Neutrinoless Double-Β Decay with Nonstandard Majoron Emission

PHYSICAL REVIEW LETTERS 122, 181801 (2019)

Neutrinoless Double-β Decay with Nonstandard Majoron Emission

† ‡ ∥ Ricardo Cepedello,1,2,* Frank F. Deppisch,2,3, Lorena González,4,2, Chandan Hati,5,§ and Martin Hirsch1, 1AHEP Group, Instituto de Física Corpuscular—CSIC/Universitat de València Edificio de Institutos de Paterna, Apartado 22085, E–46071 València, Spain 2Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom 3Institut für Hochenergiephysik, Österreichische Akademie der Wissenschaften, Nikolsdorfer Gasse 18, 1050 Wien, Austria 4Department of Physics, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile 5Laboratoire de Physique de Clermont, CNRS/IN2P3 UMR 6533, Campus des Cézeaux, 4 Avenue Blaise Pascal, F-63178 Aubière Cedex, France (Received 13 November 2018; revised manuscript received 1 February 2019; published 7 May 2019)

We present a novel mode of neutrinoless double-β decay with emission of a light Majoron-like scalar particle ϕ. We assume it couples via an effective seven-dimensional operator with a (V þ A) lepton current and (V A) quark currents leading to a long-range contribution that is unsuppressed by the light neutrino mass. We calculate the total double-β decay rate and determine the fully differential shape for this mode. β Λ ≈ 1 We find that future double- decay searches are sensitive to scales of the order NP TeV for the effective operator and a light scalar mϕ < 0.2 MeV, based on ordinary double-β decay Majoron searches. The angular and energy distributions can deviate considerably from that of two-neutrino double-β decay, which is the main background. We point out possible ultraviolet completions where such an effective operator can emerge.

DOI: 10.1103/PhysRevLett.122.181801

Introduction.—Double-β decay processes are sensitive just a charge-neutral scalar particle (Goldstone boson or probes of physics beyond the standard model (SM). not) or vector particle [3]. Originally considered to be The SM process of two-neutrino double-β (2νββ) decay massless, it may also be a light particle [4–6] that can is the rarest process ever observed with half-lives of order potentially be a dark matter candidate [7–9]. Searches for 2νββ ∼ 1021 β 0νββ extra particles in double-β decay are crucial in under- T1=2 yr. Neutrinoless double- ( ) decay, with no observation of any missing energy, is clearly the most standing neutrinos. Most importantly, violation of lepton important mode beyond the SM as it probes the Majorana number by two units and thus the Majorana nature of neutrinos can only be firmly established in the case of nature and mass mν of light neutrinos, with current 0νββ 2 26 0νββ decay. experiments sensitive as T ∼ ð0.1 eV=mνÞ × 10 yr. 1=2 Not all such emission modes have been discussed in the In general, it is a crucial test for any new physics scenario literature. Existing experimental searches so far focus on that violates lepton number by two units. the emission of one or two Majorons originating from the On the other hand, one or more exotic neutral particles intermediate neutrino exchanged in the process. The differ- may also be emitted, with a signature of anomalous missing ent Majoron scenarios have been classified into several 2νββ energy beyond that expected in decay. A well studied categories, all of which assume SM (V − A) charged set of theories involve the emission of a scalar particle, currents with the electrons and quarks. In this Letter, we called Majoron J. The first such proposed Majoron was a Goldstone boson associated with the spontaneous breaking of lepton number symmetry [1,2], coupling to a neutrino ν as gJννJ, cf. Fig. 1 (left). Current searches have a 0νββJ ∼ ð10−5 Þ2 1024 sensitivity of the order T1=2 =gJ × yr. The term Majoron has been used in a wider sense, implying

Published by the American Physical Society under the terms of 0νββ the Creative Commons Attribution 4.0 International license. FIG. 1. Feynman diagrams for ordinary J Majoron decay 0νββϕ Further distribution of this work must maintain attribution to (left), decay triggered by an effective operator of the form Λ−3 ð¯O Þð¯OνÞϕ the author(s) and the published article’s title, journal citation, NP u d e (center), and possible ultraviolet completion and DOI. Funded by SCOAP3. of the latter in a left-right symmetric model (right).

