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Neutrinoless Quadruple Beta Decay (0Ν4β) Is 2 2 2 and Zero, Respectively

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Neutrinoless Quadruple Beta Decay (0Ν4β) Is 2 2 2 and Zero, Respectively Neutrinoless Quadruple Beta Decay Julian Heeck∗ and Werner Rodejohann† Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany We point out that lepton number violation is possible even if neutrinos are Dirac particles. We illustrate this by constructing a simple model that allows for lepton number violation by four units only. As a consequence, neutrinoless double beta decay is forbidden, but neutrinoless quadruple beta decay is possible: (A, Z) → (A, Z +4)+4 e−. We identify three candidate isotopes for this decay, the 150 most promising one being Nd due to its high Q0ν4β -value of 2 MeV. Analogous processes, such as neutrinoless quadruple electron capture, are also possible. The expected lifetimes are extremely long, and experimental searches are challenging. INTRODUCTION We will first construct a simple toy model that for- bids Majorana neutrinos but allows for LNV by four Of all the open questions concerning neutrinos—mass units. Then we will search for interesting isotopes that scale and hierarchy, possible CP violation, origin of the can undergo 0ν4β and estimate the expected lifetimes. mixing pattern—the conceptually most interesting has Interestingly, all three isotopes that we identify as po- 96 136 150 to be its very nature: is the neutrino its own antiparti- tential 0ν4β-emitters ( Zr, Xe, and Nd) are fa- cle, and hence a Majorana fermion, or do neutrino and miliar from searches for neutrinoless double beta decay. 150 antineutrino differ, making the neutrino a Dirac particle The most interesting candidate is Nd, with a Q0ν4β- like all the other fermions of the Standard Model (SM). value of 2.079 MeV. We also identify four candidates for The key observation here would be neutrinoless double neutrinoless quadruple electron capture and related pro- 124 130 148 154 beta decay (0ν2β) [1], (A, Z) (A, Z+2)+2 e−, because cesses ( Xe, Ba, Gd, and Dy). More detailed the observation of this ∆L =→ 2 process unambiguously studies regarding model building aspects, collider phe- confirms the Majorana nature of neutrinos [2]. Other nomenology and cosmological aspects will be presented ∆L = 2 signatures, ranging from low-energy processes elsewhere. like neutrinoless double electron capture to collider pro- cesses, have also been proposed and tested (a collection B − L of references can be found in Refs. [1, 3]), but all experi- SIMPLE MODEL FOR ∆( ) = 4 ments came up empty so far. The necessary lepton number violation (LNV) by two We introduce three right-handed neutrinos νR (RHNs) units, ∆L = 2, can be realized directly with a tree level to the SM, which results in Dirac masses for the neutri- Majorana mass term, or indirectly via diagrams contain- nos after spontaneous electroweak symmetry breaking. ing two vertices with ∆L = 1, one example being R- A striking feature of the chiral fermion content of the parity violating supersymmetry [4]. However, it is most SM+νR is the existence of a new, accidental, anomaly- often overlooked that LNV and Majorana neutrinos are free symmetry U(1)B−L, which can therefore be consis- not necessarily connected. For instance, there are non- tently gauged in addition to the SM gauge group. Break- perturbative processes in the SM that violate lepton (and ing B L by a scalar φ with charge B L = 4 can then − | − | ZL baryon) number by three units [5], ∆L = ∆B = 3, which lead to a remaining discrete symmetry group 4 in the obviously do not lead to Majorana neutrinos, and are in lepton sector, which protects the Dirac structure of neu- fact perfectly compatible with Dirac neutrinos. trinos and still allows for LNV processes. Quartic LNV In this letter we will entertain the possibility that LNV operators for Dirac neutrinos were also mentioned in a occurs only by four units, and that ∆L = 2 processes study of anomaly-free discrete R-symmetries in Ref. [6]. For a simple realization of this idea, we work with a arXiv:1306.0580v2 [hep-ph] 25 Jul 2013 are forbidden; neutrinos are then Dirac particles. We gauged U(1) − symmetry, three RHNs ν 1, one will realize those lepton number violating Dirac neutri- B L R ∼ − nos in a simple model based on a spontaneously broken scalar φ 4, and one scalar χ 2, all of which are SM-singlets.