Searching for New Physics in Two-Neutrino Double Beta Decay
Frank F. Deppisch Lukas Graf Fedor Šimkovic
hep-ph nucl-ex nucl-th




 
f.deppisch@ucl.ac.uk

lukas.graf@mpi-hd.mpg.de

fedor.simkovic@fmph.uniba.sk


Motivated by non-zero neutrino masses and the possibility of New Physics discovery, a number of experiments search for neutrinoless double beta decay. While hunting for this hypothetical nuclear process, a significant amount of two-neutrino double beta decay data have become available. Although these events are regarded and studied mostly as the background of neutrinoless double beta decay, they can also be used to probe physics beyond the Standard Model. In this paper we show how the presence of right-handed leptonic currents would affect the energy distribution and angular correlation of the outgoing electrons in two-neutrino double beta decay. Consequently, we estimate constraints imposed by currently available data on the existence of right-handed neutrino interactions without having to assume their nature. In this way our results complement the bounds coming from the non-observation of neutrinoless double beta decay as they limit also the exotic interactions of Dirac neutrinos. We perform a detailed calculation of two-neutrino double beta decay under the presence of exotic (axial-)vector currents and we demonstrate that current experimental searches can be competitive to existing limits.


Searching for New Physics in Two-Neutrino Double Beta Decay
    Fedor Šimkovic
    December 30, 2023
===========================================================






§ INTRODUCTION



Double beta decay processes are sensitive probes of physics beyond the Standard Model (SM). The SM process of two-neutrino double beta (2νββ) decay is among the rarest processes ever observed with half lives of order T_1/2^2νββ∼ 10^19 yr and longer <cit.>. Neutrinoless double beta (0νββ) decay, with no observation of any missing energy, is clearly the most important mode beyond the SM as it probes the Majorana nature and mass m_ν of light neutrinos, with current experiments sensitive as T_1/2^0νββ∼ (0.1 eV/m_ν)^2 × 10^26 y. In general, it is a crucial test for any New Physics scenario that violates lepton number by two units <cit.>.

While 0νββ decay is the key process, experimental searches for this decay also provide a detailed measurement of the 2νββ decay rate and spectrum in several isotopes. For example, Kamland-Zen measures the 2νββ decay spectrum in ^136Xe with a high statistics <cit.>, but can only do with respect to the sum of energies of the two electrons emitted. On the other hand, the NEMO-3 experiment with the technology to track individual electrons can measure the individual electron energy spectra and the opening angle between the two electrons. This has yielded detailed measurements of the 2νββ decay spectra of ^96Zr <cit.>, ^150Nd <cit.>, ^48Ca <cit.>, ^82Se <cit.> and especially ^100Mo <cit.>, the latter with a very high statistics containing ≈ 5× 10^5 2νββ decay events. Such measurements are important for the interpretation of 0νββ decay searches as it can shed light on the value of the effective axial coupling g_A <cit.>. 

The high precision of 2νββ decay measurements, expected to continue as the experimental exposures are increased to push the sensitivity of 0νββ decay searches, begs the question whether 2νββ decay events can be directly used to search for New Physics beyond the SM. This is the focus of this work. We model such new physics effects through effective charged-current operators of the form ϵ G_F (e̅𝒪_1 ν)(u̅𝒪_2 d) with Lorentz structures 𝒪_1, 𝒪_2 other than the SM V-A type. Here, the Fermi constant G_F is introduced and the small dimensionless coupling ϵ encapsulates the New Physics effects.

Exotic charged-current operators of the above form are being searched for in nuclear, neutron β and pion decays as well as collider searches <cit.>, giving rise to limits of the order ϵ≲ 10^-4 - 10^-1, depending on the Lorentz structure and chirality of the fields involved. In this paper, we will specifically concentrate on exotic operators containing right-handed (RH) vector lepton currents. Such operators prove difficult to constrain as interference with the SM contribution is suppressed by the light neutrino masses. They are nevertheless of strong theoretical interest as their observation, along with the non-observation of lepton number violation would indicate that neutrinos are not Majorana fermions. This is because RH currents with neutrinos but in the absence of a sterile neutrino state would necessarily violate lepton number. In this work, we will show that the existing 2νββ data from the NEMO-3 experiment may set the most stringent limits on such operators which are currently only weakly constrained at the 6% level <cit.>. We thus describe a novel probe of the fundamental nature of weak interactions and the properties of neutrinos.