0031-9007=19=122(18)=181801(6) 181801-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 122, 181801 (2019) instead consider 0νββϕ decay with emission of a light broken or conserved is of crucial importance for an under- neutral scalar ϕ from a single effective dimension-seven lying model (determined by the chosen lepton numbers for Λ−3 ð¯O Þð¯OνÞϕ ν ϕ operator of the form NP u d e , cf. Fig. 1 (center), R and ) but as far as the effective interactions in Eq. (1) are with the fermion currents having a different chiral structure concerned, this does not play a role in our calculations. The from that in the SM. In the following, we will refer to the proton mass mp is introduced in the exotic interactions as light scalar as “Majoron,” independent of its origin. We ϵϕ normalization to make the effective coupling constants RL determine the sensitivity to Λ and analyze the effect on ϕ NP and ϵ dimensionless, in analogy to the effective operator the energy and angular distributions in comparison with RR treatment of 0νββ decay [10,11]. In Eq. (1), we omit exotic 2νββ decay. We also comment on ultraviolet scenarios operators with left-handed lepton currents; as in the standard underlying the effective operator. long-range case, such contributions will be additionally Effective long-range interactions.—We are interested in suppressed by the small neutrino masses [11]. We instead processes where right- and left-handed electrons are emit- focus on the process depicted in Fig. 1 (center), where the ted along with a scalar ϕ considering as a first approach, SM (V − A) Fermi interaction, the first term in Eq. (1), meets only (V þ A) and (V − A) currents. The effective one of the exotic operators. In this case, the momentum part Lagrangian can then be written as in the numerator of the neutrino propagator contributes, θ ϵϕ ϵϕ rather than the mass. In Eq. (1) we consider the first L ¼ GF pcosffiffiffi C μ þ RL μ ϕ þ RR μ ϕ 0νββϕ jLJLμ jRJLμ jRJRμ generation electron and neutrino only. Generalizing to three 2 mp mp ϕ flavors amounts to promoting the ϵ couplings to 3 × 3 þ ð Þ RX H:c:; 1 ðϵϕ Þ α ¼ μ τ matrices in generation space, RX αi ( e, , , θ ¼ ν ν ν with the Fermi constant GF, the Cabbibo angle C, and the i 1; 2; 3).P The final decay rate will then be proportional μ ¼ ¯γμð1 ∓ γ Þν jϵϕ j2 → j ðϵϕ Þ j2 leptonic and hadronic currents jL;R e 5 and to RX i RX eiUei , where U is the SM lepton μ ¼ ¯γμð1 ∓ γ Þ ν JL;R u 5 d, respectively. Here, is a four-spinor mixing matrix. field of the light electron neutrino, either defined by ν ¼ Decay rate and distributions.—We base our calculation ν þ νc 0νββϕ L L (i.e., a Majorana spinor constructed from the SM of the decay rate and kinematic distributions on Doi active left-handed neutrino νL)orν ¼ νL þ νR (a Dirac et al. [12]. The details of the calculation are given in the spinor constructed from the SM νL and a new SM-sterile Supplemental Material A [13]; here we outline the main right-handed neutrino νR). Whether the light neutrinos are of features. Summing over all intermediate nuclear states N, 0þ → 0þ 0νββϕ Majorana or Dirac type and whether total lepton number is the amplitude of I F decay can be written as

Z Z 2 X 3 ϕ ðG cos θ Þ d q M ¼ ϵ Fpffiffiffi C 3 3 ϕðyÞ iqðx−yÞ d xd y 2 e RX 2m 2π ω p N ρσ L ρσ R J ðx; yÞuρσðE1x;E2yÞ J ðx; yÞuρσðE1x;E2yÞ LX − XL − ð ↔ Þ ð Þ × 1 1 E1 E2 : 2 ω þ μN − 2 ðE1 − E2 − EϕÞ ω þ μN − 2 ðE1 − E2 þ EϕÞ

¼ ϵϕ ϵϕ μ ¼ − þ Here, X L, R correspond to RL, RR, N EN EI antisymmetry of the amplitude under the exchange of the Qββ=2 þ me with EI and EN the energies of the initial and two electrons. intermediate nucleus, respectively. The energies of the two In Eq. (2), the Majoron energy Eϕ is added or subtracted outgoing electrons and the Majoron are E1;2 and Eϕ, depending on whether the electron labeled 1 or 2 is being respectively, and the available kinetic energy release emitted from the exotic operator. The Majoron makes a Qββ. The nucleon and lepton currents are defined as crucial difference, as Eϕ goes together with ðE1 − E2Þ and not with the term proportional to the intermediate nuclei ρσ ðx yÞ¼h j ρ ðxÞj ih j σ ðyÞj i ¼ μ JYX ; F JY N N JX I ;X;Y L; R; energy N as for an ordinary Majoron. A dependence on Eϕ ð3Þ will thus appear through the matrix element in addition to that through the phase space. The differential decay rate for the 0þ → 0þ 0νββϕ decay can then be written as [12] 1 L;Rð x yÞ¼ μ ¯ð xÞγ γ γ ð1 ∓ γ Þ cð yÞ ð Þ uρσ E1 ;E2 2 q e E1 ρ μ σ 5 e E2 : 4 ð θ Þ4 2 Γ ¼ GF cos CgA me ½ ð Þ We consider that the internal neutrino propagates between d 7 2 a E1;E2 256π ðmpRÞ the interaction points x and y with momentum qμ ¼ðω; qÞ. þ ð Þ θ θ ð Þ From Eqs. (2) and (4) one can see explicitly the required b E1;E2 cos p1p2E1E2EϕdE1dE2d cos ; 5

