September 27, 2012 Ref. SISSA 26/2012/EP Ref. OU-HET 756/2012 Observables in Neutrino Mass Spectroscopy Using Atoms D. N. Dinha,b), S. T. Petcova,c) 1 N. Sasaod), M. Tanakae) and M. Yoshimuraf) 2 a)SISSA and INFN-Sezione di Trieste, 1 Via Bonomea 265, 34136 Trieste, Italy. 0 2 b)Institute of Physics, Vietnam Academy of Science and Technology, p 10 Dao Tan, Hanoi, Vietnam. e S c)Kavli IPMU, University of Tokyo (WPI), Tokyo, Japan. 6 d)Research Core for Extreme Quantum World, Okayama University, 2 Okayama 700-8530 Japan ] e)Department of Physics, Graduate School of Science, Osaka University, h p Toyonaka, Osaka 560-0043, Japan. - p f)Center of Quantum Universe, Faculty of Science, Okayama University, e Okayama 700-8530, Japan. h [ 2 v 8 0 Abstract 8 4 . 9 The process of collective de-excitation of atoms in a metastable level into emission mode of a single 0 photon plus a neutrino pair, called radiative emission of neutrino pair (RENP), is sensitive to the 2 1 absolute neutrino mass scale, to the neutrino mass hierarchy and to the nature (Dirac or Majorana) : of massive neutrinos. We investigate how the indicated neutrino mass and mixing observables can v i be determined from the measurement of the corresponding continuous photon spectrum taking the X example of a transition between specific levels of the Yb atom. The possibility of determining the r a nature of massive neutrinos and, if neutrinos are Majorana fermions, of obtaining information about the Majorana phases in the neutrino mixing matrix, is analised in the cases of normal hierarchical, inverted hierarchical and quasi-degenerate types of neutrino mass spectrum. We find, in particular, how critically the sensitivity to the nature of massive neutrinos depends on the atomic level energy difference relevant in the RENP. 1Also at: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria 1 1 Introduction Determining the absolute scale of neutrino masses, the type of neutrino mass pattern, which can be either the normal or the inverted ordering 2 (NO or IO), the nature (Dirac or Majorana) of massive neutrinos, andgettinginformationabouttheDiracandMajoranaCPviolationphases intheneutrino mixing matrix, are the most pressing and challenging problems of the future research in the field of neutrino physics (see, e.g., [1]). At present we have compelling evidence for existence of mixing of three massive neutrinos ν , i = 1,2,3, in the weak charged lepton current (see, e.g., [2]). The masses i m 0 of thethree light neutrinos ν do not exceed a value approximately 1 eV, m < 1 eV. The three i i i ≥ ∼ neutrino mixing scheme is described (to a good approximation) by the Pontecorvo, Maki, Nakagawa, Sakata (PMNS) 3 3 unitary mixing matrix, U . In the widely used standard parametrisation PMNS × [1], U is expressed in terms of the solar, atmospheric and reactor neutrino mixing angles θ , PMNS 12 θ and θ , respectively, and one Dirac (δ), and two Majorana [3, 4] (α and β) CP violation (CPV) 23 13 phases. In this parametrisation, the elements of the first row of the PMNS matrix, U , i = 1,2,3, ei which play important role in our further discussion, are given by U = c c , U = s c eiα, U = s ei(β−δ), (1) e1 12 13 e2 12 13 e3 13 where we have used the standard notation c = cosθ , s = sinθ with 0 θ π/2, 0 δ 2π ij ij ij ij ij ≤ ≤ ≤ ≤ and, in the case of interest for our analysis 3, 0 α,β π, (see, however, [5]). If CP invariance ≤ ≤ holds, we have δ = 0,π, and [6] α,β = 0,π/2,π. The neutrino oscillation data, accumulated over many years, allowed to determine the parameters which drive the solar and atmospheric neutrino oscillations, ∆m2 ∆m2 , θ and ∆m2 ⊙ ≡ 21 12 | A| ≡ ∆m2 = ∆m2 , θ , with a high precision (see, e.g., [2]). Furthermore, there were spectacular | 31| ∼ | 32| 23 developments in the last year in what concerns the angle θ (see, e.g., [1]). They culminated in a 13 high precision determination of sin22θ in the Daya Bay experiment using the reactor ν¯ [7]: 13 e sin22θ = 0.089 0.010 0.005. (2) 13 ± ± Similarly, the RENO, Double Chooz, and T2K experiments reported, respectively, 4.9σ, 3.1σ and 3.2σ evidences for a non-zero value of θ [8], compatible with the Daya Bay result. 