I Introduction

IPPP/13/90

DCPT/13/180

The Effect of Cancellation in Neutrinoless Double Beta Decay

Silvia Pascoli, Manimala Mitra, Steven Wong

Institute for Particle Physics Phenomenology, Department of Physics,

Durham University, Durham DH1 3LE, United Kingdom

Physics Department, Chinese University of Hong Kong, Shatin, N.T., Hong Kong

Abstract

In light of recent experimental results, we carefully analyze the effects of interference in neutrinoless double beta decay, when more than one mechanism is operative. If a complete cancellation is at work, the half-life of the corresponding isotope is infinite and any constraint on it will automatically be satisfied. We analyze this possibility in detail assuming a cancellation in , and find its implications on the half-life of other isotopes, such as . For definiteness, we consider the role of light and heavy sterile neutrinos. In this case, the effective Majorana mass parameter can be redefined to take into account all contributions and its value gets suppressed. Hence, larger values of neutrino masses are required for the same half-life. The canonical light neutrino contribution can not saturate the present limits of half-lives or the positive claim of observation of neutrinoless double beta decay, once the stringent bounds from cosmology are taken into account. For the case of cancellation, where all the sterile neutrinos are heavy, the tension between the results from neutrinoless double beta decay and cosmology becomes more severe. We show that the inclusion of light sterile neutrinos in this set up can resolve this issue. Using the recent results from GERDA, we derive upper limits on the active-sterile mixing angles and compare it with the case of no cancellation. The required values of the mixing angles become larger, if a cancellation is at work. A direct test of destructive interference in is provided by the observation of this process in other isotopes and we study in detail the correlation between their half-lives. Finally, we discuss the model realizations which can accommodate light and heavy sterile neutrinos and the cancellation. We show that sterile neutrinos of few hundred MeV or GeV mass range, coming from an Extended seesaw framework or a further extension, can satisfy the required cancellation.

## I Introduction

In the past fifteen years, the experimental evidence of neutrino masses and mixing has opened up a new window on the physics beyond the standard model. The solar, atmospheric and reactor neutrino oscillation (see review-osc-petcov (); mik-shap (); gelmini-rev (); nir-garcia-rev (); rev-moha-smir (); visreview () for recent reviews) experiments solar (); kl (); atm (); k2k (); t2k (); minos (); chooz () of the past decades confirmed that the standard neutrinos have very small masses in the eV range. Neutrino mixing data limits (); limitspres (); Fogli2011osc (); oscparam () is well described by the unitary PMNS matrix , parameterized by three real mixing angles, one CP violating Dirac phase and two Majorana phases. So far, the oscillation parameters, namely the solar, atmospheric mass square differences and and the three oscillation angles , and , have been measured solar (); atm (); kl (); k2k (); t2k (); minos (); chooz (); limits (); limitspres (); Fogli2011osc (); oscparam (); reno (); dayabay (); dc () upto a good accuracy. The current allowed ranges of the oscillation parameters are limitspres (); oscparam ()

 6.99×10−5eV2≤Δm221≤8.18×10−5eV2,2.17×10−3eV2≤Δm231≤2.62×10−3eV2 , (1)
 0.259≤sin2θ12≤0.359,0.331≤sin2θ23≤0.663,0.016≤sin2θ13≤0.031 . (2)

Although a lot of information on neutrino masses and mixing have been unveiled in the past decade, yet many neutrino properties remain to be determined. We still do not know the neutrino mass hierarchy, if the CP symmetry is violated in the leptonic sector, and most importantly, the nature of neutrinos - whether neutrinos are Dirac or Majorana particles. The neutrino nature is strictly related to the violation of global leptonic number and, hence, experiments in which lepton number violation can manifest itself could unveil the Majorana nature of neutrinos.

