Neutrinoless double beta decay can constrain neutrino dark matter

We examine how constraints can be placed on the neutrino component of dark matter by an accurate measurement of neutrinoless double beta ($0\nu\beta\beta$) decay and the solar oscillation amplitude. We comment on the alleged evidence for $0\nu\beta\beta$ decay.

The detection of neutrinoless double beta decay would imply the violation of lepton number conservation. The process could be induced by Majorana neutrino mass terms, or by less trivial modifications of the standard model. Here we consider the former possibility wherein there are exactly three left-handed neutrino states with Majorana masses [1]. The measurement of 0νββ decay, together with what has been learned from studies of solar and atmospheric neutrinos, has direct consequences on the spectrum of neutrino masses and therefore on the effects of neutrinos on structure formation. We define what will be necessary to determine the neutrino component of dark matter from terrestrial experiments.
The charged-current eigenstates are related to the mass eigenstates by a unitary trans- where s i and c i are the sines and cosines of θ i , and V is the diagonal matrix (1, e i φ 2 2 , e i( φ 3 2 +δ) ). In Eq. (1), φ 2 and φ 3 are additional phases for Majorana neutrinos that are not measurable in neutrino oscillations; if CP is conserved, the phases in UV are either 0 or π.
We choose the mass ordering m 1 < m 2 < m 3 with m i non-negative. There are two possible neutrino mass spectra: where in either case ∆ a ≫ ∆ s in accord with the previously described experimental data.
For the normal hierarchy (Case I), mixing is given by Eq. (1). The limit on θ 2 implies that there is very little mixing of ν e with the heaviest state. In Case I solar neutrinos oscillate primarily between the two lighter mass eigenstates. For the inverted hierarchy (Case II), solar neutrinos oscillate primarily between the two nearly degenerate heavier states. In this case the mixing is described by interchanging the roles of m 1 and m 3 . With a mixing matrix obtained from Eq. (1) by interchange of the first and third columns of UV , the parameters governing neutrino oscillations (θ i and δ) retain the same import as those in Case I. The limit on θ 2 again implies that for Case II there is very little mixing of ν e with the lightest state.
The rate of 0νββ decay depends on the magnitude of the ν e -ν e element of the neutrino mass matrix [5], which is The masses m i may be determined from the lightest mass m 1 and the mass-squared differences. Since the solar mass-squared difference is very small it can be ignored; then setting m 1 = m and ∆ a = ∆, The lightest mass is related to the sum of neutrino masses (Σ = Σm i ) via For a given value of M ee , the minimum possible value of m is obtained if the three contributions to M ee are in phase, i.e., φ 2 = φ 3 = 0. Thus The maximum possible value of m is obtained if the the two smaller contributions to M ee are out of phase with the largest contribution (i.e., φ 2 = φ 3 = π when c 3 > s 3 ). Then The allowed ranges for Σ are determined from Eqs. (8) and (9). Because θ 2 is small (sin 2 2θ 2 ≤ 0.1 or s 2 2 ≤ 0.026), its value does not significantly affect the result. (We have confirmed this result numerically). The limits on Σ (for θ 2 = 0) are where the plus sign applies to the normal hierarchy and the minus sign to the inverted hierarchy. The bounds depend on only two oscillation parameters: the scale of atmospheric neutrino oscillations (∆) and the amplitude of solar neutrino oscillations (sin 2 2θ 3 ). If the recent evidence that 0.05 eV ≤ M ee ≤ 0.84 eV at the 95% C.L. [8] is borne out, this would imply that 0.1 eV < ∼ Σ < ∼ 20 eV, which using Ω ν h 2 = Σ/(93.8 eV) translates to where Ω ν is the fraction of the critical density contributed by neutrinos and h is the dimensionless Hubble constant (H 0 = 100h km s −1 Mpc −1 ). CMB measurements and galaxy cluster surveys already constrain Σ to be smaller than 4.4 eV (Ω ν h 2 < ∼ 0.05) at the 95% C.L. [7]. Data from the MAP satellite should either determine Σ or tighten this constraint to about 0.5 eV in the near future [9]. A more stringent upper bound on Σ from terrestrial experiments must await the precise determination of θ 3 (such as is anticipated from KamLAND [10]), and a firmer measurement (or constraint) on M ee [11]. tritium beta decay [6] and cosmology [7] are shown.