# Raman stimulated neutrino pair emission

## Abstract

A new scheme using macroscopic coherence is proposed to experimentally determine 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 \(|e\rangle \) of a long lifetime to \(|g\rangle \); \(\gamma _0 + | e\rangle \rightarrow \gamma + \sum _{ij} \nu _i \bar{\nu _j} + | g\rangle \) with \( \nu _i \bar{\nu _j}\) consisting of six massive neutrino-pairs. Calculated angular distribution has six (*ij*) thresholds which show up as steps at different angles. Angular locations of thresholds and event rates of the angular distribution make it possible to experimentally determine the smallest neutrino mass to the level of less than several meV, (accordingly all three masses using neutrino oscillation data), the mass ordering pattern, normal or inverted, and to distinguish whether neutrinos are of Majorana or Dirac type. Event rates of neutrino-pair emission, when the mechanism of macroscopic coherence amplification works, may become large enough for realistic experiments by carefully selecting certain types of target. The problem to be overcome is macro-coherently amplified quantum electrodynamic background of the process, \(\gamma _0 + | e\rangle \rightarrow \gamma +\gamma _2 + \gamma _3+ | g\rangle \), when two extra photons, \(\gamma _2, \gamma _3\), escape detection. We illustrate our idea using neutral Xe and trivalent Ho ion doped in dielectric crystals.

## 1 Introduction

Remaining major problems in neutrino physics are determination of the absolute neutrino mass value and the nature of neutrino mass, either of Dirac type or of Majorana type. These problems are key important issues to clarify the origin of baryon asymmetry of our universe and to construct the ultimate unified theory beyond the standard theory. Despite of many year’s experimental efforts [1] no hint of these issues is found so far. It is necessary, in our opinion, to establish new experimental schemes based on targets besides nuclei having a few to several MeV energy release used in most of past experiments, since solution of these problems requires 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 energy levels can be chosen to be almost arbitrarily close to small neutrino masses [2, 3, 4]. The process is atomic de-excitation from a metastable state \( |e \rangle \) to the ground state \( |g \rangle \), \(|e \rangle \rightarrow | g\rangle + \gamma + \nu _i \bar{\nu _j} \) where \(\gamma \) is detected photon accompanying invisible neutrino pair \( \nu _i \bar{\nu _j} \, ( i,j = 1,2,3) \) of mass eigenstates (anti-neutrino \(\bar{\nu _i}\) is distinguishable from neutrino \(\nu _i \) in the Dirac neutrino, while \(\bar{\nu _i} = \nu _i\) in the Majorana neutrino). Necessary rate enhancement mechanism of atomic de-excitation using a coherence of macroscopic number of atoms (macro-coherence) has been proposed in [5] and its principle has been experimentally confirmed in weak QED (Quantum ElectroDynamic) process [6, 7, 8]. The enhancement factor reached \(10^{18}\) orders over the spontaneous emission rate. The coherently amplified neutrino-pair emission is called RENP (Radiative Emission of Neutrino Pair), yet to be discovered.

The paper is organized in such a way to first present the general idea and principles of macro-coherent neutrino-pair emission stimulated by Raman scattering. For brevity we call the process RANP (RAman stimulated Neutrino-Pair emission). There are three key issues to make the RANP project of neutrino mass spectroscopy successful: (1) mass determination and Majora/Dirac distinction [4] is clearly possible or not, (2) event rate is large enough or not, (3) macro-coherently amplified QED processes that may become backgrounds are controllable or not. Even if these issues are not ideally solved, the final question is (4) how technological improvements may be foreseeable. We study the general idea by using interesting examples of Xe and trivalent lanthanoid ions doped in crystals [9], for an interesting scheme of axion search in rare earth doped crystals, see [10], both of which have large target densities typically of order \(10^{20}\, \hbox {cm}^{-3}\) helping for a realistic detection and have small optical relaxation rates for the macro-coherence amplification. There may be other, hopefully better, candidate atoms or ions realizing the general idea, but the atomic or ion density in a laser excited state must be large enough, close to a value of atomic density in solids for realistic detection.