Among the different lepton number violating experiments, neutrinoless double beta decay, searching for , oli-rev (); vogel-rev (); bilenky-rev (); werner-rev (); klapdorold (); klapdor (); Gda (); gerda (); hei (); igex (); Gando:2012zm (); Auger:2012ar (); cuo (); nemo3 () is the most sensitive one. In the minimal extension of the Standard Model, augmented only by massive neutrinos, this process is mediated by light neutrino exchange 0nu2beta-old (). In this case the observation of -decay can shed some light on i) the mass hierarchy, the neutrino mass scale and, possibly, on one of the Majorana CP-violating phases, although this will be very challenging silvia0nu2beta (); Barger:2002vy (). However, in general other mechanisms could play a role in neutrinoless double beta decay. In fact, Majorana neutrino masses require further extensions of the standard model, with a new physics scale, new particles and a source of lepton number violation. The simplest realization comes from the dimension 5 operator dim5 (), which can arise as the low energy effective term from a higher energy theory with lepton number violation. The latter will typically also induce neutrinoless double beta decay directly. In most cases, such contributions are suppressed due to the heavy scale of the new mediators, but many exceptions exist feinberg (). Several detailed studies have been carried out f2005 (); ibarra (); MSV (); msvmoriond (); Pascoli (); blennow (); meroni2013 () regarding Type-I seesawM (); seesaw-Goran (); seesaw-Yan (); seesaw-Ramond (), Extended Kang-Kim (); Parida () and Inverse seesaw invo (); inv (); invdet (); inverseso10 (); inverseothers (), Left-Right symmetric LRSM (); Tello:2010 (); Nemevsek:2011aa (); Chakrabortty:2012mh (); Barry:2013xxa (); Dev:2013vxa (); Huang:2013kma (); Dev:2013oxa (), R-parity violating supersymmetric models ms0nu2beta (); mwex (); pion-ex (); hirsch (); allanach (). It is found that in the Type-I and Extended seesaw scenario sterile neutrinos with few GeV masses can give a contribution comparable to the light neutrino ones or even dominant MSV (); msvmoriond (); Pascoli (). For Left-Right symmetric models, the right handed current contribution can be significantly large, if the new gauge boson and right handed neutrino masses are in the TeV scale Tello:2010 (); Nemevsek:2011aa (); Chakrabortty:2012mh (); Barry:2013xxa (); Dev:2013vxa (); Huang:2013kma (); Dev:2013oxa (). In the case of R-parity violating supersymmetry, different lepton number violating states e.g. neutralino, squark and gluino can mediate this process, and their contributions have been analyzed in detail ms0nu2beta (); mwex (); pion-ex (); hirsch (); allanach (). The different lepton number violating states can also originate from an extra dimensional framework extra () or other possible new physics scenario vogel (); choi (); Choubey:2012ux ().

Several experiments on neutrinoless double beta decay klapdorold (); klapdor (); Gda (); gerda (); hei (); igex (); Gando:2012zm (); Auger:2012ar (); cuo (); nemo3 () have been carried out using different type of nuclei, e.g. , , , . The bounds coming from Heidelberg-Moscow hei () and IGEX igex () experiments apply to the isotope and are given by at C.L., respectively, but the most stringent bound has been recently reported by the GERDA collaboration: at C.L. gerda (). Combining the latter with the Heidelberg-Moscow and IGEX experiments, the limit improves to at C.L. gerda (). It should be pointed out that a part of the Heidelberg-Moscow collaboration, led by Klapdor-Kleingrothaus and collaborators, reported evidence of the observation of this process corresponding to the half-life klapdorold (), which was updated later to klapdor (). This claim has been constrained significantly by the recent results from GERDA gerda () but at present neither the individual nor the combined limit from GERDA [24] can conclusively rule out the updated claim klapdor (). Using the isotope, the bounds on half-life from EXO-200 and KamLAND-Zen experiments are Auger:2012ar () and Gando:2012zm () at C.L., respectively. The KamLAND-Zen collaboration has combined the two limits obtaining at C.L. Gando:2012zm (). According to the KamLAND-Zen collaboration this combined bound rules out the claim in klapdor () at C.L. but, as pointed out in Dev:2013vxa (), this conclusion depends on the nuclear matrix elements (NME) used. Future experiments will conclusively confirm or disprove the positive claim and can improve the sensitivity to the half-life by more than an order of magnitude Gda (); Cuore (); Gando:2012zm (); supernemo (); Majorana0 (); lucifer (); snop (); cobra (); next ().

The light neutrinos, if Majorana particle, will mediate the neutrinoless double beta decay. Their contribution can saturate the present limits of half-lives only in the quasi-degenerate limit. As pointed out in Ref. msvmoriond (); Dev:2013vxa (); fogli (), the bounds from cosmology put stringent constraint on neutrino masses and consequently on the interpretation of neutrinoless double beta decay mediated by light neutrino masses to satisfy the claim in klapdor (), or to saturate the experimental limits from Heidelberg-Moscow, GERDA, EXO-200 and KamLAND-Zen gerda (); hei (); Gando:2012zm (); Auger:2012ar (). The conclusion remains the same, after including the stringent cosmological bound on the sum of light neutrino masses from Planck Ade:2013lta (), as it has been explicitly shown in Dev:2013vxa ().

