arXiv:1306.0580v2 [hep-ph] 25 Jul 2013 nt,∆ units, experi- all but far. 3]), so [1, empty Refs. up collection in came (a found ments tested be and can references proposed pro- of been collider also to have capture cesses, electron double neutrinoless like oﬁm h aoaantr fnurns[] Other [2]. neutrinos of nature ∆ Majorana the conﬁrms n w etcswt ∆ with contain- vertices diagrams two via indirectly ing or term, mass Majorana h bevto fti ∆ this of observation the ayn ubrb he nt 5,∆ [5], units (and three lepton non- by violate number are that baryon) SM there the instance, in For processes are perturbative neutrinos connected. 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Higgs SM the being vleo e.Aaoospoess such processes, Analogous MeV. 2 of -value → B l.Teepce ieie r extremely are lifetimes expected The ble. ν = − p R rlpo ubrvoainb orunits four by violation number lepton or bde,btnurnls udul beta quadruple neutrinoless but rbidden, φ 124 + + IPEMDLFR∆( FOR MODEL SIMPLE L U L ν steeitneo e,acdna,anomaly- accidental, new, a of existence the is ∼ SM e . (1) 4 yascalar a by 7 e.W loietf orcniae for candidates four identify also We MeV. 079 Xe, − y β 91 edleg Germany Heidelberg, 69117 ,adoescalar one and 4, . αβ etio r ia atce.We particles. Dirac are neutrinos eitr ( -emitters B + − L ν 130 L U L 4 α kinetic β (1) Hν Ba, ymty he RHNs three symmetry, † n siaeteepce lifetimes. expected the estimate and Z B R,β ′ − φ 148 ( ν L oo,dsrbdin described boson, ν 96 ihcharge with R + hc a hrfr econsis- be therefore can which , 2 d and Gd, Zr, β ,χ φ, , κ hspoeswudhwvrvio- however would process This . αβ 136 χ + ) χ R ν e and Xe, smere nRf [6]. Ref. in -symmetries − ∼ R,α 154 L 150 Z | B ν y.Mr detailed More Dy). B ′ ,alo hc are which of all 2, R,β d iha with Nd, c − − − V L h + L 150 ν | ( 4 = ) R ,φ χ φ, H, a then can 4 = d r fa- are Nd) L . ν c − ∼ Z R Z . ′ 4 L snot is , (RHNs) , Q ) ,one 1, nthe in 0 decay ν (2) 4 β - 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. We note that our ν particular example uses a gauged B L framework; in ν general however, the observation and− the model building ν e− possibilities that might lead to lepton number violating e− Dirac neutrinos are much broader. W − d u d u CANDIDATES FOR 0ν4β Figure 2: Neutrinoless quadruple beta decay 4d → 4u + 4e− c 2 Our model from the last section gave us the eﬀec- via a ∆L = 4 operator (ν ν) (ﬁlled circle). Arrows denote c 2 ﬂow of lepton number, colors are for illustration purposes. tive dimension-six ∆L = 4 operator (νRνR) , which can 3

the emitted electrons/positrons in a 0νnβ∓ decay,

A A ∓ 0ν − Z (Z n)+ ne , (8) Q 4β → ±

2 β is given by the Q-value, and can be calculated via − + A A β Q − = M[ Z] M[ (Z + n)] , (9) 2 0νnβ − A A Mass Q + = M[ Z] M[ (Z n)] 2nm . (10) 0νnβ − − − e + The term 2nme in Q0νnβ+ already makes 0ν2β very − + Z − 2 Z Z +2 rare, but neutrinoless quadruple positron decay 0ν4β impossible. Electron capture with the emission of up to Figure 3: Three beta-stable even–even nuclei on their mass two positrons is however permitted, as the Q-value for parabola (black). The heaviest isobar (A, Z − 2) can decay the EC-process either via double beta decay into the lowest state (A, Z), or via 0ν4β into the medium state (A, Z + 2). Also shown are AZ + ke− A(Z n) + (n k) e+ (11) the “forbidden” odd–odd states in between (red). → − −

