Introduction to the SPAN Project

Transcription

1 Introduction to the SPAN Project SPAN group, Okayama University January 9, What is neutrino and what we know about neutrino? 1.1 What is neutrino? Neutrino and its family Neutrinos belong to a family of leptons; the family in which electrons, muons and tau belong. Neutrinos come in three types referred to as flavors or generations. Neutrinos interact with other particles via only weak interaction; neither strong nor electromagnetic interactions acts on neutrinos. This fact makes neutrinos very illusive and difficult to detect. Figure 1 summarizes all the elementary particles observed experimentally. Particle universe It might be surprising to know that neutrinos are the second most abundant particle in the universe. Figure 2 shows the average number density of various particles in the universe. The most abundant particle is photons: there are about 400 particles per cm 3. The second one is neutrino, and there are more than 300 particles per cm 3. The third group, protons and electrons, are down by 9-10 order of magnitude. Figure 1: Elementary particles included in the Standard Model. The three generations of fermions (quarks and leptons) in the 1st-3rd columns, gauge bosons in the 4th and the Higgs boson in the 5th. Quarks and leptons are building block of matters (atoms and molecules) while the gauge bosons are force carriers. Recently observed Higgs particle is a source-field of mass. [1] Figure 2: Particles in the universe. The vertical scale shows the average number of particles per cm 3. 1

2 1.2 What we know about neutrino? Recent advance in neutrino physics Recently neutrino physics has advanced considerably. The progress has been brought by a type of experiments called oscillation experiments. The oscillation experiments ask Is neutrino produced at some place the same type (or flavor) at different place? The answer is found to be NO in general. Fig.3 shows an example of experimental layouts. In this example, muon-type neutrino beam is produced at Tsukuba. Purity of the beam is guaranteed by selecting a π + beam decaying into ν μ and μ +. At Kamioka, about 250 km away from Tsukuba, you observe neutrino and ask Are the detected neutrinos all muon-type neutrino as were produced?. The answer we have found by the experiments is NO. In fact, some of them are found to turn into electron-type neutrinos. Here an important remark is that flavor is distinguished by interactions (in a detector) and that mass is measured by velocity (during the flight).?? e? Figure 3: The principle of neutrino oscillation experiments. Neutrinos are produced at Tsukuba as a pure ν μ state, and travel about 250 km to Kamioka where they are caught by a detector and their species are examined. How do we interpret this fact? Let s consider a simple model consisting only two flavors and two mass eigenstates. Let s also suppose neutrino flavor eigenstate is a linear combination of mass eigenstates. Then the two types of neutrinos, say ν a and ν b, are described by linear combinations of mass eigenstate neutrinos ν 1 and ν 2 ; flavor state {( }} ){ νa ν b = mixing matrix {( }} ){ cos θ sin θ sin θ cos θ ν b mass state ({}} ){ ν1. (1) ν 2 Here the mixing angle (θ) represents the degree of mixing of two neutrinos. Now let s suppose that a neutrino is produced as a pure flavor eigenstate ν a at the production point x =0. Then its wave function can be represented by ψ(x =0)= ν a =cosθ ν 1 +sinθ ν 2. (2) Note that it is a super-position of the mass-eigenstate neutrinos denoted by ν 1 and ν 2 with masses of m 1,andm 2, respectively. During the flight, each of the mass eigenstate oscillates according to its own frequency. This can be described by (in units of c = h =1) ψ(x) =cosθ ν 1 e i(e 1t p 1 x) +sinθ ν 2 e i(e 2t p 2 x), E i = m 2 i + p2 i (3) 2