In the light of the recent experimental results, in this work we carefully analyze lepton number violation in neutrinoless double beta decay for the cases in which more than one mechanism is operative Pascoli (). In presence of several left-current processes, if their contributions are comparable, they can sum up constructively in neutrinoless double beta decay or even partially or completely cancel out, making the half-life much longer than naively expected. Establishing if cancellations are at play could be of importance to conclusively determine the nature of neutrinos. In fact, if future experiments do not find neutrinoless double beta decay in contradiction with the theoretical prediction, the conclusion that neutrinos are Dirac particles is valid only if the possibility of cancellations between different mechanisms is excluded. For instance, this would be the case if no positive evidence is found down to an effective Majorana mass parameter of 10 meV and an inverted hierarchy is established in reactor, atmospheric and/or long baseline neutrino oscillation experiments. Here, we show that if both light and heavy neutrinos, compared to the momentum exchange of the process, are at work, it might be possible to test the presence of such a cancellation.

While individual contribution from different underlying mechanisms: e.g. the most popular light neutrinos, sterile neutrinos in Type-I, Extended seesaw and Inverse seesaw, gluino and squark exchange for R-parity violating supersymmetry, have been carefully analyzed in the literature, the interference effects have been neglected to a large extent (see petcov (); Faessler:2011qw (); Faessler:2011rv (); f2010 () for the few discussions on the interference). In this work, we discuss the effect of interference in detail and present simple model realizations in which such cancellations can emerge. Although our analysis is general, one immediate application would be to solve the mutual inconsistency between the positive claim in klapdor () for and the bounds from Auger:2012ar (); Gando:2012zm () in . If the found evidence klapdor () is finally refuted by future experiments, the possibility of cancellations remains open and should be tested by using different nuclei.

The paper is organized as follows. In Section II, we review the different bounds on neutrinoless double beta decay; we discuss the contribution from light neutrino exchange, the stringent bounds on neutrino masses from cosmology as well as the future bound from KATRIN katrin (). Following that, we discuss the contribution from sterile neutrinos in Section III. We discuss the cancellations in Section IV, where we carefully consider the interference between two dominant mechanisms in neutrinoless double beta decay, e.g. light neutrino-heavy sterile neutrino exchange or light neutrino-gluino/squark exchange. We show how this possibility is further constrained from beta decay as well as cosmology. Next, we consider the case in which both light and heavy sterile neutrinos are operative in neutrinoless double beta decay. This possibility allows to overcome the constraints from cosmology. We discuss the correlation of half-lives between two different isotopes in Section V. In Section VI we discuss simple model realizations which can accommodate sterile neutrinos. Finally, in Section VII we draw our conclusions.

## Ii Light Neutrino Exchange in (ββ)0ν-decay and its connection to beta decay and cosmology

Below we review the most stringent constraints on for the isotopes of interest , , , and . All bounds are reported at 90 C.L. unless otherwise specified.

1. The claim of observation of -decay by H. V. Klapdor-Kleingrothaus and collaborators for the isotope corresponds to the half-life: (the range correspond to 68 C.L.) klapdor (). This has been challenged by the previous results from Heidelberg-Moscow hei () and by the recent result from GERDA gerda (). The lower limit of half-life of that comes from GERDA gerda () is . When combined with the limits from Heidelberg-Moscow (HDM) hei () and IGEX igex () experiments, the limit is . Note that, as pointed out in Ref. Klapdor-Kleingrothaus:2013cja (); Dev:2013vxa (), the individual as well as the combined limit from GERDA does not conclusively rule out the positive claim klapdor ().

2. The bounds from EXO-200 Auger:2012ar () and KamLAND-Zen Gando:2012zm () experiments for are and , respectively. Combining the two, the lower limit becomes Gando:2012zm ().

3. The bound on the half-life of coming from CUORICINO is cuo ().

4. The lower limit on half-life of from NEMO 3 is nemo3 ().

5. The half-life of is bounded from below as nemo3 ().

Among these different bounds, those on the half-life for and are in particular quite stringent. As pointed out in Ref. Dev:2013vxa (), the claim of observation of -decay in is compatible with the individual limits from KamLAND-Zen and EXO-200 for few NME calculations, and it is in contradiction with the combined bound for most of the NME calculations, except of the calculation corresponding to Ref. engel (). For the discussion on the mutual compatibility between the positive claim klapdor () and the bounds on the half-lives, see also Ref. faess (). It should be noted that, for a given value of , the predicted value of the half-life depends strongly on the NME uncertainty. Taking this variation into account, the correlation between half-lives for two different isotopes can be used to test the positive claim klapdor (), as it has been done in Refs. Gando:2012zm (); Dev:2013vxa ().