We use the natural unit of \(\hbar = c = 1\) throughout the present paper unless otherwise stated.

## 2 Double resonance condition and Raman stimulated neutrino-pair emission

Our experimental scheme uses two counter-propagating lasers of frequencies, \(\omega _i\,, i = 1,2\) for excitation to \(| e\rangle \) from the ground state and one trigger laser of frequency \(\omega _0\) for Raman excitation, as illustrated in Fig. 2. These excitation lasers are irradiated along the same axis unit vector \(\mathbf {e}_z \), hence \(\omega _1 + \omega _2 = \epsilon _{eg} \) and \(\omega _1 - \omega _2 = r\epsilon _{eg}\,, -1 \le r \le 1 \). At excitation a spatial phase \(e^{i \mathbf {p}_{eg}\cdot \mathbf {x} }, \, \mathbf {p}_{eg} = r \epsilon _{eg} \mathbf {e}_z\), is imprinted to target atoms, each at position \(\mathbf {x}\).

*n*the assumed uniform density of excited atoms/ions. Equality to the right hand side is valid in the continuous limit of atomic distribution. This gives rise to the mechanism of macro-coherent amplification of rate \(\propto n^2 V\) 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) hold [2, 3];

*ij*) neutrino-pair emission: \( ( \omega _0 + \epsilon _{eg} - \omega )^ 2 - ( \mathbf {k}_0 + \mathbf {p}_{eg} - \mathbf {k} )^ 2 \ge (m_i+m_j)^2\). This may be regarded as a restriction to emitted photon energy \(\omega \) and its emission angle. At the location where the equality holds, the neutrino-pair is emitted at rest. On the other hand, when atomic phases of \({{{\mathcal {A}}}}_a\) at sites

*a*are random in a given target volume

*V*, the rate scales with

*nV*without the momentum conservation law, which gives much smaller rates.

*ij*) neutrino-pair emission current arising from the spatial part of axial vector charged current and neutral current interaction [2, 3]. When magnetic dipole transitions are dominant, the electric dipole operator \(\mathbf {d}\) should be replaced by the magnetic dipole operator \(\mathbf {\mu } \). 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 convenience of the level ordering \(\epsilon _p> \epsilon _q > \epsilon _e \), but other cases may also be considered. The energy conservation \(\omega - \omega _0 = \epsilon _{eg} - E_1 - E_2 \) can be used to rewrite energy denominators, for instance \(\omega - \omega _0 + \epsilon _{qe} = - ( E_1+E_2 - \epsilon _{qg})\).

We find from Eq. (3) that double resonance occurs at \(\omega _0 = \epsilon _{pe} \) and \( E_1+E_2 = \epsilon _{qg} \) giving one diagram of Fig. 1 dominant (another possibility of \(\omega - \omega _0 = \epsilon _{pg} \) is not considered due to a difficulty of meeting the condition of McQ3 rejection). The condition implies that \(\omega = \epsilon _{pq}\). Thus, the doubly resonant process occurs via a series of real transitions: first trigger photon absorption at \(|e \rangle \rightarrow | p \rangle \), followed by a photon emission at \(|p \rangle \rightarrow | q\rangle \), then by the neutrino-pair emission at \(|q \rangle \rightarrow | g\rangle \). In the double resonance scheme the energy denominator \(\epsilon _{ab} \) should include the width factor, \(\epsilon _{ab} - i (\gamma _a + \gamma _b)/2\).

Let us take the same state for \(| q\rangle = | e \rangle \). A feature of this scheme is that a macro-coherence exists for the last step of neutrino-pair emission, \(| q \rangle \rightarrow | g \rangle \). One could take a view that the process is a macro-coherent neutrino-pair emission \( | e\rangle \rightarrow | g \rangle +\nu {\bar{\nu }} \), induced by elastic Raman scattering \( \gamma _0 (\mathbf {k}_0) + | e \rangle \rightarrow \gamma (\mathbf {k}) + | e \rangle \) of frequency \(\omega _0 = \omega \), but of \(\mathbf {k}_0 \ne \mathbf {k}\).