13 A global analysis of the latest neutrino oscillation data presented at the Neutrino 2012 Inter- national Conference [2] was performed in [9]. We give below the best fit values of ∆m2 , sin2θ , 21 12 ∆m2 and sin2θ , obtained in [9], which will be relevant for our further discussion: | 31(32)| 13 ∆m2 = 7.54 10−5 eV2, ∆m2 = 2.47 (2.46) 10−3 eV2, (3) 21 × | 31(32)| × sin2θ = 0.307, sin2θ = 0.0241 (0.0244), (4) 12 13 where the values (the values in brackets) correspond to NO (IO) neutrino mass spectrum. We will neglect the small differences between the NO and IO values of ∆m2 and sin2θ and will use | 31(32)| 13 ∆m2 = 2.47 10−3 eV2, sin2θ = 0.024 in our numerical analysis. | 31(32)| × 13 After the successful measurement of θ , the determination of the absolute neutrino mass scale, 13 of the type of the neutrino mass spectrum, of the nature of massive neutrinos, as well as getting information about the status of CP violation in the lepton sector, remain the highest priority goals of the research in neutrino physics. Establishing whether CP is conserved or not in the lepton sector 2We use the convention adopted in [1]. 3Note that the two Majoranaphases α21 and α31 defined in [1] are twice the phases α and β: α21 =2α, α31 =2β. 2 is of fundamental importance, in particular, for making progress in the understanding of the origin of the matter-antimatter asymmetry of the Universe (see, e.g., [10, 11, 12]). Some time ago one of the present authors proposed to use atoms or molecules for systematic experimental determination of the neutrino mass matrix [13, 14]. Atoms have a definite advantage over conventional target of nuclei: their available energies are much closer to neutrino masses. The process proposed is cooperative de-excitation of atoms in a metastable state. For the single atom the process is e g +γ +(ν +ν ), i,j = 1,2,3, where ν ’s are neutrino mass eigenstates. If ν i j i i | i → | i are Dirac fermions, (ν + ν ) should be understood for i = j as (ν +ν¯), and as either (ν +ν¯ ) or i j i i i j (ν +ν¯) when i = j, ν¯ being the antineutrino with mass m . If ν are Majorana particles, we have j i i i i 6 ν¯ ν and (ν +ν ) are the Majorana neutrinos with masses m and m . i i i j i j ≡ The proposed experimental method is to measure, under irradiation of two counter-propagating trigger lasers, the continuous photon (γ) energy spectrum below each of the six thresholds ω cor- ij responding to the production of the six different pairs of neutrinos, ν ν , ν ν ,..., ν ν : ω < ω , ω 1 1 1 2 3 3 ij being the photon energy, and [13, 14] ǫ (m +m )2 eg i j ω = ω = , i,j = 1,2,3, m 0, (5) ij ji 1,2,3 2 − 2ǫ ≥ eg where ǫ is the energy difference between the two relevant atomic levels. eg The process occurs in the 3rd order (counting the four Fermi weak interaction as the 2nd order) of electroweak theory as a combined weak and QED process, as depicted in Fig. 1. Its effective amplitude has the form of † G a ν ~σν ~ ~ F ij ij j i ~ g d p E p S e , (6) h | | i· Pǫ ω ·h | e| i pg − 1 a = U∗U δ , (7) ij ei ej − 2 ij where U , i = 1,2,3, are the elements of the first row of the neutrino mixing matrix U , given in ei PMNS eq. (1). The atomic part of the probability amplitude involves three states e , g , p , where the two | i | i | i states e , p , responsible for the neutrino pair