aa r X i v : . [ h e p - ph ] A ug Raman stimulated neutrino pair emission
H. Hara and M. YoshimuraResearch Institute for Interdisciplinary Science, Okayama UniversityTsushima-naka 3-1-1 Kita-ku Okayama 700-8530 Japan
ABSTRACT
A new scheme using macroscopic coherence is proposed from a theoretical point to experimentally de-termine the neutrino mass matrix, in particular the absolute value of neutrino masses, and the mass type,Majorana or Dirac. The proposed process is a collective, coherent Raman scattering followed by neutrino-pair emission from an excited state | e i of a long lifetime to a lower energy state | g i ; γ + | e i → γ + P ij ν i ¯ ν j + | g i with ν i ¯ ν j consisting of six massive neutrino-pairs. Calculated angular distribution has six ( ij ) thresholds ofmassive neutrino-pair emission which show up as steps at different angles in the distribution. Angular loca-tions of thresholds and event rates of the angular distribution make it possible to experimentally determinethe smallest neutrino mass to the level of less than several meV (accordingly all three masses using neutrinooscillation data), the mass ordering pattern, normal or inverted, and to distinguish whether neutrinos are ofMajorana or Dirac type. Event rates of neutrino-pair emission, when the mechanism of macroscopic coher-ence amplification works, may become large enough for realistic experiments by carefully selecting certaintypes of target atoms or ions doped in crystals. The problem to be overcome is macro-coherently amplifiedquantum electrodynamic background of the process, γ + | e i → γ + γ + γ + | g i , when two extra photons, γ , γ , escape detection. We illustrate our idea using neutral Xe and trivalent Ho ion doped in dielectriccrystals.Keywordsneutrino mass matrix, Majorana fermion, neutrino-pair emission from atoms/ions, coherent Ramanscattering, trivalent lanthanoid ions, Xe metastable states1 Introduction
Remaining major problems in neutrino physics are determination of the absolute neutrino mass value andthe nature of neutrino mass, either of Dirac type or of Majorana type. These problems are key importantissues to clarify the origin of baryon asymmetry of our universe and to construct the ultimate unified theorybeyond the standard theory. Despite of many year’s experimental efforts [1] no hint of these issues is foundso far. It is necessary, in our opinion, to establish new experimental schemes based on targets besides nucleihaving a few to several MeV energy release used in most of past experiments, since solution of these problemsrequires high sensitivity to the expected, much smaller, sub-eV neutrino mass range.One possibility of new experimental approaches is the use of atoms/ions or molecules whose energylevels can be chosen to be almost arbitrarily close to small neutrino masses [2], [3]. The process is atomicde-excitation from a metastable state | e i to the ground state | g i , | e i → | g i + γ + ν i ¯ ν j where γ is detectedphoton accompanying invisible neutrino pair ν i ¯ ν j ( i, j = 1 , ,
3) of mass eigenstates (anti-neutrino ¯ ν i isdistinguishable from neutrino ν i in the Dirac case, while ¯ ν i = ν i in the Majorana case). Necessary rateenhancement mechanism of atomic de-excitation using a coherence of macroscopic number of atoms (macro-coherence) has been proposed in [4] and its principle has been experimentally confirmed in weak QED(Quantum ElectroDynamic) process [5]. The enhancement factor reached 10 orders over the spontaneousemission rate. The coherently amplified neutrino-pair emission is called RENP (Radiative Emission ofNeutrino Pair), yet to be discovered.One of the problems in the original RENP scheme is a difficulty of distinguishing a detected photon fromtriggered photons (necessary to stimulate the weak process) which happens to have the same frequency andthe same emitted direction. In the present work we study the process as depicted in Fig(1), γ + | e i → γ + | g i + ν i ¯ ν j , in which the detected photon γ has different