Two-Neutrino Double Beta Decaywith Sterile Neutrinos

Patrick D. Boltona Frank F. Deppischa Lukáš Gráfb Fedor Šimkovicc

aDepartment of Physics and Astronomy, University College London,London WC1E 6BT, United Kingdom

bMax-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, GermanycBLTP, JINR, 141980 Dubna, RussiaComenius University, Mlynská dolina F1, SK–842 48 Bratislava, SlovakiaIEAP CTU, 128–00 Prague, Czech Republic

E-mail: patrick.bolton.17@ucl.ac.uk, f.deppisch@ucl.ac.uk,lukas.graf@mpi-hd.mpg.de, fedor.simkovic@fmph.uniba.sk

Abstract: Usually considered a background for experimental searches for the hypothet-ical neutrinoless double beta decay process, two-neutrino double beta decay neverthelessprovides a complementary probe of physics beyond the Standard Model. In this paper weinvestigate how the presence of a sterile neutrino, coupled to the Standard Model eithervia a left-handed or right-handed current, affects the energy distribution and angular cor-relation of the outgoing electrons in two-neutrino double beta decay. We pay particularattention on the behaviour of the energy distribution at the kinematic endpoint and weestimate the current limits on the active-sterile mixing and effective right-handed couplingusing current experimental data as a function of the sterile neutrino mass. We also inves-tigate the sensitivities of future experiments. Our results complement the correspondingconstraints on sterile neutrinos from single beta decay measurements in the 0.1 – 10 MeVmass range.

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Contents

1 Introduction 1

2 Effective Interactions with Sterile Neutrinos 3

3 Constraints on Sterile Neutrinos 33.1 Neutrinoless Double Beta Decay 53.2 Beta Decay 53.3 Sterile Neutrino Decays 63.4 Cosmological and Astrophysical Constraints 6

4 Double Beta Decay Rate with a Sterile Neutrino 74.1 Purely Left-Handed Currents 84.2 Contribution with a Right-Handed Current 104.3 Decay Distributions and Total Rate 11

5 Constraints on Sterile Neutrino Parameters 145.1 Statistical Procedure 155.2 Results 19

6 Conclusions 22

1 Introduction

Sterile neutrinos are among the most sought-after candidates of exotic particles. The mainmotivation for their existence is the fact that the Standard Model (SM) does not con-tain right-handed (RH) counterparts of the left-handed neutrino states participating inelectroweak interactions, in contrast to the quarks and charged leptons. Their absence ispurely because they are required to be singlets under the weak SU(2)L and have zero weakhypercharge if they are to participate in a Yukawa interaction with the left-handed neutrinostates and the SM Higgs. They are thus truly sterile with respect to the SM gauge groupand in this paradigm can only manifest themselves through an admixture with the activeneutrinos. Thus, sterile neutrinos can in fact be considered to be any exotic fermion thatis uncharged under the SM gauge interactions; unless protected by some new symmetrythey will mix with the active neutrinos as described above. Hence, sterile neutrinos are alsooften referred to as heavy neutral leptons.

An important feature of their mixing with the active neutrinos is the resulting impacton the light neutrino masses. Neutrino oscillations [1] imply that the active neutrinos havesmall but non-zero masses. By adding a SM-singlet, RH neutrino field νR per generationto the SM, neutrinos can become massive. A so-called Dirac mass term can be generated

– 1 –

through the Yukawa interaction with the SM Higgs, though the coupling required is tinywith effects unmeasurable experimentally. In any case, the sterile states will be allowed toacquire a so-called Majorana mass, unless protected by lepton number conserving symmetry,modifying the spectrum and nature of neutrinos considerably. This of course refers to thewell-known type I seesaw mechanism [2–6]. The sterile neutrinos were initially considered tobe very heavy (mN ∼ 1014 GeV) in order to generate the correct light neutrino masses, butthere is now a strong theoretical and experimental incentive to consider sterile neutrinos ataccessible energies. Fig. 1 summarises the current constraints on the active-sterile mixingstrength |VeN |2 in the regime 1 eV < mN < 10 TeV derived from numerous experiments.The most stringent limits from fixed target and collider experiments can be found in themass range 1 GeV < mN < 100 GeV, a region motivated by leptogenesis models.

Lighter sterile neutrino masses, while challenging to accommodate due to the con-straints from astrophysics, are still of interest, especially around mN ∼ 10 keV where sterileneutrinos may act as warm dark matter. In the regime 10 eV < mN < 1 MeV, nuclear betadecays are currently the only laboratory-based experimental method able to probe sterileneutrinos. Neutrinoless double beta (0νββ) decay is an exception, setting stringent limitsover the whole range in Fig. 1, but only if the sterile neutrinos are Majorana fermions – forsterile Dirac neutrinos or quasi-Dirac neutrinos with relative splittings ∆mN/mN . 10−4

[7], the constraints vanish or become weaker respectively. In addition, if the sterile neutri-nos in question are wholly responsible for giving mass to the active neutrinos via the type Iseesaw mechanism, the contributions from active and sterile neutrinos to 0νββ decay canceleach other, see Sec. 3.1. Thus, especially around mN ∼ 1 MeV, the current constraints arerather weak, of the order |VeN |2 . few ×10−3. As mentioned above, the constraints arisefrom searches for kinks in the electron energy spectrum and measurements of the ft valueof various beta decay isotopes, see Sec. 3.2 for a brief review.

This weakening of limits motivates the use of novel methods to constrain the active-sterile mixing in this mass regime. In this work we assess the potential of 0νββ decayexperiments being sensitive to kinks in the background two-neutrino double beta (2νββ)decay spectrum caused by the presence of sterile neutrinos in the final state with massesmN . 1 MeV. This is fully analogous to the corresponding searches in single beta decaysbut 2νββ decaying isotopes typically have Q values of a few MeV and are thus expectedprobe sterile neutrinos in such a mass range. The 2νββ decay process is of course veryrare so it may at first seem difficult to achieve high enough statistics. While 2νββ decayis indeed not expected to improve the limits considerably, the 2νββ decays spectrum willbe measured to high precision in several isotopes as 0νββ decay is searched for in ongoingand future experiments. The relevant data to look for sterile neutrinos in 2νββ decays willbe available, which, generally speaking, can be used to look for signs of new physics in itsown right [8, 9].

In addition to a truly sterile neutrino, i.e. one that inherits the SM charged-currentFermi interaction albeit suppressed by the active-sterile mixing, we also consider RH currentinteractions of the ‘sterile’ neutrino, e.g. arising in left-right symmetric models. Suchinteractions change the angular distribution of the electrons emitted in 2νββ decay [8]. Weparametrise all interactions in terms of effective operators of the SM with a light sterile

– 2 –

neutrino, suitable in 2νββ decays with characteristic energies of . 10 MeV.This paper is organised as follows. In Sec. 2 we introduce the effective operators

relevant for our discussions. In Sec. 3 we briefly review the current limits on the active-sterile mixing squared |VeN |2 with a focus on the mass regimemN ∼ 1 MeV. The calculationof the 2νββ decay spectrum with the emission of one sterile neutrino is described in Sec. 4.Sec. 5 introduces our statistical procedure and presents the estimated current limits andprospective future sensitivities from sterile neutrino searches in 2νββ decay as our results.We conclude in Sec. 6.

2 Effective Interactions with Sterile Neutrinos

We consider the SM with the addition of a gauge singlet fermion N , i.e. the sterile neu-trino. As we consider the second-order weak process of 2νββ decay, we restrict ourselvesto the first generation of SM fermions. For processes with energies 100 GeV we candescribe the relevant weak processes using the effective SM Fermi interaction. The sterileneutrino inherits the Fermi interaction, but is suppressed by the active-sterile mixing VeN .In addition, we allow the sterile neutrino to participate in exotic RH V + A interactions.The effective Lagrangian taking into account the above takes the form

L =GF cos θC√

2

[(1 + δSM)jµLJLµ + VeNj

NµL JLµ + εLRj

NµR JLµ + εRRj

NµR JRµ

]+ h.c., (2.1)

with the tree-level Fermi constant GF , the Cabbibo angle θC , and the leptonic and hadroniccurrents jµL = eγµ(1 − γ5)ν, jNµL,R = eγµ(1 ∓ γ5)N and JµL,R = uγµ(1 ∓ γ5)d, respectively.The SM electroweak radiative corrections are encoded in δSM . The active-sterile mixing isVeN and the εXY encapsulate effects from integrating out new physics giving rise to V +A

currents of the sterile neutrino. We neglect any further effective operators, such as exoticcontributions to the SM Fermi interaction and RH currents with the active neutrino [8].