is given by Q0νkEC(n−k)β+ = Q0νnβ+ + 2kme, allow- ing above all for neutrinoless quadruple electron capture Q Other decays NA 0ν4β 0ν4EC in four isotopes (Tab. I). 2ν2β 96Zr 96Ru 0.629 τ 2 1019 2.8 Having identiﬁed all ∆L = 4 candidates, we discuss 40 → 44 1/2 ≃ × 136Xe 136Ce 0.044 τ 2ν2β 2 1021 8.9 their experimental prospects and challenges in more de- 54 → 58 1/2 ≃ × 2ν2β tail: Let us ﬁrst take a look at the most promising ele- 150Nd 150Gd 2.079 τ 7 1018 5.6 150 60 → 64 1/2 ≃ × ment for 0ν4β: Nd. The following decay channels are possible (see also Fig. 3): Q0ν4EC 124 124 150 150 54 Xe 50 Sn 0.577 — 0.095 60 Nd 62 Sm via 2ν2β, i.e. via the forbidden in- → • termediate→ odd–odd state 150Pm. Two neutrinos 130Ba 130Te 0.090 τ 2ν2EC 1021 0.106 61 56 → 52 1/2 ∼ and two electrons are emitted; the electrons hence 148Gd 148Nd 1.138 τ α 75 — 64 → 60 1/2 ≃ have a continuous energy spectrum and total en- 154 154 α 6 ergy Ee,1 +Ee,2 < 3.371MeV. This decay has been 66 Dy 62 Sm 2.063 τ1/2 3 10 — → ≃ × observed with a half-life of 7 1018 yrs. × Q0ν3ECβ+ 150Nd 150Gd via 0ν4β. Four electrons with 148Gd 148Nd 0.116 τ α 75 — • 60 → 64 64 → 60 1/2 ≃ continuous energy spectrum and summed energy 154Dy 154Sm 1.041 τ α 3 106 — Q = 2.079 MeV are emitted. In this special 66 → 62 1/2 ≃ × 0ν4β case, the daughter nucleus is α-unstable with half- Q0ν2EC2β+ α 150 146 6 life τ1/2(64 Gd 62 Sm) 2 10 yrs. 154 154 α 6 → ≃ × 66 Dy 62 Sm 0.019 τ1/2 3 10 — → ≃ × A sketch of the summed electron energy spectrum is shown in Fig. 4. Q will always sit somewhere in Table I: Candidates for nuclear ∆L = 4 processes neutrinoless 0ν4β the middle of the continuous spectrum,2 so one would quadruple beta decay and electron capture, the corresponding Q-values in MeV, competing (observed) decay channels with have to identify the four electrons in order to remove the j 2ν2β background. This still leaves other backgrounds half-life τ1/2 in years, and natural abundance (NA) in percent. to be considered, e.g. the scattering of the two 2ν2β electrons oﬀ of atomic electrons, which can eﬀectively that lead to four emitted electrons (and two neutrinos). Since Q0ν4β < Q2ν2β, the sum of the electron energies will M[A(Z 2)] M[A(Z + 2)] be continuously distributed and can overlap the discrete − − =2 , (7) M[A(Z 1)] M[A(Z + 1)] Q0ν4β peak. A dedicated discussion of this and other − − possible backgrounds goes far beyond the scope of this where M[AZ] denotes the mass of the neutral atom AZ letter. in its ground state. Applied to our problem, this means As an alternative to direct searches, one could even that the mass splitting of the odd–odd states in Fig. 3 omit an energy measurement and just look at the trans- 150 150 (shown in red) is expected to be smaller than the mass mutation Nd Gd using, e.g., chemical meth- ods; as the background→ for 150Nd 150Gd is basically splitting of the two ∆Z = 4 nuclei (which is just the → Q-value, see below), which implies that beta-stable 0ν4β candidates will decay into beta-stable nuclei (this simple argument is conﬁrmed with data charts [7]). 2 We note that if neutrinos are Majorana particles the decay 150 150 The Q-values in Tab. I can be readily calculated in 60 Nd → 62 Sm via 0ν2β is possible. Two mono-energetic elec- analogy to 0ν2β. In general, the total kinetic energy of trons would be emitted with total energy Q0ν2β = 3.371 MeV. 4

can be checked in the same way. Note that the energy- 0ν2β gain Q0ν4EC will here be carried away by photons instead of electrons; the captured electrons will be taken out of the K and L shells, resulting in a cascade of X-ray pho- 148 154

