Neutrinoless double-β decay 1

Neutrinoless Double-β decay
Revised August 2019 by A. Piepke (University of Alabama) and P. Vogel (Caltech).
Observation of neutrinoless double-beta (0νββ) decay would signal violation of total
lepton number conservation. The process can be mediated by an exchange of a light
Majorana neutrino, or by an exchange of other particles. However, the existence of0νββ-decay requires a nonvanishing Majorana neutrino mass, no matter what the actual
mechanism is. As long as only a limit on the lifetime is available, limits on the effective
Majorana neutrino mass, on the lepton-number violating right-handed current or otherpossible mechanisms mediating 0νββ decay can be obtained, independently of the actual
mechanism, by assuming that one of these “new physics” possibilities dominates. Theselimits are listed in the Double-β Decay Listings of the experimental measurements.
In the following we assume that the exchange of light Majorana neutrinos(mνi ≤ 10 MeV) contributes dominantly to the decay rate. Besides a dependence
on the phase space (G0ν ) and the nuclear matrix element (M 0ν ), the observable
0νββ-decay rate is proportional then to the square
Majorana mass
P of 2the effective−10ν0ν
2220ν
2= G · |M | · mee , with mee = | i Uei mνi | . The sum contains, inmee , (T1/2 )2 , i.e., cancellations may occur. For three neutrino
general, complex CP-phases in Uei
flavors there are two physical phases for Majorana neutrinos (η1 , η2 ) and one for
Dirac neutrinos (δCP ). The relevant Majorana phases affect only processes to whichlepton-number changing amplitudes contribute. Given the general 3 × 3 mixing matrix
for Majorana neutrinos,
one can construct other analogous lepton number violating
Pquantities, mℓℓ′ = i Uℓi Uℓ′ i mνi (ℓ or ℓ′ 6= e). However, these are currently much lessconstrained than mee .
Nuclear structure calculations are needed to deduce mee from the decay rate. While0ν
G can be calculated accurately, the computation of M 0ν is subject to uncertainty.
Comparing different nuclear model evaluations indicates a factor ∼2-3 spread inthe calculated nuclear matrix elements. Nuclear structure calculation consistently
overestimate Gamow-Teller (axial current) matrix elements. This inability of the nuclear
models to reproduce Gamow-Teller decay rates is often parametrized in form of a modified
coupling constant gA . Many nuclear theorists interpret this shortcoming as evidence
that important physics is missing in the modeling of weak nuclear transitions. It is
not clear how these observed uncertainties impact 0νββ-matrix elements. Nevertheless,this constitutes an additional element of uncertainty. Recent work, citeGysbers:2019uyb
shows how the discrepancy between experimental and theoretical axial current matrix
elements might be resolved. However, application of this approach to the 0νββ decay
remains to be accomplished. The particle physics quantities to be determined are thus
nuclear model-dependent, so the half-life measurements are listed first. Where possible,
we reference the nuclear matrix elements used in the subsequent analysis. Since rates forthe conventional 2νββ decay serve to constrain the nuclear theory models, results for thisprocess are also given.
Oscillation experiments utilizing atmospheric, accelerator, solar, and reactor produced
neutrinos and anti-neutrinos show that at least some neutrinos are massive. However,so far the inverted mass ordering (i.e., whether ∆m231 < 0) is disfavored only by 2-3 σcompared to the normal mass ordering (when ∆m231 > 0), while the absolute neutrino

S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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2Neutrinoless double-β decay