In Eq. (2.1), ν and N are 4-spinor fields of the light electron neutrino and the sterileneutrino. They are either defined to be Majorana fermions, ν = νL+νcL, N = N c

R+NR (i.e.a Majorana spinor constructed from the left-handed Weyl spinor and its charge-conjugate)or Dirac fermions ν = νL+νR, N = NR+NL (a Dirac spinor constructed from two differentWeyl fields). The calculation of 2νββ decay is not affected by this, i.e. it is insensitive tothe Dirac versus Majorana character. If the neutrinos are Majorana the constraints from0νββ decay must be considered.

3 Constraints on Sterile Neutrinos

In this section we review the constraints on the active-sterile mixing strength squared|VeN |2 as a function of the sterile neutrino mass mN . We mainly concentrate on limitsin the 0.1 MeV < mN < 3 MeV mass range. This is because 2νββ decay measurementsare only sensitive to sterile neutrino masses below the Q value of the 2νββ decay process,which is of order Q ∼ 1 − 3 MeV for the isotopes of interest. The relevant constraints inthis range come from the non-observation of 0νββ decay, single beta decay spectra, sterile

– 3 –

10−9 10−6 10−3 1 103

mN [GeV]

10−12

10−10

10−8

10−6

10−4

10−2

1

|VeN|2

NA62

Super−K

Rovno

Bugey

20 F

64 Cu144 Ce−144 Pr IH

EP−

JINR

Belle

NA3

CHARM

PS191

PIENU

Borexino

45 Ca

35 S63 Ni

3 H

187 Re

CMB +BAO +H0

BBN

T2K

L3

DELPHI

CMS

ATLAS

EWPD

CMB

PROSPECT

NEOS

SK+IC +DC

X−ray Seesaw

Supernovae

BESIII

LN

VD

ecay

s

Higgs

0νββ

Figure 1: Constraints on the squared mixing strength |VeN |2 of the sterile neutrino withthe electron neutrino as a function of its mass mN . For simplicity we assume νe to be theonly active neutrino. The shaded regions are excluded by the searches and observationsas labelled. They are discussed in Sec. 4 of Ref. [10]. The band labelled ‘0νββ’ denotesthe uncertainty on the current upper limit from 0νββ decay searches on a Majorana sterileneutrino. The diagonal black-dotted line labelled ‘Seesaw’ indicates the canonical seesawrelation |VeN |2 = mνe/mN with mνe = 0.05 eV.

neutrino decays and cosmological probes. We will see that the same constraints also applybroadly to the RH current couplings |εLR|2 and |εRR|2.

As an overview we show in Fig. 1 the existing |VeN |2 constraints over the mass range1 eV < mN < 10 TeV; for further information on each labelled constraint see Sec. 4 ofRef. [10] and references therein. It is interesting to note the relative weakness of the upperlimits from single beta decay experiments in the range 0.1 MeV < mN < 3 MeV. Mixingstrengths are nonetheless excluded down to |VeN |2 . 10−7−10−6 and |VeN |2 . 10−14−10−11

by 0νββ decay and cosmological probes, respectively. It is crucial though to emphasisethat the former constraints are model-dependent and can be avoided if neutrinos are Diracfermions or if the sterile neutrinos are responsible for the light neutrino mass generation.Cosmological constraints rely on modelling of the early universe and can be avoided inextended scenarios where the sterile neutrinos have exotic interactions with a dark sector[11–14]. This therefore motivates looking at the sensitivities of current and future 2νββ

decay measurements but we first look at the existing constraints within the region of interestin more detail.

– 4 –

3.1 Neutrinoless Double Beta Decay

If we consider the active and sterile neutrinos to be purely Dirac fermions, lepton numberis conserved and 0νββ decay is forbidden. Searches for this decay will thus not provideconstraints on the active-sterile mixing of Dirac neutrinos.

In the Majorana case, if nS sterile neutrinos are added to the SM with masses mNi andactive-sterile mixing strengths VeNi (we assume for simplicity a single active state νe), theinverse of the half-life T 0ν

1/2 for the 0νββ decay process can be written using the interpolatingformula

1

T 0ν1/2

= G0νg4A|M0ν |2

∣∣∣∣∣mνe

me+

⟨p2⟩

me

nS∑i=1

V 2eNi

mNi

〈p2〉+m2Ni

∣∣∣∣∣2

. (3.1)

Here G0ν is the phase space factor, gA is the axial vector coupling, M0ν is the light neu-trino exchange nuclear matrix element and

⟨p2⟩is the average momentum transfer of the

process [15, 16]. By considering a single sterile neutrino with mass mN and neglecting thecontribution from the active neutrinos, the constraint in Fig. 1 is derived using the currentexperimental bounds.

If the heavy states are related to the light state by a seesaw relation, then

(Mν)11 = mνe +

nS∑i=1

V 2eNimNi = 0 , (3.2)

must be satisfied. Thus, if the sterile states are lighter than the 0νββ decay momentumtransfer, mNi

⟨p2⟩, the 0νββ decay rate vanishes and the corresponding constraint in

Fig. 1 disappears. Sterile neutrinos have been discussed in the context of 0νββ decay indetail in Refs. [10, 17–19].

3.2 Beta Decay

Electron neutrinos are produced in the beta decays of unstable isotopes via the LH charged-current interaction. If the active-sterile mixing strength |VeN |2 or RH couplings |εLR|2,|εRR|2 are non-zero, sterile neutrinos can be produced if their masses are smaller than theQ value of the process. For a large enough mN the emission results in a distortion or ‘kink’in the beta decay spectrum and associated Kurie plot.

The beta decay spectrum with respect to the kinetic energy of the emitted electron canbe written for a single sterile neutrino with mixing as the incoherent sum

dΓβ

dEe=(1− |VeN |2

) dΓν(0)

dEe+ |VeN |2

dΓν(mN )

dEe, (3.3)

where we neglect the light neutrino masses in the standard contribution. Due to unitarity,the contribution from the light neutrinos is reduced by the active-sterile mixing strength.The sterile neutrino contribution gives rise to a kink in the spectrum of relative size |VeN |2

and at electron energies Ee = Q−mN . Alternatively, in the case the sterile neutrinos areproduced by a RH current, the SM contribution is no longer reduced as a result of unitarity.

– 5 –

This weakens the upper limits on |εLR|2 and |εRR|2 compared to |VeN |2, though the effectis negligible for upper bounds below 10−2.

Kink searches have been conducted for a variety of isotopes with different Q values,making them sensitive to a range of sterile neutrino masses. Shown in Fig. 1 are upperlimits from the isotopes 3H [20–23], 20F [24], 35S [25], 45Ca [26], 63Ni [27], 64Cu [28], 144Ce–144Pr [29] and 187Re [30], assuming there to be a single sterile state. With smaller Q values,3H and 187Re provide constraints over the range 1 eV < mN < 1 keV. It can be seen that45Ca, 64Cu, 144Ce–144Pr and 20F in the mass range of interest provide slightly weaker upperbounds (between 10−3 and 10−2) compared to 63Ni and 35S at lower masses.

3.3 Sterile Neutrino Decays

A sterile neutrino produced in the beta decay of a neutron-rich isotope in a reactor or a lightelement in the sun can decay before detection via the channels N → ννν and N → e+e−ν.The former channel is mediated by a neutral current and the latter via either a neutral orcharged current. The latter also requires the sterile neutrino mass to be mN > 2me. Attree-level (in the single-generation case) the total decay rate is given approximately by

Γtot ≈ 2×G2F

96π3|VeN |2m5

N , (3.4)

where the factor of 2 is present in the Majorana case. For RH currents, the factor |VeN |2 isreplaced by |εLR|2 or |εRR|2.