If light neutrinos are Majorana particles Majorana (), they will mediate neutrinoless double beta decay 0nu2beta-old (). The observable in -decay is the ee element of the mass matrix , known as the effective Majorana mass parameter of neutrinoless double beta decay, see e.g. Ref. visreview (); vuso (); vis-fer (); silvia0nu2beta (); steve (). Explicitly written in terms of the elements of the PMNS mixing matrix, this reads

 mνee=m1c212c213+m2s212c213e2iα2+m3s213e2i(α3+δ), (3)

where are the Majorana phases and is the Dirac phase. The half-life of -decay and the effective mass are related through the nuclear matrix element , the phase-space factor and electron mass as

 1T0ν1/2=G0ν|Mν|2∣∣∣mνeeme∣∣∣2. (4)

In Fig. 1 we show the variation of with the lightest neutrino mass , where we have used the range of oscillation parameters from oscparam (). The blue and green areas correspond to taking CP conserving values, while the red regions correspond to the violation of the CP symmetry. The dashed and dotted horizontal purple lines represent the required effective mass that will saturate the GERDA and GERDA+HDM+IGEX limits, respectively gerda (). The orange lines correspond to the positive claim (90 C.L.)klapdor (). The bands represent the NME uncertainty, taken from the compilation in Ref. Dev:2013vxa (). As the plot suggests, a measurement of will give information on masses correlated with the CP violating phases, under the assumption that light neutrino exchange is the only underlying mechanism in -decay.

In addition, the light neutrino mass is also bounded from beta decay studies as well as from cosmology. The mass probed in beta-decay is bp () and the present  C.L. limit on this observable is from MAINZ mainz () and from Troitsk troitsk () collaborations, respectively. This bound can be improved by one order of magnitude down to eV from the beta decay experiment KATRIN katrin (), which is currently under commissioning. The sum of light neutrino masses is constrained from cosmology. In the quasi-degenerate regime , that is of particular interest for -decay, beta decay as well cosmological searches, we have . The recent upper bounds on the sum of light neutrino masses coming from Planck Ade:2013lta (), which we consider in our studies, are the following: i) eV, derived from the Planck+WP+highL+BAO data (Planck1) at C.L. and ii) eV from Planck+WP+highL (AL) (Planck2) at C.L. Ade:2013lta (). As pointed out in Refs. fogli (); MSV (); msvmoriond (); Dev:2013vxa () and is evident from Fig. 1, after imposing the bounds from cosmology (assuming standard cosmology), the light neutrino contribution itself can not satisfy the claim in klapdor () or saturate the current bounds hei (); Gando:2012zm (); gerda ().

## Iii Sterile neutrino exchange in (ββ)0ν-decay

Sterile neutrinos can also give large contributions to neutrinoless double beta decay as analyzed in detail in Refs. f2005 (); ibarra (); MSV (); msvmoriond (); Pascoli (); blennow (); meroni2013 (). We assume here sterile neutrinos 111For simplicity, we call the massive states mainly in the sterile neutrino direction simply ”sterile neutrinos” as commonly done in the literature. with a mass and which mix with . The half-life is kov (); MSV ()

 1T0ν1/2=K0ν∣∣ ∣∣Θ2eiMip2−M2i∣∣ ∣∣2, (5)

where and . Here is the NME for the light neutrino exchange and is for the heavy neutrino exchange, MeV is the exchanged momentum scale in -decay, is the active-sterile neutrino mixing and is the mass of the proton. In the subsequent discussions, we denote by and by for light sterile neutrinos, i.e. when . For the heavy sterile case , and we denote them by and , respectively. For light sterile neutrinos the above equation simplifies to

 (6)

while for the heavy sterile one we have

 1T0ν1/2≃G0νM2N∣∣ ∣∣V2eNimpMNi∣∣ ∣∣2. (7)

Using the above equations and the recent result from GERDA gerda (), we derive the bound on the active-sterile mixing angle, assuming only one light or heavy sterile neutrino participates in -decay. In all our subsequent analysis, we use the values of NMEs and from Ref. petcov (), corresponding to the axial vector cut-off . We use the phase-space for : Kotila:2012zza (). In Fig. 2, we show the upper bound on the active-light sterile neutrino mixing angle from -decay, that saturates the individual limit yrs from GERDA gerda (). The gray region is due to the uncertainty introduced by the NME corresponding to the light neutrino exchange. For comparison, we also show the other different bounds, first compiled in Ref. Atre:2009rg (). For the mass of sterile neutrino , the kink searches in -decay spectrum is a sensitive probe of sterile neutrinos. The excluded regions with contours that are labelled by , , , , and refer to the bounds from kink searches Galeazzi:2001py (); Hiddemann:1995ce (); Holzschuh:1999vy (); Holzschuh:2000nj (); Deutsch:1990ut (). Note that, in addition, we have also included the bound coming from beta decay experiment of Schreckenbach:1983cg (), which was not reported in Ref. Atre:2009rg (). The reactor and solar experiments Bugey and Borexino Back:2003ae (); Hagner:1995bn () are sensitive in the region few MeV. Exclusion contours have been drawn by looking into the decay of sterile neutrino into electron-positron pairs. On the other hand, for mass few MeV, the sensitive probe is the peak search in Britton:1992pg (), where the region inside the dot-dashed black contour is excluded. As can be seen from the figure, the bound on the active-light sterile neutrino mixing coming from -decay is the most stringent for most of the parameter spaces in plane. For the mass of the light sterile neutrino GeV, the bounds from different beta decay searches are close to the ones from -decay and possibly can be improved by the future beta decay experiments. In the range GeV, the bound from -decay is the most stringent, while around GeV, the bound from peak searches, Britton:1992pg (), can almost compete with the bound from -decay.

Similarly, the upper limit on the mixing angle is shown in Fig. 3. The gray region is due to the uncertainty in the NME corresponding to the heavy neutrino exchange. In addition, we also show the other different bounds, from Ref. Atre:2009rg (). The regions inside the brown dot-dashed line is excluded from the beam dump experiment PS191 Bernardi:1987ek (). For mass of sterile neutrino MeV, the stringent bound is obtained from the electron spectrum in meson decay decay knupeak (). For heavier masses (GeV), the decays into sterile neutrinos can be used to obtain exclusion contours, labelled as DELPHI and L3 Abreu:1996pa (); Adriani:1992pq (). See Ref. Atre:2009rg () and the references therein for the detail description of other different bounds Badier:1986xz (); Bergsma:1985is (). Also in this case, for most of the parameter space, the -decay gives the most stringent limit. For the mass of the heavy sterile neutrino MeV, the bound from the beam dump experiment PS191 is competitive with the one from -decay. For the positive claim klapdor (), the results are very similar and we do not show the corresponding region explicitly.

## Iv Cancellations among different contributions in (ββ)0ν-decay

The discussion of the previous section on the effective Majorana mass relies on the assumption that either the light or heavy neutrino exchange is the only underlying mechanism in -decay. However, in an extension of the standard model leading to light Majorana masses, the lepton number violating mechanism responsible for it will also contribute to neutrinoless double beta decay directly and could potentially interfere with the light neutrino one. Below we consider this possibility in detail. This is of particular interest, as it can solve the mutual inconsistency between the positive claim klapdor () and the results from KamLAND-Zen Gando:2012zm ().

If more than one mechanism is operative at -decay, the half-life of -decay for a particular isotope will receive different contributions as

 1T0ν1/2=G0ν(|η21|M21+|η22|M22+2cosα|η1||η2|M1M2), (8)

where is the phase-space factor, are the NMEs for the two different exchange processes. Here, and are the two dimensionless quantities which contain all the information from the particle physics parameters associated with the two exchange mechanisms and is the relative phase factor between them. The different exchange mechanisms can be for e.g. light neutrino and sterile neutrino exchange, or light neutrino and squark/gluino exchange. If a complete cancellation takes place between two exchange mechanisms, then the phase and . Consequently the half-life in Eq. 8 would be infinite, and this process in a specific nucleous would never be observed. However, this does not need to be the case for another isotope. Between two isotopes (A, B), if this cancellation is effective for isotope A, then the half life for isotope B is

 1T0ν1/2(B)=GB0ν|η21|(M1,B−M1,AM2,AM2,B)2, (9)

where , are the NMEs for the two exchange processes in isotope A and , are for isotope B. As an example we consider the case when the cancellation is effective in . In this case, the bound on half-life Gando:2012zm () is automatically satisfied, irrespective of the absolute magnitude of . Denoting the nuclear matrix elements for and by , and , and the phase space of by , the half-life of is

 1T0ν1/2(76Ge)=GGe0ν|η21|(M1,Ge−M1,XeM2,XeM2,Ge)2. (10)

The value of that saturates the lower limit of half-life from GERDA gerda () and GERDA+HDM+IGEX gerda () are

 |η1|≤(2.87,2.40)×10−6∣∣(M1,Ge−M1,XeM2,XeM2,Ge)∣∣, (11)

while the range of that satisfies the positive claim ( C.L.) in klapdor () is

 |η1|=(2.42−3.18)×10−6∣∣(M1,Ge−M1,XeM2,XeM2,Ge)∣∣. (12)

As stressed before, note that, the individual or the combined limit from GERDA gerda () does not conclusively rule out the positive claim in klapdor (). Hence, in addition to the GERDA, GERDA+HDM+IGEX limits gerda (), we also carry out the discussion on the positive claim klapdor (). If the above mentioned cancellation is operative for , it would be possible to automatically satisfy the bounds obtained by EXO-200, KamLAND-Zen collaboration Auger:2012ar (); Gando:2012zm () for isotope and yet to satisfy the claim in klapdor (), irrespective of any NME uncertainty. Hence, it is possible to reconcile any mutual conflict between the results of and .

