at the same time, one should not overestimate the heuristic power of this the- u u u ?ν oretical scheme, at least within the presently available d d d e information.

direction of motion F. Neutrino nature and cosmic neutrino background u u u ν The Big Bang theory predicts that the present Uni- d d d e verse is left with a residual population of ∼ 56 non- relativistic neutrinos and antineutrinos per cm3 and per species. It constitutes a Cosmic Neutrino Background FIG. 4. Helicity of the 15 massless matter particles contained in each family of the SM (see TableI). The arrow gives the (CνB). Due to their very low energy, Eq. (9) does not direction of the momentum. hold for these neutrinos. This happens because at least two species of neutrinos are non-relativistic. The detec- tion of this CνB could therefore allow to understand E. Leptogenesis which hypothesis (Majorana or Dirac) applies for the neutrino description. 3 Although particles and antiparticles have the same im- Let us assume to have a target of 100 g of H. Electron portance in our understanding of , we neutrinos can be detected through the reaction [63, 64] know that the Universe contains mostly rather ν + 3H → 3He + e−. (11) than antibaryons.2 In 1967, Sakharov proposed a set e of necessary conditions to generate the cosmic baryon In the standard assumption of a homogeneous Fermi- asymmetry [55]. This has been the beginning of many Dirac distribution of the CνB, we expect ∼ 8 events per theoretical attempts to “explain” these observations in year if neutrinos are Majorana particles and about half terms of new physics. if the Dirac hypothesis applies [65]. Indeed, in the for- In the SM, although L and B are not conserved sepa- mer case, the states with positive helicity (by definition, rately at the non-perturbative level [56–58], the observed antineutrinos) will act just as neutrinos, since they are al- value of the Higgs mass is not big enough to account for most at rest. Instead, in the latter case, they will remain the observed baryon asymmetry [59, 60]. New violations antineutrinos and thus they will not react. of the global B or L are needed. It can be noticed that the signal rate is not pro- An attractive theoretical possibility is that RH neu- hibitively small, but the major difficulty consists in at- trinos not only enhance the SM endowing neutrinos with taining a sufficient energy resolution to keep at a man- Majorana mass, but also produce a certain amount of lep- ageable level the background from beta decay. We will tonic asymmetry in the Universe. This is subsequently not discuss further the feasibility of such an experiment, converted into a baryonic asymmetry thanks to B +L vi- and refer to Refs. [64, 65] for more details. olating effects, which are built-in in the SM. It is the so called Leptogenesis mechanism, and it can be wittingly described by asking the following question: do we all de- III. PARTICLE PHYSICS MECHANISMS FOR scend from neutrinos? The initial proposal of Leptogene- 0νββ sis dates back to 1980s [61], and there is a large consensus that this type of idea is viable and attractive. Subsequent investigators showed that the number of alternative the- oretical possibilities is very large and, in particular, that In this section, we focus on one of the most appeal- there are other possible sources of L violations besides ing lepton number violating process, the 0νββ. The ex- RH neutrinos. Conversely, the number of testable possi- change of light Majorana neutrinos is up to now the most bilities is quite limited [62]. appealing mechanism to eventually explain the 0νββ. We believe that it is important to stay aware of the pos- Some reasons justifying this statement were already men- tioned, but here a more elaborate discussion is proposed. In particular, we review the basic aspects of the light neutrino exchange mechanism for 0νββ and compare it 2 The lepton number in the Universe is probed much less pre- to other ones. Moreover, the possibility of inferring the cisely. While we know that cosmic neutrinos and antineutrinos are abundant, it is not easy to measure their asymmetry which, size of neutrino masses from a hypothetical observation according to standard cosmology, should be very small. However, of 0νββ and of constraining (or proving the correctness) we expect to have the same number of electrons and to some alternative mechanisms with searches at the accel- guarantee the overall charge neutrality. erators are also discussed. 6

TABLE I. List of the matter particles in the SM. The label “singlet” is often replaced with “right” and likewise for “doublet” it can become “left”. Hypercharge is assigned according to Q = T3L + Y . The chirality of a field (and all its U(1) numbers) c t can be exchanged by considering the charge conjugate field; e. g. eL ≡ C¯eR has electric charge +1 and leptonic charge −1.

Name Field SU(3)c SU(2)L U(1)Y lepton baryon symbol multiplicity multiplicity charge number L number B quark doublet qL 3 2 +1/6 0 1/3 singlet uR 3 1 +2/3 0 1/3 singlet down quark dR 3 1 −1/3 0 1/3 lepton doublet lL 1 2 −1/2 1 0 singlet charged lepton eR 1 1 −1 1 0

A. The neutrino exchange mechanism operator:

Z The definition of a key quantity for the description − G2 d4x J a† (x) ψ¯ (x)γ P χ (x) of the neutrino exchange mechanism needs to be intro- F hadr e a L νe duced. It is the of virtual Majorana neutri- Z d4y J b† (y) ψ¯ (y)γ P χ (y). (16) nos. Due to the reality condition, Eq. (3) can lead to new hadr e b L νe types of that do not exist within the Dirac theory. In fact, in this case we can use the antisymmetry By contracting the neutrino fields, the leptonic part of of the charge conjugation matrix and get: this operator becomes

¯ a b ¯t h0|T [χ(x)χ(y)]|0i = −∆(x − y) C (12) ψe(x) γ PL ∆(x − y)PL γ Cψe(y) (17) where ∆ denotes the usual propagator, while the ordinary propagator, sandwiched between two chiral projectors, reduces to Z 4 d q i(ˆq + m) −iqx ∆(x) ≡ e . (13) 4 4 2 2 Z d q i m (2π) q − m + i0 P ∆(x)P = P e−iqx. (18) L L L (2π)4 q2 − m2 + i0 In the low energy limit (relevant for β decay processes) the interaction of neutrinos are well described by the The momentum q represents the virtuality of the neu- current-current 4-fermion interactions, corresponding to trino, whose value is connected to the momenta of the the Hamiltonian density final state electrons and to those of the intermediate vir- tual nucleons. In particular, since the latter are confined in the nucleus, the typical 3-momenta are of the order of GF a† HFermi = √ J Ja (14) the inverse of the nucleonic size, namely 2 |~q| ∼ ~c/fm ∼ few 100 MeV (19) where GF is the Fermi coupling, and we introduced the a a a current J = Jlept +Jhadr for a = 0, 1, 2, 3, that decreases † whereas the energy (q0) is small. The comparison of this the charge of the system (its conjugate, Ja, does the con- scale with the one of neutrino mass identifies and sepa- trary). In particular, the leptonic current rates “light” from “heavy” neutrinos for what concerns 0νββ. a X ¯ a Jlept = ψ` γ (1 − γ5) ψν` (15) The most interesting mechanism for 0νββ is the one `=e,µ,τ that sees light neutrinos as mediators. It is the one orig- inally considered in Ref. [5] and it will be discussed with defines the ordinary neutrino with “flavor” `. In order to great details in the subsequent sections. In the rest of implement the Majorana hypothesis, one can use Eq. (7) this section, instead, we examine various alternative pos- ¯t and introduce the field χ = ψL + CψL. Nothing changes sibilities. in the interactions if one substitutes the field ψν` with the We have some hints, mostly of theoretical nature, that corresponding field χν` , since the chiral projector selects the light neutrinos might have Majorana mass. How- only the first piece, ψν` L. ever, the main reason for which the hypothesis that the Let us assume that the field χ is a mass eigenstate. A 0νββ receives its main contribution from light Majorana contribution to the 0νββ transition arises at the second neutrinos is the fact that experiments point out the ex- order of the Fermi interaction. Let us begin from the istence of 3 light massive neutrinos. 7

B. Alternative mechanisms to the light neutrino sterile neutrinos, or, generally, other light exchange states are added, more operators may be required. A large effective mass could also come from small adimen- 1. Historical proposals sional couplings y, e. g. 1/M = y2/µ. The number of possible mechanisms that eventually A few years after the understanding of the K0 −K¯ 0 os- can lead to the above effective operators is also very large. cillation [66–68], which led Pontecorvo to conjecture that One possible (plausible) origin of the dimension-5 oper- also neutrino oscillations could exist [17], alternative the- ator is discussed in Sec.IID. However, other cases are oretical mechanisms for the 0νββ other than the neutrino possible and the same is true for the other operators. exchange were firstly advocated. In 1959, Feinberg and Goldhaber [7] proposed the addition of the following term in the effective Lagrangian density: 3. Heavy neutrino exchange g + + t −1 Hpion = π π e C e (20) me Let us now consider the case of heavy RH neutrino where me is the electron mass and g an unspecified di- mensionless coupling. Similarly, after the hypothesis of exchange mechanism. The corresponding operator gives super-weak interactions in weak decays [69, 70], the im- rise to the effective Hamiltonian density (for heavy neu- portance for 0νββ of operators like the one of Eq. (20) trinos, the propagator of Eq. (18) is proportional to was stressed by Pontecorvo [8]. He also emphasized that δ(x − y) ): the size and the origin of these operators could be quite independent from the neutrino masses. G2 H = − F J a† ψ¯ γaγbCψ¯t J b† . (22) νheavy hadr eL eL hadr MH

2. Higher dimensional operators It is evident that this is a dimension-9 operator and it has in front a constant with mass dimension m−5, since The SM offers a very convenient language to order the MH indicates the relevant heavy neutrino mass. It has interesting operators leading to violation of L and B. It is to be noted that such a definition can be used in an possible to consider effective (non-renormalizable) opera- effective formula, but a gauge model requires to express tors that respect the gauge symmetry SU(3) × SU(2) × c L MH in terms of the single RH neutrino masses MI and of U(1)Y, but that violate L and/or B [12, 71]. Here, we the mixing between left handed neutrinos νeL and heavy consider a few representative cases (a more complete list neutrinos: can be found in Refs. [72, 73]), corresponding to the fol- lowing terms of the Lagrangian and Hamiltonian densi- 1 U 2 ties: = eI . (23) MH MI 2 c 2 (lLH) lLqLqLqL (lLqLdR) HWeinberg = + 0 2 + 00 5 . (21) M M M In particular, the mixings are small if MI is large since Dirac The matter fields (fermions) in the equation are written Uei = mei /MI . This suggests a suppression of the in the standard notation of TableI, H is the Higgs field, above effective operator with the cube of MI , whereas while the constrains on the masses are: M < 1011 TeV, the light neutrino exchange mechanism leads to a milder M 0 > 1012 TeV and M 00 > 5 TeV. In particular: suppression, linear in MI (if the mixing matrices have specific flavor structures, deviations from this generic ex- • the first (dimension-5) operator generates Majo- pectation are possible). However, it is still possible that rana neutrino masses, and the bound on M derives RH neutrinos are heavy, but not “very” heavy. Actually, from neutrino masses mν < 0.1 eV this was the first case to be considered [13], and it could be of interest both for direct searches at accelerators (see • the dimension-6 operator leads to decay and Sec.IIID) and for the 0 νββ. In fact, in this case the this implies the tight bound on the mass M 0 mixing UeI is not strongly suppressed and RH neutrinos • the dimension-9 operator contributes to the 0νββ. can give an important contribution to the transition [74]. Its role in the transition can be relevant if the scale However, two remarks on this case are in order. As it of lepton number violation is low. was argued in Ref. [75], in order to avoid fine tunings on the light neutrinos, the masses of RH neutrinos should Summarizing, if one assumes that the scale of new not be much larger than about 10 GeV. Moreover, in the physics is much higher than the electroweak scale, it is extreme limit in which the mass becomes light (i. e. it natural to expect that the leading mechanism behind the is below the value in Eq. (19)) and Type I Seesaw ap- 0νββ is the exchange of light neutrinos endowed with Ma- plies, the contribution of RH neutrinos cancels the one jorana masses. It is also worth to be noted that if light of ordinary neutrinos [76, 77]. 8