Decay rate tons. The Q-values of Gd and Dy are high enough 2ν2β to also undergo 0ν3ECβ+; 154Dy is the only isotope ca- pable of 0ν2EC2β+. This can give rise to distinguish- able signatures due to the additional 511 keV photons 0ν4β from electron–positron annihilation. The comparatively fast α-decay of 148Gd and 154Dy—and the fact that they Q0ν4β Q0ν2β Energy have to be synthesized from scratch—make them how- ever very challenging probes for ∆L = 4, despite their 124 Figure 4: Sum of kinetic electron energies in the beta decays large Q-values. Xe might then be the best element to 0ν2β, 2ν2β, and 0ν4β. Relative contribution not to scale. test for 0ν4EC; unfortunately, the enriched xenon used by EXO contains almost no 124Xe, so 0ν4EC is currently hard to test (dark matter experiments using xenon can nonexistent—the SM-allowed 4ν4β is killed by the Q- in principle be used, as they contain 124Xe). Resonant dependence of the eight-particle phase space G4ν4β enhancement of the 0ν4EC rates, as discussed for the Q23 (compared to the four-particle phase space G ∼ 0ν2EC mode [11], might boost the signal. 0ν4β ∼ Q11), and 0ν2β would be seen long before we ever see Apparently, ∆L = 4 signals are in general easier to the double 0ν2β that mimics 0ν4β. Hence, this trans- test via the 0ν4β channels, with both 96Zr and 150Nd as mutation suﬃces to test 0ν4β. In case of 150Nd, the more favorable isotopes when it comes to Q-values and instability of the daughter nucleus 150Gd can even be natural abundance. advantageous, as the resulting alpha particle provides an additional handle to look for the decay.3 The nec- essary macroscopic number of daughter elements will of course result in weak limits compared to dedicated 0ν4β RATES FOR 0ν4β searches in 0ν2β experiments. However, for elements not under consideration in 0ν2β experiments, this could be Let us estimate some rates. Similar to 0ν2β, the half- a viable and inexpensive way to test 0ν4β. life of 0ν4β can approximately be factorized as There is also the possibility of decay into an excited 150 150 ∗ −1 state, 60 Nd 64 Gd via 0ν4β. The excited ﬁnal 0ν4β 2 → τ1/2 = G0ν4β 0ν4β , (12) state will reduce the eﬀective Q-value—by 0.638 MeV h i |M | (1.207 MeV) for the lowest 2+ (0+) state—and produce where G denotes the phase space and the detectable photons. 0ν4β M0ν4β All the above holds similarly for 96Zr and 136Xe as nuclear transition matrix element (including the parti- well. Both have much smaller Q-values—which theoreti- cle physics parameters) facilitating the process. Using c 2 2 cally reduces the rate—but α-stable daughter nuclei. The an eﬀective ∆L = 4 vertex (νLνL) /Λ gives 0ν4β 4 4 2 M ∝ non-solid structure of xenon makes it in principle easier GF /pν Λ , just by counting propagators. For the virtual to check for the transmutation into cerium; furthermore, neutrino momentum pν we will use the inverse distance between the decaying nucleons, p q 1 fm−1 the EXO [8] 0ν2β experiment is currently running and ν ∼ | | ∼ ≃ could check for 0ν4β, should their detector be sensitive 100 MeV. The phase-space factor for the four ﬁnal parti- 11 at these energies and not ﬂooded by backgrounds. 96Zr cles is the same as the one in 2ν2β (proportional to Q for Q m [12]), which also tells us that each of the is a better candidate due to its higher Q-value, but there ≫ e are no dedicated 96Zr experiments planned. Still, the four electrons will be distributed just like the electrons 0.629 in 2ν2β, with a diﬀerent Q-value, of course. Purely on di- NEMO collaboration could set limits on 96Zr 96Ru mensional grounds we can then estimate the dependence by reanalyzing their data from Ref. [9]. 150Nd−−−→ is by far of the half-life on our parameters as the best candidate, due to the high Q0ν4β-value. Coinci- dentally, it also has a high Q0ν2β-value, which makes it a −1 4 2 popular isotope to test for 0ν2β, with some existing and 0ν4β 11 GF 18 τ1/2 Q 4 2 q , (13) planned experiments [1]. Once again, NEMO might al- h i ∝ q Λ ready be able to constrain 150Nd 2.079 150Gd with their data [10]. −−−→ where the last factor is included to obtain the correct The 0ν4EC channels in Tab. I lead to a similar trans- mass dimension. The above estimate is only valid for mutation behavior as discussed above for 0ν4β−, and large Q-values, as it