3 Then at x = L where neutrinos are detected, they are distinguished by flavor again. As shown in Appendix, it is a straight-forward matter to calculate the probability of finding as b which is produced as a ; P ab = ν b ψ(x = L) 2 ) =sin 2 (2θ)sin (Δm 2 2 L 21 4E Δm 2 21 = m 2 2 m 2 1, E = E 1 + E 2 (4) 2 The oscillation experiments found P ab 0, establishing that neutrinos have finite mass-squared differences (Δm ), and that neutrinos are mixed (θ 0). Figure 4 summarizes what we found by the oscillation experiments up to now. Let s look at the left panel. There are 3 horizontal boxes, each representing mass-eigenstate neutrinos. The mass of the neutrinos is shown by the vertical axis while their flavor contents by the colors. For example, the top horizontal box, the heaviest neutrino, consists of muon- and tau-types almost equally, with a tiny bit of electron type. The second heaviest one is divided into 3 types with approximately equal weights. The mass difference between the heaviest and the second heaviest neutrinos is about 50 mev. Compared to other particle, electrons for example, the neutrino s mass scale is surprisingly small. One important remark here is that the oscillation experiments are sensitive only to mass-squared difference, not to mass itself. Therefore, we cannot determine the absolute mass of neutrinos nor its mass pattern, namely which is the lightest one. Actually there are two mass patterns emerged from the measurements, they are called normal or inverted hierarchy, as shown in Fig.4. mass Normal Hierarchy mass Inverted Hierarchy m3 50 mev m2 m1 10 mev m2 m1 10 mev e m3 50 mev flavor flavor Figure 4: Summary of neutrino flavor mixing. The left (right) panel shows a mass pattern called normal (inverted) hierarchy, which has been emerged from a series of oscillation experiments. The vertical scale shows mass of mass eigenstate neutrinos, and the horizontal axis shows flavor contents. Note that only mass-squared differences can be measured with oscillation experiments. 1.3 Universe and its hitory Composition of Universe Observational cosmology has made a remarkable advance recently thanks to the space technology. The pie-chart shown in Fig. 5 is the current (energy) composition of the universe established by various observations. Surprisingly three-quarter of them are filled with unknown energy source, dubbed dark energy. We do not know what it really is. Another 22% is so-called dark matter. Its existence itself is firmly established by observing their gravitational interaction with other astronomical objects. However it cannot be observed 3

4 Figure 5: Composition of Universe: Universe is found to be composed of dark energy, dark matter and visible matter (atoms and molecules). Why matter anti-matter now while matter=antimatter at its beginning (Big-Bang)? Figure 6: Majorana vs Dirac. Like the electron and positron, a particle is different entity from its anti-particle in the Dirac theory. However, a particle is its own anti-particle in the Majorana theory. Whether neutrino is Dirac or Majorana must be determined experimentally. directly by telescopes of any wavelengths (optical, microwave, X-rays etc.), thus the name dark matter. The rest of 4% is familiar atoms. However, there are virtually no anti-atoms (or anti-matters). From view point of the particle physics, the fact is surprising because exactly the same amount of matter and anti-matter should have existed at the beginning of the universe (Big Bang). We somehow lost our partner in the course of the 14-billion-year history. Thus the fact demands physics explanation. The most promising theory is called lepto-genesis. According to the theory, when the universe was much hotter than today, a gigantic neutrino (a hypothetical partner of the standard neutrino, yet to be confirmed) made tiny imbalance between matter and anti-matter: all antimatters annihilated away with matters but small portion of the matter survived. There are two key features for this theory to be viable: one is Majorana nature of neutrinos, and the other is violation of CP-symmetry. Let s explain these key words one by one below. All the charged leptons and quarks are known to be the Dirac particles. In this case, a particle and its antiparticle is a totally different entity. The electron, for example, has a negative charge while its anti-particle, the positron, has positive charge, and they are different each other. In the case of Majorana particle, however, there is no distinction between particle and anti-particle. In another words, a particle s anti-particle is a particle itself. See Fig.6. That neutrino is Majorana particle is prerequisite for the lepto-genesis scenario. There is another reason to believe neutirnos are Majorana. As shown in Fig.7, neutrino s masses are exceptionally light compared with other fundamental particles. Again the Majorana nature of neutrinos can explain this fact very well. Masses Energy S cale 1neV 1µeV 1meV 1eV 1keV 1MeV 1GeV 1TeV Quarks u d s c b t? ν 1 ν 2 ν 3 Neutrino Leptons e μ τ Higgs H Figure 7: Masses of elementary particles in the Stadard Model. Neutrino masses are exceptionally small compared with the others. 4