4. Models with RH currents W W d u u d Another class of models of great interest are those that include RH currents and intermediate bosons. In the language of SM, the neutrino exchange leads to a core ν - - ν operator e e e e 1 FIG. 5. Diagram representing the contribution of the “black H = W +W +etC−1e (24) box” operator to the Majorana mass. Figure from Ref. [83]. W bosons M where M is a mass scale and W identify the fields of the usual W bosons. When we consider virtual W bosons, it shows that from the observation of the 0νββ, it is pos- this may eventually lead to the usual case. In principle, sible to conclude the existence of the Majorana mass. it is possible to replace the usual W bosons with the cor- The result could be seen as an application (or a general- ization) of the Symanzik’s rule as given by Coleman [82]: responding WR bosons of a new SU(2)R gauge group. In this hypothesis, the RH neutrinos play a more important if a theory predicts L-violation, it will not be possible to role and are no more subject to restrictions of the mix- screen it to forbid only a Majorana neutrino mass. ing matrix, as those of Eq. (23). However, the resulting The size of the neutrino masses is not indicated in dimension-9 operator is suppressed by 4 powers of the the original work, but a straightforward estimation of masses of the new gauge bosons. the diagram of Fig.5 shows that they are so small that Evidently, new RH gauge bosons with masses accessi- they have no physical interest, being of the order of −24 ble to direct experimental investigation are of special in- 10 eV [83]. However, what can be seen as a weak point terest (see Sec.IIID). Since to date we do not have any of the argumentation, is the concept of “natural theory”, experimental evidence, this possibility will not be empha- whose definition is not discussed in Ref. [81], but sim- sized in the following discussion. Anyway, investigations ply proclaimed. In fact, it is possible to find examples at the LHC are currently in progress and the interpre- of models where the 0νββ exists but the Majorana neu- tation of some anomalous events (among the collected trino mass contribution is zero [75], in accordance with the claim of Pontecorvo [8], but clashing with the expec- data) as a hint in favor of relatively light WR bosons has already been proposed [78–80]. tations deriving from that of Ref. [81]. We think that the (important) point made in Ref. [81] is valid not quite as a theorem (word that, anyway, the C. From 0νββ to Majorana mass: a remark on authors never use to indicate their work). We rather “natural” gauge theories believe it acts mostly as a reminder that any specific theory that includes Majorana neutrino masses will have In a well-known work, Schechter and Valle [81] employ various specific links between these masses, 0νββ and the basic concepts of gauge theories to derive some im- possibly other manifestations of L-violation. We see as portant considerations on the 0νββ. In particular, their a risk the fact that, due to the impossibility of avoiding argument proceeds as follows: the issue of model dependence, we will end up with the idea that we can accept “petition of principles”. 1. if the 0νββ is observed, there will be some process (among elementary particles) where the electron-, up- and down-fields are taken twice. This “black D. Role of the search at accelerators box” process in Ref. [81] (Fig.5) effectively resem- bles the one caused by the dimension-9 operator in There is the hope that the search for new particles at Eq. (21) the accelerators might reveal new physics relevant for the interpretation or in some way connected to the 0νββ. 2. using W bosons, it is possible to contract the two This is a statement of wide validity. For example, the quark pairs and obtain something like the operator minimal supersymmetric extension of the SM is compati- in Eq. (24) ble with new L-violating phenomena taking place already at the level of renormalizable operators [84]. Also the hy- 3. finally, the electron- and the W -fields can be con- pothesized extra-dimensions at the TeV scale might be verted into neutrino-fields. A contribution to the connected to new L-violating operators [85]. Or even, Majorana neutrino mass is therefore obtained models where the smallness of the neutrino mass is ex- 4. the possibility that this contribution could be can- plained through loop effects imply typically new particles celed by others is barred out as “unnatural”. that are not ultra-heavy [86]. Notice that these are just a few among the many theoretical possibilities to select This argument works in the “opposite direction” with which, unfortunately, lack of clear principles. respect to ones presented so far. Instead of starting from The recent scientific literature tried at least to exploit the Majorana mass to derive a contribution for the 0νββ, some minimality criteria, and the theoretical models that 9 received the largest attention are indeed those discussed mass” (it can be thought of as the “electron neutrino above. A specific subclass, named νSM [87], is found mass” that rules the 0νββ transition, but keeping in mind interesting enough to propose a dedicated search at the that it is different from the “electron neutrino mass” that CERN SPS [87, 88], aiming to find rare decays of the or- rules the β decay transition). dinary mesons into heavy neutrinos. Other models, that The previous intuitive argument in favor of this defini- foresee a new layer of gauge symmetry at accessible ener- tion is corroborated by calculating the Feynman diagram gies and, more specifically, those connected to left-right of Fig.1. Firstly, it has to be noted that the electronic gauge symmetry [89] might instead lead to impressive L- neutrino νe is not a mass eigenstate in general. Then, violation at accelerators [90–92]. This should be quite substituting Eq. (26) into Eq. (25), we see that we go analogous to the 0νββ process itself and that could be from the flavor basis to the mass basis by setting seen as manifestations of operators similar to those in X Eq. (24). ν` = U`i νi. (28) We would like just to point out that in both cases, in i=1,2,3 order to explain the smallness of neutrino masses, very small adimensional couplings are required. Although this Therefore, in the neutrino propagators of Fig.1, we will position is completely legitimate, in front of the present refer to the masses mi (that in our case are “light”) while, understanding of particle physics, it seems fair to say that in the two leptonic vertices, we will have Uei. Taking the this leaves us with some theoretical question to ponder. product of these factors, we get the expression given in Eq. (27). It should be noted that the leptonic mixing matrix U IV. PRESENT KNOWLEDGE OF NEUTRINO as introduced above differs from the ordinary one used in MASSES neutrino oscillation analyses. Indeed, the latter is given after rotating away the phases of the neutrino fields and In this section we discuss the crucial parameter de- observing that oscillations depend only upon the combi- scribing the 0νββ if the process is mediated by light Ma- nation MM †/(2E). This matrix contains only one com- jorana neutrinos (as defined in Sect.IIIA). We take into plex phase which plays a role in oscillations (the “CP- account the present information coming from the oscil- violating phase”). Instead, in the case of 0νββ the ob- lation parameters, cosmology and other data. On the servable is different. It is just |Mee|. Here, there are theoretical side, we motivate the interest for a minimal new phases that cannot be rotated away and that play interpretation of the results. a physical role. These are sometimes called “Majorana phases”. Their contribution can be made explicit by rewriting Eq. (27) as follows: A. The parameter mββ

X iξi 2 We know three light neutrinos. They are identified by mββ = e |U | mi . (29) ei their charged current interactions i. e. they have “flavor” i=1,2,3 ` = e, µ, τ. The Majorana mass terms in the Lagrangian density is described by a symmetric matrix: We can now identify Uei of Eq. (29) with the mixing matrix used in neutrino oscillation analyses.3 1 X t −1 L = ν C M 0 ν 0 + h. c. . (25) Before proceeding in the discussion, some remarks are mass 2 ` `` ` `,`0=e,µ,τ in order: The only term that violates the electronic number by • it is possible to adopt a convention for the neutrino two units is Mee, and this simple consideration motivates mixing matrix such that the 3 mixing elements Uei the fact that the amplitude of the 0νββ decay has to be are real and positive. However, in the most com- proportional to this parameters, while the width to its mon convention Ue3 is defined to be complex squared modulus. We can diagonalize the neutrino mass matrix by mean of a unitary matrix • only two Majorana phases play a physical role, the third one just being matter of convention t † M = U diag(m1, m2, m3) U (26) • it is not possible even in principle to reconstruct where the neutrino masses mi are real and non-negative. the Majorana mass matrix simply on experimen- Thus, we can define: tal bases, unless we find another observable which depends Majorana phases.

X 2 mββ ≡ U mi (27) ei i=1,2,3 where the index i runs on the 3 light neutrinos with given 3 Note that the specific choice and the symbols for these phases mass. This parameter is often called “effective Majorana may differ among authors. 10

Furthermore, a specific observation on the Type I Seesaw model is useful. Let us consider the simplest case with TABLE II. Results of the global 3ν oscillation analysis, in terms of best-fit values and allowed 1σ range for the 3ν mass- only νe and one heavy neutrino νH that mix with this mixing parameters relevant for our analysis as reported in state. The Majorana mass matrix is of the form: Ref. [94]. The last column is our estimate of the σ while ! assuming symmetric uncertainties. 0 mDirac . (30) Dirac Parameter Best fit 1σ range σsymmetric m MH NH

One should not be mislead, concluding that in this case 2 −1 −1 −1 (and, generally, in the Type I Seesaw) m is zero. In sin (θ12) 3.08 · 10 (2.91 − 3.25) · 10 0.17 · 10 ββ 2 −2 −2 −2 fact, as it is well-known, the masses of the light neutrinos sin (θ13) 2.34 · 10 (2.16 − 2.56) · 10 0.22 · 10 2 −1 −1 −1 (in this case, of νe) arise when one integrates away the sin (θ23) 4.37 · 10 (4.14 − 4.70) · 10 0.33 · 10 heavy neutrino state, getting δm2 [eV2] 7.54 · 10−5 (7.32 − 7.80) · 10−5 0.26 · 10−5 2 2 −3 −3 −3 (mDirac)2 ∆m [eV ] 2.44 · 10 (2.38 − 2.52) · 10 0.08 · 10 mνe = − . (31) MH IH 2 −1 −1 −1 As discussed in Ref. [75], we obtain in this one-flavor case sin (θ12) 3.08 · 10 (2.91 − 3.25) · 10 0.17 · 10 2 −2 −2 −2 the non-zero contribution sin (θ13) 2.39 · 10 (2.18 − 2.60) · 10 0.21 · 10 2 −1 −1 −1 sin (θ23) 4.55 · 10 (4.24 − 5.94) · 10 1.39 · 10  hq2i 2 2 −5 −5 −5 mββ = |mνe | 1 + 2 . (32) δm [eV ] 7.54 · 10 (7.32 − 7.80) · 10 0.26 · 10 MH ∆m2 [eV2] 2.40 · 10−3 (2.33 − 2.47) · 10−3 0.07 · 10−3 The second factor is the direct contribution of the heavy neutrino.4 The quantity hq2i depends on the nuclear structure and it is of the order of (100 MeV)2 and thus oscillation experiments, the squared mass splittings be- Eq. (32) is valid if we assume |mνe |  100 MeV  MH . In the above discussion, we have emphasized the three tween the three active neutrinos. In TableII, the param- flavor case. The main reason for this is evidently that we eters relevant for our analysis are reported. The mass 2 2 know about the existence of only 3 light neutrinos. It is splittings are labeled by δm and ∆m . The former is possible to test this hypothesis by searching for new oscil- measured through the observation of solar neutrino oscil- lation phenomena, by testing the universality of the weak lations, while the latter comes from atmospheric neutrino leptonic couplings and/or the unitarity of the matrix in data. The definitions of these two parameters are the fol- Eq. (28), by searching directly at accelerators new and lowing: (not too) light neutrino states, etc. . However, we believe that it is fair to state that, to date, we have no conclu- m2 + m2 δm2 ≡ m2 − m2 and ∆m2 ≡ m2 − 1 2 . (33) sive experimental evidence or strong theoretical reason 2 1 3 2 to deviate from this minimal theoretical scheme. We will 2 adopt it in the proceeding of the discussion. In this way, Practically, δm regards the splitting between ν1 and ν2, 2 we can take advantage of the precious information that while ∆m refers to the distance between the ν3 mass was collected on the neutrino masses to constrain the and the mid-point of ν1 and ν2 masses. parameter mββ and to clarify the various expectations. The sign of δm2 can be determined by observing mat- ter enhanced oscillations as explained within the MSW theory [95, 96]. It turns out, after comparing with ex- B. Oscillations perimental data, that δm2 > 0 [97]. Unfortunately, de- termining the sign of ∆m2 is still unknown and it is not simple to measure it. However, it has been argued (see e. g. Ref. [98]) that by carefully measuring the oscillation In Ref. [94], a complete analysis of the current knowl- pattern, it could possible to distinguish between the two edge of the oscillation parameters and of neutrino masses possibilities, ∆m2 > 0 and ∆m2 < 0. This is a very can be found. Although the absolute neutrino mass scale promising perspective in order to solve this ambiguity, is still unknown, it has been possible to measure, through which is sometimes called the “mass hierarchy problem”. In fact, standard names for the two mentioned possibil- ities for the neutrino mass spectra are “Normal Hierar- 2 4 chy” (NH) for ∆m > 0 and “Inverted Hierarchy” (IH) This formula agrees with the naive scaling expected from the 2 heavy neutrino contribution. But in specific three-flavor models for ∆m < 0. it is possible, at least in principle, that heavy neutrinos give The oscillation data are analyzed in Ref. [94] by writing a large and even dominating contribution to the 0νββ decay the leptonic (PMNS) mixing matrix U|osc. in terms of the rate [75]. mixing angles θ12, θ13 and θ23 and of the CP-violating 11

1

0.1

0.1 IH(Δm 2<0) IH(Δm 2<0) [ eV ] 0.01 [ eV ] ββ ββ 0.01 m m 2 NH(Δm 2>0) NH(Δm >0) 0.001

10-4 0.001 10-4 0.001 0.01 0.1 1 0.1 0.2 0.5

mlightest [eV] Σcosm [eV]

FIG. 6. Updated predictions on mββ from oscillations as a function of the lightest neutrino mass (left) and of the cosmological mass (right) in the two cases of NH and IH. The shaded areas correspond to the 3σ regions due to error propagation of the uncertainties on the oscillation parameters. Figure from Ref. [93]. phase φ according to the (usual) representation However, as already recalled, since the complex phases of the mixing parameters in Eq. (29) cannot be probed by U| = (34) osc. oscillations, the allowed region for mββ is obtained let-  −iφ c12c13 s12c13 s13e ting them vary freely. The expressions for the resulting  iφ iφ  extremes (i. e. the mββ maximum and minimum values −s12c23 − c12s13s23e c12c23 − s12s13s23e c13s23  iφ iφ due to the phase variation) can be found in App.A. We s12s23 − c12s13c23e −c12s23 − s12s13c23e c13c23 adopt the graphical representation of mββ introduced in where sij, cij ≡ sin θij, cos θij. Note the usage of the Ref. [99] and refined in Refs. [18, 100]. It consists in plot- same phase convention and parameterization of the quark ting mββ in bi-logarithmic scale as a function of the mass (CKM) mixing matrix even if, of course, the values of of the lightest neutrino, both for the cases of NH and of the parameters are different. With this convention, it is IH. The resulting plot is shown in the left panel of Fig.6. possible to obtain Eq. (29) by defining The uncertainties on the various parameters are propa- gated using the procedures described in App.B. This   results in a wider allowed region, which corresponds to U ≡ U| · diag e−iξ1/2, e−iξ2/2, eiφ−iξ3/2 . (35) osc. the shaded parts in the picture. TableII shows the result of the best fit and of the 1 σ range for the different oscillation parameters. It can be 1. Mass eigenstates composition noted that the values are slightly different depending on the mass hierarchy. This comes from the different analy- The standard three flavor oscillations involve three sis procedures used during the evaluation, as explained in massive states that, consistently with Eq. (28), are given Ref. [94]. Therefore, throughout this work the two neu- by 5 trino mass spectra are treated differently one from the X other, since we used these hierarchy-dependent param- |ν i = U |ν i . (36) eters. The uncertainties are not completely symmetric i `i ` `=e,µ,τ around the best fit point, but the deviations are quite small, as claimed by the authors themselves in the refer- Thus, it is possible to estimate the probability of finding ence. In particular, the plots in the paper show Gaussian the component ν` of each mass eigenstate νi. This prob- likelihoods for the parameters determining mββ. In or- ability is just the squared module of the matrix element der to later propagate the errors, we decided to neglect U`i, since the matrix is unitary. The result is graphically the asymmetry, which has no relevant effects on the pre- shown in Fig.7. As already mentioned, since hierarchy- sented results. We computed the maximum between the dependent parameters were used, the flavor composition distances of the best fit values and the borders of the of the various eigenstates slightly depends on the mass 1σ range (fourth column of Tab.II) and we assumed hierarchy. It is worth noting that the results also depend that the parameters fluctuate according to a Gaussian distribution around the best fit value, with a standard deviation given by that maximum. Thanks to the knowledge of the oscillation parameters, 5 Note that in this case we are in the ultra-relativistic limit. See Sec.IIC. it is possible to put a first series of constraints on mββ. 12

Normal Hierarchy It can be useful to compute the mass of the lightest neutrino, given a value of Σ. This can be convenient in ν 6 3 order to compute mββ as a function of Σ instead of m. In this way, mββ is expressed as a function of a directly Δm2 observable parameter. The close connection between the neutrino mass mea- ν 2 surements obtained in the laboratory and those probed δm2 ν by cosmological observations was outlined long ago [101]. 1 ν e Furthermore, the measurements of Σ have recently

νμ reached important sensitivities, as discussed in Sec.VII. Inverted Hierarchy In the right panel of Fig.6, an updated version of the ντ plot (mββ vs. Σ) originally introduced in Ref. [102] is ν2 2 shown. Concerning the treatment of the uncertainties, δm we use again the assumption of Gaussian fluctuations and ν 1 the prescription reported in App.B.