emission, are connected by a magnetic dipole type | i | i operator, the electron spin S~ . The g p transition involves a stronger electric dipole operator e | i− | i ~ d. From the point of selecting candidate atoms, E1 M1 type transition must be chosen between the × initial and the final states ( e and g ). The field E~ in eq. (6) is the one stored in the target by | i | i the counter-propagating fields. The formula has some similarity to the case of stimulated emission. By utilizing the accuracy of trigger laser one can decompose, in principle, all six photon energy thresholds at ω , thereby resolving the neutrino mass eigenstates instead of the flavor eigenstates. ij The spectrum rise below each threshold ω ω depends, in particular, on a 2 and is sensitive ij ij ≤ | | to the type of the neutrino mass spectrum, to the nature of massive neutrinos, and, in the case of emission of two different Majorana neutrinos, to the Majorana CPV phases in the neutrino mixing matrix (see below on further details). The disadvantage of atomic targets is their smallness of rates which are very sensitive to available energy of order eV. This can be overcome by developing, with the aid of a trigger laser, macro- coherence of atomic polarization to which the relevant amplitude is proportional, as discussed in [15, 16]. The macroscopic polarization supported by trigger field gives rise to enhanced rate n2V, ∝ where n is the number density of excited atoms and V is the volume irradiated by the trigger laser. The proposed atomic process may be called radiative emission of neutrino pair, or RENP in short. 3 Figure 1: Λ type atomic level for RENP e g + γ + ν ν with ν a neutrino mass eigenstate. i j i − | i → | i Dipole forbidden transition e g +γ +γ may also occur via weak E1 M1 couplings to p . | i → | i × | i The estimated rate roughly of order mHz or a little less makes it feasible to plan realistic RENP experiments for a target number of order of the Avogadro number, within a small region of order 1 102 cm3, if the rate enhancement works as expected. ∼ The new atomic process of RENP has a rich variety of neutrino phenomenology, since there are six independent thresholds for each target choice, having a strength proportional to different combinations of neutrino masses and mixing parameters. In the present work we shall correct the spectrum formula for the Majorana neutrino case given in [14] and also extend the discussion of the atomic spin factor. In the numerical results presented here we show the sensitivity of the RENP related photon spectral shape to various observables; the absolute neutrino mass scale, the type of neutrino mass spectrum, the nature of massive of neutrinos and the Majorana CPV phases in the case of massive Majorana neutrinos. All these observables can be determined in one experiment, each observable with a different degree of difficulty, once the RENP process is experimentally established. For atomic energy available in the RENP process of the order of a fraction of eV, the observables of interest can be ranked in the order of increasing difficulty of their determination as follows: (1) The absolute neutrino mass scale, which can be fixed by, e.g., measuring the smallest photon energy threshold min(ω ) near which the RENP rate is maximal: min(ω ) corresponds to the pro- ij ij duction of a pair of the heaviest neutrinos (max(m ) > 50 meV). j ∼ (2) The neutrino mass hierarchy, i.e., distinguishing between the normal hierarchical (NH), inverted hierarchical (IH) and quasi-degenerate (QD) spectra, or a spectrum with partial hierarchy (see, e.g., [1]). (3) The nature (Dirac or Majorana) of massive neutrinos. (4) The measurement on the Majorana CPV phases if the massive neutrinos are Majorana