## 3 Event rate of neutrino-pair emission and angular spectrum

*ij*) neutrino-pair production [2, 3]:

*ij*) neutrino-pair production thresholds. The squared neutrino pair current \(\mathbf {\mathcal{N}_{ij}}\cdot \mathbf { {{{\mathcal {N}}}}_{ij}}^{\dagger } \) is summed over neutrino helicities and their momenta. We used in the formula experimentally measurable A-coefficients \(\gamma _{ab} = (d_{ab}^2\, \mathrm{or} \,\mu _{ab}^2)\epsilon _{ab}^3/(3\pi )\) and the total width \(\gamma _{a} = \sum _b \gamma _{ab} \) instead of dipole moments. We denote the Raman trigger spectrum function by \(I(\omega _0) \) with width \(\varDelta \omega _0\) and its power \(E_0^2 = \omega _0 n_0 = \omega _0 n \eta \) where \(n_0\) is the photon number density. The dynamical factor denoted by \( \eta \) is actually time dependent, and is calculable using the Maxwell–Bloch equation [2, 3], the coupled set of partial differential equations of fields and atomic density matrix elements in the target region. This calculation is beyond the scope of this work, and we shall assume an ideal case later on.

*ij*) of neutrino pair production appear in the angular distribution at angles \(\theta _{ij}\) of \({{{\mathcal {M}}}}^2 (\omega _{pe}, \theta _{ij}) =(m_i +m_j)^2 \).

rate = Raman scattering rate (\( \gamma _{pe}^2/(\sqrt{ \gamma _{p}^2 + \gamma _{e}^2 + (\varDelta \omega _0)^2} )\,) \times \) lifetime of \(|e \rangle ( = | q\rangle )\) state (\(1/\gamma _e\)) \(\times \) neutrino-pair emission rate (\(G_F^2\epsilon _{eg}^2/ \epsilon _{pe}^3) \times \) coherence amplification factor (\(n^3 V \eta \)),

in the neutrino-pair emission stimulated by elastic Raman scattering. From Eq. (8) it becomes very important for target selection how large a dimensionless combination of decay rate, lifetime and energy differences, \(\gamma _{pe}^2 \epsilon _{eg}^2/ (\gamma _e \epsilon _{pe}^3 ) \) is.

## 4 Amplified QED backgrounds

The macro-coherent amplification necessary for RANP rate enhancement may also amplify QED processes which may give rise to serious backgrounds. These amplified QED processes are termed as McQn (macro-coherent QED n-th order photon emission) [12]. We shall first consider how to get rid of MacQn (n \(=\) 2, 3) backgrounds.

*r*and trigger frequency \( \omega _0\), one may readily work out the parameter region that excludes QED backgrounds of McQ3. The other background, McQ2 (Paired Super-Radiance) \( |e\rangle \rightarrow |g \rangle + \gamma _0 +\gamma \), is rejected unless \(r = - 1 + 2\omega _0/\epsilon _{eg}\), which we shall assume to be valid in the following.

*d*replaced by the magnetic dipole \(\mu \) for E1 \(\times \) M1) \(I _{2\gamma } \) to be compared with the factor of RANP function \(I _{2\nu }\) is

^{1}

## 5 Numerical rate estimate for Xe and \(\hbox {Ho}^{3+}\) doped in crystals

*f*electrons shielded by outer 6

*s*electrons, giving rise to sharp line widths, when they are doped in dielectric crystals [9]. Xe has two metastable excited states, \(2^-\) (lifetime \(\sim 43\) s) and \(0^-\) (lifetime \(\sim 0.13\) s), suitable for RENP [2, 3] and RANP. Both targets can be prepared to have large number densities.

