Searching for New Physics in Two-Neutrino Double Beta Decay

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 has become available. Although these events are regarded and studied mostly as the background of neutrinoless double beta decay, they can be also 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.


I. 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 2νββ 1/2 ∼ 10 19 yr and longer [1].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 0νββ 1/2 ∼ (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 [2][3][4].
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 136 Xe with a high statistics [5], 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 96 Zr [6], 150 Nd [7], 48 Ca [8], 82 Se [9] and especially 100 Mo [10], 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 [11].
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 chargedcurrent operators of the form G F (ēO 1 ν)(ūO 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 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 [12], giving rise to limits of the order 10 −4 − 10 −2 , depending on the Lorentz structure and chirality of the fields involved.
In this paper, we will specifically concentrate on exotic right-handed vector 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 right-handed currents with neutrinos but in the absence of a sterile neutrino state would necessarily violate lepton number.

II. EXOTIC CHARGED-CURRENT INTERACTIONS
We are interested in processes where right-and left-handed electrons are emitted considering, as a first approach, only (V + A) and (V − A) currents.The effective Lagrangian can then be written as with the tree-level Fermi constant G F , the Cabbibo angle θ C , and the leptonic and hadronic )d, respectively.The SM electroweak radiative corrections are understood to be encoded in δ SM and the XY encapsulate new physics effects.We here concentrate on the latter two operators with right-handed 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 [13] and further, exotic particles may participate [14].
In Eq. ( 1), ν is a 4-spinor field of the light electron neutrino, either defined by ν = ν L +ν c L (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 right-handed 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. ( 1) are concerned, this does not play a role in our calculations.If the neutrino in Eq. ( 1) is a Majorana particle, the operators associated with LR and RR violate total lepton number by two units and they will give rise to extra contributions to 0νββ decay [15].In this case, severe limits are set by 0νββ decay searches of the order LR 3 × 10 −9 , RR 6 × 10 −7 [2].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 right-handed current interactions in Eq. ( 1) 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 [16] with unbroken lepton number [17].The observation of the effect of right-handed neutrino operators without the observation of lepton number violation would thus 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, LL , RL 5 × 10 −4 , LR , RR 6 × 10 −2 [12,13].The limits on the right-handed neutrino 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 [18], pp → eX + MET, may also be used to constrain the above operators, LL 4.5 × 10 −3 , RR 2.2 × 10 −3 [19].While the constraints are stringent and the sensitivity is expected to improve to LL ≈ 10 −5 [20], 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 right-handed current mediators at the LHC [21], 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 XY ≈ 5 × 10 −4 − 5 × 10 −5 .An indirect limit on LR can be set from the fact that the associated operator contributes to the Dirac neutrino mass at the second loop order [22].Using current direct neutrino mass bounds this results in LR 10 −2 [23].