energy and different emitted direction fromthe trigger photon γ . Doubly resonant neutrino-pair emission becomes possible and massive neutrino-pairthresholds appear in the angular distribution of detected photon γ .The paper is organized in such a way to first present the general idea and principles of macro-coherentneutrino-pair emission stimulated by Raman scattering. For brevity we call the process RANP (RAmanstimulated Neutrino-Pair emission). There are three key issues to make the RANP project of neutrinomass spectroscopy successful: (1) mass determination and Majora/Dirac distinction [3] is clearly possible ornot, (2) event rate is large enough or not, (3) macro-coherently amplified QED processes that may becomebackgrounds are controllable or not. Even if these issues are not ideally solved, the final question is (4) howtechnological improvements may be foreseeable. We study the general idea by using interesting examplesof Xe and trivalent lanthanoid ions doped in crystals [6], [7], both of which have large target densitiestypically of order 10 cm − helping for a realistic detection and have small optical relaxation rates for themacro-coherence amplification. There may be other, hopefully better, candidate atoms or ions realizing thegeneral idea, but the atomic or ion density in a laser-excited macroscopic state must be large enough, closeto a value of atomic density in solids for realistic detection.We use the natural unit of ~ = c = 1 throughout the present paper unless otherwise stated. Our experimental scheme uses two counter-propagating lasers of frequencies, ω i , i = 1 , | e i from the ground state and one trigger laser of frequency ω for Raman excitation, as illustrated inFig(2). These excitation lasers are irradiated along the same axis unit vector ~e z , hence ω + ω = ǫ eg and ω − ω = rǫ eg , − ≤ r ≤
1. At excitation a spatial phase e i~p eg · ~x , ~p eg = rǫ eg ~e z , is imprinted to target atoms,each at position ~x .Suppose that a collective body of atoms, when they have a common spatial phase imprinted at excitation[8], de-excite emitting plural particles, which can be either photons or neutrino-pair. Quantum mechanical2 | e i | p i | q i | g i γ γ ν i ν j (a) (b) | p i| q i| g i γ ν i ν j | e i Figure 1: (a) Feynman diagram of γ + | e i → γ + | g i + ν i ¯ ν j . There are five more diagrams that contributeoff resonances, as in eq.(3). (b) Corresponding energy levels indicating absorption and emission of photonsand a neutrino-pair. γ excitation1 excitation2trigger γ detectedphoton crystal ν ν θ neutrinopair Figure 2: Schematics of experimental layout.3ransition amplitude, if the phase of atomic part of amplitudes, A a = A , is common and uniform, is givenby a formula, X a e i ( ~p eg + ~k − ~k − ~p − ~p ) · ~x a A a ≃ n (2 π ) δ ( ~p eg + ~k − ~k − ~p − ~p ) A , (1)with n the assumed uniform density of excited atoms/ions. Equality to the right hand side is valid in thecontinuous limit of atomic distribution. This gives rise to the mechanism of macro-coherent amplificationof rate ∝ n V with V the volume of target region. Thus, in the macro-coherent process depicted in Fig(1),both the energy and the momentum conservation (equivalent to the spatial phase matching condition inatomic physics terminology) hold [2]; ω + ǫ eg = ω + E + E , ~k + ~p eg = ~k + ~p + ~p , (2)where E i = q p i + m i with m i , i = 1 , , ij ) neutrino-pair emission: ( ω + ǫ eg − ω ) − ( ~k + ~p eg − ~k ) ≥ ( m i + m j ) . This may be regarded as a restriction to emitted photon energy ω and its emission angle.At the location where the equality holds, the neutrino-pair is emitted at rest. On the other hand, whenatomic phases of A a at sites a are random in a given target volume V , the rate scales with nV without themomentum conservation law, which gives much smaller rates.The amplitude corresponding to Fig(1) and related five more diagrams is given by A ν ( ω ) = − ω − ǫ pe ( ~E · ~d ep ~E · ~d pq ~ N ij · ~σ qg ω − ω + ǫ qe − ~E · ~d ep ~ N ij · ~σ pq ~E · ~d qg ω − ǫ qg )+ 1 ω + ǫ pe ( ~E · ~d ep ~E · ~d pq ~ N ij · ~σ qg ω − ω + ǫ qe + ~ N ij · ~σ pq ~E · ~d qg ~E · ~d ep ω + ǫ qg )+ 1 ω − ω − ǫ pg ( ~ N ij · ~σ ep ~E · ~d pq ~E · ~d qg ω − ǫ qg + ~ N ij · ~σ ep ~E · ~d pq ~E · ~d qg ω − ǫ qg ) , (3)neglecting coupling constant factors G F / √