### 5.1 Lanthanoid case: example of \(\hbox {Ho}^{3+}\) doped in YLF

We first comment on the important quantum number of state classification in solids. Without a magnetic field application (and even in the presence of an internal magnetic field of nucleus) time-reversal symmetry holds, but parity may be violated in the presence of the crystal field. Unlike the state classification in terms of parity in the free space (vacuum) one should use time-reversal quantum number, even or odd, or T quantum number in short. Hence optical transitions between two Stark states of definite T quantum numbers, 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 emission goes via E1 or E2 (electric quadrupole). Transitions among states made of 4*f* electrons in the free space are mainly M1, but parity violating effects caused by crystal field make E1 often dominant in crystals, as shown in [14, 15, 16, 17], the idea of relevant parity violating effect in crystals goes back to [18].

The last step in the resonant path, \(|q \rangle \rightarrow | g \rangle \), must be M1 due to the nature of neutrino-pair emission operator, the spin of electron \(\mathbf {S}_e \). In the trivalent Ho ion transition paths of M1 nature are limited:

\(^5\mathrm{\!I}_7 \rightarrow ^5\mathrm{\!\!I}_8\,, ^5\mathrm{\!\!I}_6 \rightarrow ^5\mathrm{\!\!I}_7 \,, ^5\mathrm{\!I}_5 \rightarrow ^5\mathrm{\!\!I}_6, ^5\mathrm{\!I}_4 \rightarrow ^5\mathrm{\!\!I}_5, ^5\mathrm{\!F}_4 \rightarrow ^5\mathrm{\!\!F}_5\) from the list of [14, 15]. Other steps, \(| e \rangle \rightarrow | p \rangle \) and \(|p \rangle \rightarrow | q \rangle \), should be chosen from large listed A-coefficients, often from E1 transitions.

We first discuss McQ4 background whose amplitude is obtained by replacing neutrino-pair emission at \(|q \rangle \rightarrow | g \rangle \) by two-photon emission. T-odd two-photon emission occurs dominantly via M1 \(\times \) E1. Parity violating effect due to crystal field and consequent weak E1 decay rate calculation was formulated in [16, 17, 18], and decay rates among J-manifolds has been given in [14, 15] where we can find almost all data we need for our calculation. In trivalent Ho ion there are not many common levels \(|n \rangle \) that have E1 and M1 coupling to \(|q \rangle \,, | g \rangle \). From the point of background rejection it is desirable to search for \(|q \rangle , | g \rangle \) which have no sizable M1 \(\times \) E1. Indeed, there are a few candidates of this property. Another consideration we have to focus on is to choose the rate factor, \(\gamma _{pe}\gamma _{pq}/\sqrt{\gamma _e^2 + \gamma _q^2}\), as large as possible.

We first show the angular distribution given by

\(\sum _{ij} F_{ij} (\omega _{pe}, \cos \theta ) \) of Eq. (6) with \({{{\mathcal {M}}}}^2\) of Eq. (9). The angular spectrum is sensitive to an adopted value of the imprinted phase factor *r*. We investigated this dependence for a few trivalent lanthanoid ions doped in crystals by calculating the squared mass \({{{\mathcal {M}}}}^2 (\theta ; r)\) and searched for the parameter *r* to optimize the shape of angular spectrum clearly showing neutrino-pair threshold rises. The search is illustrated in Fig. 4, which gives an optimal value, \( r= -\,0.36532\). We show for this *r* choice the squared mass \({{{\mathcal {M}}}}^2\) distribution in Fig. 5 and the angular distribution in Figs. 6, 7 and 8. The two largest threshold rises appear at the pairs, (12) and (33), where \(|b_{12}|^2 = 0.405\,, |b_{33}|^2 = 0.227 \), making up most of the weight sum \(\sum _{ij} | b_{ij}|^2 = 3/4\).^{2} The smallest neutrino mass of order 1 meV is best determined by measurements around (12) threshold, as is made evident in Figs. 5 and 6, while the Majorana/Dirac distinction is better studied after (33) threshold opens. The mass ordering pattern [1], normal ordering (NO) vs inverted ordering (IO) distinction, is relatively easy as seen in Fig. 7.