Reactor experiments with neutrino energies ∼ 10 MeV are sensitive to sterile neutrinoswith masses in the range 1 MeV < mN < 10 MeV. Limits have been set by searches at theRovno [31] and Bugey [32] reactors. Sterile neutrino decays were also searched for by theBorexino experiment [33] which was sensitive to heavy neutrinos with masses up to 14 MeVproduced in the decays of solar 8B nuclei. Borexino enforces the relatively stringent limit|VeN |2 . 10−6 − 10−5 for mN ∼ 10 MeV.

3.4 Cosmological and Astrophysical Constraints

The presence of sterile states with mixing strengths |V`N |2 (and/or the presence of RHcurrents) has wide-ranging consequences for early-universe observables. These include theabundances of light nuclei formed during Big Bang Nucleosynthesis (BBN), temperatureanisotropies in the Cosmic Microwave Background (CMB) radiation and the large-scaleclustering of galaxies [34]. Deviations from the standard smooth, isotropic backgroundevolution (and perturbations around this background) impose severe constraints – the regionbetween the grey lines labelled CMB+BAO+H0 (an upper limit) and BBN (a lower limit) isexcluded. These limits are highly sensitive however to the production and decay mechanismof the sterile state and can be relaxed in certain models.

The main constraint to consider in the 0.1 MeV < mN < 3 MeV mass range is the upperlimit labelled CMB+BAO+H0. Via the active-sterile mixing or RH current, sterile statesare populated in the early-universe and they decouple when the Hubble expansion overcomesthe interaction rate with the SM particles. It is then possible for these states to decay atlater times to produce non-thermally distributed active neutrinos, modifying the amount of

– 6 –

extra radiation measured at recombination, ∆Neff , beyond the usual value including activeneutrino oscillations, Neff ' 3.046. Useful probes include the CMB shift parameter RCMB,the first peak of the Baryon Acoustic Oscillations (BAO) and the Hubble parameter H(z)

inferred from type Ia supernovae, BAO and Lyman-α data. These exclude values of mN

and |VeN |2 corresponding to lifetimes up to the present day, where the condition that Ndoes not make up more than the observed matter density Ωsterile < ΩDM ≈ 0.12h−2 alsoapplies. This constraint can be evaded in exotic models [11–14], for example those thatinject additional entropy and dilute the dark matter (DM) energy density.

4 Double Beta Decay Rate with a Sterile Neutrino

Considering one sterile neutrino N with mass mN < Qββ . few MeV and a SM charged-current as in Eq. (2.1) with additional suppression by the active-sterile mixing strengthVeN allows for the possibility that in 2νββ decay one N is emitted (νNββ) instead of aνe (we assume that N is long-lived and does not decay within the detector, thus beinginvisible). The final state is different from the standard 2νββ decay and thus there is nointerference between νNββ and 2νββ. There is also no anti-symmetrisation with respect tothe two different neutrinos in νNββ. Moreover, a RH lepton current can be also assumedto be associated with the emission of the sterile neutrino, which further affects the 2νββ

observables, mainly the angular correlation of the outgoing electrons.In order to write down expressions for the 2νββ and νNββ decay rates, including the

possibility of RH currents, let us start with the general expression [35]

dΓ = 2(2− δνiνj )πδ(Ee1 + Ee2 + Eν1 + Eν2 + Ef − Ei)∑spins

|R2ν |2dΩe1dΩe2dΩν1dΩν2 ,

(4.1)

where Ei, Ef , Eei =√p2ei +m2

e and Eνi =√p2νi +m2

νi (i = 1, 2) denote the energies ofinitial and final nuclei, electrons and antineutrinos, respectively. The magnitudes of theassociated spatial momenta are pei = |pei | and pνi = |pνi | and me and mνi denote theelectron and neutrino masses. The phase space differentials are dΩe1 = d3pe1/(2π)3, etc..The symmetry factor in Eq. (4.1) is (2− δνiνj ) = 1 if identical neutrinos are being emittedin the process and (2−δνiνj ) = 2 if they are distinguishable, i.e. in the case of νNββ. Here,the amplitude R2ν contains the average contribution from two diagrams with the neutrinosinterchanged, with a relative minus sign if the neutrinos are identical. Note that in ourcalculations we neglect the mass of the light neutrino being emitted and we retain only themass mN of the heavy neutrino.

After integrating over the phase space of the outgoing neutrinos, the resulting differen-tial 2νββ decay rate can be generally written in terms of the energies 0 ≤ Ee1 , Ee2 ≤ Q+me

of the two outgoing electrons, with Q = Ei − Ef − 2me, and the angle 0 ≤ θ ≤ π betweenthe electron momenta pe1 and pe2 as [35]

dΓ2ν

dEe1dEe2dcos θ=c2ν

2

(A2ν +B2ν cos θ

)pe1Ee1pe2Ee2 , (4.2)

– 7 –

where

c2ν = (2− δνiνj )G4βm

9e

8π7, (4.3)

with Gβ = GF cos θC (GF is the Fermi constant and θC is the Cabbibo angle).The quantities A2ν and B2ν in Eq. (4.2), generally functions of the electron energies,

include the integration over the neutrino phase space,

A2ν =

∫ Ei−Ef−Ee1−Ee2

mν1

A2ν√E2ν1 −m2

ν1

√(Ei − Ef − Ee1 − Ee2 − Eν1)2 −m2

ν2

× Eν1(Ei − Ef − Ee1 − Ee2 − Eν1) dEν1 , (4.4)

B2ν =

∫ Ei−Ef−Ee1−Ee2

mν1

B2ν√E2ν1 −m2

ν1

√(Ei − Ef − Ee1 − Ee2 − Eν1)2 −m2

ν2

× Eν1(Ei − Ef − Ee1 − Ee2 − Eν1) dEν1 , (4.5)

where we have used Eν2 = Ei − Ef − Ee1 − Ee2 − Eν1 due to energy conservation andkept the dependence on the neutrino masses, although in the SM case they can be safelyneglected. In turn, the quantities A2ν and B2ν , generally functions of the electron andneutrino energies, are calculated below using the nuclear and leptonic matrix elements.

The rate corresponding to νNββ decay then differs only by the non-negligible massof the sterile neutrino entering the neutrino energy and, most importantly, the integrationbounds. Consequently, the corresponding rate can be obtained from the above by a simplesubstitution ν1 → N , ν2 → ν and neglecting the mass mν . As shown later in this section,in the standard case with only LH lepton currents the quantities A2ν and B2ν do notdepend on neutrino masses; hence, the main effect of the sterile neutrino mass is the shrunkelectron energy distribution given by the effectively smaller Q value, now given by Q =

Ei − Ef − 2me −mN .In our calculations we take the S1/2 spherical wave approximation for the outgoing

electrons, i.e.