### iv.1 Light active and heavy sterile neutrinos

We first discuss the case when the two interfering mechanisms correspond to light active and heavy sterile neutrino exchange. We also include the discussion when the cancellation is operative between light neutrino exchange and squark/gluino exchange mechanisms, for e.g. in R-parity violating supersymmetry.

First, we study the case of light active neutrinos and heavy sterile with mass , larger than the typical momentum exchange in -decay: . We consider maximum destructive interference between the two mechanisms, i.e. . A cancellation in isotope A will lead to the following relation,

 |ηN|=|ην|Mν,AMN,A. (13)

Here, we have replaced of the previous section by , respectively, where correspond to light neutrino exchange and correspond to the heavy sterile neutrino exchange. The nuclear matrix elements and in this case correspond to light and heavy neutrino exchange and have been denoted as and , respectively. In the above, the particle physics dimensionless parameters and are given by

 ην = mνeeme, (14) ηN = ∑jV2ejmpMNj. (15)

The half life for any other isotope B is predicted to be

 (16)

It can be rewritten in terms of an effective mass, where the redefined effective mass is

 ∣∣meffee∣∣=∣∣∣mνee(1−Mν,AMN,AMN,BMν,B)∣∣∣. (17)

Hence, if the light and heavy exchange contributions cancel each other for isotope A, for any other isotope B the effect would manifest itself by increasing the half-life. Below, as a relevant example, we again focus on the case in which the cancellation is present in and we explore its effect on the half-life of .

Using Eq. 16, the different values of redefined effective mass that is required to saturate the individual and combined limits of half-life from GERDA gerda () and to satisfy the positive claim ( C.L.) klapdor () are given in Table 1. The redefined effective mass is smaller than the true effective mass , as expected. We show the variation of the effective mass with the lightest neutrino mass scale in Fig. 4. The horizontal purple bands show the effect of NME uncertainties and correspond to the two different ranges of required effective masses eV (dashed purple band) and eV (dotted purple band) to saturate the GERDA and GERDA+HDM+IGEX gerda () limits, respectively. The horizontal dashed orange lines represent the minimum and maximum of the required ranges of effective mass that satisfies the positive claim klapdor (). In both of the figures, the vertical black solid line represents the future sentivity of KATRIN  eV katrin () and the other two vertical lines represent the bound and , following the two extreme bounds from Planck data set (Planck1) and (Planck2) Ade:2013lta (), respectively.

As can be seen from the figure, the effective mass can saturate the required values only in the quasi-degenerate regime. However, this possibility can be severely constrained by the future sensitivity from KATRIN katrin (), which does not depend on any particular physics model. In particular, for the bound from KATRIN katrin (), the effective mass can not reach the required value of . The bound from cosmology is even more stringent compared to the case when the light neutrinos are the only mediators and therefore the tension between cosmology and the possible claim in neutrinoless double beta decay is more severe. We also show the future sensitivity for by the horizontal brown and black lines that correspond to half-lives for GERDA Phase-II gerdafuture () and werner-rev (), respectively. It is evident from Fig. 4, that the effective mass can saturate the future limit from GERDA Phase-II around eV. This possibility is unconstrained from the most stringent limit from Planck and marginally constrained by the future sensitivity of KATRIN. For the half-life , the effective mass can saturate the limit even for as low as eV. This possibility is not at reach for future cosmological observations and beta decay experiments.

The cancellation between light contribution and heavy contribution can also be realized in other new physics scenarios, for e.g. R-parity violating supersymmetry. In this framework, the gluino and squarks can give large contribution in -decay. Below, we discuss the case when the cancellation is effective between light neutrino exchange and gluino/squark exchange. We denote the NMEs corresponding to the gluino exchange by and the squark exchange by and parameterize their contributions by and , respectively. The detail description of and on the fundamental parameters of the theory has been described in detail in Ref. f2010 (); allanach (), and we do not repeat them here. Like the previous case, the cancellation between light neutrino exchange and squark/gluino exchange in isotope A will result in a reduction of effective mass for any other isotope. The left and right panels of Fig. 5 corresponds to the two different cases, when the cancellation is effective between light neutrino-gluino and light neutrino-squark exchanges, respectively. The NMEs have been used from Ref. petcov (). The horizontal dashed and dotted purple lines represent the required effective mass that will saturate GERDA and GERDA+HDM+IGEX gerda () limits. They have been derived using Eq. 16 and includes the effect of cancellation in . The horizontal orange lines correspond to the required ranges of effective mass , that will satisfy the positive claim klapdor ().