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TABLE I. Current limits and expected future sensitivity on the focus on two isotopes: (i) 136Xe, for which the KamLAND- ϵϕ ϵϕ 0νββϕ 82 effective couplings RL and RR of decay for Se and Zen collaboration [39] currently provides the most stringent 136Xe. The limits are estimated based on the experimental half-life constraints; (ii) 82Se used by NEMO-3 and the upcoming constraints for ordinary Majoron emission (spectral index n ¼ 1) SuperNEMO experiments [38] that can measure the as given. Nuclear matrix elements from Refs. [35–37] where used detailed electron topology. for this estimate. For all experimental searches, the crucial distribution is

ϕ ϕ with respect to the sum of the kinetic energies of the Isotope T1 2 [yr] jϵ jjϵ j = RL RR detected electrons. With the SM 2νββ decay as irreducible 82Se 3.7 × 1022 [38] 4.1 × 10−4 4.6 × 10−2 background to any exotic signal, it is important to calculate 136Xe 2.6 × 1024 [39] 1.1 × 10−4 1.1 × 10−2 it precisely. In Fig. 2 (left), we compare the normalized total 82Se 1.0 × 1024 8.0 × 10−5 8.8 × 10−3 electron kinetic energy distribution of 0νββϕ decay with 136Xe 1.0 × 1025 5.7 × 10−5 5.8 × 10−3 that of 2νββ decay and ordinary 0νββJ Majoron decay (with spectral index n ¼ 1) for the isotope 136Xe. The ϵϕ distribution associated with RL is very similar to ordinary with the axial coupling gA of the nucleon and the radius R 0νββJ decay, while the introduction of a hadronic right- of the nucleus. The magnitudes of the electron spatial ϵϕ 0 ≤ θ ≤ π handed current in the RR term changes considerably the momenta are denoted p1;2 and is the angle shape of the distribution. In both cases, the spectral index between the emitted electrons. In Eq. (5), the Majoron still corresponds to n ¼ 1 with the characteristic onset near ¼ þ 2 − − energy is determined as Eϕ Qββ me E1 E2 by the kinematic endpoint. We emphasize that because of the energy conservation. Definitions for the coefficients different shape, a dedicated signal over background analy- aðE1;E2Þ, bðE1;E2Þ in the decay rate can be found in sis is required to determine the experimental sensitivity on the Supplemental Material A [13], where we show in detail ϵϕ ϵϕ the effective parameters RL and RR precisely. the derivation of the differential decay rate. Therein we use NEMO-3 and SuperNEMO are able to measure the the nuclear matrix elements listed in Table I and Coulomb- individual electron energies. In right-handed current sce- corrected relativistic electron wave functions. narios without emission of a Majoron, the single energy From Eq. (5), the total decay rate and thus the half-life is distribution exhibits a distinctive valley-type shape. This calculated by performing the integration of aðE1;E2Þ over occurs as the dominant term is proportional to ðE1 − E2Þ for all energies within the allowed phase space limits E1, the corresponding ϵRR term, as a result of the antisymmetry E2 ≥ 0 and E1 þ E2 ≤ Qββ þ 2me. In addition, we deter- with respect to electron exchange. (For the ϵRL term with a mine and discuss several distributions below. We will show left-handed hadronic current, P-wave and nuclear recoil ϵϕ ϵϕ results for the RL and RR versions of the effective contribute constructively, giving a dominant contribution operators where we consider only one of these to be proportional to ðE1 þ E2Þ [12].) In our case, depicted in present at a time. We assume the exotic ϕ Majoron to Fig. 2 (right), part of the energy is being carried away by the be massless in our calculations and comment on massive ϕ Majoron, shifting the distribution towards lower electron in the discussion below. For our numerical evaluation we energies and softening the characteristic valley-type

FIG. 2. Left: Normalized 0νββϕ decay distributions in the total kinetic energy of the electrons for 136Xe. Right: Normalized 0νββϕ 82 ϵϕ decay distribution in the single electron kinetic energy distribution for Se. The blue solid and red dashed lines correspond to the RR ϵϕ 2νββ 0νββ and RL cases, respectively. The corresponding distributions for the SM decay and ordinary J Majoron decay (spectral index n ¼ 1) are given for comparison.