ψs(pe) =

(g−1(Ee)χs

f+1(Ee) (σ · pe)χs

). (4.6)

Here, pe = pe/|pe| denotes the direction of the electron momentum, χs is a two-componentspinor and g−1(Ee) and f+1(Ee) stand for the radial electron wave functions depending onthe electron energy Ee. As commonly done, we approximate them with their values at thenucleus’ surface, i.e. at distance R from the centre of the nucleus. The neutrinos, beingneutral, can be simply described as plane waves in the long-wave approximation,

ψ(pν) =

√Eν +mν

2Eν

(χs

(σ·pν)Eν+mν

χs

). (4.7)

4.1 Purely Left-Handed Currents

The standard contribution to 2νββ decay given by the first term in the Lagrangian inEq. (2.1) has been studied in great detail [36, 37]. Sticking to the formalism outlined

– 8 –

above, the decay rate is described by the functions

A2νSM =

1

4

[g2V

(MKF +ML

F

)− g2

A

(MKGT +ML

GT

)]2+

3

4

[g2V

(MKF −ML

F

)+

1

3g2A

(MKGT −ML

GT

)]2× [g2

−1(Ee1) + f21 (Ee1)][g2

−1(Ee2) + f21 (Ee2)] , (4.8)

and

B2νSM =

1

4

[g2V

(MKF +ML

F

)− g2

A

(MKGT +ML

GT

)]2− 1

4

[g2V

(MKF −ML

F

)+

1

3g4A

(MKGT −ML

GT

)]2× 4f1(Ee1)f1(Ee2)g−1(Ee1)g−1(Ee2) , (4.9)

where we define Fermi and Gamow-Teller nuclear matrix elements

MK,LF,GT = me

∑n

MF,GT (n)En − (Ei + Ef )/2

[En − (Ei + Ef )/2]2 − ε2K,L

. (4.10)

The electron massme in the above expression is inserted conventionally to make the nuclearmatrix elements dimensionless. The lepton energies enter in Eq. (4.10) through the terms

εK =1

2(Ee2 + Eν2 − Ee1 − Eν1) , εL =

1

2(Ee1 + Eν2 − Ee2 − Eν1) , (4.11)

which satisfy −Q/2 ≤ εK,L ≤ Q/2. In case of 2νββ decay with energetically forbiddentransitions to the intermediate states, En − Ei > −me, the quantity En − (Ei + Ef )/2 =

Q/2 +me + (En − Ei) is always larger than Q/2.The above expressions may be further simplified using several well-motivated approxi-

mations.

Isospin Invariance: Neglecting the isospin non-conservation in the nucleus, the doubleFermi nuclear matrix elements vanish, i.e. MK

F = MLF = 0. Therefore, Eqs. (4.12) and

(4.13) then respectively acquire the approximate form

A2νSM ≈

1

4g4A

[(MKGT +ML

GT

)2+

1

3

(MKGT −ML

GT

)2]× [g2

−1(Ee1) + f21 (Ee1)][g2

−1(Ee2) + f21 (Ee2)] , (4.12)

and

B2νSM ≈

1

4g4A

[(MKGT +ML

GT

)2+

1

9

(MKGT −ML

GT

)2]× 4f1(Ee1)f1(Ee2)g−1(Ee1)g−1(Ee2) . (4.13)

– 9 –

Nuclear matrix element dependence on lepton energies: If we neglect the depen-dence of nuclear matrix elements on εK,L, the nuclear and leptonic parts can be separatedand we get

A2νSM ≈ g4

AM2GT [g2

−1(Ee1) + f21 (Ee1)][g2

−1(Ee2) + f21 (Ee2)] , (4.14)

B2νSM ≈ g4

AM2GT 4f1(Ee1)f1(Ee2)g−1(Ee1)g−1(Ee2) , (4.15)

with the Gamow-Teller nuclear matrix element now defined as

MGT = me

∑n

〈0+f |∑

m τ+mσm|1+

n 〉〈1+n |∑

m τ+mσm|0+

i 〉En − (Ei + Ef )/2

. (4.16)

A better approximation is obtained by Taylor expansion of the nuclear matrix elementsin the small parameters εK,L [37]. Keeping terms up to the fourth power in εK,L gives

A2νSM ≈ g4

A

[(MGT−1)2 + (ε2K + ε2L)MGT−1MGT−3 +

1

3ε2Kε

2L(MGT−3)2

+ (ε4K + ε4L)

(MGT−1MGT−5 +

1

3(MGT−3)2

)]× [g2

−1(Ee1) + f21 (Ee1)][g2

−1(Ee2) + f21 (Ee2)] , (4.17)

and

B2νSM ≈ g4

A

[(MGT−1)2 + (ε2K + ε2L)MGT−1MGT−3 +

4

9ε2Kε

2L(MGT−3)2

+ (ε4K + ε4L)

(MGT−1MGT−5 +

5

18(MGT−3)2

)]× 4f1(Ee1)f1(Ee2)g−1(Ee1)g−1(Ee2) . (4.18)

Here, the nuclear matrix elements introduced are defined as

MGT−1 = MGT , (4.19)

MGT−3 = m3e

∑n

4MGT (n)

(En − (Ei + Ef )/2)3, (4.20)

MGT−5 = m5e

∑n

16MGT (n)

(En − (Ei + Ef )/2)5. (4.21)

This is the approximation we employ in our later numerical analyses.

4.2 Contribution with a Right-Handed Current

The non-standard contribution to 2νββ decay involving the RH currents proportional tothe εXR coupling, as appearing in the Lagrangian in Eq. (2.1), was calculated in Ref. [8].

– 10 –

The corresponding functions A2ν and B2ν entering Eq. (4.2) read

A2νε = 4

[g4V (MK

F −MLF )2 +

1

3g4A(MK

GT −MLGT )2

]+

[g4V (MK

F +MLF )2 +

1

3g4A(MK

GT +MLGT )2

]×

[g2−1(Ee1) + f2

1 (Ee1)][g2−1(Ee2) + f2

1 (Ee2)]

+ [g2−1(Ee1)− f2

1 (Ee1)][g2−1(Ee2)− f2

1 (Ee2)]mνmN

Eν1Eν2

+ 2

[g4V (MK

F −MLF )2 − 1

3g4A(MK

GT −MLGT )2

]−[g4V (MK

F +MLF )2 − 1

3g4A(MK

GT +MLGT )2

]+ 2g2

V g2A

[(MK

F −MLF )(MK

GT −MLGT ) + (MK

F +MLF )(MK

GT +MLGT )

]×

[g2−1(Ee1)− f2

1 (Ee1)][g2−1(Ee2)− f2

1 (Ee2)]

+ [g2−1(Ee1) + f2

1 (Ee1)][g2−1(Ee2) + f2

1 (Ee2)]mνmN

Eν1Eν2

. (4.22)

Here, the dependence on the electron radial wave functions has been made explicit. Like-wise, the terms proportional to p1 · p2 = cos θ combine to give

B2νε =

2g4V

[(MK

F +MLF )2 − (MK

F −MLF )2] mνmN

Eν1Eν2

+8

9g4A

[(MK

GT −MLGT )2 + (MK

GT +MLGT )2

]+

10

9g4A

[(MK

GT +MLGT )2 − (MK

GT −MLGT )2

] mνmN

Eν1Eν2

+4

3g2V g

2A

[(MK

F −MLF )(MK

GT −MLGT ) + (MK

F +MLF )(MK

GT +MLGT )

] mνmN

Eν1Eν2

− 8

3g2V g

2A

[(MK

F −MLF )(MK

GT −MLGT ) + (MK

F +MLF )(MK

GT +MLGT )

]× 4f1(Ee1)f1(Ee2)g−1(Ee1)g−1(Ee2) , (4.23)

In Eqs. (4.22) and (4.23), the terms proportional to mνmN are small, as one of the emittedneutrinos is still assumed to be the light with mν . 0.1 eV. As in the SM case, for thepurpose of numerical computations we approximate the above expressions with their Taylorexpansions up to the fourth power in the small parameters εK,L.

4.3 Decay Distributions and Total Rate

The kinematics of the electrons emitted in the decay is captured by the fully differentialdecay rate expressed in Eq. (4.2) depending on the (in principle) observable electron energiesEe1 , Ee2 and the angle θ between the electron momenta. All the information is contained

– 11 –

Isotope M2νGT−1 M2ν

GT−3 M2νGT−5

76Ge 0.111 0.0133 0.0026382Se 0.0795 0.0129 0.00355

100Mo 0.184 0.0876 0.0322136Xe 0.0170 0.00526 0.00169

Table 1: Nuclear matrix elements calculated within the pn-QRPA with partial isospinrestoration [37] assuming the effective axial coupling gA = 1.0.

by the quantities A2ν and B2ν presented above both for the standard LH (Eqs. (4.12)and (4.13)) and the exotic RH (Eqs. (4.17) and (4.18)) case. The following values of thephysical constant are used in our numerical computations: Gβ = 1.1363 × 10−11 MeV−2,α = 1/137, me = 0.511 MeV, mp = 938 MeV, R = 1.2A1/3 fm (nucleon number A = 100

for Molybdenum), Q(100Mo) = 3.03 MeV, gV = 1. Since quenching of the axial coupling gAis expected in the nucleus [38], we take gA = 1 instead of the usual value gnucleon

A = 1.269

for a free neutron. Further, we use the 2νββ decay nuclear matrix elements from Ref. [37],as shown in Tab. 1.