Δm2

ν3 2. Constraints from cosmological surveys FIG. 7. Graphic view of the probability of finding one of the flavor eigenstates if the neutrino is in a certain mass eigen- state. The value φ = 0 for the CP-violating phase is assumed. The indications for neutrino masses from cosmology has kept changing for the last 20 years. A comprehensive review on the topic can be found in Ref. [107]. In Fig.8 on the possible choices of φ, while they do not depend the values for Σ given in Refs. [103–106] are shown. The on the eventual Majorana phases. TableIII reports the scientific literature contains several authoritative claims calculation for the cases φ = 0 and φ = 1.39π (1.31π), for a non-zero value for Σ but, being different among each best fit value for the NH (IH) according to Ref. [94]. others, these values cannot be all correct (at least) and

C. Cosmology and neutrino masses TABLE III. Flavor composition of the neutrino mass eigen- states. The two cases refer to the values for the CP-violating phase φ = 0 and φ = 1.39π (1.31π), best fit value in case of 1. The parameter Σ NH (IH) according to Ref. [94]

The three light neutrino scenario is consistent with all Eigenstate NHIH known facts in particle physics including the new mea- (φ = 0) (φ = 1.39π)(φ = 0) (φ = 1.31π) surements by Planck [34]. In this assumption, the phys- ν ical quantity probed by cosmological surveys, Σ, is the 1 sum of the masses of the three light neutrinos: νe .676 .676 .675 .675 νµ .254 .160 .252 .141

Σ ≡ m1 + m2 + m3. (37) ντ .070 .164 .073 .184 ν Depending on the mass hierarchy, is it possible to express 2 Σ as a function of the lightest neutrino mass m and of νe .301 .301 .301 .301 the oscillation mass splittings. In particular, in the case νµ .331 .425 .322 .432 of NH one gets: ντ .368 .274 .378 .267

 ν3  m1 = m  ν .023 .023 .024 .024 p 2 2 e m2 = m + δm (38) νµ .415 .415 .426 .426  p 2 2 2  m3 = m + ∆m + δm /2 ντ .562 .562 .550 .550 while, in the case of IH:

 p 2 2 2  m1 = m + ∆m − δm /2  6 In App.C, an approximate (but accurate) alternative method p 2 2 2 m2 = m + ∆m + δm /2 (39) for the numerical calculation needed to make this conversion is  given. m3 = m. 13

10 recently, a very stringent limit, Σ < 146 meV (2σ C. L.), was set by Palanque-Delabrouille and collaborators [106]. 1 New tight limits were presented after the data release by the Planck Collaboration in 2015 [34]. Some of the most 10-1 significant results are reported in TableIV. The bounds

Σ [eV] on Σ indicated by these post-Planck studies are quite small, but they are still larger than the final sensitivi- 10-2 ties expected, especially thanks to the inclusion of other cosmological data sets probing smaller scales (see e. g. 10-3 Refs. [112, 113] for review works). Therefore, these small 1995 2000 2005 2010 2015 values cannot be considered surprising and, conversely, Publication year margins of further progress are present. In our view, this situation should be considered as fa- FIG. 8. Evolution of some significant values for Σ as indicated vorable since more proponents are forced to carefully ex- by cosmology, based on well-known works [103–106]. Since amine and discuss all the available hypotheses. In view of the error for the first value is not reported in the reference, this discussion, in Sec.VII we consider two possible sce- we assumed an error of 50% for the purpose of illustration. narios and discuss the implications from the cosmological The yellow region includes values of Σ compatible with the investigations for the 0νββ in both cases. NH spectrum, but not with the IH one. The gray band in- cludes values of Σ incompatible with the standard cosmology and with oscillation experiments. D. Other non-oscillations data this calls us for a cautious attitude in the interpretation. Referring to the most recent years, two different posi- For the sake of completeness, we mention other two po- tions emerge: on one side, we find claims that cosmology tential sources of information on neutrinos masses. They provides us a hint for non-zero neutrino masses. On the are: other, we have very tight limits on Σ. • the study of kinematic effects (in particular of su- In the former case, it has been suggested [105, 108] pernova neutrinos) that a total non zero neutrino mass around 0.3 eV could alleviate some tensions present between cluster number • the investigation of the effect of mass in single beta counts (selected both in X-ray and by Sunyaev-Zeldovich decay processes. effect) and weak lensing data. A sterile neutrino parti- cle with mass in a similar range is sometimes also ad- The first type of investigations, applied to SN1987A, pro- vocated [109, 110]. However, evidence for non-zero neu- duced a limit of about 6 eV on the electron antineutrino trino masses either in the active or sterile sectors seems mass [118, 119]. The perspectives for the future are con- to be claimed in order to fix the significant tensions be- nected to new detectors, or to the existence of antineu- tween different data sets (cosmic microwave background trino pulses in the first instants of a supernova emission. (CMB) and baryonic acoustic oscillations (BAOs) on one The second approach, instead, is presently limited to side and weak lensing, cluster number counts and high about 2 eV [120, 121], even having the advantage of be- values of the Hubble parameter on the other). ing obtained in controlled conditions – i. e. in laboratory. In the latter case, the limit on Σ is so stringent, that Its future is currently in the hands of new experiments 3 it better agrees with the NH spectrum, rather than with based on a H source [122] and on the electron capture IH one (see the discussion in Sec.VIIA). 7 The tightest experimental limits on Σ are usually obtained by combin- ing CMB data with the ones probing smaller scales. In TABLE IV. Tight constraints on Σ obtained in 2015, by ana- this way, their combination allows a more effective inves- lyzing the data on the CMB by the Planck Collaboration [34], tigation of the neutrino induced suppression in terms of polarization included, along with other relevant cosmological matter power spectrum, both in scale and redshift. Quite data probing smaller scales. upper bound included dataset on Σ (2σ C. L.)

7 a Actually, it has been shown in Ref. [111] that the presence in 153 meV, [34] SNe, BAO, H0 prior the nuclear medium of L-violating four-fermion interactions of 120 meV, [114] Lyman-α neutrinos with quarks from a decaying nucleus could account for an apparent incompatibility between the 0νββ searches in the 126 meV, [115] BAO, H0, τ priors , Planck SZ clusters laboratory and the cosmological data. In fact, the net effect of 177 meV, [116] BAO these interactions (not present in the latter case) would be the 110 meV, [117] BAO, galaxy clustering, lensing generation of an effective “in-medium” Majorana neutrino mass a matrix with a corresponding enhancement of the 0νββ rate. Results as reported in wiki.cosmos.esa.int/planckpla2015, page 311. 14 of 163Ho [123–125], which have the potential to go below Even Mass Odd Mass the eV in sensitivity. Number Number N,Z odd Nuclear Mass Nuclear Mass ββ- ββ+

Suppressed ~1021 E. Theoretical understanding ββ- ββ+ N,Z even Theorists have not been very successful in anticipating - + the discoveries on neutrino masses obtained by means of β β oscillations. The discussion within gauge models clarified Z-2 Z-1 Z+1 Z+2 Z-2 Z-1 Z+1 Z+2 Z Z that it is possible or even likely to have neutrino masses in Atomic Number Atomic Number gauge models (compare with Sec.IID). However, a large part of the theoretical community focused for a long time FIG. 9. Nuclear mass as a function of the atomic number Z on models such as “minimal SU(5)”, where the neutrino in the case of an isobar candidate with A even (left) and A masses are zero, emphasizing the interest in proton decay odd (right). search rather than in neutrino mass search. On top of that, we had many models that aimed to predict e. g., the trino mass matrix correct solar neutrino solution or the size of θ13 before the measurements, but none of them were particularly M = m × diag(ε, 1, 1) C diag(ε, 1, 1) (40) convincing. More specifically, a lot of attention was given neutrino to the “small mixing angle solution” and the “very small where the flavor structure is dictated by a diagonal ma- θ13 scenario”, that are now excluded from the data. trix that acts only on the electronic flavor and suppresses Moreover, it is not easy to justify the theoretical posi- the matrix elements Meµ,Meτ and Mee (twice). The di- tion where neutrino masses are not considered along the mensionful parameter (the overall mass scale) is given by masses of other fermions. This remark alone explains ∆ ≡ p∆m2 ≈ 50 meV. We thus have a matrix of co- the difficulty of the theoretical enterprise that theorists atm efficients C with elements C``0 = O(1) that are usually have to face. For the reasons commented in Sec.IID, the treated as random numbers of the order of 1 in the ab- SO(10) models are quite attractive to address a discus- sence of a theory. A choice of ε that suggested values sion of neutrino masses. However, even considering this of θ and θ in the correct region (before their mea- specific class of well-motivated Grand Unified groups, it 12 13 surement) is ε = θ or pm /m [129]. Within these remains difficult to claim that we have a complete and C µ τ assumptions, the matrix element in which we are inter- convincing formulation of the theory. In particular, this ested is holds for the arbitrariness in the choice of the representa- tions (especially that of the Higgs bosons), for the large 2 mββ = m ε O(1) ≈ (2 − 4) meV. (41) number of unknown parameters (especially the scalar po- tential), for the possible role of non-renormalizable oper- Finally, we note that the SM of the ele- ators, for the uncertainties in the assumption concerning ments of the neutrino mass matrix is multiplicative. The low scale supersymmetry, for the lack of experimental effect of renormalization is therefore particularly small tests, etc. . Note that, incidentally, preliminary investiga- for mββ (see e. g. Eq. (17) of Ref. [130] and the discus- tions on the size of mββ in SO(10) did not provide a clear sion therein). In other words, the value mββ = 0 (or evidence for a significant lower bound [126]. Anyway, values close to this one) should be regarded as a stable even the case of an exactly null effective Majorana mass point of the renormalization flow. does not increase the symmetry of the Lagrangian, and Let us conclude repeating that, anyway, there are thus does not forbid the 0νββ, as remarked in Ref. [127]. many reasons to consider the theoretical expectations Here, we just consider one specific theoretical scheme, with detachment, and the above theoretical scheme is not for illustration purposes. This should not be considered an exception to this rule. It is very important to keep in a full fledged theory, but rather it attempts to account mind this fact in order to properly assess the value of the for the theoretical uncertainties in the predictions. The search for the 0νββ and to proceed accordingly in the hierarchy of the masses and of the mixing angles has sug- investigations. gested the hypothesis that the elements of the Yukawa couplings and thus of the mass matrices are subject to some selection rule. The possibility of a U(1) selection rule has been proposed in Ref. [128] and, since then, it V. THE ROLE OF NUCLEAR PHYSICS has become very popular. Immediately after the first strong evidences of atmo- 0νββ is first of all a nuclear process. Therefore, the spheric neutrino oscillations (1998) specific realizations transition has to be described properly, taking into ac- for neutrinos have been discussed in various works (see count the relevant aspects that concern nuclear structure Ref. [129] for references). These correspond to the neu- and dynamics. In particular, it is a second order nuclear 15 weak process and it corresponds to the transition from a 7 IBM 2 nucleus (A, Z) to its isobar (A, Z + 2) with the emission Pd of two electrons. In principle, a nucleus (A, Z) can decay 6 Mo QRPA Tü Ge ISM via double beta decay as long as the nucleus (A, Z + 2) 5 Se is lighter. However, if the nucleus can also decay by sin- Te Cd Xe Gd U gle beta decay, (A, Z + 1), the branching ratio for the 4 Te 0 Sn Th 0νββ will be too difficult to be observed due to the over- Xe Sm M 3 Zr Nd whelming background rate from the single beta decay. Nd Pt Therefore, candidate isotopes for detecting the 0νββ are 2 Ca even-even nuclei that, due to the nuclear pairing force, are lighter than the odd-odd (A, Z + 1) nucleus, making 1 single beta decay kinematically forbidden (Fig.9). It is 0 worth noting that, since the 0νββ candidates are even- 20 40 60 80 100 120 140 even nuclei, it follows immediately that their spin is al- number ways zero. The theoretical expression of the half-life of the process FIG. 10. Most updated NMEs calculations for the 0νββ with in a certain nuclear species can be factorized as: the IBM-2 [138], QRPA-T¨u[139] and ISM [140] models. The results somehow differ among the models, but are not too far 1/2 −1 2 2 [t ] = G0ν |M| |f(mi,Uei)| (42) away. Figure from Ref. [138]. where G0ν is the phase space factor (PSF), M is the nuclear matrix element (NME) and f(mi,Uei) is an adi- Recent developments in the numerical evaluation of mensional function containing the particle physics be- Dirac wave functions and in the solution of the Thomas- yond the SM that could explain the decay through the Fermi equation allowed to calculate accurately the PSFs neutrino masses mi and the mixing matrix elements Uei. both for single and double beta decay. The key ingredi- In this section, we review the crucial role of nuclear ents are the scattering electron wave functions. The new physics in the expectations, predictions and eventual calculations take into account relativistic corrections, the understanding of the 0νββ, also assessing the present finite nuclear size and the effect of the atomic screening knowledge and uncertainties. We mainly restrict to the on the emitted electrons. The main difference between discussion of the light neutrino exchange as the candi- these calculations and the older ones is of the order of a date process for mediating the 0νββ transition, but the few percent for light nuclei (Z = 20), about 30% for Nd mechanism of heavy neutrino exchange is also considered. (Z = 60), and a rather large 90% for U (Z = 92). In the former case (m . 100 MeV, see Eq. (19)), the In Refs. [135–137], the most up to date calculations of factor f is proportional mββ: the PSFs for 0νββ can be found. The results obtained in these works are quite similar. Throughout this paper,