2. Here ~σ = 2 ~S is the electron spin operator, ~d the electric dipoleoperator, and ~ N ij = ν † i ~σ (1 − γ ) ν j is the ( ij ) neutrino-pair emission current arising from the spatial partof axial vector charged current and neutral current interaction [2]. When magnetic dipole transitions aredominant, the electric dipole operator ~d in eq.(3) should be replaced by the magnetic dipole operator ~µ .Note that the magnetic dipole operator is odd under time reversal, while the electric dipole operator is even.The formula, eq.(3), is written for the level ordering ǫ p > ǫ q > ǫ e , but other cases of ordering may also beconsidered. The energy conservation ω − ω = ǫ eg − E − E can be used to rewrite energy denominators,for instance ω − ω + ǫ qe = − ( E + E − ǫ qg ).We find from eq.(3) that double resonance occurs at ω = ǫ pe and E + E = ǫ qg giving one diagramof Fig(1) dominant (another possibility of ω − ω = ǫ pg is not considered due to a difficulty of meeting thecondition of McQ3 rejection later discussed). This condition implies that ω = ǫ pq . Thus, the doubly resonantprocess occurs via a series of real transitions: at first, trigger photon absorption at | e i → | p i , followed bya photon emission at | p i → | q i , then by the neutrino-pair emission at | q i → | g i . In the double resonancescheme the energy denominator ǫ ab should include the width factor, ǫ ab − i ( γ a + γ b ) / | q i = | e i . A feature of this scheme is that a macro-coherence exists for thelast step of neutrino-pair emission, | q i → | g i . One could take a view that the process is a macro-coherentneutrino-pair emission | e i → | g i + ν ¯ ν , induced by elastic Raman scattering γ ( ~k ) + | e i → γ ( ~k ) + | e i offrequency ω = ω , but of ~k = ~k . 4 Event rate of neutrino-pair emission and angular spectrum
The differential spectrum rate in the double resonance scheme consists of a sum over contributions of massive( ij ) neutrino-pair production [2]: d Γ ν dωd Ω = 4 π G F ω E I ( ω ) (cid:0) ( ω − ǫ pe ) + ( γ p + γ e + (∆ ω ) ) / (cid:1) (cid:0) ( ω − ǫ pq ) + ( γ e + γ q ) / (cid:1) γ pe γ pq ǫ pe ǫ pq n V × X ij F ij ( ω pe , cos θ )Θ (cid:0) M ( ω pe , θ ) − ( m i + m j ) (cid:1) , (4) F ij = Z d p d p (2 π ) δ ( ǫ eg + ω − ω − E − E ) δ ( ~p eg − ~k − ~p − ~p ) ~ N ij · ~ N ij † . (5)Dependence n V on the excited target number density is a result of macro-coherence amplification. Theatomic part and the neutrino-pair emission part F ij are factorized in the differential rate formula, eq.(4). Thestep function Θ (cid:0) M ( ω pe , θ ) − ( m i + m j ) (cid:1) determines locations of ( ij ) neutrino-pair production thresholds.The squared neutrino pair current ~ N ij · ~ N ij † is summed over neutrino helicities and their momenta. Weused in the formula experimentally measurable A-coefficients γ ab = ( d ab or µ ab ) ǫ ab / (3 π ) and the total width γ a = P b γ ab instead of dipole moments. We denote the Raman trigger spectrum function by I ( ω ) withwidth ∆ ω and its power E = ω n = ω nη where n is the photon number density. The dynamical factordenoted by η is actually time dependent, and is calculable using the Maxwell-Bloch equation [2], the coupledset of partial differential equations of fields and atomic density matrix elements in the target region. Thiscalculation is beyond the scope of this work, and we shall assume an ideal case later on.The quantity that appears in the formula, eq.