With all the above ingredients we can now calculate the the total decay rate as well asvarious decay distributions potentially observable in 2νββ decay experiments.

Total electron energy and single electron energy: The 2νββ decay experimentsmeasure primarily the distribution with respect to the total kinetic energy of the outgoingelectrons, i.e. dΓ2ν/dEK with EK = Ee1 + Ee2 − 2me − mν1 − mν2 . Here, mν1,2 denotethe masses of the emitted neutrinos, which can be safely neglected in the SM case, butwe consider also a contribution involving a heavy sterile neutrino, in which case one of themasses becomes non-negligible and we denote it mN . We also neglect the recoil of the finalstate isotope which would change the endpoint by ∼ Q2/M . 0.1 keV, with the Q . 3 MeVand the mass of the nucleus M ≈ 76 − 136 GeV. Some experiments capture the energiesand tracks of individual electrons, thus allowing for study of the single electron energydistribution dΓ2ν/dEe1 (the symmetry of the process ensures the distribution with respectto the second electron is identical) and the double differential distribution dΓ2ν/(dEe1dEe2).These distributions are calculated from Eq. (4.2) as

dΓ2ν

dEe1dEe2=

∫ 1

−1d cos θ

dΓ2ν

dE1dE2d cos θ,

dΓ2ν

dEe1=

∫ Ei−Ef−mν1−mν2−Ee1

me

dEe2dΓ2ν

dEe1dEe2,

dΓ2ν

dEK=

EKEmaxK

∫ EmaxK

0dE

dΓ2ν

dEe1dEe2, (4.24)

where in the latter

Ee1 = EK −EKEmaxK

E +me, Ee2 =EKEmaxK

E +me, (4.25)

– 12 –

and EmaxK = Ei − Ef − 2me − mν1 − mν2 . We neglect the light neutrino masses mν1 =

mν2 = 0 in the SM case and retain only the heavy neutrino mass in the sterile contribution,mν1 = mN ,mν2 = 0. Given the fact that most experiments provide only the 2νββ decaydistribution in dependence on the total kinetic energy of the electrons, in the followinganalysis we focus primarily on this observable.

The kinematic endpoint of the summed electron energy spectrum of νNββ decay withan emission of sterile neutrino is of primary interest, as it leads to a distortion in thespectrum as the main experimental signal. Here we note that the quantity A2ν in Eq. (4.17)depends only weakly on the heavy neutrino mass mN as in the Taylor expansion in theparameters εK,L the leading term, which is free of mN and the lepton energies, is thedominant one. By restricting our consideration only to this leading term for the sterileneutrino with left-handed current we can express the energy spectrum as

dΓ2ν

dEK=

EKEmaxK

f(EK)FN (EK ,mN ), (4.26)

with

f(EK) = c2νg4AM

2GT−1

∫ EmaxK

0pe1Ee1pe2Ee2

(g2−1(Ee1) + f2

1 (Ee1)) (g2−1(Ee2) + f2

1 (Ee2))dE,

(4.27)

where Ee1 and Ee2 are expressed in terms of EK and E according to Eq. (4.25). The shapeof the distribution near the endpoint is determined by the function

FN (EK ,mN ) =1

60

√(y +mN )2 −m2

N

[2(y +mN )4 − 9(y +mN )2m2

N − 8m4N

]+

1

4(y +mN )m4

N ln

∣∣∣∣∣∣ ymN+ 1 +

√(y

mN+ 1

)2

− 1

∣∣∣∣∣∣, (4.28)

with y = EmaxK − EK (0 < y < Emax

K = Q −mN ). For mN = 0, this function reduces toFN (EK , 0) = (Emax

K −EK)5/30, leading to the well known scaling of standard 2νββ decaynear the endpoint.

In analogy to the construction of the Kurie function in single β decay we introduce theνNββ decay equivalent

K(EK ,mN ) =

(dΓ2ν/dEKf(EK)

EmaxK

EK

)1/5

= (FN (EK ,mN ))1/5 , (4.29)

which is plotted in Fig. 2 as a function of −y −mN near the endpoint for various neutrinomasses. We see that K(EK) is linear near the endpoint for zero neutrino mass (mN = 0).However, the linearity of the Kurie plot is lost if the sterile neutrino has a non-zero masswith the deviation from the straight line depending on the magnitude of mN .

Near the kinematic endpoint EK . EmaxK = Q−mN , the function FN (EK ,mN ) asymp-

totically approaches

dΓ2ν

dEK∝ FN (EK ,mN ) −−−−−−→

0<ymN

16√

2

105m

3/2N (Emax

K − EK)7/2 . (4.30)

– 13 –

-3 -2.5 -2 -1.5 -1 -0.5 0

-y-mN

[MeV]

0.0

0.5

1.0

1.5

(FN

(y,m

N))

1/5

[M

eV]

mN

=0

mN

=0.2 MeV

mN

=0.5 MeV

mN

=1.0 MeV

mN

=1.5 MeV

Figure 2: Kurie-type expression K(EK ,mN ) = F1/5N (EK ,mN ) for νNββ decay as a func-

tion of −y −mN for various values of the neutrino mass, mN = 0, 0.2, 0.5, 1.0, 1.5 MeV.

Hence, the total electron energy spectrum of νNββ is rather smooth near the endpoint,unlike in the case of single β decay. Therefore, no sharp kink is expected to appear in thetotal energy spectrum including both the SM and the sterile neutrino contributions.

Angular correlation factor and total decay rate: The integration over the electronenergies leads to the equation

dΓ2ν

d cos θ=

Γ2ν

2

(1 +K2ν cos θ

), (4.31)

describing the angular distribution of the decay. Here, Γ2ν denotes the total 2νββ decayrate and K2ν = Λ2ν/Γ2ν stands for the angular correlation factor, which are given by(

Γ2ν

Λ2ν

)=

c2ν

m11e

∫ Ei−Ef−me

me

dEe1pe1Ee1

∫ Ei−Ef−Ee1

me

dEe2pe2Ee2

(A2ν

B2ν

). (4.32)

As the inclusion of RH current leads to the opposite sign of the angular correlation of theemitted electrons [8], it can be also used to distinguish the corresponding contributions, asanalysed in the following section.

5 Constraints on Sterile Neutrino Parameters

We will now use the differential 2νββ decay rates derived in Sec. 4 to exclude regions ofthe sterile neutrino parameter space – namely, the sterile neutrino mass mN and mixingwith the electron neutrino |VeN |2. To do this we will first outline a simple frequentist limitsetting method. We will then use the non-observation of deviations from the SM 2νββ

decay spectrum by 0νββ decay search experiments such as GERDA-II, CUPID-0, NEMO-3 and KamLAND-Zen to put upper limits on |VeN |2 as a function of mN . We will alsoestimate upper limits from the forecasted sensitivities of future 0νββ decay experimentssuch as LEGEND, SuperNEMO, CUPID and DARWIN. Finally, we will compare these

– 14 –

0.2

0.4

0.6

(dΓ

2ν/dEK)/

Γ2ν

SM

100 Mo136 Xe

dΓ2ν /dEK

|VeN|2 dΓ2ν

N /dEK

0.0 0.5 1.0 1.5 2.0 2.5 3.0

EK [MeV]

−60

−40

−20

0

Dev

iation

[%]

Figure 3: Total differential 2νββ decay rate (solid) and the sterile neutrino contribution(dashed) with mN = 1.0 MeV and |VeN |2 = 0.5 for the two isotopes 100Mo (purple) and136Xe (blue). Both distributions are normalised to the SM decay rate. The vertical dottedlines indicate the respective Q values and the panel at the bottom shows the correspondingpercentage deviations from the SM rate.

upper limits to existing constraints in the 0.1 MeV < mN < 3 MeV range from singlebeta decay probes (64Cu, 144Ce−144Pr and 20F) and sterile neutrino decays (Borexino) asdiscussed in Sec. 3.