§ EXOTIC CHARGED-CURRENT INTERACTIONS








We are interested in processes where right- and left-handed electrons are emitted considering only (V+A) and (V-A) currents. The effective Lagrangian is written as


    ℒ = 
    	G_Fcosθ_C/√(2)(
    	  (1+δ_SM + ϵ_LL) j^μ_L J^μ_Lμ 
    	                      + ϵ_RL  j_L^μ J^μ_Rμ
    		                  + ϵ_LR  j_R^μ J^μ_Lμ
    		                  + ϵ_RR  j_R^μ J^μ_Rμ) + h.c.,


with the tree-level Fermi constant G_F, the Cabbibo angle θ_C, and the leptonic and hadronic currents j_L,R^μ = e̅γ^μ(1∓γ_5)ν and J^μ_L,R = u̅γ^μ(1∓γ_5)d, respectively. The SM electroweak radiative corrections are encoded in δ_SM and the ϵ_XY encapsulate new physics effects. We here concentrate on the latter two operators with RH lepton currents as they are expected to change the 2νββ decay kinematic spectra more significantly. Extensions of the above set of operators can be considered; for example, currents other than vector and axial-vector can be included <cit.> and further, exotic particles may participate <cit.>. 

In Eq. (<ref>), ν is a 4-spinor field of the light electron neutrino, either defined by ν = ν_L + ν_L^c (i.e. a Majorana spinor constructed from the SM active left-handed neutrino ν_L and its charge-conjugate) or ν = ν_L + ν_R (a Dirac spinor constructed from the SM ν_L and a new SM-sterile RH neutrino ν_R). Whether the light neutrinos are of Majorana or Dirac type and whether total lepton number is broken or conserved is of crucial importance for an underlying model but as far as the effective interactions in Eq. (<ref>) are concerned, this does not play a role in our calculations. If the neutrino in Eq. (<ref>) is a Majorana particle, the operators associated with ϵ_LR and ϵ_RR violate total lepton number by two units and they will give rise to extra contributions to 0νββ decay <cit.>. In this case, severe limits are set by 0νββ decay searches of the order ϵ_LR≲ 3× 10^-9, ϵ_RR≲ 6× 10^-7 <cit.>. On the other hand, if there exists a sterile neutrino Weyl state ν_R that combines with ν_L to form a Dirac neutrino, the RH current interactions in Eq. (<ref>) do not necessarily violate lepton number which, in fact, can remain an unbroken symmetry of the underlying model. For example, such effective interactions can emerge in Left-Right symmetric models (LRSMs) <cit.> with unbroken lepton number <cit.>. The observation of the effect of RH neutrino operators without the observation of lepton number violation would thus strongly suggest that neutrinos are Dirac fermions.