### iv.2 Light and heavy sterile neutrinos

The tension discussed above between cosmology and neutrinoless double beta decay can be avoided if, in addition to the heavy sterile neutrinos, we also have light sterile neutrinos. The latter, depending on their mass and mixing, can give a large contribution even compared to the light active ones and can infact saturate the required value of . On the other hand, the bounds from cosmology is only relevant if the masses of the sterile neutrinos are very small and they were copiously produced in the Early Universe contributing to hot dark matter. For heavier masses, , the mixing angles of interest are very large and would lead to fast decays of sterile neutrinos and consequently to no bounds from cosmology. Hence, adding light sterile neutrinos in addition to heavy sterile ones can solve the mutual inconsistency between the positive claim in klapdor () and KamLAND-Zen Gando:2012zm (), can saturate the upper limits of effective masses for GERDA and GERDA+HDM+IGEX gerda () and can be in accordance with the bounds coming from cosmology. Here, we study in detail this case .

We assume both Majorana light sterile neutrinos with mass and heavy sterile neutrinos with mass . In this case, the half-life of any isotope is

 1T0ν1/2=G0ν(|η2l|M2ν+|η2N|M2N+2cosα|ηl||ηN|MlMN), (18)

where the parameters and correspond to the contributions from light and heavy neutrinos as

 ηl=(ΣimiU2ei+Σkm4kU2e4k)me,ηN=∑jmpV2eNjMNj. (19)

For simplicity we consider the case in which only one light sterile and one heavy sterile neutrinos are present. If the cancellation between light and heavy neutrino contribution is effective for isotope A, then following the discussions of previous sections, and are related as . For any other isotope B, the redefined effective mass is

 meffee=(mνee+m4U2e4)×(1−Mν,AMN,AMN,BMν,B). (20)

In the above we have dropped the generation index and denotes the mass of the light sterile neutrino, while is the active-light sterile mixing. We again assume a cancellation for and we examine its implications on . From Table 1, it is evident that to satisfy/saturate either the positive claim klapdor () or the limits from GERDA gerda (), a large effective mass eV is required. We denote the limiting values of effective masses of Table 1 by for GERDA, GERDA+HDM+IGEX gerda () and the minimum and maximum values of the required by and for the positive claim klapdor (). Following the stringent constraint from cosmology, the effective neutrino mass corresponding to the light neutrino exchange is extremely small eV (see Fig. 1) and we will neglect it in the following. Hence, if the total contribution saturates the limits from GERDA and GERDA+HDM+IGEX gerda (), the active-light sterile neutrino mixing is bounded as follows

 |U2e4|≲κm41∣∣∣(1−Mν,XeMN,XeMN,GeMν,Ge)∣∣∣ . (21)

On the other hand, in order to explain the positive claim in Ref.  klapdor () we need,

 κ1m41∣∣(1−Mν,XeMN,XeMN,GeMν,Ge)∣∣≲|U2e4|≲κ2m41∣∣(1−Mν,XeMN,XeMN,GeMν,Ge)∣∣. (22)

In Fig. 6, we show the upper bound on the active-light sterile neutrino mixing angle corresponding to the individual (solid lines) and combined (dashed lines) limits of half-life for from GERDA gerda (). The area in the plane, that is above this line is excluded.

The red, blue lines have been derived using the NMEs corresponding to the Argonne potential (intermediate and large size single-particle spaces, respectively) between two different nucleons and the purple and orange lines are using the NMEs corresponding to the CD-Bonn potential (intm and large, respectively). For the positive claim klapdor (), the variation of the active sterile mixing with mass of the sterile neutrino is quite similar and hence we do not show it separately. In this case, the cancellation for is operative mostly between the light sterile and heavy sterile neutrino contributions. For comparison we also show the other different bounds, first compiled in Ref. Atre:2009rg (). By comparing Fig. 6 with Fig. 2, it is evident that in the presence of cancellation, a larger mixing is required to give the same value of the half-life. Also, as compared to Fig. 2, in this case the bound on active-sterile mixing angle from can compete with the bound from -decay.

As we are assuming a cancellation in , the heavy sterile neutrino contribution is also constrained and a bound in the mass-mixing plane can be obtained. Using the cancellation relation and the values of as given in Table 1, the active-heavy sterile neutrino mixing angle corresponding to the GERDA and GERDA+HDM+IGEX limits gerda () is bounded as

 |V2eN|≲κMNmempMν,XeMN,Xe1∣∣(1−Mν,XeMN,XeMN,GeMν,Ge)∣∣, (23)

while for the positive claim klapdor (), it is

 κ1MNmempMν,XeMN,Xe1∣∣(1−Mν,XeMN,XeMN,GeMν,Ge)∣∣≲|V2eN|≲κ2MNmempMν,XeMN,Xe1∣∣(1−Mν,XeMN,XeMN,GeMν,Ge)∣∣. (24)