5.1 Statistical Procedure

To obtain upper limits on the mixing |VeN |2 we follow the standard frequentist approach ofRefs. [1, 39]. Firstly, we define the total differential 2νββ decay rate as the incoherent sumof the sterile neutrino and SM rates for a given sterile mass mN and total kinetic energyEK = Ee1 + Ee2 − 2me,

dΓ2ν(ξ)

dEK= (1− |VeN |2)2dΓ2ν

SMdEK

+ (1− |VeN |2)|VeN |2dΓ2ν

N (mN )

dEK, (5.1)

explicitly writing the dependence on active-sterile mixing |VeN |2. The total differentialrate depends on the sterile neutrino parameters ξ ≡ (mN , |VeN |2) and EK . Here, thecontribution dΓ2ν

N /dEK due to the sterile neutrino includes a factor of two compared to theSM contribution, as two distinguishable neutrinos are emitted in the process, cf. Eq. (4.1).

– 15 –

In Fig. 3, the total differential decay rate in Eq. (5.1) is compared to the sterile neutrinocontribution |VeN |2 ·dΓ2ν

N /dEK (where both are normalised to the total SM decay rate Γ2νSM)

for the isotopes 100Mo and 136Xe. The respective Q values of the isotopes are indicated bythe vertical dotted lines and the values mN = 1.0 MeV and |VeN |2 = 0.5 are chosen. In thepanel below we show the corresponding percentage deviation of the total differential ratefrom the SM rate,(

dΓ2ν

dEK−dΓ2ν

SMdEK

)/Γ2νSM

dEK= |VeN |2

(dΓ2ν

N

dEK

/Γ2νSM

dEK− 1

). (5.2)

It can be seen that the magnitude of dΓ2ν/dEK decreases with respect to the dΓ2νSM/dEK as

the total kinetic energy increases, eventually plateauing at around −10%. This is becausethe sterile neutrino contribution |VeN |2 dΓ2ν

N /dEK falls as EK increases above ∼ 1.0 MeV.Eventually its contribution is negligible, but there remains a suppression from the (1 −|VeN |2) factor multiplying the SM contribution, which is particularly sizeable for the choice|VeN |2 = 0.5. It is apparent from Eq. (5.2) that the deviation tends to a factor of −|VeN |2.The characteristic signature of the sterile neutrino is a relative increase of the differentialrate for EK . Q−mN .

Any experiment measuring the 2νββ decay spectrum will count a number of eventsNevents distributed over a number of bins Nbins in the total kinetic energy EK . In thepresence of a sterile neutrino, the expected fraction of events ∆N

(i)exp per bin will be the

integral of dΓ2ν/dEK over the width of the bin from the total kinetic energy Ei to Ei+1,

∆N (i)exp =

1

N

∫ Ei+1

Ei

dEKdΓ2ν

dEK, (5.3)

where the normalisation factor N is

N =

∫ Emax

Emin

dEKdΓ2ν

dEK, (5.4)

i.e. the total area enclosed by dΓ2ν/dEK between kinetic energies Emin and Emax. Thetotal number of expected events per bin will then be

N (i)exp = N

(i)sig +N

(i)bkg = Nevents ·∆N (i)

exp , (5.5)

where we have also split the expected number of events into the number of signal andbackground events as

N(i)sig =

Nevents

N|VeN |2

∫ Ei+1

Ei

dEK

(dΓ2ν

N

dEK−dΓ2ν

SMdEK

), (5.6)

N(i)bkg =

Nevents

N

∫ Ei+1

Ei

dEKdΓ2ν

SMdEK

. (5.7)

The probability of the experiment observing N (i)obs events per bin given N (i)

exp expected eventsis the Poisson probability P (N

(i)obs|N

(i)exp). The likelihood of the data D given the sterile

– 16 –

neutrino hypothesis, L(D|ξ), is defined as the product of the Poisson probabilities over allbins. It is more convenient to write the log-likelihood

−2 logL(D|ξ) = 2

Nbins∑i

N (i)

exp(ξ)−N (i)obs +N

(i)obs log

(N

(i)obs

N(i)exp(ξ)

)

≈Nbins∑i

(N

(i)obs −N

(i)exp(ξ)

)2

N(i)exp(ξ)

, (5.8)

where the second equality holds via Wilks’ theorem if there are a large number of eventsper bin [40]. From this we can construct the test-statistic

qξ = −2(

logL(D|ξ)− logL(D|ξ)), (5.9)

where ξ are the values of the sterile neutrino parameters that minimise the log-likelihoodfunction. The quantity qξ is expected to follow a χ2 distribution with one degree of freedom.

We assume that the experiment does not observe a spectrum deviating significantlyfrom the SM prediction. We therefore set the number of observed events in Eq. (5.8) toN

(i)obs = N

(i)exp(ξ) with ξ = (mN , 0). In reality, however, the experiment could be repeated

many times and record a different value of N (i)obs each iteration. This fluctuation can be

imitated by running a series of toy Monte Carlo simulations of the experiment. For everytoy Monte Carlo there is a value of qξ, with the relevant test-statistic becoming the medianof these values. A representative data set is commonly used as a good approximation of theMonte Carlo method in the large sample limit [41]. This is the so-called Asimov data setDA for which the observed number of events per bin N (i)

obs equals the number of backgroundevents N (i)

bkg [42]. The ξ that minimises the log-likelihood to −2 logL(DA|ξ) = 0 is thensimply ξ = (mN , 0) which matches our initial approach.

The magnitude of the test-statistic qξ = −2 logL(DA|ξ) translates to a degree of com-patibility between the Asimov data set and the sterile neutrino hypothesis with parametersξ = (mN , |VeN |2). For example, if both parameters are allowed to vary, combinations ofthe parameters giving qξ & 4.61 are excluded at 90% confidence level (CL). Rather thanperforming a two-dimensional scan of the parameters, we instead fix mN for values over therange ∼ 0.1− 3 MeV and find the value of |VeN |2 for which qξ = 2.71, corresponding to the90% CL upper limit on the mixing.

Finally we note that we have not yet included the effect of systematic uncertainties.Systematics altering the total number of observed events without leading to distortions inthe spectrum can be accounted for by introducing the nuisance parameter η

−2 logL(D|ξ, η) ≈Nbins∑i

(N

(i)bkg − (1 + η)N

(i)exp(ξ)

)2

(σ(i)stat)

2 + (σ(i)sys)2

+

(η

ση

)2

, (5.10)

where ση is a small associated uncertainty. The remaining systematic uncertainties areincluded in the quantity σ(i)

sys = σfN(i)exp which adds in quadrature with the statistical un-

– 17 –

certainty (σ(i)stat)

2 = N(i)exp in the denominator of Eq. (5.10). The test-statistic becomes

qξ = −2(

logL(D|ξ, ˆη)− logL(D|ξ, η)), (5.11)

where ˆη minimises the log-likelihood for a given ξ while ξ and η are the values at theglobal minimum of the log-likelihood. For the Asimov data set the parameters at the globalminimum are ξ = (mN , 0) and η = 0 such that −2 logL(DA|ξ, η) = 0. The test-statisticthen reduces to

qξ = minη

Nbins∑i

(N

(i)bkg − (1 + η)N

(i)exp(ξ)

)2

(σ(i)stat)

2 + (σ(i)sys)2

+

(η

ση

)2

, (5.12)

which will be used to derive constraints in the next subsection.We note that the critical uncertainty is that of the experimental measurement of the

2νββ decay rate and not that in theoretical calculation of the corresponding nuclear matrixelements. This is because both the SM 2νββ decay and the one involving a sterile neutrino(νNββ) have the same nuclear matrix element and depend e.g. on the axial couplingstrength gA in the same way, at least to a very good approximation as detailed below.Thus, while the individual decay rates have a large theoretical uncertainty, e.g. considering arange of 0.7 . gA . 1.27, their ratio is largely unaffected and one may use the experimentalmeasurement to set the overall scale.