III. DECAY RATE AND DISTRIBUTIONS
We have calculated the differential rate of 2νββ decay under the presence of the exotic interactions in Eq. (1).Because 2νββ decay is possible in the SM, arising in second order perturbation theory of the first term in Eq. ( 1), interference between SM and exotic contributions is in principle possible.In general, the amplitude of 2νββ decay is calculated as a coherent sum of the Feynman diagrams in decay energy release Q.For light eV-scale neutrinos it is thus utterly negligible. 1 Contributions to second-order ∝ 2 XR 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 2νββ transition can thus be written as the incoherent sum where R 2ν SM is the matrix element for standard SM 2νββ decay and R 2ν is the exotic contribution.As discussed in the Appendix, the latter may be expressed as 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 ), 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 ∆E n (J are the energies of the intermediate nuclear states with respect to the initial ground state.Overall energy conservation is implied, 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. [11], the calculation of the 2νββ decay rate and distributions is detailed in the Appendix.We use nuclear matrix elements in the QRPA formalism from Ref. [11] 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 LR and RR are identical; both cases yield the same rates and distributions.As a result, we calculate the full differential 2νββ decay rate in a given 0 + → 0 + double beta decaying isotope with respect to the two electron energies 0 ≤ E e 1 ,e 2 ≤ Q + m e and the angle 0 ≤ θ ≤ π between the emitted electrons, which may be written as Because interference effects between the SM and the right-handed current diagram are negligible, the differential rate is simply the incoherent sum of both.Specifically, for 100 Mo the total decay rate Γ 2ν = ln 2/T 2ν 1/2 associated with the 2νββ half-life T 2ν 1/2 may be approximated as Γ 2ν ≈ Γ 2ν SM (1 + 6.11Given the uncertainties in nuclear matrix elements, the change of the total decay rate due to the presence of a right-handed 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, 100 Mo, it is shown in Fig. 2 (left), comparing the 2νββ decay distributions in the SM case (dashed) and for the exotic leptonic right-handed 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 doublydifferential energy spectrum shown in Fig. A.1 in the Appendix is in principle measurable.
Alternatively, the spectrum with respect to the kinetic energy of a single electron is shown in Fig. 2 (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 case.
This behaviour can be traced to the kinematic differences.In the presence of a righthanded lepton current in 2νββ decay, the electrons are preferably emitted collinearly and the energy-dependent correlation factor is always κ 2ν > 0 whereas in the SM case the electrons are preferably emitted back-to-back with κ 2ν SM < 0, cf.
with the angular correlation factor K 2ν .For 100 Mo, we calculate K 2ν SM = −0.63 in the SM and K 2ν = +0.37 for the exotic contribution.This deviation is clearly the most striking consequence of a right-handed lepton current on 2νββ decay.The general dependence of K 2ν on XR is displayed in Fig. A.4 of the Appendix.In the limit XR 1, the angular correlation factor K 2ν under the presence of the SM and an exotic right-handed current is given by For 100 Mo, the coefficient α turns out to be α ≈ 6.1.Despite the small correction expected if XR ≈ 10 −2 as indicated in current bounds, searches for 2νββ 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 2νββ decay events with the electrons being emitted with a relative angle As shown, the asymmetry is simply related to the angular correlation factor K 2ν and it is clearly independent of the overall 2νββ 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 = −0.63 ± 0.0027.No significant deviation from this SM expectation should then constrain XR 2.7 × 10 −2 at 90% confidence level.This would already improve on the single β decay constraint of 6×10 −2 [13].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 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 [10,24].
As detailed in the Appendix, our results were calculated within the nuclear structure framework of the pn-QRPA with partial isospin restoration [11].We nevertheless expect no significant difference for the angular correlation in other nuclear matrix element calculations as it is largely insensitive to the nuclear part of the amplitude.On the other hand, as can be seen in Fig. 2 (right), the effect of right-handed currents is similar to that of varying the contribution of intermediate nuclear states as described in [11].Compare for example with Fig. 4 in [10], which 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.This has the benefit that experimental searches for these effects, such as described in [5,10,25], could be adapted to our scenario.

IV. CONCLUSIONS
Nuclear double beta decay with the emission of two neutrinos and nothing else was proposed over 80 years ago [26] 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 2νββ 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 2νββ decay can be competitive to existing limits.This provides a motivation to utilize the already large set of observed 2νββ 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 right-handed 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 right-handed 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 right-handed 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 right-handed 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 right-handed 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.