∼ The Lagrangian takes∼ −the form U(1)B−L. As an interesting consequence, neutrinoless quadruple beta decay (0ν4β), = SM + kinetic(νR,φ,χ)+ Z′ V (H,φ,χ) − L L L L − (2) (A, Z) (A, Z +4)+4 e , (1) c → + yαβLαHνR,β + καβχ νR,ανR,β + h.c. , is allowed. This novel nuclear decay process plays for our H being the SM Higgs doublet. The phenomenology of framework the role that neutrinoless double beta decay ′ the accompanying Z boson, described in ′ , is not plays for Majorana neutrinos: it will be the dominant Z important here. Working in the diagonal chargedL lep- possible LNV process for Dirac neutrinos, which is surely ton basis, the neutrinos obtain the Dirac mass matrix of great conceptual interest even if the decay rates that we estimate are tiny.1 noless beta decay after 0ν2β. This process would however vio- late Lorentz symmetry, similar to neutrinoless single beta decay 1 One might think that 0ν3β should be the next probable neutri- n → p + e−. 2 M H y upon electroweak symmetry breaking. νc φ ν αβ αβ R h i R A bi-unitary≡ |h i| transformation can be used to diagonalize χ this mass matrix via U †M V = diag(m ,m ,m ), where + 1 2 3 χ χ φ U is the lepton mixing matrix relevant for electroweak h i c χ charged-current interactions. Contrary to other models νR νR with Dirac neutrinos, the right-handed transformation Figure 1: Tree-level realization of the ∆L = 4 operator matrix V does not drop out, but can be absorbed by the c 2 c c → (νRνR) describing the scattering νRνR νRνR. complex symmetric Yukawa coupling matrix καβ = κβα, which is non-diagonal in general. The scalar potential of our model is of the simple form lead to an interesting signature in beta decay measure- V (H,φ,χ) µ2 X 2 + λ X 4 ments: four nucleons undergo beta decay, emitting four ≡ X | | X | | X=XH,φ,χ neutrinos; these four meet at the effective ∆L = 4 ver- (3) + λ H 2 φ 2 + λ H 2 χ 2 + λ χ 2 φ 2 tex and remain virtual. We only see four electrons go- Hφ| | | | Hχ| | | | χφ| | | | − 2 ing out, so at parton level we have 4d 4u + 4e , µφχ + h.c. − → − and on hadron level 4n 4p + 4e (Fig. 2). Obvi- → Here, the coefficients µj and λj have mass dimension one ously this neutrinoless quadruple beta decay (0ν4β) is 2 2 2 and zero, respectively. Assuming µH ,µφ < 0 < µχ and highly unlikely—more so than 0ν2β, as it is of fourth appropriate signs and magnitudes of the λj , we can easily order—but one can still perform the exercise of identi- construct a potential that is bounded from below and fying candidate isotopes for the decay and estimating ZL breaks SU(2)L U(1)Y U(1)B−L to U(1)EM 4 . In the lifetime; constraining the lifetime experimentally is order to forbid Majorana× × neutrinos, it is imperative× that of course also possible. Besides 0ν4β, one can imagine χ does not acquire a vacuum expectation value; without analogous processes such as neutrinoless quadruple elec- the last line in Eq. (3), the necessary condition for this tron capture (0ν4EC), neutrinoless quadruple positron would be decay (0ν4β+), neutrinoless double electron capture dou- ble positron decay (0ν2EC2β+), etc. We will find po- m2 µ2 + λ H 2 + λ φ 2 > 0 , (4) c ≡ χ Hχh i χφh i tential candidates for 0ν4β, 0ν2EC2β+, 0ν3ECβ+, and but the µ term modifies this condition. To see how, let us 0ν4EC. first note that we can chose µ and φ real and positive We will now identify those candidate isotopes for ∆L = h i w.l.o.g. using phase and B L gauge transformations. 4 processes. We need to find isotopes which are more sta- − The µ term will then induce a mass splitting between ble after the flip (A, Z) (A, Z 4). Normal beta decay the properly normalized real (pseudo)scalar fields Re (χ) has to be forbidden in order→ to± handle backgrounds and and Im (χ) make the mother nucleus sufficiently stable. Using nu- m2 = m2 2µ φ , m2 = m2 +2µ φ , (5) clear data charts [7], we found seven possible candidates: Re (χ) c − h i Im(χ) c h i three for 0ν4β, four for neutrinoless quadruple electron 2 so the condition χ = 0 becomes equivalent to mRe (χ) > capture and related decays. They are listed in Tab. I, 0, which can beh easilyi satisfied. together with their Q-values, competing decay channels, Neutrinos are hence Dirac particles, but we also obtain and natural abundance. It should be obvious that not effective ∆L = 4 four-neutrino operators by integrating all 0ν2β candidates (A, Z) make good 0ν4β candidates, out χ at energies E mRe (χ), mIm(χ): as (A, Z + 4) can have a larger mass than (A, Z); it is ≪ less obvious that there exist no 0ν4β candidates with ∆L=4 1 −2 −2 c 2 eff mIm(χ) mRe (χ) καβνR,ανR,β + h.c., beta-unstable daughter nuclei. Using the semi-empirical L ⊃ 2 − (6) Bethe–Weizs¨acker mass formula, one can however show see Fig. 1 for the relevant Feynman diagrams. For sim- plicity, we will assume physics at the TeV scale as the d u source of our four-neutrino operators throughout this pa- d u per; a discussion of more constrained light mediators, as well as of other and more complicated models that gen- W − e− erate effective four-neutrino operators with left-handed ν e− neutrinos, will be presented elsewhere.
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