The heavier mass of the sterile neutrino does influence the energy denominators inEq. (4.11) which changes the matrix elements as a sub-leading effect. This mostly affectsdifferential decay properties, such as the electron energy spectrum, but it is essentiallynegligible for the sterile neutrino case with a left-handed current. This is because the dis-tinctive feature, the different energy threshold for the νNββ case, is unaffected: its locationis determined by kinematics and its shape is already smooth, ∝ (Q − mN − E)7/2, withsmall corrections having no discernable effect within the experimental energy resolutionsconsidered. In other words, there is no sharp threshold (as in single β decay) which is indanger of being washed out due to corrections.

The same procedure can be applied to place upper limits on the RH current couplings|εLR|2 and |εRR|2. As seen in the previous sections, the RH current modifies the total kineticenergy distribution to

dΓ2ν(ξ)

dEK=dΓ2ν

SMdEK

+ |εXR|2dΓ2ν

N (mN )

dEK, (5.13)

where the SM contribution is no longer reduced by the sterile neutrino mixing. The RHcurrent also modifies the angular distribution to Eq. (4.31) with the total rate Γ2ν and theangular correlation factor K2ν given in terms of SM and RH current contributions as

Γ2ν(ξ) = A2νSM +A2ν

N (mN )|εXR|2 , K2ν(ξ) =B2νSM +B2ν

N (mN )|εXR|2

A2νSM +A2ν

N (mN )|εXR|2. (5.14)

Assuming |εXR|2 1, K2ν can be Taylor expanded as

K2ν(ξ) ≈ K2νSM + α(mN )|εXR|2 , (5.15)

– 18 –

0.0 0.5 1.0 1.5 2.0 2.5 3.0

mN [MeV]

0.0

1.0

2.0

3.0

4.0

5.0

6.0

α(m

N)

α(0) = 6.114 (82 Se)

α(0) = 6.082 (100 Mo)

82 Se100 Mo

Figure 4: The approximate factor α(mN ) multiplying the RH current coupling |εXR|2

yielding the sterile neutrino contribution to the angular correlation factor K2ν for 82Se(red) and 100Mo (blue).

where the SM contribution and RH current contributions, respectively, are

K2νSM =

B2νSM

A2νSM

, α(mN ) =B2νN (mN )−K2ν

SMA2νN (mN )

A2νSM

, (5.16)

The SM values areK2νSM = −0.627 for 100Mo andK2ν

SM = −0.631 for 82Se (the isotopes ofexperiments that are sensitive to the angular correlation factor, NEMO-3 and SuperNEMO,respectively). The α(mN ) factors are plotted for 82Se (red) and 100Mo (blue) in Fig. 4, whichalso indicates the values at mN = 0. The factor α(mN ) is positive, indicating a change ofthe angular distribution away from the back-to-back configuration of electrons in the SMV − A case. It is maximal for mN = 0 and is suppressed to zero as mN approaches the Qvalue.

Using the measured total kinetic energy distributions from all 2νββ decay experiments,the ξ = (mN , |εXR|2) parameter space can be constrained in the same was as (mN , |VeN |2)

described above, i.e. using the test-statistic in Eq. (5.12). In addition, the experimentsNEMO-3 and SuperNEMO will measure a certain number of events N (i)

obs distributed inbins of the cosine of the angle, cos θ. We can estimate the total number of signal plusbackground events N (i)

exp in each bin by integrating over the angular distribution Eq. (4.31).We can then compute the test-statistic in Eq. (5.12) to put an additional constraint on theξ parameter space.

5.2 Results

A selection of current and next generation 0νββ decay search experiments measuring the2νββ decay of isotopes 76Ge, 82Se, 100Mo and 136Xe are shown in Tab. 2. Listed are theexposures, total number of events Nevents, energy resolutions ∆E and estimates for theparameters ση and σf quantifying the uncertainties on the nuisance parameter η and from

– 19 –

other systematic effects, respectively. Values are taken from the list of references given forthe experiments. For each experiment we make use of Eq. (5.12) to set an upper limit onthe active-sterile mixing |VeN |2 as a function of the sterile neutrino mass mN .

Fig. 5 (left) shows the 90% CL upper limits derived from the current generation ex-periments GERDA II (76Ge, grey), CUPID-0 (82Se, red), NEMO-3 (100Mo, purple) andKamLAND-Zen (136Xe, blue). We also show a combined constraint (black dashed) foundby summing the log-likelihoods of the experiments (each minimised with respect to a sep-arate nuisance parameter η). It can be seen that the upper limits worsen for smaller andlarger values of the sterile mass in the range 0.1 MeV < mN < 3 MeV, with the moststringent upper bound being found at mN similar to the peak energy of the associatedspectrum. The constraints are compared to pre-existing constraints (shaded areas) fromsingle beta decay experiments and sterile neutrino decays. While NEMO-3 and KamLAND-Zen provide the best individual constraints (|VeN |2 . 0.02), they are not as competitiveas previous limits. However, it is promising that 2νββ decay is more sensitive for sterilemasses 0.3 MeV < mN < 0.7 MeV where existing constraints are less stringent.

Fig. 5 (right) shows the corresponding sensitivities estimated for the next generationof 0νββ decay experiments. The forecasted range of exposures given by the collaborationsare often one or two orders of magnitude larger than those of the current generation.We estimate the total number of events Nevents seen in future by multiplying the currentvalues by the ratio of future to current exposures. Energy resolutions are taken from thereferences in Tab. 2 and we assume an optimistic value of ση ∼ σf ∼ 0.5% for the systematicuncertainties. We compute the 90% CL sensitivity for both the higher and lower forecastednumber of events in Tab. 2, shown as bands for LEGEND (76Ge, grey), SuperNEMO (82Se,red), CUPID (100Mo, purple) and DARWIN (136Xe, blue). Also shown is the combinedsensitivity (black dashed) using the largest predicted exposure of each experiment. Fora given experiment the upper bounds exhibit the same improvement for sterile masses

Isotope Experiment Exposure [kg · y] Nevents ∆E [keV] (ση, σf ) [%]

76GeGERDA II [43] 103.7 3.63× 104 15 (4.6, 1.9)

LEGEND [44] 103–104 105–106 2.5 (0.5, 0.5)

82SeCUPID-0 [45] 9.95 5.8× 103 50 (1.5, 1.0)

SuperNEMO [46] 102–103 104–105 50 (0.5, 0.5)

100MoNEMO-3 [47] 34.3 4.95× 105 100 (5.4, 1.8)

CUPID-Mo [48] 0.116 3.9× 104 20 (1.4, 0.5)

CUPID [49] 102–103 106–107 5 (0.5, 0.5)

136XeKamLAND-Zen [50] 126.3 9.83× 104 50 (3.1, 0.3)

DARWIN [51] (2–5)× 104 106–107 5 (0.5, 0.5)

Table 2: Current and next generation 0νββ decay search experiments measuring the 2νββ

decay spectrum of the isotopes considered in this work. Shown are the current and fore-casted exposures, total number of events Nevents, energy resolutions ∆E and parameters(ση, σf ) estimating the effect of systematic errors on the log-likelihood function.

– 20 –

0.1 0.2 0.5 1 2mN [MeV]

0.001

0.01

0.1

1|VeN|2

GERDA II, 76 Ge

CUPID−0, 82 Se

NEMO−3, 100 Mo

KamLAND−Zen, 136 Xe

Combined 2νββ

0.1 0.2 0.5 1 2mN [MeV]

0.001

0.01

0.1

1

|VeN|2

LEGEND, 76 Ge

SuperNEMO, 82 Se

CUPID, 100 Mo

DARWIN, 136 Xe

Combined Future 2νββ

Figure 5: Upper limits and sensitivities at 90% CL on the squared mixing |VeN |2 betweenthe electron and sterile neutrino as a function of the sterile neutrino mass mN from 2νββ

in current (left) and future (right) experiments. Shown are the individual constraints asindicated in the legend as well as a combined constraint (black dashed). The bands inthe right plot correspond to the possible future exposures in Tab. 2. The combined futuresensitivity uses the maximum forecasted exposure of each experiment.

close to the maximum of the total differential decay rate. The most stringent upper limitscome from CUPID and DARWIN, |VeN |2 . 2.5× 10−3, which would exclude the currentlyunconstrained region in the 0.3 MeV < mN < 0.7 MeV range.