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ϵϕ Discussion.—Searches for Majorons or Majoron-like distribution for RR. The distribution does not vanish for 1 particles are a staple in double-β decay experiments. So E1 − me ¼ 2 Qββ (as in the ordinary right-handed current case), but is still significantly different from that of ordinary far, they only cover the case where the neutrino involved − Majoron emission. The distribution with respect to both couples via the SM (V A) charged current interaction. electron energies is depicted in Fig. 1 (top panel) in the This is clearly a well-motivated minimal choice but it is Supplemental Material A [13]. It exhibits an even more worthwhile to explore other scenarios. In this Letter, we ϵϕ 2νββ have discussed one such alternative where a Majoron-like pronounced difference between the RR mode and . particle ϕ is emitted from effective operators with (V þ A) This may be used experimentally to improve the sensitivity leptonic currents, cf. Fig 1 (center). The future sensitivities through kinematic selection criteria, counteracting the effect ϵϕ ϵϕ on the effective couplings RL and RR shown in Table I of the less peaked total energy distribution, cf. Fig. 2 (left). Λ ≈ One can use also angular correlations to distinguish may be translated into effective operator scales NP 1 3 1 Λ3 ¼ between left-handed and right-handed currents [40,41], see . TeV and 270pffiffiffi GeV, respectively, using = NP ϵϕ θ ð 2 Þ Fig. 1 (bottom panel) in the Supplemental Material A [13]. RXGF cos C= mp . As noted before, we assume a Integrating over the electron energies one obtains the massless ϕ in deriving these limits; they remain essentially average angular distribution which takes the simple form unchanged for masses small compared to Qββ, mϕ ≲ ðdΓ=d cos θÞ¼ðΓ=2Þð1 þ k cos θÞ. The coefficient k is 0.2 MeV and are of the same order for mϕ ≲ 1 MeV, ϕ ¼þ0 70 mϕ → Qββ kRL . (electrons are dominantly emitted colline- but will deteriorate as (for a recent analysis in ϕ ¼ −0 05 ordinary Majoron emission, see Ref. [9]). Constraints on arly) and kRR . (electrons are emitted nearly iso- 0νββϕ ϵϕ ϵϕ our operators may also be set from other processes, such as tropically) in our scenarios with RL and RR, − − 82 exotic decay modes of the pion, π → e ν¯eϕ.Aswe respectively, for Se. For comparison, the angular corre- þ 2νββ 2νββ ¼ −0 66 consider only V A currents, helicity suppression will lation factor for SM decay is k . and still apply and the limits are expected to be correspondingly kJ ¼ −0.80 for ordinary Majoron emission; i.e., the elec- Λ ≳ 15 weak, we roughly estimate NP GeV. trons are dominantly emitted back to back. An ultraviolet scenario generating the effective operators Finally, we estimate the sensitivity of existing and in Eq. (1) is suggested in left-right symmetric models β planned future double- decay searches on the effective [44–47] where the SM W and ν are replaced by their ϵϕ ϵϕ 0νββϕ coupling strength RL and RR of decay. We would right-handed counterparts WR and N. The heavy neutrino N like to emphasize again that due to the different total then mixes with ν via a Yukawa coupling yν once the electron energy distribution, a dedicated signal over back- SM Higgs boson acquires its vacuum expectation value ground analysis is required to determine the constraints hHi¼174 GeV. A massless or light scalar ϕ is not part of precisely. Experiments such as NEMO-3 and SuperNEMO the minimal left-right symmetric model which thus needs to ð1Þ can also improve their sensitivity due to the nonstandard be modified, e.g., by keeping the U B−L symmetry global ϵϕ ϕ decay topology, especially for RR. As detailed in the or by extending its scalar sector. Charging under lepton Supplemental Material A [13], a requirement that any number allows coupling to N with a strength yN. The one electron has a kinetic energy of Ei − me >Qββ=2 corresponding diagram is shown in Fig. 1 (right). We can can for example reduce the 2νββ background by an order of then identify magnitude. Here, we simply estimate the sensitivity by 2 R 0νββϕ G cos θ ϕ g y yνhHi cos θ comparing our predictions for the decay half-life pF ffiffiffi C ϵ ¼ R N C ; ð6Þ ¼ 2 Γ 2 RR 8m2 m2 T1=2 ln = with the experimental constraints on ordi- mp WR N nary (n ¼ 1) Majoron emission. We use the most stringent limits for 82Se by NEMO-3 [38] and for 136Xe by leading to the estimate KamLAND-Zen [39]. For future prospects, we estimate Xe −4 2 4 4 that experimental Majoron search sensitivities may reach T1=2 3.5 × 10 mW m ≈ R N ; Se ≈ 1024 1025 2 θR 4 100 T1=2 yr (e.g., with the help of angular and energy yr gRyNyν cos C TeV MeV Xe ≈ 1025 selection cuts at SuperNEMO) and T1=2 yr. (The ð7Þ corresponding 0νββ decay sensitivities of the planned θR SuperNEMO [42] and nEXO experiments [43] may where gR is the gauge coupling constant and C the improve by Oð100Þ, but this requires an experimental equivalent of the Cabibbo angle, both associated with ð2Þ approach that is essentially background-free. This is not the SU R of the left-right symmetric model. possible for Majoron emission with a continuous total ϵϕ Alternatively, it is also possible to trigger the RL mode ϵϕ θ electron energy spectrum.) The corresponding limits on RL through the WR-W mixing . Its value is generically ϵϕ expected to be θ ¼ κg m2 =ðg m2 Þ where κ ¼ Oð1Þ. and RR are shown in Table I, where only one effective R W L WR operator is assumed to present at a time. In this case one has