we use the values from the first reference. mββ 1 X 2 f(mi,Uei) ≡ = Uekmk (43) me me k=1,2,3 B. Models for the NMEs where the electron mass me is taken as a reference value. In the scheme of the heavy neutrino exchange Let us suppose that the decay proceeds through an s- (m & 100 MeV), the effective parameter is instead: wave. Since we have just two electrons in the final state, we cannot form an angular momentum greater than one. Therefore, usually only 0νββ matrix elements to final 0+ −1 X 2 1 f(mi,Uei) ≡ mp MH = mp UeI (44) states are considered. These can be the ground state, MI I=heavy + + 01 , or the first excited state, 02 . Of course, we consider as a starting state just a 0+ state, since the double beta where the proton mass mp is now used, according to the decay is possible only for (Z,A) even-even isobar nuclei. tradition, as the reference value. The calculation of the NMEs for the 0νββ is a diffi- cult task because the ground and many excited states of open-shell nuclei with complicated nuclear structure A. Recent developments on the phase space factor have to be considered. The problem is faced by using calculations different approaches and, especially in the last few years, the reliability of the calculations improved a lot. Here, The first calculations of PSFs date back to the late a list of the main theoretical models is presented. The 1950s [131] and used a simplified description of the wave most relevant features for each of them are highlighted. functions. The improvements in the evaluation of the PSFs are due to always more accurate descriptions and • Interacting Shell Model (ISM), [140, 141]. In the less approximations [132–134]. ISM only a limited number of orbits around the 16

Fermi level is considered, but all the possible corre- C. Theoretical uncertainties lations within the space are included and the pair- ing correlations in the valence space are treated ex- 1. Generality actly. Proton and neutron numbers are conserved and angular momentum conservation is preserved. Following Eq. (42), an experimental limit on the A good spectroscopy for parent and daughter nuclei 0νββ half-life translates in a limit on the effective Majo- is achieved rana mass:

me mββ ≤ √ . (45) • Quasiparticle Random Phase Approximation 1/2 M G0ν t (QRPA), [139, 142]. The QRPA uses a large valence space and thus it cannot comprise all the From the theoretical point of view, in order to constrain possible configurations. Typically, single particle mββ, the estimation of the uncertainties both on G0ν states in a Woods-Saxon potential are considered. and M is crucial. Actually, the PSFs can be assumed The proton-proton and neutron-neutron pairings quite well known, being the error on their most recent are taken into account and treated in the BCS calculations around 7% [135]. approximation (proton and neutron numbers are A convenient parametrization for the NMEs is the fol- not exactly conserved) lowing [145]:  2 ! (0ν) gV (0ν) (0ν) • Interacting Model (IBM-2), [138] In the M ≡ g2 M = g2 M − M + M A 0ν A GT g F T IBM, the low-lying states of the nucleus are mod- A eled in terms of bosons. The bosons are in either s (46) boson (L = 0) or d boson (L = 2) states. There- where gV and gA are the axial and vector coupling con- + + (0ν) fore, one is restricted to 0 and 2 neutron pairs stants of the nucleon, MGT is the Gamow-Teller (GT) transferring into two protons. The bosons inter- operator matrix element between initial and final states (0ν) act through one- and two-body forces giving rise to (spin-spin interaction), MF is the Fermi contribution bosonic wave functions. (0ν) (spin independent interaction) and MT is the tensor operator matrix element. The form of Eq. (46) empha- • Projected Hartree-Fock Bogoliubov Method sizes the role of gA. Indeed, M0ν mildly depends on (PHFB), [143] In the PHFB, the NME are calcu- gA and can be evaluated by modeling theoretically the lated using the projected-Hartree-Fock-Bogoliubov nucleus. Actually, it is independent on gA if the same wave functions, which are eigenvectors of four quenching is assumed both for the vector and axial cou- different parameterizations of a Hamiltonian with pling constants, as we do here for definiteness, following pairing plus multipolar effective two-body interac- Ref. [146]. tion. In real applications, the nuclear Hamiltonian is restricted only to quadrupole interactions 2. Is the uncertainty large or small?

• Energy Density Functional Method (EDF), [144]. The main sources of uncertainties in the inference on The EDF is considered to be an improvement with mββ are the NMEs. A comparison of the calculations respect to the PHFB. The state-of-the-art density from 1984 to 1998 revealed an uncertainty of more than functional methods based on the well-established a factor 4 [100]. A similar point of view comes out from Gogny D1S functional and a large single particle the investigation of Ref. [147], where the results of the basis are used. various calculations were used to attempt a statistical inference. The most common methods are ISM, QRPA and IBM- An important step forward was made with the first 2. In Fig. 10, a comparison among the most recent NME calculations of M0ν that estimated also the errors, see calculations computed with these three models is shown. Refs. [148, 149]. These works, based on the QRPA model, It can be seen that the disagreement can be generally assessed a relatively small intrinsic error of ∼ 20%. The quantified in some tens of percents, instead of the factors validity of these conclusions have been recently supported 2−4 of the past. This can be quite satisfactory. As it will by the (independent) calculation based on the IBM-2 de- be discussed in Sec.VC, the main source of uncertainty scription of the nuclei [135, 138], which assesses an in- in the inference does not rely in the NME calculations trinsic error of 15% on M0ν . However, the problem in anymore, but in the determination of the quenching of assessing the uncertainties in the NMEs is far from being the axial vector . For this reason, in solved. Each scheme of calculation can estimate its own the subsequent discussion we will restrict to one of the uncertainty, but it is still hard to understand the differ- considered models, namely the IBM-2 [138], without sig- ences in the results among the models (Fig. 10) and thus nificant loss of generality. give an overall error. Notice also that when a process 17

“similar” to the 0νββ is considered (single beta decay, 400 electron capture, 2νββ) and the calculations are com- ●● IBM-2 QRPA-Tü pared with the measured rates, the actual differences are ■ ■■ much larger than 20% [146]. This suggests that it is not ■ 300 ■ cautious to assume that the uncertainties on the 0νββ are ■ instead subject to such a level of theoretical control. ■ ■ Recently, there has been a lively interest in a specific ( 0 ν ) I 200 ■ and important reason of uncertainty, namely the value M ■ ■ ● of the axial coupling constant gA. This has a direct im- ● ■ plication on the issue that we are discussing, since any ● 100 ● ● ● ● uncertainty on the value of gA reflects itself into a (larger) ● ● ● uncertainty factor on the value of the matrix element M. ■● We will examine these arguments in greater details in the 0 rest of this section. 48Ca 76Ge 82Se 96Zr 100Mo 110Pd 116Cd 124Sn 128Te 130Te 136Xe It is important to appreciate the relevance of these con- Isotope siderations for the experimental searches. If the value of the axial coupling in the nuclear medium is decreased by FIG. 11. Most updated NMEs calculations for the 0νββ via a factor δ, namely gA → gA ·(1−δ), the expected decay heavy neutrino exchange with the IBM-2 [138] and QRPA- rate and therefore the number of signal events S will also T¨u[150] models. In both cases, the value gA = gnucleon for the decrease, approximatively as S · (1 − δ)4. This change axial coupling constant and the Argonne parametrization for can be compensated by increasing the time of data tak- the short-range correlations are assumed. The results show a continuous overestimation of the QRPA estimations over the ing or the mass of the√ experiment. However, the figure of merit, namely S/ B, which quantifies the statistical IBM-2 ones. significance of the measurement, changes only with the square root of the time or of the mass, in the typical 1970s [156–159], when it was argued that gA ∼ 1. case in which there are also background events B. For In heavy nuclei, the question of quenching was first instance, if we have a decrease by δ = 10 (20)% of the ax- discussed in Ref. [156]. In this case, g was found ial coupling, we will obtain the same measurement after A 8 to be even lower than 1, thus stimulating the state- a time that is larger by a factor of 1/(1 − δ) = 2.3 (6). ment that massive renormalization of g occurs; In other words, an effect that could be naively consid- A ered small has instead a big impact for the experimental • the contribution of non-nucleonic degrees of free- search for the 0νββ. dom. This effect does not depend much on the nuclear model adopted, but rather on the mecha- nism of coupling to non-nucleonic degrees of free- 3. The size of the axial coupling dom. It was extensively investigated theoretically in the 1970s [160–162]. Recently, it has been inves- It is commonly expected that the value gA ' 1.269 tigated again within the framework of the chiral Ef- measured in the weak interactions and decays of nucle- fective Field Theory (EFT) [163]. It turns out that ons is “renormalized” in the nuclear medium towards the it may depend on momentum transfer and that it value appropriate for quarks [148, 149, 151]. It was ar- may lead in some cases to an enhancement rather gued in Ref. [146] that a further modification (reduction) than a quenching; is rather plausible. This is in agreement with what was stated some years before in Ref. [152], where the possibil- • the renormalization of the GT operator due to two- body currents. The first calculations for GT tran- ity of a “strong quenching” of gA (i. e. gA < 1) is actually favored. The same was also confirmed by recent study sitions for the 0νββ operator based on the chiral on single beta decay and 2νββ [153]. It has to be noticed EFT [163] showed the importance of two-body cur- that within the QRPA framework, the dependence of M rents for the effective quenching of gA. This was later confirmed in independent works [164, 165] upon gA is actually milder than quadratic, because the model is calibrated through the experimental 2νββ decay and, more recently, by the use of a no-core- rates using also another parameter, the particle-particle configuration-interaction formalism within the den- sity functional theory [155]. strength gpp [154]. There could be different causes for the quenching of It is still not clear if the quenching in both the tran- g . It was found that it can be attributed mainly to the A sitions (0νββ and 2νββ) is the same. One argument following issues [146, 155]: which suggests that this is not unreasonable, consists in • the limited model space (i. e. the size of the basis noting that the 2νββ can occur only through a GT (1+) of the eigenstates) in which the calculation is done. transition. Instead, the 0νββ could happen through all This problem is by definition model dependent and the possible intermediate states, so it is possible to ar- it was extensively investigated in light nuclei in the gue that the transitions through states with spin parity 18

+ -4 different from 1 can be unquenched or even enhanced. 10 DELPHI Incidentally, it turns out that the dominant multipole -5 in the 0νββ transition is the GT one, thus making the 10 TRIUMF CHARM hypothesis that the quenching in 2νββ and 0νββ is the quenching PS 191 gA same quite solid. Following Ref. [138], we adopt this as a 10-6 working hypothesis in our discussion, however keeping in 2 0νββ( 76Ge) eI -7

mind that some indications that the quenching might be U 10 different in the 0νββ and 2νββ transitions are present in other models [140, 165]. 10-8 It would be extremely precious if these theoretical SHiP questions could be answered by some experimental data. 10-9 It has been argued that the experimental study of nu- clear transitions where the nuclear charge is changed by 10-10 0.01 0.10 1 10 100 two units leaving the mass number unvaried, in anal- M [GeV] ogy to the 0νββ decay, could give important informa- I tion. Despite the Double Charge Exchange reactions and FIG. 12. Bounds on the mixing between the electron neutrino 0νββ processes are mediated by different interactions, and a single heavy neutrino from the combination of bounds some similarities between the two cases are presents. obtained with Ge 0νββ experiments [167] using the repre- These could be exploited to assess effectively the NME for sentation introduced in Ref. [168]. The bands correspond to the 0νββ (and, more specifically, the entity of the quench- the uncertainties discussed in the text. The dashed contours ing of gA). In the near future, a new project will be indicate the mass regions excluded by some of the accelera- started at the Laboratori Nazionali del Sud (Italy) [166] tor experiments considered in Ref. [74]: CHARM (90% C. L., with the aim of getting some inputs to deepen our theo- [169]), DELPHI (95% C. L., [170]), PS 191 (90% C. L., [171]), retical understanding of this nuclear process. TRIUMF (90% C. L., [172, 173]). The continuous contour in- dicate the expected probed region by the new proposed SHiP experiment at the CERN SPS [87]. Figure from Ref. [87]. 4. Quenching as a major cause of uncertainty ing, since the specific behavior is different in each nucleus In view of the above considerations, we think that cur- and it somewhat differs from this parametrization [146]. rently the value of g in the nuclear medium cannot be A The question which is the “true value” of gA is still regarded as a quantity that is known reliably. It is rather open and introduces a considerable uncertainty in the in- an important reason of uncertainty in the predictions. In ferences concerning massive neutrinos. The implications a conservative treatment, we should consider at least the are discussed in Secs.VIF andVIG. following three cases,  g = 1.269  nucleon D. The case of heavy neutrino exchange gA = gquark = 1 (47)  −0.18 gphen. = gnucleon · A As already discussed in Sec.III, it is possible to at- tribute the 0νββ decay rate to the same particles that where the last formula includes phenomenologically the are added to the SM spectrum to explain oscillations, effect of the atomic number A. It represents the e. g. heavy neutrinos. In this context one can assume that worst possible scenario for the 0νββ search. The the exchange of MH > 100 MeV saturates the 0νββ de- gphen. parametrization as a function of A comes directly cay rate, also reproducing the ordinary neutrino masses. from the comparison between the theoretical half-life for Heavy neutrino masses and mixing angles, compatible 2νββ and its observation in different nuclei, as reported with the rate of 0νββ, depend on the NMEs of the tran- in Ref. [146]. From the comparison between the theoret- sition (compare e. g. [74] and [75]). Thus, nuclear physics ical half-life for the process and the experimental value has an impact also on the limits that are relevant for a it was possible to extract an effective value for gA, thus direct search for heavy neutrinos with accelerators. Each determining its quenching. The assumption that gA de- scheme of nuclear physics calculation can estimate its in- pends only upon the atomic number A is rather conve- trinsic uncertainty. This is usually found to be small nient for a cursory exploration of the potential impact of in modern computations (about 28% for heavy neutrino unaccounted nuclear physics effects on 0νββ, but most exchange [138]). In a conservative treatment, this uncer- likely it is also an oversimplification of the truth, as sug- tainty plus the already discussed unknown value of gA gested by the residual difference between the calculated should be taken into account. It has to be noticed that 2νββ rates. Surely, it cannot replace an adequate theo- if the 0νββ is due to a point-like (dimension-9) operator, retical modeling, that in the light of the following discus- as for heavy neutrino exchange, two nucleons are in the sion has become rather urgent. Anyway, we stress that same point. Therefore, the effect of a hard core repulsion, this is just a phenomenological description of the quench- estimated modeling the “short-range correlations”, plays 19 an important role in the determination of the uncertain- ties. A significant step forward has been recently made, pushing down this source of theoretical error of about an order of magnitude [138]. The most updated NMEs for the 0νββ via heavy neu- trino exchange are evaluated within the frames of the IBM-2 [138] and QRPA [150] models. A comparison be- 2νββ tween these results is shown in Fig. 11. It can be seen that Counts the values obtained within the QRPA model are always larger than those obtained with the IBM-2. The differ- 0νββ ence is quite big for many of the nuclei and might be due to the different treatment of the intermediate states. Also Q in this case, we use the NMEs evaluated with the IBM-2 Total electron energy ββ model. This allows us to keep a more conservative ap- proach by getting less stringent limits. Considering, for FIG. 13. Schematic view of the 2νββ and the 0νββ spectra. example, the case of 76Ge, we have:

( 104 ± 29 gA = gnucleon four different mechanisms are possible: M0ν (Ge) = . (48) 22 ± 6 g = g A phen. (A, Z) → (A, Z + 2) + 2e− (β−β−) (A, Z) → (A, Z + 2) + 2e+ (β+β+) From the experimental point of view, the limits on (50) 2 − 0νββ indicate that the mixings of heavy neutrinos |UeI | (A, Z) + 2e → (A, Z − 2) (ECEC) are small. Using the current values for the PSF, NME (A, Z) + e− → (A, Z − 2) + e+ (EC β+). and sensitivity for the isotope [174], we get: Here, β− (β+) indicate the emission of an electron 1 U 2 7.8 · 10−8  104  3 · 1025 yr 2 () and EC stands for electron capture (usually a X eI < · · 0ν K-shell electron is captured). MI mp M0ν (Ge) τ I 1/2 The explicit violation of the number of electronic lep- (49) tons e, ¯e, νe orν ¯e appears evident in each process in where mp is the proton mass and the heavy neutrino Eq. (50). A large number of experiments has been and masses MI are assumed to be & GeV. is presently involved in the search for these processes, Fig. 12 illustrates the case of a single heavy neutrino especially of the first one. mixing with the light ones and mediating the 0νββ tran- In this section, we introduce the experimental as- sition. In particular, the plot shows the case of the mixing pects relevant for the 0νββ searches and we present an for 76Ge assuming that a single heavy neutrino dominates overview of the various techniques. We review the sta- the amplitude. The two regimes of heavy and light neu- tus of the past and present experiments, highlighting the trino exchange are matched as proposed in Ref. [168]. main features and the sensitivities. The expectations The colored bands reflect the different sources of theo- take into account the uncertainties coming from the theo- retical uncertainty. retical side and, in particular, those from nuclear physics. As it is clear from Fig. 12, the bound coming from The requirements for future experiments are estimated 0νββ searches is still uncertain. It weakens by one or- and finally, the new constraints from cosmology are used der of magnitude if the axial vector coupling constant is as a complementary information to that coming from the strongly quenched in the nuclear medium. 0νββ experiments. The potential of the 0νββ sensitivity to heavy neutri- nos is therefore weakened and very sensitive to theoreti- cal nuclear physics uncertainties. For some regions of the A. The 0νββ signature parameter space, even the limits obtained more than 15 years ago with accelerators are more restrictive than the From the experimental point of view, the searches for a current limits coming from 0νββ search. 0νββ signal rely on the detection of the two emitted elec- trons. In fact, being the energy of the recoiling nucleus negligible, the sum of kinetic energy of the two electrons is equal to the Q-value of the transition. Therefore, if we VI. EXPERIMENTAL SEARCH FOR THE 0νββ consider these as a single body, we expect to observe a monochromatic peak at the Q-value (Fig. 13). The process described by Eq. (1) is actually just one of Despite this very clear signature, because of the rar- the forms that 0νββ can assume. In fact, depending on ity of the process, the detection of the two electrons is the relative numbers of the nucleus protons and , complicated by the presence of background events in the 20

large mass. With the only exception of the 130Te, TABLE V. Isotopic abundance and Q-value for the known all the relevant isotopes have a natural isotopic 2νββ emitters [175]. abundance < 10%. This practically means that the

Isotope isotopic abundance (%) Qββ [MeV] condition translates into ease of enrichment for the material; 48Ca 0.187 4.263 76Ge 7.8 2.039 • compatibility with a suitable detection technique. It 82Se 9.2 2.998 has to be possible to integrate the isotope of inter- 96Zr 2.8 3.348 est in a working detector. The source can either be separated from the detector or coincide with it. 100Mo 9.6 3.035 Furthermore, the detector has to be competitive in 116Cd 7.6 2.813 130 providing results and has to guarantee the potential Te 34.08 2.527 for the mass scalability. 136Xe 8.9 2.459 150Nd 5.6 3.371 This results in a group of “commonly” studied iso- topes among all the possible candidate 0νββ emitters. It includes: 48Ca, 76Ge, 82Se, 96Zr, 100Mo, 116Cd, 130Te, 136 150 same energy region, which can mask the 0νββ signal. Xe and Nd. TableV reports the Q-value and the The main contributions to the background come from isotopic abundance for the mentioned isotopes. the environmental radioactivity, the cosmic rays, and the From the theoretical side, referring to Eq. (42), one 2νββ itself. In particular, the last contribution has the should also try to maximize both the PSF and the NME problematic feature of being unavoidable in presence of in order to get more strict bounds on mββ with the same finite energy resolution, since it is originated by the same sensitivity in terms of half-life time. However, as recently isotope which is expected to undergo 0νββ. discussed in Ref. [176], a uniform inverse correlation be- In principle, any event producing an energy deposition tween the PSF and the square of the NME emerges in similar to that of the 0νββ decay increases the back- all nuclei (Fig. 14). This happens to be more a coin- ground level, and hence spoils the experiment sensitivity. cidence than something physically motivated and, as a The capability of discriminate the background events is consequence, no isotope is either favored or disfavored thus of great important for this kind of search. for the search for the 0νββ. It turns out that all isotopes have qualitatively the same decay rate per unit mass for any given value of mββ. B. The choice of the isotope In recent time, also another criterion is becoming more and more relevant. This is simply the availability of the isotope itself in view of the next generations of 0νββ ex- periments, which will have a very large mass. In fact, The choice for the best isotope to look for 0νββ is the once the 0νββ isotope mass for an experiment will be of first issue to deal with. From one side, the background the order of some tons, a non negligible fraction of the level and the energy resolution need to be optimized. annual world production of the isotope of interest could From the other, since the live-time of the experiment can- be needed. This is e. g. the case of 136Xe, where the re- not exceed some years, the scalability of the technique, quests from the 0νββ experiments also “compete” with i. e. the possibility to build a similar experiment with en- those from the new proposed dark matter ones. The con- larged mass and higher exposure, is also fundamental. sequences are a probable price increase and a long storage This translates in a series of criteria for the choice of the for the isotope that needs to be taken into account. isotope:

• high Q-value (Qββ). This requirement is proba- bly the most important, since it directly influences C. Sensitivity the background. The 2615 keV line of 208Tl, which represents the end-point of the natural gamma ra- In the fortunate event of a 0νββ peak showing up in dioactivity, constitutes an important limit in terms the energy spectrum, starting from the law of radioactive of background level. Qββ should not be lower than decay, the decay half-life can be evaluated as ∼ 2.4 MeV (the only exception is the 76Ge, due N to the extremely powerful detection technique, see t1/2 = ln 2 · T · ε · ββ (51) Sec.VID). The ideal condition would be to have it Npeak even larger than 3270 keV, the highest energy beta among the 222Rn daughters (238U chain), coming where T is the measuring time, ε is the detection effi- from 214Bi; ciency, Nββ is the number of ββ decaying nuclei under observation, and Npeak is the number of observed decays • high isotopic abundance. This is a fundamental in the region of interest. If we assume to know exactly requirement to have experiments with sufficiently the detector features (i. e. the number of decaying nuclei, 21 the efficiency and the time of measurement), the uncer- TABLE VI. 1σ ranges both for Gaussian and Poisson distri- tainty on t1/2 is only due to the statistical fluctuations of butions for two different values of Npeak. In the former case, the counts: p we assumed a standard deviation equal to Npeak. To com- δt1/2 δN pute the error columns, we halved the total width of the range = peak . (52) 1/2 and divided it by Npeak. t Npeak It seems reasonable to suppose Poisson fluctuations on Distribution Npeak range relative error (%) Npeak. Since the expected number of events is “small”, Gauss 5 2.8 - 7.2 44.7 the Poisson distribution differs in a non negligible way 20 15.5 - 24.5 22.4 from the Gaussian. In order to quantify this discrepancy, Poisson 5 3.1 - 7.6 45.0 we consider two values for N , namely N = 5 and peak peak 20 15.8 - 24.8 22.5 Npeak = 20. In Tab.VI we show the confidence intervals at 1σ for the counts both considering a purely Poisson distribution (with mean equal to Npeak) and a Gaussian p 0ν one (with mean Npeak and standard deviation Npeak). for S : Notice that, even if the number of counts is just 5, the r Poisson and Gaussian distributions give almost the same nββ 1 x η NA M · T S0ν = ln 2 · T · ε · = ln 2 · ε · · · relative uncertainties. nσ · nB nσ MA B · ∆ If no peak is detected, the sensitivity of a given (54) 0νββ experiment is usually expressed in terms of “de- where B is the background level per unit mass, energy, tector factor of merit”, S0ν [25]. This can be defined as and time, M is the detector mass, ∆ is the FWHM energy the process half-life corresponding to the maximum sig- resolution, x is the stoichiometric multiplicity of the ele- nal that could be hidden by the background fluctuations ment containing the ββ candidate, η is the ββ candidate nB (at a given statistical C. L.). To obtain an estimation isotopic abundance, NA is the Avogadro number and, fi- 0ν for S as a function of the experiment parameters, it nally, MA is the compound molecular mass. Despite its is sufficient to require that the 0νββ signal exceeds the simplicity, Eq. (54) has the advantage of emphasizing the standard deviation of the total detected counts in the in- role of the essential experimental parameters. teresting energy window. At the confidence level nσ, this Of particular interest is the case in which the back- means that we can write: ground level B is so low that the expected number of p background events in the region of interest along the ex- nββ ≥ nσ nββ + nB (53) periment life is of order of unity: where nββ is the number of 0νββ events and Poisson statistics for counts is assumed. If one now states that M · T · B · ∆ . 1. (55) the background counts scale linearly with the mass of the detector,8 from Eq. (51) it is easy to find an expression This is called as the “zero background” experimental con- dition and it is likely the experimental condition that next generation experiments will face. Practically, it ] 4 -2 10 means that the goal is a great mass and a long time of 10 meV eV data taking, keeping the background level and the energy -1

48 yr resolution as little as possible. 3 Ca 150 -1 10 Nd 96 In this case, nB is a constant, Eq. (54) is no more valid Zr 100 100 meV 116Cd Mo and the sensitivity is given by: [ Mg 136 82 2 Xe Se 0 ν 10 124 130Te Sn G 110Pd N x η N MT 76Ge 0ν ββ A S0B = ln 2 · T · ε · = ln 2 · ε · · . (56) nσ · nB MA NS 101 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 (specific) (specific) 0.1 1 10 100 2 The constant NS is now the number of observed events M0ν in the region of interest.

FIG. 14. Geometric mean of the squared M0ν considered in Ref. [176] vs. the specific G0ν . The case gA = gquark is assumed. Adapted from Ref. [176].