mass values or the properties of neutrinos under CPT-conjugation (Dirac or Majorana)
remain undetermined. All confirmed oscillation experiments can be consistently describedusing three interacting neutrino species with two mass splittings and three mixing angles.
(For values of the mixing angles and mass square differences see the corresponding tables.)Based on the 3-neutrino analysis:
2mee = | cos2 θ13 cos2 θ12 m1 + e2i(η2 −η1 ) cos2 θ13 sin2 θ12 m2 + e−2i(η1 +δCP ) sin2 θ13 m3 |2 ,valid for both mass orderings. Given the present knowledge of the neutrino oscillation
parameters one can derive a relation between the effective Majorana mass and the massof the lightest neutrino, as illustrated in Figure 14.11 in the Neutrino Masses, Mixing andOscillations review. The three mass orderings allowed by the oscillation data: normal(m1 < m2 ≪ m3 ), inverted (m3 ≪ m1 < m2 ), and degenerate (m1 ≈ m2 ≈ m3 ), result indifferent projections. The width of the colored bands reflects the uncertainty introducedby the unknown Majorana and Dirac phases as well as the experimental errors of theoscillation parameters. The latter causes only minor broadening of the bands. Because of
the overlap of the different mass scenarios, a measurement of mee would not reveal whichmass ordering is applicable, provided the value of mee is in the overlapping range.
Analogous plots depict the relation of mee with the summed neutrino mass
mtot = m1 + m2 + m3 , constrained by observational cosmology, and mee as a function ofef f
the average mass mνe = [Σ|Uei |2 m2νi ]1/2 determined through the analysis of the electronenergy distribution in low energy beta decays. (See Fig. 1 of [2]. ) The oscillation data
ef fthus allow to test whether observed values of mee and mtot or mνe are consistent within
the 3 neutrino framework. The rather large intrinsic width of the ββ-decay constraints
essentially does not allow to positively identify the mass ordering, and thus the signof ∆m231 , even in combination with these other observables. Naturally, if a value of
0 < mee ≤ 0.01 eV is ever established, then the normal mass ordering becomes the only
possible scenario.
It should be noted that systematic uncertainties of the nuclear matrix elements andpossible quenching of the axial current matrix elements are sometimes not folded into
the mass limits reported by ββ-decay experiments. Taking this additional uncertainty
into account would further widen the projections. The plots are based on a 3-neutrino
analysis. If it turns out that additional, i.e. sterile light neutrinos exist, the allowed
regions would be modified substantially.
If neutrinoless double-beta decay is observed, it will be possible to fix a range ofabsolute values of the masses mνi . Unlike the direct neutrino mass measurements,
however, a limit on mee does not allow one to constrain the individual mass values mνieven when the mass differences ∆m2ij are known.
Neutrino oscillation data imply the existence of a lower limit ∼ 0.014 eV for theMajorana neutrino mass for the inverted mass ordering pattern, while mee could, by finetuning, vanish in the case of the normal mass ordering. Several new double-beta searches
have been proposed to probe the interesting mee mass range, with the prospect of fullcoverage of the inverted mass ordering region within the next decade.
The 0νββ decay mechanism discussed so far is not the only way in which the decay
can occur. Numerous other possible scenarios have been proposed, however, all of them
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Neutrinoless double-β decay 3
requiring new physics. It will be a challenging task to decide which mechanism wasresponsible once 0νββ decay is observed. LHC experiments may reveal corresponding
signatures for new physics of lepton number violation. If lepton-number violatingright-handed weak current interactions exist, its strength can be characterized by thephenomenological coupling constants η and λ (η describes the coupling between the righthanded lepton current and left-handed quark current while λ describes the coupling
P whenboth currentsP
are right-handed). The 0νββ decay rate then depends on hηi = η i Uei Vei
and hλi = λ i Uei Vei that vanish for massless or unmixed neutrinos (Vℓj is a matrix
analogous to Uℓj but describing the mixing with the hypothetical right-handed neutrinos).
The observation of the single electron spectra could, in principle, allow to distinguish thismechanism of 0νββ from the light Majorana neutrino exchange driven mode. The limitson hηi and hλi are listed in a separate table. The reader is cautioned that a number ofearlier experiments did not distinguish between η and λ. In addition, see the section on
Majoron searches for additional limits set by these experiments.
References:
1. P. Gysbers et al., Nature Phys. 15, 5 (2019); [arXiv:1903.00047].
2. M.J. Dolinski, A.W.P. Poon and W. Rodejohann, Ann. Rev. Nucl. Part. Sci. 49,219 (2019); [arXiv:1902.04097].

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