The unique feature of RENP and RANP experiments is that there is a sensitivity to determine CP violation phases intrinsic to Majorana neutrinos. We have not, however, studied this sensitivity in the present work.

With the *r* choice of these figures, frequencies of two excitation lasers are \(\omega _1 = 202. 366\) meV \(, \omega _2 =435.329\) meV, while the Raman trigger and detected photon have energies, \(\omega _0 = \omega = 435.33 \, \)meV. In Fig. 8 we show contributions from inelastic RANP paths arising from different, wide spread, Stark states for \(|q \rangle \) which should be separately detectable with a high resolution of detected photon energy. These inelastic Raman paths (contributions beside the one in solid black of Fig. 8) give rise to pair production at finite neutrino velocities. These contributions refer to production far away from thresholds, hence they are not sensitive to neutrino mass determination. But they are important to identify the process of macro-coherent neutrino-pair emission in atoms/ions. Other paths from Stark states in manifolds, \(^5\hbox {I}_7\) and \(^5\hbox {I}_8\), of the same T quantum numbers should equally contribute to RANP photon angular distributions.

The RANP rate scale \(\varGamma _0\) can be calculated using the formula, Eq. (8), with

### 5.2 Xe case

*LS*coupling scheme, although energy spacings are better described by intermediate coupling scheme close to

*JJ*scheme. The following RANP path is used:

The problem of Xe scheme is a large McQ4 event rate. Relevant two-photon emission at \(|q\rangle \rightarrow | g \rangle \) occurs via E1 \(\times \) E1 unlike smaller E1 \(\times \) M1 in \(\hbox {Ho}^{3+}\) doped crystal. Xe value of McQ4 integral is \(I_{2\gamma } = 3.1\times 10^{-22}\,\hbox {eV}^{-2}\) (zero at calculation accuracy for \(\hbox {Ho}^{3+}\) case) to be compared the RANP value, \(I_{2\nu } = 2.3 \times 10^{-49}\,\hbox {eV}^{-2}\). A solution is to use photonic crystal for suppression of strayed McQ4 event [20]. Due to a large level spacing the excitation to Xe \(0^-\) is more complicated than a simple two-photon excitation, which has to be studied.

## 6 Summary

In summary, we proposed a general scheme of macro-coherent neutrino-pair emission stimulated by Raman scattering, in order to measure important neutrino properties, the unknown smallest neutrino mass to the level of less than 1 meV, NO/IO distinction, and

Majorana/Dirac distinction. The general scheme was illustrated using \(\hbox {Ho}^{3+}\) doped crystal and Xe atom. Xe has a larger rate than lanthanoid ions, while its QED background is much more severe. Both theoretical and experimental works on QED background rejection are needed to make the general scheme realistic.

## Footnotes

- 1.
We assume that major McQn \((n=5, 6)\) events of smaller rates can be identified by extra photons and are subtracted from data for RANP, while McQn (\(n\ge 7\))’s are negligible with very small rates. Needless to say, higher order spontaneous emission rates without macro-coherence amplification are completely negligible.

- 2.
Parameters determined from neutrino oscillation experiments [1] and used this work are squared mixing matrix elements, \( |U_{ei}|^2, i = 1,2,3\) directly derived from experimental data, and mass differences, \(\delta m^2_{ij} = m_i^2 - m_j^2 \). The smallest neutrino mass is assumed in each calculation. Three CP violation phases are not known in oscillation experiments, and assumed vanishing, for simplicity, in the present work.

## Notes

### Acknowledgements

We thank S. Uetake at Okayama, and C. Braggio, G. Carugno, and F. Chiossi at Padova for useful discussions. 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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