D. Experimental techniques

8 This is reasonable since, a priori, impurities are uniform inside the detector but, of course, this might not be always the case The experimental approach to search for the 0νββ con- e. g. if the main source of background is removed with volume sists in the development of a proper detector, able to re- fiducialization. veal the two emitted electrons and to collect their sum 22

20 20 20 1% FWHM 3.5% FWHM 10% FWHM 18 18 18 16 16 16 counts / bin counts / bin counts / bin 14 14 14 12 12 12 10 10 10 8 8 8 6 6 6 4 4 4 2 2 2 0 0 0 0.94 0.96 0.98 1.00 1.02 1.04 1.06 0.94 0.96 0.98 1.00 1.02 1.04 1.06 0.94 0.96 0.98 1.00 1.02 1.04 1.06 E/Q E/Q E/Q

FIG. 15. Signal and background (red and grey stacked histograms, respectively) in the region of interest around Qββ for 3 Monte Carlo experiments with the same signal strength (50 counts) and background rate (1 count keV−1), but different energy resolution: top: 1% FWHM, centre: 3.5% FWHM, bottom: 10% FWHM. The signal is distributed normally around Qββ , while the background is assumed flat. Figure from Ref. [177].

energy spectrum (see Sec.VIA). 9 The desirable features It has to be noted that it is impossible to optimize the for such a detector are thus: listed features simultaneously in a single detector. There- fore, it is up to the experimentalists to choose which one • good energy resolution. This is a fundamental re- to privilege in order to get the best sensitivity. quirement to identify the sharp 0νββ peak over an The experiments searching for the 0νββ of a certain almost flat background, as shown in Fig. 15, and isotope can be classified into two main categories: de- it is also the only protection against the (intrinsic) tectors based on a calorimetric technique, in which the background induced by the tail of the 2νββ spec- source is embedded in the detector itself, and detector trum. Indeed, it can be shown that the ratio R0ν/2ν using an external source approach, in which source and of counts due to 0νββ and those due to 2νββ in the detector are two separate systems (Fig. 16). peak region can be approximated by [178]:

 6 1/2 Qββ t2ν 1. Calorimetric technique R0ν/2ν ∝ 1/2 . (57) ∆ t0ν The calorimetric technique has already been imple- This expression clearly indicates that a good energy mented in various types of detectors. The main advan- resolution is critical. But it also shows that the tages and limitations for this technique can be summa- minimum required value actually depends on the rized as follows [25]: chosen isotope, considered a strong dependence of 1/2 Eq. (57) upon the 2νββ half-life t2ν ; (+) large source masses are achievable thanks to the intrinsically high efficiency of the method. Exper- • very low background. Of course 0νββ experiments iments with masses up to ∼ 200 kg have already have to be located underground in order to be pro- proved to work and ton-scale detectors seem possi- tected from cosmic rays. Moreover, radio-pure ma- ble terials for the detector and the surrounding parts, as well as proper passive and/or active shielding (+) very high resolution is achievable with the proper are mandatory to protect against environmental ra- type of detector (∼ 0.1% FWHM with Ge diodes dioactivity;10 and bolometers)

• large isotope mass. Present experiments have (−) severe constraints on detector material (and thus masses of the order of some tens of kg up to a few on the isotope that can be investigated) arise from hundreds kg. Tons will be required for experiments the request that the source material has to be em- aiming to cover the IH region (see Sec.VIG) bedded in the structure of the detector. However, this is not the case for some techniques (e. g. for bolometers and loaded liquid scintillators)

9 Additional information (e. g. the single electron energy or the (−) the event topology reconstruction is usually diffi- initial momentum) can also be provided sometimes. cult, with the exception of liquid or gaseous Xe 10 The longest natural radioactivity decay competing to the TPC. However, the cost is paid in terms of a lower 9 10 25 0νββ are of the order of (10 −10 ) yr versus lifetimes & 10 yr. energy resolution. 23

Among the most successful examples of detectors using detector the calorimetric technique, we find: e- • Ge-diodes. The large volume, high-purity, and source high-energy resolution achievable make this kind of detector suitable for the 0νββ search, despite the e- 76 low Qββ of Ge detector • bolometers. Macro-calorimeters with masses close to 1 kg very good energy resolution (close to that of Ge-diodes) are now available for many compounds including 0νββ emitters. The most significant case e- 130 is the search for the 0νββ of Te with TeO2 e- bolometers

• Xe liquid and gaseous TPC. The lower energy reso- source = detector lution is “compensated” by the capability of recon- structing the event topology FIG. 16. Schematic representation of the two main experi- mental categories for the 0νββ search: calorimetric technique • liquid scintillators loaded with the 0νββ isotope. (source ≡ detector) and external source approach (source 6= These detectors have a poor energy resolution. detector). However, a huge amount of material can be dis- solved and, thanks to the purification processes, very low backgrounds are achievable. They are external source detector type has provided excellent re- ideal detectors to set very stringent limits on the sults on the studies of the 2νββ. Moreover, in case of decay half-life. discovery of a 0νββ signal, the event topology recon- struction could represent a fundamental tool for the un- derstanding of the mechanism behind the 0νββ. 2. External source approach

Also in the case of the external source approach, dif- E. Experiments: a brief review ferent detection techniques have been adopted, namely scintillators, solid state detectors, and gas chambers. The main advantages and limitations for this technique can The first attempt to observe the 0νββ process dates be summarized as follows: back to 1948 [200, 201]. Actually, the old experiments aiming to set a limit on the double beta decay half-lives (+) the reconstruction of the event topology is possible, did not distinguish between 2νββ and 0νββ. In the case thus making in principle easier the achievement of of indirect investigations through geochemical observa- the zero background condition. However, the poor tion, this was not possible even in principle. energy resolution does not allow to distinguish be- However, the importance that the 0νββ was acquiring tween 0νββ events and 2νββ events with total elec- in particle physics provided a valid motivation to contin- tron energy around Qββ. Therefore the 2νββ rep- uously enhance the efforts in the search for this decay. resent an important background source On the experimental side, the considerable technological improvements allowed to increase the half-life sensitiv- (−) the energy resolutions are low (of the order of 10%). ity of several orders of magnitude.11 The long history of The limit is intrinsic and it mainly due to the elec- 0νββ measurements up to about the year 2000 can be tron energy deposition in the source itself found in Refs. [203–205]. Here, we concentrate only on a (−) large isotope masses are hardly achievable due to few experiments starting from the late 1990s. self-absorption in the source. Up to now, only TableVII summarizes the main characteristics and masses of the order of some tens of kg have been performances of the selected experiments. It has to be possible, but an increase to about 100 kg target noticed that, due to their different specific features, the seems feasible actual comparison among all the values is not always pos- sible. We tried to overcome this problem by choosing a (−) the detection efficiencies are low (of the order of common set of units of measurement. 30%). So far, the most stringent bounds come from the calori- metric approach which, anyway, remains the one promis- 11 The 2νββ was first observed in the laboratory in 82Se in ing the best sensitivities and it is therefore the chosen 1987 [202], and in many other isotopes in the subsequent years. technique for most of the future projects. However, the See Ref. [175] for a review on 2νββ. 24

TABLE VII. In this table, the main features and performances of some past, present and future 0νββ experiments are listed.

0ν Experiment Isotope Techinique Total mass Exposure FWHM @ Qββ Background S (90% C. L.) [kg] [kg yr] [keV] [counts/keV/kg/yr] [1025 yr]

Past

130 Cuoricino, [179] Te bolometers 40.7 (TeO2) 19.75 5.8 ± 2.1 0.153 ± 0.006 0.24 130 CUORE-0, [180] Te bolometers 39 (TeO2) 9.8 5.1 ± 0.3 0.058 ± 0.006 0.29 Heidelberg-Moscow, [181] 76Ge Ge diodes 11 (enrGe) 35.5 4.23 ± 0.14 0.06 ± 0.01 1.9 76 IGEX, [182, 183] Ge Ge diodes 8.1 (enrGe) 8.9 ∼ 4 . 0.06 1.57 GERDA-I, [167, 184] 76Ge Ge diodes 17.7 (enrGe) 21.64 3.2 ± 0.2 ∼ 0.01 2.1 NEMO-3, [185] 100Mo tracker + 6.9 (100Mo) 34.7 350 0.013 0.11 calorimeter Present

EXO-200, [186] 136Xe LXe TPC 175 (enrXe) 100 89 ± 3 (1.7 ± 0.2) · 10−3 1.1 KamLAND-Zen, [187, 188] 136Xe loaded liquid 348 (enrXe) 89.5 244 ± 11 ∼ 0.01 1.9 scintillator Future

130 CUORE, [189] Te bolometers 741 (TeO2) 1030 5 0.01 9.5 GERDA-II, [174] 76Ge Ge diodes 37.8 (enrGe) 100 3 0.001 15 LUCIFER, [190] 82Se bolometers 17 (Zn82Se) 18 10 0.001 1.8 MAJORANA D., [191] 76Ge Ge diodes 44.8 (enr/natGe) 100 a 4 0.003 12 NEXT, [192, 193] 136Xe Xe TPC 100 (enrXe) 300 12.3 − 17.2 5 · 10−4 5

100 enr −4 AMoRE, [194] Mo bolometers 200 (Ca MoO4) 295 9 1 · 10 5 nEXO, [195] 136Xe LXe TPC 4780 (enrXe) 12150 b 58 1.7 · 10−5 b 66 PandaX-III, [196] 136Xe Xe TPC 1000 (enrXe) 3000 c 12 − 76 0.001 11 c SNO+, [197] 130Te loaded liquid 2340 (natTe) 3980 270 2 · 10−4 9 scintillator SuperNEMO, [198, 199] 82Se tracker + 100 (82Se) 500 120 0.01 10 calorimeter a our assumption (corresponding sensitivity from Fig. 14 of Ref. [191]). b we assume 3 tons fiducial volume. c our assumption by rescaling NEXT.

1. The claimed observation F. Present sensitivity on mββ

Once the experimental sensitivities are known in terms of S0ν , by using Eq. (45), it is possible to correspondingly In 2001, after the publication of the experiment final find the lower bounds on m . results [181], a fraction of the Heidelberg-Moscow Col- ββ Fig. 17 shows the most stringent limits up to date. laboration claimed to observe a peak in the spectrum, They come from 76Ge [174], 130Te [180] and 136Xe [187]. whose energy corresponded to the 76Ge 0νββ transition In particular, the combined sensitivity from the single Q-value [206]. After successive re-analysis (by fewer and experimental limits is taken from the corresponding ref- fewer people), the final value for the half-life was found erences. to be: t1/2 = (2.23+0.44)·1025 yr [207]. This claim and the −0.31 In the left panel of the figure, the case g = g subsequent papers by the same authors aroused a number A nucleon (unquenched value) is assumed. The uncertainties on of critical replies (see e. g. Refs.[24, 100, 208, 209]). Many NME and PSF are taken into account according to the of the questions and doubts still remain unanswered. To procedure shown in App.B, and they result in the broad- summarize, caution suggests that we disregard the claim ening of the lines describing the limits. As the plot shows, that the transition was observed. the current generation of experiments is probing the quasi degenerate part of the neutrino mass spectrum. 76 Anyway, to date, the limit on the Ge 0νββ half-life The effect of the quenching of gA appears evident in is more stringent than the reported value [174]. the right panel: the sensitivity for the same combined 25

1 1 gphen.

130Te(Cuoricino+ CUORE-0) 76 Ge(IGEX+ HdM+ GERDA-I) gquark

136Xe(KamLAND-Zen+ EXO-200) gnucleon 0.1 0.1

IH(Δm 2<0) IH(Δm 2<0)

[ eV ] 0.01 [ eV ] 0.01 ββ ββ m m NH(Δm 2>0) NH(Δm 2>0) 0.001 0.001

10-4 10-4 10-4 0.001 0.01 0.1 1 10-4 0.001 0.01 0.1 1

mlightest [eV] mlightest [eV]

FIG. 17. The colored regions show the predictions on mββ from oscillations as a function of the lightest neutrino mass with the relative the 3σ regions. The horizontal bands show the experimental limits with the spread due to the theoretical uncertainties on the NME [138] and PSF [135][187]. (Left) Combined experimental limits for the three isotopes: 76Ge [174], 130Te [180] and 136 136 Xe. The case gA = gnucleon. (Right) Combined experimental limit on Xe for the three different values for gA, according to Eq. (47).

136 Xe experiment in the two cases of gnucleon and gphen. we require a sensitivity mββ = 8 meV. The mega experi- differs of a factor & 5. It is clear from the figure that this ment is the one that satisfies this requirement in the most is the biggest uncertainty, with respect to all the other favorable case, namely, when the quenching of gA is ab- theoretical ones. sent. Instead, the ultimate experiment assumes that gA is The single values for the examined cases are reported maximally quenched. We chose the 8 meV value because, in TableVIII. even taking into account the residual uncertainties on the NME and on the PSF, the overlap with the allowed band for mββ in the IH is excluded at more than 3σ. Notice that we are assuming that at some point the issue of the quenching will be sorted out. Through Eq. (45), we ob- G. Near and far future experiments tain the corresponding value of t1/2 and thus we calculate the needed exposure to accomplish the task. Referring to Eq. (56), if we suppose ε ' 1 (detector efficiency of 100% and no fiducial volume cuts), x ' η ' 1 It is also possible to extract the bounds on mββ com- (all the mass is given by the candidate nuclei), and we ing from the near future experiments starting from the assume one observed event (i. e. NS = 1) in the region of expected sensitivities and using Eq. (45). The results are interest, we get the simplified equation: shown in TableIX. It can be seen the mass region below 100 meV will begin to be probed in case of unqueched 0ν MA · S value for gA. But still we will not enter the IH region. M · T = . (58) ln 2 · NA In case gA is maximally quenched, instead, the situation is much worse. Indeed, the expected sensitivity would This is the equation we used to estimate the product M·T correspond to values of mββ which we already consider (exposure), and thus to assess the sensitivity of the mega probed by the past experiments. and ultimate scenarios. The key input is, of course, the Let us now consider a next generation experiment (call theoretical expression of t1/2. The calculated values of the it a “mega” experiment) and a next-to-next generation exposure are shown in TableX for the three considered one (an “ultimate” experiment) with enhanced sensitiv- nuclei: 76Ge, 130Te and 136Xe. The last column of the ity. To define the physics goal we want to achieve, we table gives the maximum allowed value of the product refer to Ref. [93]. B · ∆ that satisfies Eq. (55). The most honest way to talk of the sensitivity is in Fig. 18 compares (in a schematic view) the masses of terms of exposure or of half-life time that can be probed. 76Ge and 136Xe corresponding to the present sensitiv- From the point of view of the physical interest, however, ity [174, 187] assuming zero background condition and besides the hope of discovering the 0νββ, the most ex- 5 years of data acquisition to those of the “mega” and citing investigation that can be imagined at present is “ultimate” experiments with the same assumptions. the exclusion of the IH case. This is the goal that most of the experimentalists are trying to reach with future 0νββ experiments (see e. g. Ref. [210]). For this reason, 26

76 130 136 TABLE VIII. Lower bounds for mββ for Ge, Te and Xe. The sensitivities were obtained by combining the most stringent limits from the experiments studying the isotopes. Refs. [135] and [138] were used for the PSFs and for the NME, respectively. The different results correspond to different values of gA according to Eq. (47).