(5), is calculated as F ij ( θ ) = 18 π (cid:26)(cid:18) − ( m i + m j ) M ( θ ) (cid:19) (cid:18) − ( m i − m j ) M ( θ ) (cid:19)(cid:27) / × (cid:20) | b ij | (cid:0) M ( θ ) − m i − m j (cid:1) − δ M ℜ b ij m i m j (cid:21) , b ij = U ∗ ei U ej − δ ij . (6)using methods of [2]. δ M = 0 for Dirac neutrino and = 1 for Majorana neutrino due to identical fermioneffect [3]. The 3 × U ei ) , i = 1 , , , refers to the neutrino mass mixing [1]. In the doubleresonance scheme the detected photon energy is fixed at ω = ǫ pq , and six thresholds ( ij ) of neutrino pairproduction appear in the angular distribution at angles θ ij of M ( ω pe , θ ij ) = ( m i + m j ) .A more practical formula for rate estimate is obtained by integrating over the detected photon energyand convoluting with the trigger laser power. The convolution integral over a power spectrum I ( ω ) is Z ∞ dω I ( ω ) ω ( ω − ǫ pe ) + ( γ p + γ e + (∆ ω ) ) / ≃ πI ( ǫ pe ) ǫ pe q γ p + γ e + (∆ ω ) , with E = ω × the trigger photon number density, while the integration over the detected photon energy is Z ǫ pq +∆ ω/ ǫ pq − ∆ ω/ dω ω ( ω − ǫ pq ) + ( γ e + γ q ) / ≃ Z ∞−∞ dω ω ( ω − ǫ pq ) + ( γ e + γ q ) / πǫ pq q γ e + γ q , since the region of detected photon energy ∆ ω ≫ q γ e + γ q . We obtain the practical formula, d Γ ν d Ω ≃ π G F γ pe γ pq q γ e + γ q q γ p + γ e + (∆ ω ) ǫ pe n V η π X ij F ij ( ω pe , θ ) . (7)5e shall estimate RANP rate stimulated by elastic Raman scattering. Taking partial decay rates andtotal decay rates to be of the same order, one may derive a total RANP rate scale Γ by taking a typicalvalue of the neutrino-pair phase space integration ǫ eg /
24 from eq.(6) approximated in the massless neutrinolimit, Γ π = 32 π √ γ pe q γ p + γ e + (∆ ω ) γ e G F ǫ eg ǫ pe n V η . (8)The rate formula calculated this way contains, besides detector and laser related quantities, four impor-tant factors, and these appear in the rate asrate = Raman scattering rate ( γ pe / ( q γ p + γ e + (∆ ω ) ) ) × lifetime of | e i (= | q i ) state (1 /γ e ) × neutrino-pair emission rate ( G F ǫ eg /ǫ pe ) × coherence amplification factor ( n V η ),in the neutrino-pair emission stimulated by elastic Raman scattering. From eq.(8) it becomes veryimportant for target selection how large a dimensionless combination of decay rate, lifetime and energydifferences, γ pe ǫ eg / ( γ e ǫ pe ) is. The macro-coherent amplification necessary for RANP rate enhancement may also amplify QED processeswhich may give rise to serious backgrounds. These amplified QED processes are termed as McQn (macro-coherent QED n-th order photon emission) [9]. We shall first consider how to get rid of MacQn (n=2,3)backgrounds.The quantity M = ( ω + ǫ eg − ω ) − ( ~k + ~p eg − ~k ) is equal to ( E + E ) − ( ~p + ~p ) using variablesof unseen ( ν , ν ) and is important for discussion of backgrounds. In terms of Raman scattering variables itreads as M ( θ ) = (1 − r ) ǫ eg − − r ) ǫ eg ( ω − ω ) − ω ( rǫ eg + ω ) sin θ , (9)where θ is the γ emission angle measured from the excitation axis. When a photon of 4-momentum k = ( ω, ~k )is emitted instead of the neutrino-pair, this quantity vanishes at some angle satisfying M ( θ )(= ω − ~k ) = 0.In this case a serious amplified McQ3 background exists. On the other hand, if this quantity is arranged tobe positive at all angles and is taken close to neutrino mass thresholds, ( m i + m j ) , neutrino-pair emissionoccurs without the McQ3 background. With a proper choice of the imprinted phase r and trigger frequency ω , one may readily work out the parameter region that excludes QED backgrounds of McQ3. The otherbackground, McQ2 (Paired Super-Radiance) | e i → | g i + γ + γ , is rejected unless r = − ω /ǫ eg , whichwe shall assume to be valid in the following.In the case of elastic RANP ( ω = ω ) the first angular threshold rise occurs at pair production of smallestneutrino mass m at an angle, sin θ − r ) ǫ eg − m ω ( rǫ eg + ω ) . (10)The formula may be used in other situations. A light hypothetical particle X such as axion and hiddenphoton [10] can be searched as an angular peak given by the angle θ X obtained by replacing 4 m in eq.