particles. The last item is particularly challenging. The importance of getting information about the Ma- jorana CPV violation phases in the proposed RENP experiment stems, in particular, from the possibility that these phases play a fundamental role in the generation of the baryon asymmetry of the Universe [11]. The only other experiments which, in principle, might provide information about the Majorana CPV phases are the neutrinoless double beta ((ββ) -) decay experiments (see, e.g., 0ν [17, 18]). The paper is organized as follows. In Section 2 the basic RENP spectral rate formula is given along with comments on how the Majorana vs Dirac distinction arises. We specialize to rates under 4 no magnetic field so that the experimental setup is simplest. In Section 3 we discuss the physics potential of a RENP experiment for measuring the absolute neutrino mass scale and determining the type of neutrino mass spectrum (or hierarchy) and the nature (Dirac or Majorana) of massive neutrinos. This is done on the examples of a candidate transition of Yb J = 0 metastable state and of a hypothetical atom of scaled down energy of the transition in which the photon and the two neutrinos are emitted. Section 4 contains Conclusions. 2 Photon Energy Spectrum in RENP When the target becomes macro-coherent by irradiation of trigger laser, RENP process conserves boththemomentumandtheenergywhicharesharedbyaphotonandtwoemittedneutrinosresulting in the threshold relation (5) [15]. The atomic recoil can be neglected to a good approximation. Since neutrinos are practically impossible to measure, one sums over neutrino momenta and helicities, and derives the single photon spectrum as a function of photon energy ω. We think of experiments that do not apply magnetic field and neglect effects of atomic spin orientation. The neutrino helicity (denoted by h ,r = 1,2) summation in the squared neutrino current jk = a ν†σ ν gives bilinear r ij i k j terms of neutrino momenta (see [13] and the discussion after eq. (17)): KS jk(jn)† kn ≡ X h1,h2 m m (Im(a ))2 1 = a 2 1 δ i j 1 2 ij δ + pkpn +pkpn δ p~ p~ . (8) | ij| (cid:20)(cid:18) − M E E (cid:18) − a 2 (cid:19)(cid:19) kn E E i j j i − kn i · j (cid:21) i j | ij| i j (cid:0) (cid:1) The case δ = 1 applies to Majorana neutrinos, δ = 0 corresponds to Dirac neutrinos. The term M M m m (1 2(Im(a ))2/ a 2 is similar to, and has the same physical origin as, the term M M i j ij ij i j ∝ − | | ∝ in the production cross section of two different Majorana neutralinos χ and χ with masses M and i j i M in the process of e−+e+ χ +χ [19]. The term M M of interest determines, in particular, j i j i j → ∝ the threshold behavior of the indicated cross section. Thesubsequent neutrinomomentumintegration(withE = p~2 +m2 beingtheneutrino energy) i i i p d3p d3p 1 1 2δ3(~k +~p +p~ )δ(ǫ ω E E )KS d KS , (9) Z (2π)2 1 2 eg − − 1 − 2 ij ≡ 2π Z Pν ij can be written as a second rank tensor of photon momentum, k k G(1) +δ ~k2G(2) from rotational i j ij covariance. Two coefficient functions G(i) are readily evaluated by taking the trace and a i=j product with kikj and using the energy-momentum conservation. But their explicit foPrms are not necessary in subsequent computation. We now consider sum over magnetic quantum numbers of E1 M1 amplitude squared: × R = d Me gM d~ E~ pM pM S~ ~j eM 2. (10) Z Pν2JP+1 | h g| · | pi·h p| e · ν| ei| e X X Mg Mp The field E~ is assumed to be oriented along the trigger axis taken parallel to 3 axis. Since there is − no correlation of neutrino pair emission to the trigger axis, one may use the isotropy of space and replace (S~e ~k)ni(S~e ~k)in′ by (S~e)ni (S~e)in′~k2/3. Using the isotropy, we define the atomic spin factor · · · C (X) of X atom by ep 2PJ M+e1hpMp|S~e|eMei·heMe|S~e|pMp′i = δMpMp′Cep(X). (11) e 5 This means that only the trace part of eq. (8), 4KS/3, is relevant for the neutrino phase space ii integration. The result is summarized by separating the interference term relevant to the case of Majorana neutrinos ν : i 3n2VG2γ ǫ n Γ (ω) = Γ I(ω)η (t), Γ = F pg eg (2J +1)C , (12) γ2ν 0 ω 0 2ǫ3 p ep pg 1 I(ω) = a 2∆ (ω) I (ω) δ m m BM , (13) (ǫ ω)2 | ij| ij ij − M i j ij pg − Xij (cid:0) (cid:1) (a2) (Im(a ))2 1 BM = ℜ ij = 1 2 ij , a = U∗U δ , (14) ij a 2 (cid:18) − a 2 (cid:19) ij ei ej − 2 ij ij ij | | | | 1 ∆ (ω) = ǫ (ǫ 2ω) (m +m )2 ǫ (ǫ 2ω) (m m )2 1/2 , (15) ij eg eg i j eg eg i j ǫ (ǫ 2ω) − − − − − eg eg − (cid:8)(cid:0) (cid:1)(cid:0) (cid:1)(cid:9) 1 1 1 1 1 (ǫ ω)2 I (ω) = ǫ (ǫ 2ω)+ ω2 ω2∆2 (ω) (m2 +m2) eg − (m2 m2)2 . ij (cid:18)3 eg eg − 6 − 18 ij − 6 i j − 6ǫ2 (ǫ 2ω)2 i − j (cid:19) eg eg − (16) The term δ m m appears only for the Majorana case. We shall define and discuss the dynamical M i j ∝ dimensionless factor η (t) further below. The limit of massless neutrinos gives the spectral form, ω ω2 6ǫ ω +3ǫ2 I(ω;m = 0) = − eg eg , (17) i 12(ǫ ω)2 pg − wheretheprefactor of a 2 = 3/4iscalculated using theunitarityoftheneutrino mixing matrix. ij| ij| On the other hand, nePar the threshold these functions has the behavior √ωij ω. ∝ − We will explain next the origin of the interference term for Majorana neutrinos. The two- component Majorana neutrino field can be decomposed in terms of plane wave modes as ψM(~x,t) = u(p~)e−iEit+ip~·~xb (p~)+uc(p~)eiEit−ip~·~xb†(p~) , (18) i i X(cid:16) (cid:17) i,p~ where the annihilation b (p~) and creation b†(p~) operators appears as a conjugate pair of the same i i type of operator b in the expansion (the index i gives the i th neutrino of mass m , and the helicity i − summation is suppressed for simplicity). The concrete form of the 2-component conjugate wave function uc iσ u∗ is given in [13]. A similar expansion can be written in terms of four component 2 ∝ field if one takes into account the chiral projection (1 γ )/2 in the interaction. The Dirac case is 5 −† different involving different type of operators b (p~) and d (p~): i i ψD(~x,t) = u(p~)e−iEit+ip~·~xb (p~)+v(p~)eiEit−ip~·~xd†(p~) . (19) i i X(cid:16) (cid:17) i,p~ Neutrino pair emission amplitude of modes ip~ ,j~p contains two terms in the case of Majorana 1 2 particle: b†b†(a u∗(p~ )uc(p~ ) a u∗(p~ )uc(p~ )) , (20) i j ij 1 2 − ji 2 1 and its rate involves 1 1 (a u∗(p~ )uc(p~ ) a u∗(p~ )uc(p~ ) 2 = a 2 ψ(1,2) 2+ ψ(2,1) 2 (a2)(ψ(1,2)ψ(2,1)∗) , 2| ij 1 2 − ji 2 1 | 2| ij| | | | | −ℜ ij (cid:0) (cid:1) (21) 6 where the relation a = a∗ is used and ψ(1,2) = u∗(p~ )uc(p~ ). The result of the helicity sum ji ij 1 2 (ψ(1,2)ψ(2,1)∗) is in [13], which then gives the interference term BM in the formula (14). ∝ ij P Weseefromeqs. (12)and(13)thattheoveralldecayrateisdeterminedbytheenergyindependent Γ , while the spectral information is in the dimensionless function I(ω). The rate Γ given here is 0 0 obtained by replacing the field amplitude E of Eq.6 squared by ǫ n, which is the atomic energy eg density stored in the upper level e . | i The dynamical factor η (t) is defined by a space integral of a product of macroscopic polarization ω squared times field strength, both in dimensionless units, 1 αmL/2 r (ξ,α t)2 +r (ξ,α t)2 η (t) = dξ 1 m 2 m e(ξ,α t) 2. (22) ω m α L Z 4 | | m −αmL/2 Here r ir is the medium polarization normalized to the target number density. 