The most stringent direct limits on the above operators for process energies E≈ MeV are set by fitting experimental results of neutron and various nuclear single β decays, ϵ_LL, ϵ_RL≤ 5× 10^-4, ϵ_LR, ϵ_RR≤ 6× 10^-2 <cit.>. The limits on the RH lepton currents are much less severe due to the absence of an interference with the SM contribution. Searches at the Large Hadron Collider (LHC) for single electron and missing energy signatures <cit.>, pp→ eX + MET, may also be used to constrain the above operators, ϵ_LL≲ 4.5× 10^-3, ϵ_RR≲ 2.2× 10^-3 <cit.>. While the constraints are stringent and the sensitivity is expected to improve to ϵ_LL≈ 10^-5 <cit.>, the LHC operates at a much higher energy and the effective operator analysis is only applicable if the new physics mediators integrated out are much heavier than this. More model-dependent limits can also be set by direct searches for RH current mediators at the LHC <cit.>, from considerations of sterile neutrino thermalization and the resulting increase of the effective number of light degrees of freedom in the early universe and supernova cooling. The associated new physics scales probed range between Λ≈ 5 - 20 TeV, corresponding to ϵ_XY≈ 5× 10^-4 - 5× 10^-5. An indirect limit on ϵ_LR can be set from the fact that the associated operator contributes to the Dirac neutrino mass at the second loop order <cit.>. Using current direct neutrino mass bounds this results in ϵ_LR≲ 10^-2 <cit.>. Especially the direct limit ϵ_LR≤ 6× 10^-2 is rather feeble and motivates the need to probe for admixtures of exotic currents in the SM Fermi interaction. While underlying scenarios are expected to trigger the other, better constrained operators as well, it is not difficult to envision cases where ϵ_LR or ϵ_RR are dominant. For example, in LRSMs, the operator associated with ϵ_RL is mediated at lowest order by the SM W boson and involves the mixing θ_LR with an exotic W_R boson. This mixing is a priori unrelated to the m_W_R scale and ϵ_RL can thus be suppressed compared to ϵ_RR if θ_LR is small. It is also not difficult to think of extensions of the minimal LRSM where exotic copies of quarks are charged under the LRSM SU(2)_R but not the SM quarks. The exotic quarks instead mix with the SM quarks and the latter will inherit a suppressed RH current, suppressing ϵ_RL with respect to ϵ_LR.





§ DECAY RATE AND DISTRIBUTIONS



We have calculated the differential rate of  decay under the presence of the exotic interactions in Eq. (<ref>). Because  decay is possible in the SM, arising in second order perturbation theory of the first term in Eq. (<ref>), interference between SM and exotic contributions is in principle possible. In general, the amplitude of   decay is calculated as a coherent sum of the Feynman diagrams in Fig. <ref>. To lowest order in ϵ_XR, exotic effects occur from the interference of the SM diagram Fig. <ref> (left) and Fig. <ref> (center). Due to the RH nature of the exotic current, such an interference is helicity suppressed by the masses of the emitted electron and neutrino as m_e m_ν / Q^2, with the  decay energy release Q. For light eV-scale neutrinos it is thus utterly negligible.[This is not necessarily the case if currents other than V± A vector currents are considered in Eq. (<ref>).] Contributions to second-order ∝ϵ_XR^2 come from the center diagram and the interference of the SM contribution (left) with the second-order exotic diagram (right). The latter is suppressed even more strongly by the neutrino mass and thus negligible. To lowest order in the exotic coupling, the squared matrix element for ground state to ground state  transition can thus be written as the incoherent sum


    |ℛ^2ν|^2 = |ℛ^2ν_ SM|^2 + |ϵ_XR|^2 |ℛ^2ν_ϵ|^2,


where ℛ^2ν_ SM is the matrix element for SM  decay and ℛ^2ν_ϵ is the exotic contribution. As discussed in detail in the  <cit.>, the latter may be expressed as
Doi:1985dx, Haxton:1985am, Gysbers:2019uyb


    ℛ^2ν_ϵ   = i (1/√(2))^2 (G_Fcosθ_W/√(2))^2
            [1-P(e_1,e_2)] [1-P(ν̅_1,ν̅_2)] 
       ×[ ψ(p_e_1)γ^μ(1+γ_5)ψ^c(p_ν̅_1) ψ(p_e_2)γ^ν(1-γ_5)ψ^c(p_ν̅_2) . 
         + .ψ(p_e_1)γ^ν(1-γ_5)ψ^c(p_ν̅_1) ψ(p_e_2)γ^μ(1+γ_5)ψ^c(p_ν̅_2) ] 
       ×( g_μ 0g_ν 0  g_V^2 M_F
            ∓1/3 g_μ k g_ν k  g_A^2 M_GT),


where ψ(p_f) is the wave function of the emitted fermion f with momentum p_f and we consider here the commonly used approximation of the S_1/2 wave evaluated at the nuclear surface. The nuclear matrix elements between the initial 0_i^+, the intermediate 0^+_n (1^+_n) and the final 0^+_f states of the nucleus are generally of Fermi (Gamow-Teller) type with the associated nucleon-level vector (effective axial-vector) coupling g_V (g_A),