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GF cos θC ϕ gRgLθyNyνhHi cos θC couple to it via thevector field itself as well as its field strength pffiffiffi ϵ ¼ ; ð Þ μν RL 8 2 2 8 tensor f . An even number of exotic neutral fermions χ may 2mp mWmN also be emitted but this would quickly increase the dimension resulting in the estimate of the corresponding effective operator. Instead, they may also originate from the internal neutrino via a dimension-six −2 Xe −4 2 4 4 operator of the form Λ ννχχ [51]. Exploring such alter- T1 2 1.4 × 10 m m NP = ≈ WR N : ð9Þ natives to the well-studied neutrinoless double-β decay is 1025 2 κ 25 100 yr gR yNyν TeV MeV imperative in order to be able to draw reliable conclusions on the nature of neutrino mass generation. ϵϕ This is more stringent due to the better sensitivity on RL in The work of R. C. and M. H. was supported by Table I. Choosing the right-handed neutrino mass mN to be as low as 100 MeV is strictly speaking not allowed the Spanish Grants No. FPA2017-85216-P (MICINN), in the effective operator treatment which requires No. SEV-2014-23510398 (AEI/FEDER, UE), No. PROMETEO/2018/165 (Generalitat Valenciana), mN ≫ pF ≈ 100 MeV, but it may be more natural in a scenario where the mass of N is generated through the Red Consolider MultiDark No. FPA2017-90566-REDC (MICINN), No. FPU15/03158, and No. EST17/00328. vacuum expectation value of ϕ, mN ¼ yNhϕi. In fact, The work of F. F. D. was supported by the UK STFC choosing mN to be smaller and abandoning the effective operator treatment may be more natural; the qualitative and a Royal Society Exchange Grant. The work of L. G. arguments should hold as above though a dedicated was supported by Conicyt Chile under Grant No. 21160645 calculation of 0νββϕ would be required. In addition, the and DGIIP of UTFSM. C. H. acknowledges support within ’ contribution to 0νββϕ via a heavy neutrino is expected to the framework of the European Union s Horizon 2020 research and innovation programme under the Marie peak at mN ≈ pF with the above estimates applying to a good approximation [48]. (In addition to the operators Sklodowska-Curie Grant Agreements No. 690575 and discussed here, the left-right symmetric scenario will also No. 674896. R. C. and L. G. would like to thank the induce a standard Majoron interaction ϕνν (leading to UCL Department of Physics for its hospitality. standard Majoron emission with spectral index n ¼ 1) after electroweak symmetry breaking from an operator of the form ϕHHνν. It is suppressed relative to our contributions *[email protected] † by an additional power of yν but does not suffer from [email protected] ‡ suppression by the heavy WR mass or the small WR − W [email protected] §[email protected] mixing.) 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