Note that the equations Eqs. 21, 22, 23 and 24 are only valid for the light and heavy sterile masses smaller and larger than the exchange momentum scale, MeV, respectively. We show the generic equation that is valid for all mass scales in the Appendix. Following Eq. 23 and the formalism given in Appendix, we show in Fig. 7 the upper bound on the active-heavy sterile mixing corresponding to the individual and combined limits of half-life from GERDA gerda (). The description of the different color coding is the same as in Fig. 4. The region above the different contours is excluded by -decay. In this figure, for comparison we also show the bounds coming from other experiments Atre:2009rg (). Again comparing Fig. 7 with Fig. 3, one can see a larger mixing angle required to saturate the limits on the half-life from -decay for the case of cancellation. For the mass of the heavy sterile neutrino MeV, the bound from the beam dump experiment PS191 Bernardi:1987ek () is even stronger than the -decay one. In the range GeV, the bound from CHARM Bergsma:1985is () can compete with the bound from -decay. For the positive claim klapdor (), the result is similar and we do not show the corresponding region explicitly.

## V Correlation between half-lives

In this section, we extend our discussion of the effects of cancellations to other isotopes. To this aim, for definiteness, we investigate how the cancellation between active and sterile neutrinos in would influence the half-life of other isotopes, such as , and as well as . The ratio of half-lives in two isotopes, isotope A and isotope B, is

 T0ν1/2(A)T0ν1/2(B)=GB0νGA0ν(Mν,B−Mν,XeMN,XeMN,B)2(Mν,A−Mν,XeMN,XeMN,A)2. (25)

Using Eq. 25, we show the correlations between half-lives of Te, -, Mo and Se in Figs. 8 and 9, respectively. We use different values of the NMEs which correspond to the various lines in the figures, as specified in the captions. The region within the two horizontal black dashed lines correspond to positive claim (90 C.L.) klapdor (). The horizontal black solid line corresponds to the individual limit from GERDA gerda (), where the region below this line is excluded. We also show the combined GERDA+HDM+IGEX limit gerda () by the green horizontal line.

Also in this case we can express the half-life in terms of the redefined effective mass which depends on the half-life and the NMEs as

 |meffee|=me√GA0νT0ν1/2(A)M2ν,A=∣∣∣mνee(1−Mν,XeMN,XeMN,AMν,A)∣∣∣ (26)

and similarly for the isotope B. The different numerical values shown in the figures represent the required effective mass in eV for a particular set of half-lives of the two isotopes. Finally, we conclude this section by showing the individual prediction of half-lives of and in Table 2 and Table 3, respectively.

## Vi Model-Seesaw realizations

As discussed above, the cancellation between light and heavy contributions to neutrinoless double beta decay in one isotope requires very specific values of neutrino masses and mixing angles. In this section we discuss how such values can emerge from theoretical models. The most natural framework embedding sterile neutrinos is the Type-I seesaw mechanism. Typically, heavy sterile neutrinos are introduced at or just below the GUT scale leading to light neutrino masses. If their mass is larger than few tens of TeV, the contribution in -decay would be negligibly small ibarra (); MSV (); Pascoli (). However, sterile neutrino can have much smaller masses, even well below the electroweak scale, e.g. in low energy see-saw models lowscaleseesaw (); lowscaleseesaw1 (). A lot of attention has been recently devoted to sterile neutrino states with masses lighter than TeV scale in -decay in Refs. ibarra (); MSV (); Pascoli (); blennow (); meroni2013 (). Below we discuss specific models which can accommodate light as well as heavy sterile neutrinos and lead to the cancellations we are interested in.

### vi.1 Model A - Light Active and Heavy Sterile Neutrinos

We consider first the case in which all sterile neutrinos are heavy, having masses larger than the momentum exchange scale MeV, see Section IV.1. We consider generations of sterile neutrinos denoted in the flavor basis. In the basis, the mass matrix of active+sterile neutrinos has the following form

 Mn=⎛⎜ ⎜⎝0αTmTDαμmTSmDmSmR⎞⎟ ⎟⎠ , (27)

where and are two lepton number violating parameters 222Depending on the choice of the lepton number assignment for the fields, different parameters in the mass matrix will be lepton number violating. Here, we adopt a common choice in which and are large masses and is very small.. Particularly interesting phenomenology arises for the hierarchy and which will lead to the Extended seesaw scenario Kang-Kim (); Parida (). We denote the mass basis as . The mass of the sterile neutrinos , are obtained by diagonalizing Eq. 27 and are given by

 mN ≃ −mTSm−1RmS, (28) mN′ ≃ mR. (29)

Let us note that for simplicity we call sterile neutrinos both the flavor states and the massive states which are mainly in the sterile neutrino direction. From the inequality it follows that . In the following discussion we consider the simplest case in which is negligibly small. The mass matrix of the active neutrino depends on the small lepton number violating parameter and is

 mν≃mTD(mTS)−1μ