1 2 ± The dimensionless field strength e(ξ,τ) 2 = E(ξ = α x,τ = α t) 2/(ǫ n) is to be calculated m m eg | | | | using the evolution equation for field plus medium polarization in [16], where ξ = α x (α = m m ǫ µ n/2 with µ the off-diagonal coefficient of AC Stark shifts [14]) is the atomic site position in eg ge ge dimensionless unit along the trigger laser direction ( L/2 < x < L/2 with L the target length), − and τ = α t is the dimensionless time. The characteristic unit of length and time are α−1 m m ∼ 1cm(n/1021cm−3)−1 and 40ps(n/1021cm−3)−1 for Yb discussed below. We expect that η (t) in the ω formula given above is roughly of order unity or less.4 We shall have more comments on this at the end of this section. Note that what we calculate here is not the differential spectrum at each frequency, instead it is the spectral rate of number of events per unit time at each photon energy. Experiments for the same target atom are repeated at different frequencies ω ω in the NO case (or ω ω in 1 11 1 33 ≤ ≤ the IO case) since it is irradiated by two trigger lasers of different frequencies of ω (constrained by i ω +ω = ǫ ) from counter-propagating directions. 1 2 eg As a standard reference target we take Yb atom and the following de-excitation path, Yb; e = (6s6p)3P , g = (6s2)1S , p = (6s6p)3P . (23) 0 0 1 | i | i | i Relevant atomic parameters are as follows [20]: ǫ = 2.14349 eV, ǫ = 2.23072 eV, γ = 1.1 MHz. (24) eg pg pg The notation based on LS coupling is used for Yb electronic configuration, but this approximation must be treated with care, since there might be a sizable mixing based on jj coupling scheme. The relevant atomic spin factor C (Yb) is estimated, using the spin Casimir operator within an ep irreducible representation of LS coupling. Namely, 1 2 3P S~ 3P ,M 3P ,M S~ 3P = 3P S~ 3P ,M 3P ,M S~ 3P = , (25) 0 e 1 1 e 0 0 e 1 1 e 0 h | | i·h | | i 3 h | | i·h | | i 3 X M since S~ S~ = 2 for the spin triplet. This gives C (Yb)= 2/3 for the intermediate path chosen. e e ep · We also considered another path, taking the intermediate state of Yb, 1P with ǫ = 3.10806 eV, 1 pg γ = 0.176 GHz. Using a theoretical estimate of A-coefficient 4.6 10−2 Hz for 1P 3 P transition pg 1 1 × → 4There is a weak dependence of the dynamical factor ηω(t) on the photon energy ω, since the field e in Eq.22, a solution of the evolution equation, is obtained for the initial-boundary condition of frequency ω dependent trigger laser irradiation. 7 given in NIST [20] and taking the estimated Lande g-factor [21], 3/2 for the 3P case, we calculate 1 the mixed fraction of jj coupling scheme in LS forbidden amplitude squared 1P S~ 3P 2, to give 1 e 1 |h | | i| C 1 10−4. ep ∼ × Summarising, the overall rate factor Γ is given by 0 3n2VG2γ ǫ n n V Γ = F pg eg (2J +1)C 0.37 mHz( )3 , (26) 0 2ǫ3 p ep ∼ 1021 cm−3 102 cm3 pg wherethenumber isvalidfortheYbfirstexcited stateofJ = 0. Ifonechooses theother intermediate path, 1P , the rate Γ is estimated to be of order, 1 10−3 mHz, a value much smaller than that 1 0 × of the 3P path. The denominator factor 1/(ǫ ω)2 is slightly larger for the 3P path, too. We 1 pg 1 − consider the intermediate 3P path alone in the following. 1 The high degree of sensitivity to the target number density n seems to suggest that solid envi- ronment is the best choice. But de-coherence in solids is fast, usually sub-pico seconds, and one has to verify how efficient coherence development is achieved in the chosen target. Finally, we discuss a stationary value of time independent η (t) (22) some time after trigger ω irradiation. The stationary value may arise when many soliton pairs of absorber-emitter [16] are created, since the target in this stage is expected not to emit photons of PSR origin (due to the macro-coherent e g +γγ), or emits very little only at target ends, picking up an exponentially | i → | i small leakage tail. This is due to the stability of solitons against two photon emission. Thus the PSR background is essentially negligible. According to [22], the η (t) integral (22) is time dependent ω in general. Its stationary standard reference value may be obtained by taking the field from a single created soliton. This quantity depends on target parameters such as α and relaxation times. m Moreover, a complication arises, since many solitons may be created within the target, and the number of created solitons should be multiplied in the rate. This is a dynamical question that has to be addressed separately. In the following sections we compute spectral rates, assuming η (t) = 1. ω 3 Sensitivity of the Spectral Rate to Neutrino Mass Ob- servables and the Nature of Massive Neutrinos We will discuss in what follows the potential of an RENP experiment to get information about the absolute neutrino mass scale, the type of the neutrino mass spectrum and the nature of massive neutrinos. We begin by recalling that the existing data do not allow one to determine the sign of ∆m2 = ∆m2 andinthe case of3-neutrino mixing, thetwo possible signs of∆m2 corresponding A 31(2) 31(2) to two types of neutrino mass spectrum. In the standard convention [1] the two spectra read: i) spectrum with normal ordering (NO): m < m < m , ∆m2 = ∆m2 > 0, ∆m2 > 0, m = 1 2 3 A 31 21 2(3) 1 (m21 + ∆m221(31))2; ii) spectrum with inverted ordering (IO): m3 < m1 < m2, ∆m2A = ∆m232 < 0, 1 1 ∆m2 > 0, m = (m2 + ∆m2 )2, m = (m2 + ∆m2 ∆m2 )2. Depending on the values of the 21 2 3 23 1 3 23 − 21 smallest neutrino mass, min(m ) m , the neutrino mass spectrum can also be normal hierarchical j 0 ≡ (NH), inverted hierarchical (IH) and quasi-degenerate (QD): 1 1 NH : m m < m , m = (∆m2 )2 = 0.009eV, m = (∆m2 )2 = 0.05eV, (27) 1 ≪ 2 3 2 ∼ 21 ∼ 3 ∼ 31 ∼ 1 IH : m m < m , m = ∆m2 2 = 0.05 eV, (28) 3 ≪ 1 2 1,2 ∼ | 32| ∼ QD : m = m = m = m, m2 ∆m2 , m > 0.10 eV. (29) 1 ∼ 2 ∼ 3 ∼ j ≫ | 31(32)| ∼ All three types of spectrum are compatible with the existing constraints on the absolute scale of neutrino masses m . j 8 Table 1: The quantity a 2 = U∗U 1δ 2 | ij| | ei ej − 2 ij| a 2 = c2 c2 1 2 a 2 = c2 s2 c4 a 2 = c2 s2 c2 | 11| | 12 13 − 2| | 12| 12 12 13 | 13| 12 13 13 0.0311 0.2027 0.0162 a 2 = s2 c2 1 2 a 2 = s2 s2 c2 a 2 = s2 1 2 | 22| | 12 13 − 2| | 23| 12 13 13 | 33| | 13 − 2| 0.0405 0.0072 0.2266 3.1 General features of the Spectral Rate The first thing to notice is that the rate of emission of a given pair of neutrinos (ν +ν ) is suppressed, i j in particular, by the factor a 2, independently of the nature of massive neutrinos. The expressions ij | | for the six different factors a 2 in terms of the sines and cosines of the mixing angles θ and θ , as ij 12 13 | | well as their values corresponding to the best fit values of sin2θ and sin2θ quoted in eq. (3), are 12 13 given in Table 1. It follows from Table 1 that the least suppressed by the factor a 2 is the emission ij | | of the pairs (ν +ν ) and (ν +ν ), while the most suppressed is the emission of (ν +ν ). The values 3 3 1 2 2 3 of a 2 given in Table 1 suggest that in order to be able to identify the emission of each of the six ij | | pairs of neutrinos, the photon spectrum (i.e., the RENP spectral rate) should be measured with a relative precision not worse than approximately 5 10−3. × As it follows from eqs. (13) and (14), the rate of emission of a pair of Majorana neutrinos with masses m and m differs from the rate of emission of a pair of Dirac neutrinos with the same masses i j by the interference term m m BM. For i = j we have BM = 1, the interference term is negative ∝ i j ij ij and tends to suppress the neutrino emission rate. In the case of i = j, the factor BM, and thus the 6 ij rate of emission of a pair of different Majorana neutrinos, depends on specific combinations of the Majorana and Dirac CPV phases of the neutrino mixing matrix: from