A. CALCULATION OF TWO-NEUTRINO DOUBLE BETA DECAY
The 2νββ decay rate can be calculated using the expression [27] The quantities A 2ν and B 2ν in Eq. (A.2), generally functions of the electron energies, are determined by integrating over the neutrino phase space where we used E ν2 = E i − E f − E e 1 − E e 2 − E ν1 due to energy conservation.In turn, the quantities A 2ν and B 2ν , generally functions of the electron and neutrino energies, are calculated below using the nuclear and leptonic matrix elements.In the context of our calculation, they may be expressed as expanded in terms of the small exotic coupling coefficients RX = RL , RR of the exotic right-handed currents in Eq. ( 1) of the main text.Here, we assume that only one exotic contribution is present at a given time.In the following, we will also take the exotic coupling coefficient to be real.The zero order terms A 2ν SM and B 2ν SM correspond to the standard 2νββ decay mechanism, cf.In principle, the chirality of the quark current involved in the considered effective interaction ( LR or RR ) does affect the resulting 0νββ decay contribution.However, this difference would manifest only as an opposite sign of the Gamow-Teller part of the amplitude.Hence, in the well-motivated approximation of a vanishing double Fermi NME, which we will apply later on, the resulting expressions for the decay rate and the angular correlation of the emitted electrons will not depend on chirality of the considered quark current.Thus, our conclusions will be generally applicable to both effective couplings LR and RR , collectively denoted as RX .
A.1 First-order contribution in the exotic coupling We here describe the calculation of 2νββ under the presence of exotic right-handed vector currents.We follow the formalism in [27] and adapt it to our scenario.Considering the Lagrangian in Eq. ( 1) of the main text, 2νββ decay occurs at second order of the perturbative expansion; namely, the matrix element is in general given by and it contains both the SM contribution and the exotic contribution proportional to XR .
Further, T denotes the time-ordered product and the initial and final states are composed of the decaying nucleus |i and the final nucleus |f together with the emitted electrons e 1,2 and antineutrinos ν1,2 .The integrations are over the space-time coordinates x and y of the two interactions involved.
We here concentrate on the case with one SM interaction and one exotic right-handed interaction.The matrix element can then be expressed as Writing the time dependence of the wave functions and currents explicitly allows performing the integration over time variables x 0 and y 0 with the result Here, E x denotes the energy of particle x or respective nucleus, and the delta function guaranteeing energy conservation and energy denominator appear as a result of the integration over the time components.Now we employ two approximations.First, we take the non-relativistic expansion of the nuclear currents, where we ignored the induced currents for their negligible contribution.Here, g V and g A are the vector and effective axial-vector coupling constants, respectively.Second, for the purpose of a factorization of nuclear matrix elements and phase space integral calculation we assume a standard approximation in which lepton wave functions are replaced with their values ψ(p) = ψ(p, R) at the nuclear surface.For a 0 + → 0 + , ground state to ground state, transition we get The sign of the g 2 A -proportional part depends on the chirality X of the quark current appearing in the exotic effective interaction XR -it is negative (positive) for a left-handed (right-handed) quark current.Further, we specify the angular momentum and parity of the nuclear states with |0 + i , |0 + f denoting the 0 + ground states of the initial and final even-even nuclei, respectively.The intermediate nucleus states are denoted |0 + n (|1 + n ) for all possible levels n with angular momentum and parity J π = 0 + (J π = 1 + ) with the corresponding energy E n .The isospin-raising operators for a given nucleon j is denoted as τ + j , summed over all nucleons in the initial and final states.Likewise, σ j stands for the spin operator of nucleon j.
By writing out explicitly all terms in Eq. (A.11) we find where we define Fermi and Gamow-Teller nuclear matrix elements Here, we conventionally put the electron mass m e to make the NMEs dimensionless.The lepton energies enter in Eq. (A.14) through the terms .15)which range between −Q/2 ≤ ε K,L ≤ Q/2.For 2νββ decay with energetically forbidden transitions to the intermediate states, We first focus on the leptonic part of the total matrix element.to all four permuted terms in Eq. (A.13), and using the identity γ α γ µ γ α = −2γ µ one obtains the reaction matrix element in the following form In the following, we consider the S 1/2 spherical wave approximation for the outgoing electrons, i.e.
where χ s is a two-component spinor, pe = p e /|p e | stands for the direction of the electron momentum and g −1 (E e ) and f +1 (E e ) are the radial electron wave functions depending on the electron energy E e and evaluated at the nucleus' surface, i.e. at distance R from the centre of the nucleus.On the other hand, as neutrinos do not feel the electromagnetic potential of the nucleus, they are considered to be plane waves in long-wave approximation, We now take the square of the absolute value of the matrix element in Eq. (A.18), using the wave functions in Eqs.(A. 19) and (A.20), and sum over the spins.After evaluating those and keeping only the terms which do not vanish when integrating over neutrino momenta, we are left with a somewhat lengthy expression, for the Gamma function.The results obtained using this approximation do not deviate from the more accurate radial electron wave functions coming from the numerical solution of the Dirac equation by more than ∼ 10 − 15% [28].
Equation (A.21) can now be mapped to the coefficients A 2ν and B 2ν entering the differential decay rate Eq.(A.2) of the process.For the terms independent of the scalar product of the spatial electron momenta this gives Here, the dependence on the electron radial wave functions has been made explicit.Likewise, the terms proportional to pe 1 • pe 2 = cos θ combine to give Electron energy total and single electron energy: The main observable in double beta decay experiments is the distribution with respect to the total kinetic energy of the two electrons, dΓ 2ν /dE K , E K = E e 1 + E e 2 − 2m e .In experiments where the individual electrons can be tracked and their energies measured individually, the single electron energy distribution dΓ 2ν /dE e 1 (by symmetry, the distribution with respect to the second electron is identical) and the double differential distribution dΓ 2ν /(dE e 1 dE e 2 ) are relevant as well.The bottom panels show the relative deviation of the exotic distribution from the SM case.for both the SM contribution and the exotic contribution.The correlation κ 2ν is negative for all energies in the SM case, thus indicating that the electrons are preferably emitted back-to-back.On the contrary, the correlation is positive for the exotic scenario meaning that the electrons prefer to escape from the nucleus in the same direction.
Angular correlation factor and total decay rate: One can further proceed and integrate over the electron energies which yields the general form

FIG. 1 .
FIG. 1. Feynman diagrams for ordinary 2νββ decay via the second-order transition through the SM V − A interaction with strength given by the Fermi constant G F (left), a transition involving one exotic interaction XR G F with a V + A lepton current of the form (ē R O 1 ν)(ūO 2 d) (center) and a second-order transition through the same exotic interaction (right).