0ν Experiment Isotope S (90% C. L.) Lower bound for mββ [eV] 25 [10 yr] gnucleon gquark gphen. IGEX + HdM + GERDA-I, [174] 76Ge 3.0 0.25 ± 0.02 0.40 ± 0.04 1.21 ± 0.11 Cuoricino + CUORE-0, [180] 130Te 0.4 0.36 ± 0.03 0.58 ± 0.05 2.07 ± 1.05 EXO-200 + KamLAND-ZEN, [187] 136Xe 3.4 0.15 ± 0.02 0.24 ± 0.03 0.87 ± 0.10

TABLE IX. Lower bounds for mββ for the more (upper group) and less (lower group) near future 0νββ experiments. Refs. [135] and [138] were used for the PSFs and for the NME, respectively. The different results correspond to different values of gA according to Eq. (47).

0ν Experiment Isotope S (90% C. L.) Lower bound for mββ [eV] 25 [10 yr] gnucleon gquark gphen. CUORE, [189] 130Te 9.5 0.073 ± 0.008 0.14 ± 0.01 0.44 ± 0.04 GERDA-II, [174] 76Ge 15 0.11 ± 0.01 0.18 ± 0.02 0.54 ± 0.05 LUCIFER, [190] 82Se 1.8 0.20 ± 0.02 0.32 ± 0.03 0.97 ± 0.09 MAJORANA D., [191] 76Ge 12 0.13 ± 0.01 0.20 ± 0.02 0.61 ± 0.06 NEXT, [193] 136Xe 5 0.12 ± 0.01 0.20 ± 0.02 0.71 ± 0.08

AMoRE, [194] 100Mo 5 0.084 ± 0.008 0.14 ± 0.01 0.44 ± 0.04 nEXO, [195] 136Xe 660 0.011 ± 0.001 0.017 ± 0.002 0.062 ± 0.007 PandaX-III, [196] 136Xe 11 0.082 ± 0.009 0.13 ± 0.01 0.48 ± 0.05 SNO+, [197] 130Te 9 0.076 ± 0.007 0.12 ± 0.01 0.44 ± 0.04 SuperNEMO, [198] 82Se 10 0.084 ± 0.008 0.14 ± 0.01 0.41 ± 0.04

VII. INTERPLAY WITH COSMOLOGY The authors of Ref. [106] computed a probability for Σ that can be summarized to a very a good approximation Here, we want to assess the possibility of taking advan- by: tage of the knowledge about the neutrino cosmological 2 2 (Σ − 22 meV) mass to make inferences on some 0νββ experiment results ∆χ (Σ) = 2 . (59) (or expected ones). In particular, we follow Ref. [211]. As (62 meV) already discussed in Sec. IV C 2, we consider two possible Starting from the likelihood function L ∝ exp −(∆χ2/2) scenarios. Firstly, we assume only upper limits both on with ∆χ2 as derived from Fig. 7 in the reference, one can Σ and mββ, without any observation of 0νββ. Later, we obtain the following limits: imagine an observation of 0νββ together with a non zero measurement of Σ (in both cases, we consider the un- Σ < 84 meV (1σ C. L.) quenched value gA = gphen. for the axial vector coupling Σ < 146 meV (2σ C. L.) (60) constant). Σ < 208 meV (3σ C. L.) which are very close to those predicted by the Gaus- A. Upper bounds scenario sian ∆χ2 of Eq. (59). In particular, it is worth noting that, even if this measurement is compatible with zero at The tight limit on Σ in Ref. [106] was obtained by less than 1σ, the best fit value is different from zero, as combining the Planck 2013 results [110] with the one- expected from the oscillation data and as evidenced by dimensional flux power spectrum measurement of the Eq. (59). We want to remark that, despite the impact rel- Lyman-α forest extracted from the BAO Spectroscopic ative impact of systematic versus statistical errors on the Survey of the Sloan Digital Sky Survey [212]. In par- estimated flux power is considered and discussed [212], it ticular, the data from a new sample of quasar spectra is anyway advisable to take these results from cosmology were analyzed and a novel theoretical framework which with the due caution. incorporates neutrino non-linearities self consistently was The plot showing mββ as a function of Σ which was al- employed. ready shown in the right panel of Fig.6, is again useful for 27

TABLE X. Sensitivity and exposure necessary to discriminate between NH and IH: the goal is mββ = 8 meV. The two cases refer to the unquenched value of gA = gnucleon (mega) and gA = gphen. (ultimate). The calculations are performed assuming zero background experiments with 100% detection efficiency and no fiducial volume cuts. The last column shows the maximum value of the product B · ∆ in order to actually comply with the zero background condition.

0ν Experiment Isotope S0B [yr] Exposure (estimate) −1 −1 M · T [ton·yr] B · ∆ (zero bkg) [counts kg yr ] mega Ge 76Ge 3.0 · 1028 5.5 1.8 · 10−4 mega Te 130Te 8.1 · 1027 2.5 4.0 · 10−4 mega Xe 136Xe 1.2 · 1028 3.8 2.7 · 10−4 ultimate Ge 76Ge 6.9 · 1029 125 8.0 · 10−6 ultimate Te 130Te 2.7 · 1029 84 1.2 · 10−5 ultimate Xe 136Xe 4.0 · 1029 130 7.7 · 10−6 the discussion. A zoomed version of that plot (with lin- inequality: ear instead of logarithmic scales for the axis) is presented (y − m (Σ))2 (Σ − Σ(0))2 in the left panel of Fig. 19. As already mentioned, the ββ + < 1 (62) (n σ[m (Σ)])2 (Σ − Σ(0))2 extreme values for mββ after variation of the Majorana ββ n phases can be easily calculated (see App.A). This vari- where mββ(Σ) is the Majorana Effective Mass as a func- ation, together with the uncertainties on the oscillation tion of Σ and σ[mββ(Σ)] is the 1σ associated error, com- parameters, results in a widening of the allowed regions. puted as discussed in Ref. [93]. Σn is the limit on Σ It is also worth noting that the error on Σ contributes derived from Eq. (59) for the C. L. n = 1, 2, 3,... By to the total uncertainty. Its effect is a broadening of the solving Eq. (62) for y, it is thus possible to get the al- light shaded area on the left side of the minimum allowed lowed contour for mββ considering both the constraints value Σ(m = 0) for each hierarchy. In order to compute from oscillations and from cosmology. In particular, the this uncertainty, we considered Gaussian errors on the Majorana phases are taken into account by computing y oscillation parameters, namely max along the two extremes of mββ(Σ), namely mββ (Σ) and mmin(Σ), and then connecting the two contours. The re- s ββ ∂ Σ 2  ∂ Σ 2 sulting plot is shown in the right panel of Fig. 19. δΣ = σ(δm2) + σ(∆m2) . ∂ δm2 ∂ ∆m2 The most evident feature of Fig. 19 is the clear dif- ference in terms of expectations for both m and Σ in (61) ββ the two hierarchy cases. The relevant oscillation param- It is possible to include the new cosmological con- eters (mixing angles and mass splittings) are well known straints on Σ from Ref. [106] considering the following and they induce only minor uncertainties on the expected value of mββ. These uncertainties widen the allowed con- tours in the upper, lower and left sides of the picture. The boundaries in the rightmost regions are due to the new information from cosmology and are cut at various confidence levels. It is notable that at 1σ, due to the 136 Xe exclusion of the IH, the set of plausible values of mββ is highly restricted. The impact of the new constraints on Σ appears even more evident by plotting mββ as a function of the mass 76Ge of the lightest neutrino. In this case, Eq. (62) becomes:

2 2 (y − mββ(m)) m 2 + 2 < 1. (63) (n σ[mββ(m)]) m(Σn) The plot in Fig. 20 globally shows that the next gener- ation of experiments will have small possibilities of de- FIG. 18. Masses corresponding to present, mega and ultimate tecting a signal of 0νββ due to light Majorana neutrino exposures, assuming zero background condition and 5 years exchange. Therefore, if the new results from cosmology of data acquisition. The cubes represent the amount of 76Ge, are confirmed or improved, ton or even multi-ton scale the (150 bar) bottles the one of 136Xe. The smallest masses detectors will be needed [93]. depict the present exposure, while the biggest bottle is out of On the other hand, a 0νββ signal in the near future scale. could either disprove some assumptions of the present 28

0.08 0.08

3σ

0.06 0.06 2σ

[ eV ] 0.04 IH [ eV ] 0.04 IH ββ ββ m m

0.02 0.02 NH NH 3σ 2σ 1σ (95% C.L.) 0 0 0.05 0.10 0.15 0.20 0.05 0.10 0.15 0.20

Σcosm [eV] Σcosm [eV]

FIG. 19. (Left) Allowed regions for mββ as a function of Σ with constraints given by the oscillation parameters. The darker regions show the spread induced by Majorana phase variations, while the light shaded areas correspond to the 3σ regions due to error propagation of the uncertainties on the oscillation parameters. (Right) Constraints from cosmological surveys are added to those from oscillations. Different C. L. contours are shown for both hierarchies. Notice that the 1σ region for the IH case is not present, being the scenario disfavored at this confidence level. The dashed band signifies the 95% C. L. excluded region coming from Ref. [106]. Figure from Ref. [211]. cosmological models, or suggest that a different mech- written as following: anism other than the light neutrino exchange mediates meas 2 " meas 2 # the transition. New experiments are interested in test-  (Σ − Σ )  (mββ − m ) L ∝ exp − exp − ββ . ing the latter possibility by probing scenarios beyond the meas 2 meas 2 2σ(Σ ) 2σ(mββ ) SM [87, 91, 213]. (65) Recalling the relation between the χ2 and the likelihood, 2 namely L ∝ e−χ /2, we obtain: B. Measurements scenario meas 2 meas 2 (Σ − Σ ) (mββ − m ) χ2 = + ββ (66) σ(Σmeas)2 σ(mmeas)2 Here we consider the implications of the following non- ββ zero value of Σ [105]: which represents an elliptic paraboloid. Since we are dealing with a two parameter χ2, we need to find the Σ = (0.320 ± 0.081) eV. (64) appropriate prescription to define the confidence inter- vals. At the desired confidence level, we get: We focus on the light neutrino exchange scenario and 2 ZZ x2 y assume that 0νββ is observed with a rate compatible 1 − 2 − 2 C. L. = dx dy e 2σx 2σy (67) with: 2 2 2πσxσy χ <χ0

1. the present sensitivity on mββ. In particular, we and thus 136 use the limit coming from the combined Xe- 2 based experiments [187]. We refer to this as to the χ0 = −2 ln (1 − C. L.). (68) “present” case This defines the value for χ2 correspondent to the confi- dence level C. L. . 2. a value of mββ that will be likely probed in the In order to write down the likelihood we need to eval- next few years. In particular, we use the CUORE uate the standard deviations both on Σ and on mββ. experiment sensitivity [189], as an example of next While the error on Σ comes directly from the cosmo- generation of 0νββ experiments. We refer to this logical measurement, the one on mββ has to be deter- as to the “near future” case. mined. It has two different contributions: one is statis- tical and comes from the Poisson fluctuations on the ob- For the sake of completeness, it is useful to recall a served number of events (see Sec.VIC), while the other few definitions and relations. The likelihood of a simul- comes from the uncertainties on the nuclear physics (see taneous observation of some values for Σ and mββ (re- Sec.VC). Actually, a greater effect would rise if we took meas meas spectively with uncertainties σ(Σ ) and σ(mββ ) and into account the error on gA, but here we assume that distributed according to Gaussian distributions) can be the quenching is absent. 29

0.25

��� 0.20 �σ �σ �� 0.15 Present �σ [ eV ] [ �� ] ���� �σ ββ ββ m 0.10 � Near future �σ �� 0.05 IH ����� NH 0 0.1 0.2 0.3 0.4 ����� ���� ��� Σ[eV] ��������� [��]

FIG. 21. The plots show the allowed regions for mββ as a FIG. 20. Constraints from cosmological surveys are added function of the neutrino cosmological mass Σ. The ellipses to those from oscillations in the representation of mββ as a show the 90% C. L. regions in which a positive observation function of the lightest neutrino mass. The dotted contours of 0νββ could be contained, according to the experimental represent the 3σ regions allowed considering oscillations only. uncertainties and 5 (solid) and 20 (dashed) actually observed The shaded areas show the effect of the inclusion of cosmo- events. In particular, the upper ellipse assumes the present logical constraints at different C. L. . The horizontal bands limit from the combined 136Xe experiments [187]. The lower correspond to the expected sensitivity for future experiments. one assumes the sensitivity of CUORE [189]. Figure from Ref. [211].