assumes massless electrons; the low Q0ν4β of most elements in Tab. I render (some of) the four electrons non-relativistic and make necessary a more accurate calculation of the phase space. To partially can- 3 The alpha decay is however too slow to be used in coincidence cel the uncertainties, we can approximate that the phase with 0ν4β. space for 0ν4β and 2ν2β is overall similar and consider 5 the ratio (for 150Nd and q 100 MeV) CONCLUSION | |≃ Contrary to popular belief, Majorana neutrinos are not τ 0ν4β 11 4 4 a prerequisite for lepton number violation, and we have 1/2 Q0ν2β Λ 46 Λ 2ν2β 12 4 10 . (14) given a simple counterexample of lepton number violat- τ ≃ Q0ν4β q G ≃ TeV 1/2 F ing Dirac neutrinos in this work. This gives rise to previ- ously undiscussed ∆L = 4 processes, the most striking of which would be neutrinoless quadruple beta decay, which This is of course a rough estimate, and a better calcu- can in principle be observable in three nuclei. The most 150 lation, dropping the implicitly used closure approxima- promising isotope is Nd due to its high Q0ν4β-value tion, including eﬀects of the nuclear Coulomb ﬁeld etc., and natural abundance (see Tab. I), and existing exper- will certainly change this rate. To this eﬀect we stress iments could already be used to test 0ν4β. a diﬀerence between 0ν2β and 0ν4β: while the former Let us stress that the decay should be constrained ex- decay proceeds via a kinematically forbidden intermedi- perimentally, as our theoretical estimates for TeV-scale ate state, the latter also features an energetically pre- physics induced 0ν4β might be too conservative. Not ferred intermediate state X, only to rush past it on the only is it a novel possible decay channel on the nuclear mass parabola (see Fig. 3). Since excited states of X can physics side, but it contains very interesting conceptual still have a lower mass than our initial nucleus, the sum- information about the fate of the classically conserved mation over all these states is important and cannot be lepton number symmetry. approximated away as easily as the excited states of an The authors thank Kai Zuber for discussions and com- already forbidden intermediate state. ments. J.H. thanks Sebastian Lindemann for experimen- tal insights and acknowledges support by the IMPRS- Finally, in our simple model from above, we generate PTFS. This work was supported by the Max Planck So- c 2 the ∆L = 4 operator with RHNs, (νRνR) , so each of the ciety in the project MANITOP. neutrinos in Fig. 2 requires a mass-ﬂip in order to couple to the W bosons. The particle physics amplitude is there- 4 −37 fore further suppressed by a factor (mν /q) 10 , making this process all the more unlikely. These≃ mass- ﬂips can be avoided in left–right-symmetric extensions of ∗ Electronic address: [email protected] † our model, at the price of replacing the four W bosons Electronic address: [email protected] [1] For a recent review see W. Rodejohann, Int. J. Mod. in Fig. 2 with their heavier WR counterparts. Phys. E 20, 1833 (2011) [arXiv:1106.1334 [hep-ph]]. Even with all our approximations leading to the above [2] J. Schechter and J. W. F. Valle, Phys. Rev. D 25, 2951 estimates, one can safely conclude that the half-life for (1982). [3] A. Atre, T. Han, S. Pascoli, and B. Zhang, JHEP 0905, neutrinoless quadruple beta decay is very large, at least 030 (2009) [arXiv:0901.3589 [hep-ph]]. if physics at the TeV scale is behind it in any way. [4] For a review, see R. Barbier et al., Phys. Rept. 420, 1 This may be a too conservative approach, because four- (2005) [hep-ph/0406039]. neutrino interactions do not suﬀer from such stringent [5] G. ’t Hooft, Phys. Rev. Lett. 37, 8 (1976); V. A. 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