eqs. (14) and (1) we get BM = cos2α, BM = cos2(β δ), BM = cos2(α β +δ). (30) 12 13 − 23 − Incontrast, therateofemissionofapairofDiracneutrinosdoesnotdependontheCPVphasesofthe PMNS matrix. In the case of CP invariance we have α,β = 0,π/2,π, δ = 0,π, and, correspondingly, BM = 1 or +1, i = j. For BM = +1, the interference term tends to suppress the neutrino emission ij − 6 ij rate, while for BM = 1 it tends to increase it. If some of the three relevant (combinations of) ij − CPV phases, say α, has a CP violating value, we would have 1 < BM < 1; if all three are CP − 12 violating, the inequality will be valid for each of the three factors BM: 1 < BM < 1, i = j. Note, ij − ij 6 however, that the rates of emission of (ν +ν ) and of (ν +ν ) are suppressed by a 2 = 0.016 and 1 3 2 3 13 | | a 2 = 0.007, respectively. Thus, studying the rate of emission of (ν +ν ) seems the most favorable 23 1 2 | | approach to get information about the Majorana phase α, provided the corresponding interference term m m BM is not suppressed by the smallness of the factor m m . The mass m can be very ∝ 1 2 12 1 2 1 small or even zero in the case of NH neutrino mass spectrum, while for the IH spectrum we have m m > ∆m2 = 2.5 10−3 eV2 . We note that all three of the CPV phases in eq. (30) enter into 1 2 ∼ | 32| ∼ × the expression for the (ββ) decay effective Majorana mass as their linear combination (see, e.g., 0ν − [17, 23]): m U2 2 = m2s4 +m2s4 c4 +m2c4 c4 | i ei| 3 13 2 12 13 1 12 13 Xi +2m m s2 c2 c4 cos(2α)+2m m s2 c2 c2 cos2(β δ)+2m m s2 s2 c2 cos2(α β +δ). (31) 1 2 12 12 13 1 3 13 12 13 − 2 3 13 12 13 − 9 YbRENPglobalspectrum 0.20 ΩL 0.15 HI m u ctr 0.10 e p S 0.05 0.00 0.0 0.2 0.4 0.6 0.8 1.0 Ω @eVD Figure 2: Global feature of photon energy spectrum I(ω) for Yb 3P 1S . Two cases of m = 0 0 0 → 20meV in black and the massless limit in (red) are degenerate in this figure. In the case of m < m < m (NO spectrum), the ordering of the threshold energies at ω = ω 1 2 3 ij ji is the following: ω > ω > ω > ω > ω > ω . For NH spectrum with negligible m which can 11 12 22 13 23 33 1 be set to zero, the factors (m +m )2 κ in the expression (5) for the threshold energy ω are given i j ij ij ≡ by: κ = 0, κ = ∆m2 , κ = 4∆m2 , κ = ∆m2 , κ = ( ∆m2 + ∆m2 )2, κ = 4∆m2 . It 11 12 21 22 21 13 31 23 31 21 33 31 follows from eq.(5) and the expressions for κ that ω , ω apnd ω arepvery close, ω and ω are ij 11 12 22 13 23 somewhat more separated and the separation is the largest between ω and ω , and ω and ω : 22 13 23 33 1 1 NH : ω ω = (ω ω ) = ∆m2 = 1.759 (8.794) 10−5 eV, (32) 11 − 12 3 12 − 22 2ǫ 21 ∼ × eg 1 NH : ω ω = (2 ∆m2 ∆m2 +∆m2 ) = 0.219 (1.095) 10−3 eV, (33) 13 − 23 2ǫeg q 21q 31 21 ∼ × 1 NH : ω ω = (∆m2 4∆m2 ) = 0.506 (2.529) 10−3 eV, (34) 22 − 13 2ǫ 31 − 21 ∼ × eg 1 NH : ω ω = (3∆m2 2 ∆m2 ∆m2 ∆m2 ) = 1.510 (7.548) 10−3 eV,(35) 23 − 33 2ǫeg 31 − q 21q 31 − 21 ∼ × where the numerical values correspond to ∆m2 given in eq. (3) and ǫ = 2.14349 ( numbers in 21 eg parenthesis corresponding to the 1/5 of Yb value, namely 0.42870) eV. We get similar results in what concerns the separation between the different thresholds in the case of QD spectrum and ∆m2 > 0: 31 1 QD : ω ω = ω ω = ω ω = ∆m2 = 3.518 (17.588) 10−5 eV, (36) 11 − 12 ∼ 12 − 22 ∼ 13 − 23 ∼ ǫ 21 ∼ × eg 1 1 QD : ω ω = ω ω ∆m2 = (∆m2 2∆m2 ) = 1.082 (5.410) 10−3 eV(.37) 22 − 13 ∼ 23 − 33 − ǫ 21 ǫ 31 − 21 ∼ × eg eg For spectrum with inverted ordering, m < m < m , the ordering of the threshold energies is 3 1 2 different: ω > ω > ω > ω > ω > ω . In the case of IH spectrum with negligible m = 0, 33 13 23 11 12 22 3 we have: κ = 0, κ = ∆m2 ∆m2 , κ = ∆m2 , κ = 4(∆m2 ∆m2 ), κ = ( ∆m2 + 33 13 23 − 21 23 23 11 23 − 21 12 23 p 10

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