    M_F    = ∑_n
    		⟨ 0^+_f|∑_jτ^+_j |0^+_n⟩⟨ 0^+_n|∑_kτ^+_k |0^+_i⟩/Δ E_n(0_n^+) + E_e_2+E_ν̅_2,
    
    	M_GT   = ∑_n
    		⟨ 0^+_f|∑_jτ^+_j σ_j|1^+_n⟩·⟨ 1^+_n|∑_kτ^+_k σ_k | 0^+_i⟩/Δ E_n(1_n^+) + E_e_2+E_ν̅_2.


The summations are over all intermediate 0^+_n, 1^+_n states and all nucleons j,k inside the nucleus where τ^+_j,k is the isospin-raising operator transforming a neutron into a proton and σ_j,k represents the nucleon spin operator. Assuming isospin invariance, the Fermi matrix elements vanish. The energy denominators arise due to the second-order nature of the above matrix element where Δ E_n(J^π_n) = E_n(J^π_n) - E_i (J^π_n = 0^+_n and 1^+_n) are the energies of the intermediate nuclear states with respect to the initial ground state. Overall energy conservation is implied, E_i = E_f + E_e_1 + E_e_2 + E_ν̅_1 + E_ν̅_2, and, as indicated by the particle exchange operator P(a,b), the matrix element is anti-symmetrized with respect to the exchange of the identical electrons and antineutrinos (the corresponding anti-symmetrization over the nucleons is implicitly included in the nuclear states).

Following Ref. <cit.>, the calculation of the  decay rate and distributions is detailed in the . We use nuclear matrix elements in the QRPA formalism from Ref. <cit.> assuming isospin invariance with M_F = 0 and including higher order corrections from the effect of the final state lepton energies. Because of M_F = 0 and negligible SM – exotic interference effects, the calculations for ϵ_LR and ϵ_RR are identical; both cases yield the same rates and distributions. As a result, we calculate the full differential  decay rate in a given 0^+ → 0^+ double beta decaying isotope with respect to the two electron energies m_e ≤ E_e_1,e_2≤ Q + m_e and the angle 0 ≤θ≤π between the emitted electrons, which may be written as


    dΓ^2ν/dE_e_1dE_e_2dcosθ = Γ^2ν/2dΓ^2ν_norm/dE_e_1dE_e_2(1 + κ^2ν(E_e_1,E_e_2)cosθ).


Because interference effects between the SM and the RH current diagram are negligible, the differential rate is simply the incoherent sum of both. In the  we describe in detail the calculation of the above differential decay rate and the derived energy distributions, angular correlations and total rate. Specifically, for ^100Mo the total decay rate Γ^2ν = ln 2/T_1/2^2ν associated with the  half-life T_1/2^2ν may be approximated as Γ^2ν≈Γ^2ν_ SM(1 + 6.11 ϵ^2_XR), where Γ^2ν_SM is the SM rate. The experimentally accessible kinematic information is contained in the normalized double-differential energy distribution dΓ^2ν_norm/(dE_e_1dE_e_2) and the energy-dependent angular correlation -1 < κ^2ν(E_e_1,E_e_2) < 1. The latter determines whether the two electrons are preferably emitted back-to-back (κ^2ν≈ -1), in the same direction (κ^2ν≈ 1) or in intermediate configurations.




Given the uncertainties in nuclear matrix elements, the change of the total decay rate due to the presence of a RH current contribution is not expected to be measurable. Instead, differences in spectral shape of either the energy or angular distributions may be more sensitive. All double beta decay experiments measure the spectrum of events with respect to the sum of the electron kinetic energies, E_K = E_e_1 + E_e_2 - 2m_e. For ^100Mo, it is shown in Fig. <ref> (left), comparing the  decay distributions in the SM case (dashed) and for the exotic leptonic RH current operators (solid). The deviation is sizeable leading to a shift of the spectrum to smaller energies and a flatter profile near the endpoint E_K / Q = 1. We find that relative deviations of the order of 10% for small energies and near the endpoint are expected to occur. In experiments that are able to track and measure the individual electrons, such as NEMO-3 and SuperNEMO, the full doubly-differential energy spectrum is in principle measurable. Alternatively, the spectrum with respect to the kinetic energy of a single electron is shown in Fig. <ref> (right). It helps explain the shift of the energy sum spectrum in the exotic case as each electron receives on average less energy than in the SM.

