Analysis of light neutrino exchange and short-range mechanisms in $0\nu \beta \beta$ decay

Neutrinoless double beta decay ($0\nu \beta \beta$) is a crucial test for lepton number violation. Observation of this process would have fundamental implications for neutrino physics, theories beyond the Standard Model and cosmology. Focusing on so-called short-range operators of $0\nu \beta \beta$ and their potential interplay with the standard light Majorana neutrino exchange, we present the first complete calculation of the relevant nuclear matrix elements, performed within the interacting boson model (IBM-2). Furthermore, we calculate the relevant phase space factors using exact Dirac electron wave functions, taking into account the finite nuclear size and screening by the electron cloud. The obtained numerical results are presented together with up-to-date limits on the standard mass mechanism and effective $0\nu \beta \beta$ short-range operators in the interacting boson model framework. Finally, we interpret the limits in the particle physics scenarios incorporating heavy sterile neutrinos, left-right symmetry and $R$-parity violating supersymmetry.

Figure 1

Contributions to $0\nu \beta \beta$ decay from effective LNV operators. (a) Standard light neutrino exchange via dim-5 operator; (b) long-range contribution via dim-7 operator; (c) short-range contribution via dim-9 operator. Adapted from [26].

Figure 2

Comparison between the light neutrino exchange IBM-2 NMEs ${\mathcal{M}}_{\nu }$ calculated in this work (red circles) and the ones calculated in [30] (solid blue squares), assuming the quenched value $g_A=1.0$. We show also the old total NME ${\tilde{\mathcal{M}}}_{\nu }^{\mathrm{old}}$ incorporating the (old) tensor part but with the correct sign (empty blue squares).

Figure 3

As Fig. 2, but showing the comparison for the heavy-neutrino-exchange NME ${\mathcal{M}}_{{\nu }_h}$.

Figure 4

Normalized single electron energy distributions ${\mathrm{\Gamma }}^{-1}d\mathrm{\Gamma }/d{E}_1^{\mathrm{kin}}$ (left) and angular correlation $\alpha (E_1^{\mathrm{kin}})$ (right) for ${}^{76}\mathrm{Ge}$ as a function of the kinetic energy $E_1^{\mathrm{kin}}=E_1-m_e$. Shown are the phase space factors for Eq. (50) in the former and for Eq. (51) in the latter.

Figure 5

Relation between the $0\nu \beta \beta$ mass $|m_{\beta \beta }|$ and the sum of neutrino masses $\mathrm{\Sigma }{m}_{\nu }$ for normally (NO) and inversely ordered (IO) neutrinos with the oscillation parameters fixed to the current best fit values. The dark shaded regions denote the parameter space allowed by the limits on $\mathrm{\Sigma }{m}_{\nu }$ at 95% C.L. from cosmological searches. The horizontal bars indicate the current upper limit on $|m_{\beta \beta }|$ and future sensitivities of $0\nu \beta \beta$ decay searches with an unquenched ($g_A=1.27$, bottom edge) and quenched ($g_A=1.0$, top edge) value of the axial coupling.

Figure 6

Lower limits on the effective short-range operator scales ${\mathrm{\Lambda }}_I$ defined at $m_W$ and assuming all other contributions are zero. The limits are from the current bounds (dark shade) and two future sensitivities (lighter shades) in ${}^{76}\mathrm{Ge}$ at $(1.8,10,100)\times {10}^{26}\mathrm{yr}$ (left, blue), ${}^{100}\mathrm{Mo}$ at $(0.011,5,10)\times {10}^{26}\mathrm{yr}$ (middle, orange) and ${}^{136}\mathrm{Xe}$ at $(1.1,5,9.2)\times {10}^{26}\mathrm{yr}$ (right, green).

Figure 7

Constraints on the effective $0\nu \beta \beta$ decay mass $|m_{\beta \beta }|$ and the short-range operator coupling ${\epsilon }_3^{LLL}$ (left) as well as the associated operator scale ${\mathrm{\Lambda }}_3^{LLL}$ (right). All other effective couplings are set to zero. The highlighted regions denote the allowed parameter space from the current limits $T_{1/2}({}^{76}\mathrm{Ge})>1.8\times {10}^{26}\mathrm{yr}$ (light blue) and $T_{1/2}({}^{136}\mathrm{Xe})>1.1\times {10}^{26}\mathrm{yr}$ (light green), as well as the future sensitivity $T_{1/2}({}^{76}\mathrm{Ge})={10}^{28}\mathrm{yr}$ (dashed blue). The grey shaded area on the right is excluded assuming the bound on the sum of neutrino masses of $\mathrm{\Sigma }{m}_{\nu }>290\mathrm{meV}$ from cosmological observations for an inverse neutrino mass ordering.

Figure 8

As Fig. 7, but for the short-range operator coupling ${\epsilon }_5^{RR}$ and associated scale ${\mathrm{\Lambda }}_5^{RR}$, also showing constraints from the current limit and future sensitivity in ${}^{100}\mathrm{Mo}$.

Figure 9

Upper limit on the active-sterile neutrino mixing strength $|V_{eN}{|}^2$ as a function of the sterile neutrino mass $m_N$ from current $0\nu \beta \beta$ decay searches (solid curves) and at future sensitivities with $T_{1/2}={10}^{28}\mathrm{yr}$ (dashed curves). The sterile neutrino is assumed to be of Majorana or quasi-Dirac nature as indicated and contributions from light neutrinos are neglected. The blue shaded area is excluded by current data from neutrino oscillations, beta decays, meson decays, colliders, and electroweak precision measurements. The dashed contours indicate the estimated future sensitivity in Tritium decays (KATRIN), long-lived particle (LLP) searches and at colliders (FCC-hh, ILC, CLIC). The diagonal line gives the seesaw relation of light neutrino mass generation, $m_{\nu }=|V_{eN}{|}^2{m}_N=0.01\mathrm{eV}$.

Figure 10

Lower limit on the right-handed $W_R$ boson mass $m_{W_R}$ as a function of the right-handed neutrino mass $m_N$ from current $0\nu \beta \beta$ decay searches (solid curve) and the corresponding future sensitivities with $T_{1/2}({}^{76}\mathrm{Ge})={10}^{28}\mathrm{yr}$ (dashed curve) in the LRSM with negligible $W$ boson mixing. The dotted curve indicates the future sensitivity on the scenario where the $W$ boson mixing is $\sin {\theta }_{LR}^W=m_W^2/m_{W_R}^2$. The blue shaded area is excluded by current data from the LHC and the dashed contours indicate the estimated future sensitivity at the LHC with $300{\mathrm{fb}}^{-1}$ and at SHiP.