(10)by the squared X-mass m X Without McQ2 and McQ3 events the largest amplified QED background arises from McQ4, γ + | e i → γ + γ + γ + | g i , obtained by replacing the neutrino-pair ν i ¯ ν j in | q i → | g i by two photons, γ γ . It is difficultto kinetically reject McQ4 events when two extra photons escape detection, since the two-photon system has6ts squared pair-mass coincident to RANP events. The crucial question is how big the McQ4 backgroundrate is. Disregarding the macro-coherent amplification factor common to RANP and McQ4 event rates,8 π ω E I ( ω ) (cid:0) ( ω − ǫ pe ) + ( γ p + γ e + (∆ ω ) ) / (cid:1) (cid:0) ( ω − ǫ pq ) + ( γ e + γ q ) / (cid:1) γ pe γ pq ǫ pe ǫ pq n V , one should compare two quantities, the RANP rate function I ν = G F P ij F ij / × M1 (magnetic dipole) or E1 × E1.Macro-coherent E1 × E1 rate function (one electric dipole d replaced by the magnetic dipole µ for E1 × M1) I γ to be compared with the factor of RANP function I ν is I γ = Z d k d k (2 π ) X n d ne d ng ( ω + ǫ ne ) ω ω δ ( ω + ω − ǫ eg ) δ ( ~k + ~k − ~P ) , (11)with ~P = ~p eg + ~k − ~k . The result of calculation is given by I γ ( M ) = π X n γ ne γ ng ǫ ne ǫ ng q ǫ eg − M Z ω + ω − dω ω ( ǫ eg − ω ) ( ω + ǫ ne ) , ω ± = 12 (cid:16) ǫ eg ± q ǫ eg − M (cid:17) , (12)where the relation ~P = ǫ eg − M was used. Upper energy states | n i in eq.(12) must be connected to lowerenergy states by the combination of E1 × E1 for Xe case and E1 × M1 for trivalent lanthanoid ion, whichrestricts states of large contributions [11]. doped in crystals In this section we apply theoretical formulas given in the preceding sections to real target atoms/ions. Anexhaustive study of RANP and background rates is beyond the scope of this work. We shall restrict totwo interesting cases of a trivalent lanthanoid ion and Xe atom whose relevant energy levels are depicted inFig(3). As shown there, there are a number of Stark states (degenerate states lifted by crystal field) that maycontribute to RANP process: 11 theoretically expected and 9 experimentally detected levels for | p i = I ,and 13 theoretically expected and 9 experimentally detected levels for | g i = I [12]. Trivalent lanthanoidions have a number of 4 f electrons shielded by outer 6 s electrons, giving rise to sharp line widths, whenthey are doped in dielectric crystals [6]. Xe has two metastable excited states, 2 − (lifetime ∼
43 sec) and 0 − (lifetime ∼ .
13 sec), suitable for RENP [2] and RANP. Both targets can be prepared to have large numberdensities. doped in YLF We first comment on the important quantum number of state classification in solids. Without a magneticfield application (and even in the presence of an internal magnetic field of nucleus) time-reversal symmetryholds, but parity may be violated in the presence of the crystal field. Unlike the state classification interms of parity in the free space (vacuum) one should use time-reversal quantum number, even or odd, orT quantum number in short. Hence optical transitions between two Stark states of definite T quantumnumbers, either inter- or intra-J manifolds, should be classified according to relative T quantum numbers,even or odd. Following this classification T-odd single photon emission goes via M1, while T-even emissiongoes via E1 or E2 (electric quadrupole). Transitions among states made of 4 f electrons in the free spaceare mainly M1, but parity violating effects caused by crystal field make E1 often dominant in crystals, asshown in [12], [13]. 7he last step in the resonant path, | q i → | g i , must be M1 due to the nature of neutrino-pair emissionoperator, the spin of electron ~S e . In the trivalent Ho ion transition paths of M1 nature are limited: I → I , I → I , I → I , I → I , F → F from the list of [12]. Other steps, | e i → | p i and | p i → | q i , should be chosen from large listed A-coefficients, often from E1 transitions. (a) (b) | p i | q i| g i| e i I I I S ( P ) 6s [3/2] [3/2] ( P ) 6s [1/2] [1/2] ( P ) 6p [3/2] [1/2] [5/2] [3/2] [1/2] [5/2] | g i| q i| e i | p i parity odd even Figure 3: Relevant energy levels of trivalent Ho ion in YLF and neutral Xe. (a) Ho in the unit of cm − (10 cm − = 1 . | q i and | g i . (b) Xe in eV unit.We first discuss McQ4 background whose amplitude is obtained by replacing neutrino-pair emission at | q i → | g i by two-photon emission. T-odd two-photon emission occurs dominantly via M1 × E1. Parityviolating effect due to crystal field and consequent weak E1 decay rate calculation was formulated in [13],and decay rates among J-manifolds has been given in [12] where we can find almost all data we need forour calculation. In trivalent Ho ion there are not many common levels | n i that have E1 and M1 couplingto | q i , | g i . From the point of background rejection it is desirable to search for | q i , | g i which have no sizableM1 × E1. Indeed, there are a few candidates of this property. Another consideration we have to focus onis to choose the rate factor, γ pe γ pq / q γ e + γ q , as large as possible.A choice of RANP path considering McQ3 rejection and large RANP rate is for neutrino-pair emissionstimulated by elastic Raman scattering, | e i = I − → | p i = I − → | q i = | e i → | g i = I . − . (13)8he useful relation in the natural unit is 10 cm − = 1 . × E1 two-photon emissionin | e i → | g i transition is forbidden, hence there is no McQ4 background to this accuracy.We first show the angular distribution given by P ij F ij ( ω pe , cos θ ) of eq.(6) with M of eq.(9). Theangular spectrum is sensitive to an adopted value of the imprinted phase factor r . We investigated thisdependence for a few trivalent lanthanoid ions doped in crystals by calculating the squared mass M ( θ ; r )and searched for the parameter r to optimize the shape of angular spectrum clearly showing neutrino-pairthreshold rises. The search is illustrated in Fig(4), which gives an optimal value, r = − . r choice the squared mass M distribution in Fig(5) and the angular distribution in Fig(6) ∼ Fig(8).The two largest threshold rises appear at the pairs, (12) and (33), where | b | = 0 . , | b | = 0 . P ij | b ij | = 3 / r choice of these figures, frequencies of two excitation lasers are ω = 202 .
366 meV , ω =435 .
329 meV, while the Raman trigger and detected photon have energies, ω = ω = 435 .
33 meV. In Fig(8)we show contributions from inelastic RANP paths arising from different, wide spread, Stark states for | q i which should be separately detectable with a high resolution of detected photon energy. These inelasticRaman paths (contributions beside the one in solid black of Fig(8) ) give rise to pair production at finiteneutrino velocities. These contributions refer to production far away from thresholds, hence they are notsensitive to neutrino mass determination. But they are important to identify the process of macro-coherentneutrino-pair emission in atoms/ions. Other paths from Stark states in manifolds, I and I , of the sameT quantum numbers should equally contribute to RANP photon angular distributions.The RANP rate scale Γ can be calculated using the formula, eq.(8), with ǫ eg = 631 meV , ǫ pe = 431 meV , γ pe = 14 . − , γ e = 69 . − (the lifetime 1 /γ e including non-radiativecontributions at 10 K) for the relevant path. The result isΓ (Ho) = 0 . × − sec − π ω n V η (10 ) cm − . (14)We assumed that laser related factors can give the combination n V η of order (10 ) cm − , having inmind 10 mJ (= 1 . × eV) lasers. Whether this n V η value is a reasonable assumption or not has tobe verified by detailed simulations. The rate unit for the Ho figures in the present work is 24Γ /ǫ eg =5 . × − sec − eV − , assuming n V η of order (10 ) cm − and ∆ ω / π = 100Hz. The very small ratevalue is due to small combination factors, γ pe /γ e , in particular a small A-coefficient γ pe . A possible wayto get larger rates is to use multiple identical laser systems of order 10, each system capable of producing(10 ) cm − of n V η , within the allowed range of target excitation to (10 ) cm − , which may enhance therate unit 24Γ /ǫ eg to 5 . × − sec − eV − due to Γ ∝ n V η . A search for better lanthanoid candidates ishighly recommended.