This behaviour can be traced to the kinematic differences. In the presence of a RH lepton current in  decay, the electrons are preferably emitted collinearly and the electron energy-dependent correlation factor is always κ^2ν_ϵ > 0 whereas in the SM case the electrons are preferably emitted back-to-back with κ^2ν_ SM < 0. This behaviour can be understood from angular momentum considerations when the two electrons are produced with opposite dominant helicities. Integrating Eq. (<ref>) over the electron energies one arrives at the angular distribution,


    dΓ^2ν/dcosθ = 
    	Γ^2ν/2(1 + K^2νcosθ),


with the angular correlation factor K^2ν. For ^100Mo, we calculate K^2ν_ SM = -0.626 in the SM and K^2ν_ϵ = +0.370 for the exotic contribution. This deviation is clearly the most striking consequence of a RH lepton current on  decay. For small ϵ_XR≪ 1, the angular correlation factor K^2ν can be expanded as 


    K^2ν = K^2ν_SM + α ϵ_XR^2 + 𝒪(ϵ_XR^4).


For ^100Mo, the coefficient α turns out to be α = 6.078. Despite the small correction expected if ϵ_XR≈ 10^-2 as indicated in current bounds, searches for  decay can be sensitive in this regime. A simple signature is to look for the forward-backward asymmetry A^2ν_θ, comparing the number of  decay events with the electrons being emitted with a relative angle θ < π/2 and θ > π/2,


    A^2ν_θ = N_θ>π/2 - N_θ<π/2/N_θ>π/2 + N_θ<π/2 = 1/2K^2ν.


As shown, the asymmetry is simply related to the angular correlation factor K^2ν and it is clearly independent of the overall  decay rate. Considering only the statistical error, with N_events = 5× 10^5 2νββ decay events at NEMO-3, the angular correlation coefficient should be measurable with an uncertainty K^2ν_SM = -0.6260 ± 0.0027. No significant deviation from this SM expectation should then constrain ϵ_XR≲ 2.7× 10^-2 at 90% confidence level. This would already improve on the single β decay constraint of 6× 10^-2 <cit.>. If an experiment such as SuperNEMO were able to achieve an increase in exposure by three orders of magnitude, the expected future sensitivity, scaling as 1/√(N_events), would be ϵ_XR≲ 4.8× 10^-3. This only gives a very rough order of magnitude estimate and a dedicated experimental analysis is required to verify the sensitivity. For example, at NEMO-3 and SuperNEMO, detector effects will result in a reduced acceptance for small electron angles thus affecting the systematic uncertainty <cit.>. We note though that it is not strictly necessary to measure the forward-backward asymmetry in Eq. (<ref>). Instead, even if only including events with cosθ < 0, where the majority of 2νββ events occur, will allow to fit the angular distribution in Eq. (<ref>), albeit with a lower statistical significance. In this back-to-back region with cosθ≲ 0 the existing NEMO-3 data is well within the statistical fluctuations <cit.>. As a very rough but conservative estimate, dropping half of the events will give a dataset limited by statistics. This would weaken our estimated sensitivity by a factor of √(2).

We must also consider the theoretical uncertainty in predicting the angular correlation. Our results were calculated within the nuclear structure framework of the pn-QRPA with partial isospin restoration <cit.>. We consider the three main sources of theoretical errors: 


  (i) The spectrum of intermediate nuclear states as calculated in different nuclear structure models has a small but potentially significant impact on the external lepton phase space and thus the angular correlation. To conservatively model this, we vary the effective axial coupling g_A between g_A = 0.8 and g_A = 1.269 as described in Ref. <cit.>. This drastically changes the associated ^100Mo matrix element by a factor of 2.6 and thus the decay rate by a factor of ≈ 6.8 but the SM angular correlation changes only as K^2ν_SM = -0.6260± 0.0030. Thus the very conservatively estimated theoretical error is of the same order as the current statistical error. It will be crucial to reduce it to match the improved future statistical uncertainty, though. 