For a few observed events, let us say less than 10 erarchy, some conclusions could be carried out regarding events, the global error is dominated by the statistical the Majorana phases. fluctuations. The error on the nuclear physics becomes This simple analysis shows that, thanks to the great the main contribution only if many events (more than a efforts done in the NME and PSF calculations, it is most few tens) are detected. Using the described procedure likely that the biggest contribution to the error will come from the statistical fluctuations of the counts. However, and for the present case, we find an uncertainty on mββ of about 31 meV for 5 observed events, which reduces to the theoretical uncertainty from the nuclear physics could 24 meV for 10 events. If we neglect the statistical un- make the picture really hard to understand because, up to now, it is a source of uncertainty of a factor 4 − 8 on certainty, e. g. we put Nevents, the uncertainty becomes 14 meV. This means that the Poisson fluctuations ef- mββ. fect is not negligible at all. Similarly, repeating the same work for the near future case, we obtain an uncertainty of 17 meV for 5 events, 13 meV for 10 events and 8 meV for N . C. Considerations on the information from events cosmological surveys Let us now concentrate on the case of 5 0νββ observed events. If we cut the χ2 at the 90% C. L. and we con- sider the data previously mentioned, we obtain the big- The newest results reported in TableIV confirms and ger, solid ellipses drawn in Fig. 21. This shows that in strengthens the cosmological indications of upper limits the near future case, a detection of 0νββ would allow to on Σ, and it is likely that we will have soon other sub- say nothing neither about the mass hierarchy nor about stantial progress. Moreover, the present theoretical un- the Majorana phases. Interestingly, if 0νββ were actu- derstanding of neutrino masses does not contradict these ally discovered with a mββ a little bit lower than the one cosmological indications. These considerations empha- probed in the present case, some conclusions about the size the importance of exploring the issue of mass hi- Majorana phases could be carried out. In any case, in erarchy in laboratory experiments and with cosmological order to state anything precise about mββ and the Ma- surveys. However, as already stated, a cautious approach jorana phases, even assuming the discovery of 0νββ, the in dealing with the results from cosmological surveys is uncertainty on the quenching of the axial vector coupling highly advisable. constant has to be dramatically decreased. From the point of view of 0νββ, these results show that If we repeat the same exercise assuming an observed ton or multi-ton scale detectors will be needed in order number of events of 20, we obtain the smaller, dashed el- to probe the range of mββ now allowed by cosmology. lipses of Fig. 21. In this case, an hypothetical observation Nevertheless, if next generation experiments see a signal, coming from the present case is highly disfavored while in it will likely be a 0νββ signal of new physics different the future case, even if nothing can be said about the hi- from the light Majorana neutrino exchange. 30

VIII. SUMMARY to be real. It thus follows that also the other two terms must be real: this is equivalent to considering the sum of In this review, we analyzed the 0νββ process under the modules of the single terms. many different aspects. We assessed its importance to To prove the second statement, let us consider the gen- test lepton number, to determine the nature of neutrino eral case mββ ∼ |z1 + z2 + z3| ≡ r where zi are complex mass and to probe its values. Various particle physics numbers. We want to minimize r, by keeping fixed the mechanisms that could contribute to the 0νββ were ex- |zi|. Let us define: amined, although with the conclusion that from the the-  r = |z | − |z | − |z | oretical point of view the most interesting and promising  1 1 2 3 remains the light Majorana neutrino exchange. We stud- r2 = |z2| − |z1| − |z3| (A4)  ied the current experimental sensitivity, focussing on the  r3 = |z3| − |z1| − |z2| critical point of determining the uncertainties in the the- oretical calculations and predictions. In view of all these and considerations, the prospects for the near future experi-  q = |z | − |z + z | mental sensitivity were presented and the main features  1 1 2 3 of present, past and future 0νββ experiments were dis- q2 = |z2| − |z1 + z3| (A5)  cussed. Finally, we stressed the huge power of cosmolog-  q3 = |z3| − |z1 + z2|. ical surveys in constraining neutrino masses and conse- quently the 0νββ process. It is worth noting that only one of the ri can be positive, at most. Therefore, it is possible to distinguish 4 cases:

ACKNOWLEDGMENTS i) r1 > 0;

ii) r2 > 0; We wish to acknowledge extensive discussions with Professor F. Iachello and thank him for stimulating this iii) r3 > 0; study. F. V. also thanks E. Lisi. iv) ri ≤ 0 for i = 1, 2, 3. min In the first one, it is possible to show that r = r1. In Appendix A: Extremal values of mββ fact, we can write:

Recalling the definition of Eq. (27) for the Majorana r = |z1 + z2 + z3| = |z1 − (−z2 − z3)|

effective mass: ≥ |z1| − | − z2 − z3| (A6)

3 = |z1| − |z2 + z3| = |q1| X 2 mββ = Ueimi (A1) i=1 and, since

it is possible to demonstrate that the extreme values as- q1 = |z1| − |z2 + z3| ≥ |z1| − |z2| − |z3| = r1 > 0 (A7) sumed by this parameter due to free variations of the phases are:12 we obtain:

3 r ≥ |q1| ≥ q1 ≥ r1. (A8) max X 2 mββ = Uei mi (A2) min min Similarly, r2 > 0 ⇒ r = r2 and r3 > 0 ⇒ r = r3 i=1 in the second and in the third cases, respectively. In min n 2 max o mββ = max 2 Uei mi − mββ , 0 i = 1, 2, 3. (A3) the last case, it is necessary to observe that, if one of min the ri = 0, then r = 0. Therefore, only the case in which ri < 0 ∀i must be considered. In this case, q1 1. Formal proof goes from negative when arg(z2) = arg(z3), to positive, when arg(z2) = −arg(z3). By continuity, this implies that a proper phase choice such that q1 = 0 must exist. Regarding the first assertion, it is obvious that the min sum of n complex numbers has the biggest allowed mod- Thus, one can conclude also in this case that r = 0 ule when those numbers have aligned phases. Since the (by choosing r = |q1|). 2 In synthesis, the single case analysis leads to: physical quantities depend on mββ, without any loss of generality it is possible to choose the first term (U 2 m ) min e1 1 r = max{ri, 0}. (A9)

This proves the original statement, since ri = |zi|−|zj|− P3 |zk|+|zi|−|zi| = 2|zi|− l=1 |zl| for i 6= j 6= k, {i, j, k} = 12 The proof shown here is based on the work reported in Ref. [99]. {1, 2, 3}. 31

mββ=m3 from Eq. (A10) we have:

2 max 2 Uei mi − mββ 2 2 2 2 = 2 Uei mi − Ue1 m1 − Ue2 m2 − Ue3 m3 2 2 = 2 Uei mi − 2 Ue1 m1. (A13)

Substituting the possible values i = 1, 2, 3, and recalling min that the condition to get mββ is expressed by Eq. (A3), 2 we obtain: |Ue1| min n 2 o mββ = max −2 Ue1 m1, 0 = 0. (A14) 2 |Ue2| The same argument can be applied also for the other two conditions. It is therefore possible to identify a region in- min side the triangle where mββ is zero. The experimental mββ=m1 |U 2 | mββ=m2 constraints on the oscillation parameters limit the possi- e3 min bility of mββ = 0, only to the case of NH. Of course, min FIG. 22. Representation of mββ in the unitarity triangle. the position and the extension of this region depends on The internal point in the middle of the small colored bar is the lightest neutrino mass. identified by the constraints from the oscillation parameters. Instead of choosing one particular value for the lightest The colored regions correspond to the 1σ (red), 2σ (orange) neutrino mass, it is more convenient to plot the super- and 3σ (yellow). The distance from a side represent the size 2 position of the regions obtained for increasing values of of the corresponding mixing element Uei . The inner shaded this parameter. In Fig. 22, in orange we show the region min regions of the triangle enclose the areas where mββ = 0 for obtained varying m from 10−5 eV, up to the 90% C. L. −5 1 a lightest neutrino mass that can vary from 10 eV to the maximum value it can have considering the limit on Σ value which corresponds to a cosmological mass Σ = 0.14 eV from Ref. [106], according to Eq. (59). The gray region (orange, 90% C. L. current bound) and Σ = 0.06 eV (gray, for shows the superposition obtained when m ∼ 0, namely purpose of illustration). 1 we show what happens if it turns out that the cosmo- logical mass is close to its lower limit (. 0.06 eV for the min NH case). 2. Remarks on the case mββ = 0 min The existence of a mββ = 0 region implies that, in principle, 0νββ could be forbidden just by particular 2 The three mixing elements Uei are constrained by the combinations of the phases, even if the neutrino is a Ma- P 2 unitarity: i Uei = 1. This condition can be graphi- jorana particle. cally pictured by using the inner region of an equilateral triangle with unitary height, where the distance from the i-th side corresponds to the value of U 2 (see Ref. [99] ei Appendix B: Error propagation for details). The result is displayed in Fig. 22. The experimental constraints on the oscillation param- 2 It is convenient and usually appropriate to adopt sta- eters make it possible to evaluate the elements Uei and, therefore, to identify a point inside the triangle, which is tistical procedures that are as direct and as practical as placed at the center of the colored bar in Fig. 22. The possible. We are interested in the following situation. For different colors of the bar correspond to the 1σ, 2σ and any choice of the Majorana phases, the massive parame- 3σ regions. ter that regulates the 0νββ can be thought as M(m, x). min It is a function of the parameters that are determined At each vertex, the value of mββ coincides with m ββ by oscillation experiments up to their experimental error, and with one of the mass eigenstates (νe ≡ νi). Then, min xi ±∆xi, and of another massive parameter m. Here a re- the value of mββ decreases moving from one vertex to- wards the inner part of the triangle, until it becomes zero mark is necessary. When in the literature we found max- inside the region delimited by the vertices defined by the imal or systematic uncertainties, in order to propagate conditions: their effects in our calculations, we decided to interpret them as the semi-widths of√ flat distributions and thus, 2 2 2 dividing these numbers by 3 we could get the standard U m1 = U m2 when U = 0 (A10) e1 e2 e3 deviations of those distributions. Then, we considered 2 2 2 Ue1 m1 = Ue3 m3 when Ue2 = 0 (A11) those values as standard deviations for Gaussian fluctu- 2 2 2 ations of the parameters around the given values. Ue2 m2 = Ue3 m3 when Ue1 = 0. (A12) For any fixed value of m, and for the other parameters In fact, if we consider, for example, the first condition, set to their best fit values xi, we can attach the following 32 error to M: with Σ ≥ b. In the case when a ≈ b, instead, which v corresponds to the IH case, it is convenient to write the u  2 uX ∂M quartic equation as ∆M| = t ∆x2. (B1) m ∂x i i i (3m2 + 2mΣ − λ2)(m − Σ)2 − (a2 − b2)2 = 0 (C5)

When we want to consider the prediction and the error where for a fixed value of another massive parameter Σ(m, x), we have to vary also m, keeping δΣ = ∂Σ/∂m δm + λ2 ≡ Σ2 − 2(a2 + b2). (C6) ∂Σ/∂xi δxi = 0. Therefore, in this case we find: Again, we see that this quartic equation has spurious so- v lutions in the limit a ≈ b, e. g. m ≈ Σ. We are interested u  2 uX ∂M ∂Σ/∂xi ∂M 2 to the one that in the case a = b reads ∆M|Σ = t − ∆xi . (B2) ∂xi ∂Σ/∂m ∂m √ i −Σ + 2 Σ2 − 3b2 m = m (Σ, b) ≡ (C7) IH 3 Of course, we will calculate m by inverting the Σ(m, x) = Σ (here, the symbol Σ denotes the function and also its with Σ ≥ 2b. value; however, this abuse of notation is harmless in prac- Finally, we discuss useful approximate formulae for the tice). specific parameterization suggested in Ref. [94], namely ( a = δm2 Appendix C: Σ = f(mlightest), analitical solution (C8) b = ∆m2 + δm2/2 Let us write in full generality the three flavor relation for the mass probed in cosmology as for the NH case and

p p ( 2 2 Σ = m + m2 + a2 + m2 + b2 (C1) a = ∆m − δm /2 (C9) b = ∆m2 + δm2/2 where m, Σ, a and b are masses, i. e. non-negative pa- rameters. It is possible to obtain m as a function of Σ in for the IH one. the physical range In the latter case, the approximation obtained by Σ ≥ a + b Eq. (C7), namely,

2 simply by solving a quartic equation. Since we are inter- m = mIH(Σ, ∆m ) (C10) ested in certain specific cases (NH or IH) we specify the discussion further. is already excellent, being better than 3 µeV in the whole When a  b, corresponding to the NH case, it is range of masses. Instead, Eq. (C4) implies a maximum convenient to write the quartic equation as error that can reach 5 meV for NH. Although this is quite adequate for the present and near future sensitivity, (3m2 − 4mΣ + λ2)(m2 − λ2) + 4a2b2 = 0 (C2) it is possible to improve the approximation also in the case of NH by using where δm2 2 2 2 2 2 λ ≡ Σ − (a + b ). (C3) m = mNH(Σ, ∆m ) − 2 . (C11) 4 mNH(Σ, ∆m ) Indeed, we see that this quartic equation has spurious This formula is obtained by linearly expanding in δm2 solutions in this limit, e. g. those for m ≈ ±λ. Instead, the relation that links Σ and m, Eq. (C1), around the we are interested in the one that (for a = 0) reads point m = m (Σ, ∆m2). The error is remarkably small √ NH 2Σ − Σ2 + 3b2 error and more than adequate for the present sensitivity: m = m (Σ, b) ≡ (C4) less than 0.2 meV. NH 3

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CONTENTS B. Models for the NMEs 15 C. Theoretical uncertainties 16 I. Introduction1 1. Generality 16 2. Is the uncertainty large or small? 16 II. The total lepton number2 3. The size of the axial coupling 17 A. B and L symmetries in the SM3 4. Quenching as a major cause of B. Majorana neutrinos3 uncertainty 18 C. Ultrarelativistic limit and massive D. The case of heavy neutrino exchange 18 neutrinos3 D. Right-handed neutrinos and unified groups4 VI. Experimental search for the 0νββ 19 E. Leptogenesis5 A. The 0νββ signature 19 F. Neutrino nature and cosmic neutrino B. The choice of the isotope 20 background5 C. Sensitivity 20 D. Experimental techniques 21 III. Particle physics mechanisms for 0νββ 5 1. Calorimetric technique 22 A. The neutrino exchange mechanism6 2. External source approach 23 B. Alternative mechanisms to the light E. Experiments: a brief review 23 neutrino exchange7 1. The claimed observation 24 1. Historical proposals7 F. Present sensitivity on mββ 24 2. Higher dimensional operators7 G. Near and far future experiments 25 3. Heavy neutrino exchange7 4. Models with RH currents8 VII. Interplay with cosmology 26 C. From 0νββ to Majorana mass: a remark on A. Upper bounds scenario 26 “natural” gauge theories8 B. Measurements scenario 28 D. Role of the search at accelerators8 C. Considerations on the information from cosmological surveys 29 IV. Present knowledge of neutrino masses9 VIII. Summary 30 A. The parameter mββ 9 B. Oscillations 10 Acknowledgments 30 1. Mass eigenstates composition 11 C. Cosmology and neutrino masses 12 A. Extremal values of mββ 30 1. The parameter Σ 12 1. Formal proof 30 2. Constraints from cosmological surveys 12 2. Remarks on the case mmin = 0 31 D. Other non-oscillations data 13 ββ E. Theoretical understanding 14 B. Error propagation 31

V. The role of nuclear physics 14 C. Σ = f(mlightest), analitical solution 32 A. Recent developments on the phase space factor calculations 15 References 32