5 Xe h Energy Level [ev] p> e> p ( P 1/2)6s [1/2] p ( P )6s [1/2] 1/ p ( P 3/2)6s [3/2] p ( P )6s [3/2] 3/2 2 0 g> 5p 6 1 S0 Figure 8: RENP Diagram Figure 9: Xe energy levels relevant to the RENP process. The other key feature is the CP-symmetry nature, which is the symmetry between particle and its anti-particle times the space inversion symmetry (parity symmetry). Quarks are known to violate this symmetry; it is explained by the Kobayashi-Maskawa CP-violation mechanism (by the CP-violating phase δ). At present, whether or not the CP-violation exits in the neutrino sector is unknown. If neutrinos are Majorana type, there are other CP-violation sources (by CPviolation phases α, β) in addition to a similar CP-violation phase δ to the Kobayashi-Maskawa mechanism in the quark sector. The question whether neutrino is Dirac or Majorana has a profound impact on the elementary particle physics as well as cosmology, and it must be answered by experiments. 2 Spectroscopy with Atomic Neutrino 2.1 Basic Principle of SPAN The SPAN project aims to determine neutrino s most important properties such as their absolute masses, mass type (Majorana/Dirac nature), CP-violating phases etc. SPAN stands for SPectroscopy with Atomic Neutrino. As the name suggests we employ atoms or molecules as a source of neutrinos instead of more traditional nuclear/particle decays. Why do we use atoms, instead of more traditional and popular particle/nuclear decays? How do we do actual experiments with atoms? How sensitive are the experiments to the neutrino properties of interests? We will answer these questions below. (See reference [2] for more details.) RENP process We focus on the process e g + γ + ν ν, where e and g represent an excited and a ground state of atoms, γ is a photon, and ν ν is a pair of neutrinos. It is a radiative emission of neutrino pair process, and we call it RENP in short. Actually we detect the emitted photons and measure their energies (neutrinos are impossible to detect). The photon spectra contain information on the neutrino s absolute masses, the mass type (Majorana/Dirac distinction), and CP phases, as shown below. Use of atoms as neutrino sources has disadvantages as well as advantages. One advantage is closeness of two energy scales between atomic levels ( 1eV) and neutrino masses (< 1eV). This fact permits us to determine all desired neutrino properties. On the other hand the biggest disadvantage is smallness of rate. We plan to overcome this disadvantage by a new amplification mechanism called macro-coherent amplification. This amplification is the most crucial concept in our project, and will be explained in

6 RENP rate and spectrum terms, and is expressed by [2] The rate of the RENP process is conveniently factorized by three Γ γ2ν =Γ 0 I(ω)η ω (t), Γ 0 G 2 F n3 V. (5) The first factor, Γ 0, is an over all rate. It is proportional to the product of the Fermi weak coupling constant (G F ), and target parameters such as its number density (n) andvolume(v ) etc. The second one, I(ω), represents photon energy spectrum containing the physics information about neutrinos: it can be calculated reliably with the Standard Model. The last factor, η ω,is a product of the target coherence and field energy stored in the atomic system. Below we show RENP spectra of Xe target as an example, assuming the macro-coherent amplification functions as expected. The Xe state in interests is the metastable state of 5p 5 ( 2 P 3/2 )6s[ ] 2(see Fig. 9). It is an E1-forbidden state, and has a natural life time of 40 seconds. Figure 10 shows the over all spectrum. The spectrum starts at the half of the energy gap between the metastable and ground states ( 8.4 ev). Real physics information exists near threshold. Figure 11 shows an expanded view around the threshold region. Now many structures can be seen clearly. In particular, the sudden rises (kinks) in the spectrum indicate that new channels are opening for a particular combination of neutrino pairs. For example, the largest rise, on the left, comes from the heaviest neutrinos m 3 and m 3. The threshold positions are expressed by (in units of c = h =1) ω ij = E eg 2 (m i + m j ) 2 (6) 2E eg where E eg is the energy gap between e and g. In this plot, the lightest neutrino mass is assumed to be 2 (in blue) and 20 mev (in red). The solid line or dashed line show normal or inverted hierarchy mass pattern. In real experiment, we turn the argument around. We determine the photon spectrum, or locate the threshold position, and then infer the neutrinos absolute mass from it. In a similar way, we will be able to determine Majorana/Dirac mass type, as well as CP-violating phases. These are, however, much more difficult tasks because the differences in spectra are more subtle. In summary, SPAN project provides us with a systematic approach to studying neutrino properties that are undetermined at present. Xe, Dirac NH vs IH: m0 2,20meV 0.25 Xe NH and IH,m0 20meV ev Figure 10: Xe RENP spectrum I(ω). ev Figure 11: Xe RENP spectrum I(ω) in the the threshold region. The smallest neutrino massisassumedtobe2and20mevfortwo cases of mass patterns, IH (in dashed curves) and NH (in solid). 6