  (ii) In Eq. (<ref>) we only include the fundamental parton-level interactions and we neglect higher-order nuclear currents, namely the induced weak magnetism and pseudo-scalar currents. Their dominant effect on the amplitude will occur in the interference between the latter and the axial-vector nuclear current, which is suppressed by ≲ Q^2 / (3 m_π^2) ≈ 2× 10^-4 <cit.>, where m_π is the pion mass. This results in a currently negligible correction.


  (iii) For simplicity, we analytically treat the outgoing electron wave functions in the so called Fermi approximation. The proper Coulomb interaction with the nucleus and the electron cloud can be calculated numerically <cit.>, leading to a 15% correction in the resulting phase space factor but only a negligible shift in the SM angular correlation of 0 < Δ K_SM^2ν≲ 10^-4 <cit.>.


As can be seen in Fig. <ref> (right), the effect of RH currents is similar to that of varying the contribution of intermediate nuclear states as described in <cit.>. It exhibits a similar variation for small electron energies near the peak, depending on single state dominance (SSD) vs. higher state dominance (HSD) modelling of the intermediate nuclear state contributions <cit.>. This has the benefit that experimental searches for these effects, such as described in <cit.>, could be adapted to our scenario.




§ CONCLUSIONS



Nuclear double beta decay with the emission of two neutrinos and nothing else was proposed over 80 years ago <cit.> as a consequence of the Fermi theory of single β decay. Its main role for particle physics has largely been confined to being an irreducible background to the exotic and yet unobserved lepton number violating neutrinoless (0νββ) mode. We have demonstrated here, for the first time to our knowledge, that  decay may be used in its own right as a probe of new physics. Our result shows that searches for deviations in the spectrum of  decay can be competitive to existing limits. This provides a motivation to utilize the already large set of observed  decay events to probe exotic scenarios. The number of events will necessarily increase in the future by one to two orders of magnitude, as 0νββ decay is being searched for in future experiments.

We have here focussed on the case of effective operators with RH chiral neutrinos where the interference with the SM contributions is negligible due to the suppression by the neutrino mass. The exotic contribution to observables is therefore proportional to the square of the small New Physics parameter. As a result, such operators are comparatively weakly constrained. They still play an important role in our understanding of neutrinos as the RH nature can be accommodated in one of two ways: (i) through the right-chiral part of the SM neutrino as a Majorana fermion in which case the associated operators will also induce the lepton number violating 0νββ decay mode at a level that is already ruled out; or (ii) through the presence of a separate RH neutrino state that, while sterile under the SM gauge interactions, participates in exotic interactions beyond the SM. In the latter case, neutrinos are expected to be Dirac fermions and the observation of RH neutrino currents while the lepton number violating 0νββ decay is not observed would indicate this scenario.

If other operators such as scalar currents are considered, interference can be sizeable and even larger effects may be seen, although existing limits such as those from single β decay are expected to be more restrictive as well. As we have demonstrated in the example of exotic RH vector currents, while the search for 0νββ decay and thus the Majorana nature of neutrinos is the main motivation, the properties of the second-order SM process of 2νββ decay can also contain potential hints for New Physics.


FFD acknowledges support from the UK Science and Technology Facilities Council (STFC) via a Consolidated Grant (Reference ST/P00072X/1). FFD is grateful to HEPHY Vienna and together with LG to the Comenius University Bratislava where part of the work was completed. FFD would also like to thank Morten Sode for collaboration in the early stages of the project. FŠ acknowledges support by the VEGA Grant Agency of the Slovak Republic under Contract No. 1/0607/20 and by the Ministry of Education, Youth and Sports of the Czech Republic under
the INAFYM Grant No. CZ.02.1.01/0.0/0.0/16_019/0000766.