Four lowest excited energy states of Xe are P , , and P in LS coupling scheme, although energy spacingsare better described by intermediate coupling scheme close to J J scheme. The following RANP path is9 .0 0.5 0.5 1.0 r1.51.00.5eV ^2
Mass squared vs r I_7 I_6 I_7Ho^(3+):
Figure 4: Mass squared M ( θ ; r ) vs r for three angles: θ = 0 in solid black, θ = π/ θ = π in dash-dotted blue. The line M ( θ ; r ) = 4(60meV) (expected (33) mass square M assuming thesmallest neutrino mass ≪
10 meV ) is shown in dotted black as a guide. It is concluded that no McQ3 isexpected for r ≤ − . M ( θ ; r ) > r = − ǫ pe /ǫ eg there is no dependence of rates on emission angle. Rate in threshold region : rate/au
Ho^(3+)(12) (33) pair-mass/eV
Figure 5: Rate vs neutrino-pair mass p M ( θ ; r ) with r = − . .8 2.0 2.2 2.4 2.6 2.80.0050.0100.015au Angular distribution : Ho^(3+) angle/rad (12)(33)
Figure 6: RANP angular distribution with r = − . . × − sec − assuming n V η = (10 ) cm − . This rate unit is applied to Fig(5) and Fig(7) ∼ Fig(8) aswell.
Angular distribution : Ho^(3+) angle/radNO DiracNO MajoranaIO Majorana IO Dirac
Figure 7: RANP angular distribution with r = − . .0 2.2 2.4 2.6 2.8 3.00.0050.0100.015au Angular distribution : Ho^(3+) angle/rad655.0 meV647.6 meV638.2 meV637.7 meV
Figure 8: RANP angular distribution with r = − . | q i = I , . | e i = 5 p ( P / )6 s [1 / . → | p i = 5 p ( P / )6 p [3 / . → | q i = | e i→ | g i = 5 p ( P / )6 s [3 / . , (15)where energy values in the free space are taken from NIST data [15]. We have in mind using Xe in thefree space so that parity is a good quantum number, and spin-parity J P changes in Xe RANP path are0 − → + → − → − . In the proposed scheme of r = 0 . ω = 6 . , ω = 3 . ω = ω = 0 . F = γ pe ǫ eg / ( γ e ǫ pe ) are F (Xe) = 0 . × − , F (Ho) = 0 . × − . (16)Xe RANP units are Γ = 5 × − sec − and 24Γ /ǫ eg = 0 .
12 sec − eV − using the same value of n V η =(10 ) cm − . Angular distributions are shown in Fig(9) and Fig(10). Sensitivity to the neutrino mass andDirac/Majorana distinction is better than the ordinary 2 − RENP [2].The problem of Xe scheme is a large McQ4 event rate. Relevant two-photon emission at | q i → | g i occurs via E1 × E1 unlike smaller E1 × M1 in Ho doped crystal. Xe value of McQ4 integral is I γ =3 . × − eV − (zero at calculation accuracy for Ho case) to be compared the RANP value, I ν =2 . × − eV − . A solution is to use photonic crystal for suppression of strayed McQ4 event [16]. Due to alarge level spacing the excitation to Xe 0 − is more complicated than a simple two-photon excitation, whichhas to be studied.In summary, we proposed a general scheme of macro-coherent neutrino-pair emission stimulated byRaman scattering, in order to measure important neutrino properties, the unknown smallest neutrino massto the level of less than 1 meV, NO/IO distinction, and Majorana/Dirac distinction. The general scheme12 .0 2.5 3.0 th (cid:144) rad0.020.040.060.080.100.120.14au Angular distribution: Xe
Figure 9: Xe RANP angular distribution with r = 0 . .
12 sec − assuming n V η = (10 ) cm − . (cid:144) rad0.0050.0100.015au Angular distribution: Xe
Figure 10: Xe RANP angular distribution with r = 0 . .
12 sec − assuming n V η = (10 ) cm − . 13as illustrated using Ho doped crystal and Xe atom. Xe has a larger rate than lanthanoid ions, whileits QED background is much more severe. Both theoretical and experimental works on QED backgroundrejection are needed to make the general scheme realistic. It would be interesting to extend the RANPscheme also for search of other elusive particles such as axion and hidden photon. Acknowledgements
We thank S. Uetake at Okayama, and C. Braggio, G. Carugno, and F. Chiossi at Padova for usefuldiscussions. This research was partially supported by Grant-in-Aid 17K14363(HH) and 17H02895(MY)from the Ministry of Education, Culture, Sports, Science, and Technology.
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