7 2.2 Macro-coherent Amplification The next topic is macro-coherent amplification and its experimental proof by paired superradiance, or PSR in short. Before we go into the detail, let s clarify why rate amplification is necessary. Raw rate, or rate without amplification for the RENP process is extremely small. For a typical weak process, the rate is roughly expressed by Γ G2 F 30π 3 Q5 1/(10 26 year) for Q =1eV (7) where Q is available energy. If we take 1 ev for Q, then we must wait for year to observe one event! This is longer than the age of our universe. Thus we definitely need some amplification mechanism. Our proposal is to use coherence or cooperative process. Fortunately, there exists a good example in atomic physics: it is the super-radiance. 3 excited atoms confined within Figure 12: Rudimentary explanation of the super-radiance with 3 excited atoms. Figure 13: The macro-coherent state may be regarded as one large quantum system of atoms and fields, in which the excited and ground states become coherent by continuous exchanges. Below we give a very simple explanation of the super-radiance. Suppose we have 3 identical atoms as shown in Fig.12. They will be de-excited by emitting photons with a wavelength of λ. Suppose also that they are confined within λ. Then it is impossible to tell which of the 3 atoms actually has emitted photons. Figure 12 shows all possible ways by which 3 atoms go to the ground state. Now the quantum mechanics tells us to sum up all possible amplitudes and then square it to obtain the rate. A simple calculation (see Appendix) leads us to the conclusion of increase in rates. When there are N atoms initially, then the rate is found to be proportional to N 2. The N 2 dependence of the super-radiance can also be understood by the formulae N 2 R γ exp(i k r a )M a a where exp(i k r a ) denotes the plane wave of the emitted photon by the atom located at r a, k =2π/λ represents its wave vector, and Ma is the atomic part matrix element. If M a is common to all (in another words atoms are coherent), then the amplitude is proportional to N because exp (i k r a ) 1 within the wavelength λ, thus the rate is proportional to N 2. One caution must be kept in mind. For this coherent phenomenon to happen, the relaxation time (or dephasing time) must be longer than the phenomenon s characteristic time. Otherwise, the key assumption that M a is common to all fails. The situation can only be realized with a proper target preparation. (8) 7

8 Now we come to a crucial point. Suppose that we have a process which emits multiple outgoing particles, say a photon and a pair of neutrinos for clarity. Then the rate is represented by N R γν ν exp (i( k 1 + k 2 + ) 2 k 3 ) r a M a (9) a where k i denotes the wave vectors of emitted particles. If k 1 + k 2 + k 3 =0,ormomentum conservation law holds, then a coherent amplification mechanism takes place in a similar way to the super-radiance. In this case, the coherent volume is not limited by wavelength λ because exp (i( k 1 + k 2 + ) k 3 ) r a = 1 always holds for k 1 + k 2 + k 3 = 0. The macro-coherence may be visualized in Fig.13 where the atomic excited and ground states become coherent by exchanging photons continuously. Then the atoms and fields as a whole may be regarded as one large quantum system. 2.3 Paired Super-Radiance (PSR) Macro-coherent amplification is a new type of coherent phenomenon. Thus it is highly desirable to prove it experimentally before actual applications. Although it should work for any process, we plan to do it with a process shown in Fig.14, in which two photons are emitted in a deexcitation process from an excited atom. Figure 14: Paired super-radiance process Figure 15: Illustration of PSR outputs. We call this process paired super-radiance (PSR in short). PSR has the following advantages compared with RENP. It is a QED process so that the rate is much larger compared with much weaker weak interaction involving neutrinos. In addition, there is a good experimental signature; namely back-to-back radiations with the same color. Illustration of expected outputs is shown in Fig.15. Note that PSR needs trigger laser to assist (induce) the process because spontaneous emission is usually strictly forbidden. This point is clear contrast to the superradiance in which spontaneous emission (due usually to an electric-dipole transition E1) initiates the process. Para-hydrogen molecules are identified as one of the best PSR targets. We briefly explain properties of this target because numerical simulations shown below as well as the actual experiment now underway (see 3.1) employ this target. The process of interest is the transition from vibrational level Xv = 1 to the ground level Xv = 0. Here X means electronically-ground state. The reasons we use hydrogen molecules are as follows. First, electric-dipole transition E1 is forbidden because of homo-nuclear di-atomic molecule. Second, its de-coherence time is long enough. Third, high density target can be prepared without difficulties. Figure 17 shows an example of numerical simulation results. It is an output signal from the target ends as a function of time. Three color lines correspond to different trigger laser power. In this plot, the trigger laser intensities are varied by 12 order of magnitudes from pw/mm 2 to 8