Fig. 1 .
To lowest order in XR , exotic effects occur arising from the interference of the SM diagram Fig. 1 (left) and the diagram in the center.Due to the right-handed 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 2νββ

FIG. 2 .
FIG. 2. Left: Normalized 2νββ decay distributions with respect to the total kinetic energy E K = E e 1 + E e 2 − 2m e of the emitted electrons for standard 2νββ decay through SM V − A currents (dashed) and a pure right-handed lepton current (solid).Right: Normalized 2νββ decay distributions with respect to the energy of a single electron in the same scenarios.Both plots are for the isotope 100 Mo and the energies are normalized to the Q value.The bottom panels show the relative deviation of the exotic distribution from the SM case.
Fig. A.2 in the Appendix.Integrating over the electron energies one arrives at the angular distribution,

Fig. 1 (
left) in the main text.The terms A 2ν and B 2ν quadratic in XR arise from the exotic 2νββ decay mechanism involving one right-handed vector lepton current, cf.Fig. 1 (center) 2 .Finally, the terms A 2ν SM and B 2ν SM linear in XR correspond to the interference between the two mechanisms.Because of the different electron and neutrino chiralities involved in the standard V − A and the exotic V + A currents, the interference is suppressed as ≈ m ν /Q and for | XR | m ν /Q, the linear terms are negligible.This is certainly the case for the emission of light active neutrinos with m ν 0.1 eV.

8 )
where G β = G F cos Θ C and P(a, b) is the permutation operator interchanging the particles a and b.Further, ψ(p, x) stands for the electron or antineutrino wave function with four momentum p = (E, p) and position x = (x 0 , x), J µ (x) X denotes the nuclear current with chirality X and |n is the intermediate nucleus state.For calculating the matrix element of the SM contribution, one would only need to replace in the above expression the righthanded projector (1 + γ 5 ) in the first lepton current by a left-handed one and follow the subsequent derivation in an analogous manner.

1 .
FIG. A.1.Normalized double energy distributions as functions of electron energies for SM 2νββ decay (left) and for the exotic scenario incorporating a right-handed lepton current (right).Both plots are for 100 Mo and the energies are normalized to the Q value.

2 .
FIG. A.2. Angular correlation κ 2ν as a function of the electron kinetic energies for SM 2νββ decay (left) and for the exotic scenario incorporating a right-handed lepton current (right).Both plots are for 100 Mo and the energies are normalized to the Q value.

FIG. A. 3 .
FIG. A.3.Left: Normalized 2νββ total electron kinetic energy decay distributions for SM 2νββ (dashed) and the right-handed lepton current case (solid), for the isotopes 76 Ge, 82 Se and 136 Xe (top to bottom).Right: Likewise, the normalized single electron energy 2νββ decay distributions.
given by Eq. (A.4) applied on the SM, exotic-SM interference and exotic contributions.The resulting angular correlation is plotted in Fig. A.2 e 1 dΩ e 2 dΩ ν1 dΩ ν2 , (A.1) where E i , E f , E e i = p 2 e i + m 2 e and E νi = p 2 νi + m 2 ν (i = 1, 2) denote the energies of initial and final nuclei, electrons and antineutrinos, respectively.The magnitudes of the associated spatial momenta are p e i = |p e i | and p νi = |p νi | and m e and m ν denote the electron and neutrino masses.The phase space differentials are dΩ e 1 = d 3 p e 1 /(2π) 3 , etc.After integrating over the phase space of the outgoing neutrinos, the resulting differential 2νββ decay rate can be generally written in terms of the energies 0 ≤ E e 1 , E e 2 ≤ Q + m e of the two outgoing electrons, with Q = E i − E f − 2m e , and the angle 0 ≤ θ ≤ π between the electron momenta p e 1 and p e 2 as [27] dΓ 2ν dE e 1 dE e 2 dcos θ = c 2ν A 2ν + B 2ν cos θ p e 1 E e 1 p e 2 E e 2 , Q+me medE e 1 dE e 2 δ(E K − E e 1 − E e 2 + 2m e ) FIG. A.4. Angular correlation factor K 2ν for 100 Mo as a function of the the new-physics coupling 2ν (E e 1 , E e 2 ) = B 2ν A 2ν p e 1 p e 2 E e 1 E e 2 = B 2ν SM + 2 LR B 2ν SM + 2 LR B 2ν A 2ν SM + 2 LR A 2ν SM + 2 LR A 2ν p e 1 p e 2 E e 1 E e 2 , (A.39) with A 2ν SM , A 2ν SM , A 2ν , B 2ν SM , B 2ν SM , B 2ν XR .byκ 1 − K 2ν cos θ , (A.40)where Γ 2ν is the total 2νββ decay rate andK 2ν = Λ 2ν /Γ 2ν is the angular correlation factor, −E f −me me dE e 1 p e 1 E e 1 E i −E f −Ee 1 me dE e 2 p e 2 E e 2In the case of 100 Mo, the total 2νββ decay rate may be approximately expressed as