Likewise, we estimate the current limits and future sensitivity on the RH couplings|εLR|2 and |εRR|2 from measuring the 2νββ decay energy distribution and angular correla-tion. In Fig. 6 we plot the upper limits at 90% CL on |εLR|2 and |εRR|2 as a function of thesterile neutrino mass mN . The blue solid line is the combined constraint from current 2νββ

decay experiments using the total kinetic energy distribution, while the red solid line is theupper limit derived from the angular distribution measurement of NEMO-3 (100Mo). Theblue dashed line is the combined sensitivity from future 2νββ decay experiments, while thered dashed band indicates the sensitivity range from the angular distribution measurementof SuperNEMO (82Se). The latter does not improve over the current limit as SuperNEMOis not expected to have a significantly increased exposure compared to NEMO-3, see Tab. 2.We therefore also indicate the sensitivity of a hypothetical 82Se angular measurement withan exposure of 107 events (red dot dashed).

Due to the different total kinetic energy distribution for the RH current in Eq. (5.13)(no suppression of the SM rate), the combined constraints on |εLR|2 and |εRR|2 (dashedlines) are slightly weaker than the equivalent constraints on |VeN |2. The constraints fromthe NEMO-3 angular distribution are generally better, tending to a constant upper bound|εXR|2 . 10−3 for mN . 0.2 MeV. This roughly agrees with the result εXR < 2.7× 10−2 inthe massless case found in Ref. [8].

– 21 –

0.1 0.2 0.5 1 2

mN [MeV]

0.001

0.01

0.1

1

|ε XR|2

NEMO−3, 100 Mo, dΓ2ν

dcosθ

SuperNEMO, 82 Se, dΓ2ν

dcosθ

Future Angular, 82 Se, dΓ2ν

dcosθ

Combined 2νββ, dΓ2ν

dEe1

Combined Future 2νββ, dΓ2ν

dEe1

Figure 6: Current upper limits and future sensitivities at 90% CL on the RH coupling|εXR|2 as a function of the sterile neutrinos mass mN . The solid (dashed) blue line showsthe combined constraint from current (future) 2νββ decay experiments measuring the totalkinetic energy distribution. The solid red line is the upper limit derived from the angulardistribution measurement of NEMO-3 (100Mo). The dashed red band indicates the range ofupper limits expected from the angular distribution measurement of SuperNEMO (82Se).The dot-dashed red line shows the upper limit from a future 82Se experiment with anexposure of 107 events.

6 Conclusions

Measuring the kinematic endpoint in single beta decay is arguably the cleanest means todetermine the absolute neutrino masses in a model-independent fashion. For the light activeneutrinos in the SM, the most promising isotope for this is tritium (3H) and its beta decay iscurrently measured in the KATRIN experiment [52] as well as the future Project 8 [53] andCRESDA [54] efforts. The same method can be applied to search for sterile neutrinos, notonly in Tritium but in a host of beta decay isotopes where masses smaller than the respectiveQ value of the decay can be probed. The limits on the active-sterile mixing strength |VeN |2

from such searches are summarised in Fig. 7. They are comparatively weak, of the order|VeN | . 2× 10−2 − 2× 10−3, in the sterile neutrino mass range 0.1 MeV < mN < 1 MeV.

In this work, we have analysed the prospects to search for sterile neutrinos using thesame principle in 2νββ decay. If one of the two neutrinos emitted in the process is a heavier,sterile neutrino it will likewise affect the distribution with respect to the kinetic energy ofthe two electrons observed in the decay: the kinematic endpoint is shifted to lower valuesdepending on the sterile neutrino mass and the active-sterile mixing will reduce the usualSM contribution. This is expected to be challenging because of the very long 2νββ decay

– 22 –

10−4 10−3 10−2 10−1 1 10mN [MeV]

10−5

10−4

10−3

10−2

10−1

1

|VeN|2,|ε X

R|2

KATRIN

3 H

187 Re

Rovno

Bugey

20 F

64 Cu

144 Ce −144 Pr

Borexino

45 Ca

35 S63 Ni

|VeN|2 (2νββ Energy)

|εXR|2 (2νββ Angular)

Figure 7: Current upper limits (solid blue) and future sensitivities (dashed blue) on themixing strength |VeN |2 between the electron and sterile neutrino as a function of the sterilemass mN . Likewise, the red curves give the current limit and future sensitivity on theRH coupling |εXR|2 using a measurement of the angular distribution in 2νββ decay. Theshaded regions are excluded by existing searches in single beta decay and sterile decays inreactor and solar neutrino oscillation experiments.

half lives and small rates compared to single beta decay. Nevertheless, future searches forthe lepton number violating 0νββ decay will push the envelope in terms of exposure andallow measuring 2νββ decay with up to 107 events. These data can then be used to probeexotic physics with 2νββ decay in its own right. Apart from sterile neutrino searches, otherexamples include exotic neutrino self interactions [9] and RH leptonic currents [8]. Wehave extended the latter analysis here to consider a RH V +A current for a sterile neutrinorather than the SM electron neutrino. As in Ref. [8], this gives rise to an anomalous angulardistribution of the electrons in 2νββ decay.

To summarise the sensitivity we compare in Fig. 7 the current limits on |VeN |2 fromexisting 2νββ decay (solid blue) to constraints from single beta decays and sterile neutrinodecays over a wider range of masses, 100 eV < mN < 10 MeV. The blue curve usesthe combined constraints from measurements of 2νββ decay electron energies. The redcurve shows the current constraint on the effective RH coupling |εXR|2 using the NEMO-3angular distribution measurement. The dashed curves indicate the corresponding futuresensitivities. At lower masses both the current and future upper limits on |VeN |2 cannotcompete with existing constraints from 64Cu and 144Ce−144Pr beta decays. At highermasses they are also less stringent than constraints from Borexino, Bugey and Rovno. It isthe 0.3 MeV < mN < 0.7 MeV range where 2νββ decay can provide competitive constraints

– 23 –

in the future, though we expect that similar improvements from 20F and 144Ce−144Pr betadecays are also possible. The constraints on the RH coupling |εXR|2 using an angulardistribution measurement in 2νββ decay is most sensitive for light sterile neutrino massesmN . 0.1 MeV as the effect is phase space suppressed otherwise. We note, though, thatthe limits from single beta decays and the other processes shown strictly speaking applyto |VeN |2 only and need to be re-evaluated for a heavy neutrino coupling through a RHcurrent.

Our analysis demonstrates that 2νββ decay can be used to search for sterile neutrinoswith masses lighter than mN ∼ 1 MeV. While current searches are not competitive withlimits from single beta decays, future searches will have a much more increased statisticswhere effects of new physics can be tested. While sterile neutrinos in this mass range arealso heavily constrained from astrophysical measurements and cosmological considerations,it is important to improve our understanding using all available data.

Acknowledgements

The authors would like to thank Alexander Derbin for useful discussions on the constraintsfrom single beta decay. The authors would also like to thank Matteo Agostini, ElisabettaBossio, Alejandro Ibarra and Xabier Marcano for useful discussions on the revision of themanuscript. F. F. D. and P. D. B. acknowledge support from the UK Science and Technol-ogy Facilities Council (STFC) via a Consolidated Grant (Reference ST/P00072X/1). FŠacknowledges support by the VEGA Grant Agency of the Slovak Republic under ContractNo. 1/0607/20 and by the Ministry of Education, Youth and Sports of the Czech Republicunder the INAFYM Grant No. CZ.02.1.01/0.0/0.0/16_019/0000766.

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