9 nuclear spin: totally zero Flux W mm^ Figure 16: The first excited state of vibrational level of para-hydrogen is employed in our PSR experiment. Nuclear spins are anti-parallel in the para-hydrogen as opposed to the ortho-hydrogen whose nuclear spins are parallel t ns Figure 17: Output flux vs time. The transition Xv = 1 Xv = 0 of ph2 is considered assuming the target density of n = cm 3, target length l =30cm,relaxation times T 2 =10ns,T 1 =10 3 ns, and the initial coherence of r 1 = 1. Approximately 70 % of energy stored in the initial metastable state is released. W/mm 2. As seen, they change delay time, but not the output signal size. In this particular example, about 70% of stored energy, stored in atomic system initially, are released within a few nano-second. Events look like an explosion illustrated in Fig.15. The natural life time of 2-photon emitting process can be calculated and is found to be seconds. So, the PSR event has shortened the life time, the required time necessary to observe such rare process, by One important prerequisite, we have not mentioned so far, is initial coherence of atoms. In the calculation, we have assumed a complete coherent condition between the excited and ground state. We describe the current status of PSR experiment in the following section. 3 Current Status of the SPAN project 3.1 Search for PSR with para-hydrogen molecules The aim of the experiment is to experimentally prove the principle of the macro-coherence amplification. The setup is shown in Fig.18 (upper part). The target is gas-phase para-hydrogen (ph 2 ) at the liquid nitrogen temperature. It observes PSR emissions from ph 2 vibrational excited states (Xv = 1) to its ground state (Xv = 0). We employ adiabatic Raman process to excite the Xv = 1 state by a pair of lasers with λ = 532 nm and λ = 683 nm. The adiabatic Raman process with appropriate detuning is known to produce coherence between the excited and ground states. The coherence development may be judged by observing the higher order Stokes or anti-stokes emissions whose frequencies are expressed by Ω n = n(ω 0 Ω 1 )+Ω 0. (10) Here Ω 0 and Ω 1 are the two pump laser frequencies (in our case, Ω 0 corresponds to λ = 532 nm and Ω 1 to λ = 683 nm), and n is an integer representing the order of Stokes/anti-Stokes emissions. As seen in Fig.18 (lower part), we are able to observe up to the 8-th anti-stokes emission (λ = 192 nm), indicating good coherence between the levels. The experiment is now in progress, hoping to observe signals soon. 9

10 871LD+TA 1064LD+TA Nd:YAG WG-PPLN 532 nm PPSLT+LBO 4.8 µm Gas ph 2 Cryostat 683 nm Prism Photo Detectors para H nm 683 nm (1583) Figure 18: PSR experiment with ph 2. (upper left) The experimental layout. (upper right) The vibrational level Xv = 1 is excited by adiabatic Raman process. (lower part) The observed higher-order Stokes/anti-Stokes emissions. 3.2 Coherence in solid para-hydrogen matrix Solid material with large coherence is an ideal target to realize neutrino mass spectroscopy with atoms. One potential problem in solid targets lies in its short coherence time. In this experimental research, we have studied the coherence time of the solid ph 2 in the Xv =2state. The principle of the measurement is time-resolved coherent anti-stokes Raman spectroscopy. It utilizes the fact that the intensity of the anti-stokes emission is proportional to the coherence between the Xv =0adXv = 2 states. Figure 19 shows the results of the measurements, indicating that coherence time is longer than 20 nsec. This is exceptionally long for the solid materials whose typical coherence time is much less than 1 nsec. As seen in the plot, the orthohydrogen acts as impurity and reduces the coherence time. See reference [5] for the detail. CARS intensity (au) delay t (ns) o/p 1.9 % 0.23 % 0.08 % 0.01 % 600 Figure 19: Coherence decay curves at T = 4.2 K with various o/p ratios. Figure 20: SR peak height vs n 2 e. The solid circles (black) and open squares (red) are, respectively, the experimental and MB simulation results. The error bars represent rms fluctuations of the data. 10

11 3.3 Coherence development by SR We have performed an experiment with atomic Barium in gas phase. The goal of the experiments is twofold: (i) fast and efficient production of a high-density atomic target suited to the observation of PSR, and (ii) investigation into the coherence development at the initial stage of super-radiance (SR). We adopt the following production method: we first bring the ground state atoms to the intermediate state of 6s6p 1 P 1 by illuminating 553.7nm laser light. Then they are transferred to the 1 D 2 state via radiative transition in the SR mode, anticipating otherwise slow transition rate ( 1/4 μs 1 ) to be accelerated. One salient feature of the experiment is use of a Stokes laser; nm CW laser light corresponding to the 1 P 1 1 D 2 energy difference. Figure 20 shows the density dependence of SR peak pulse. As seen SR peak pulse is proportional to n 2 e where n e is the number density of Ba. The curve looks like saturating: this is due to the slow response of the detector. When the Stokes laser was injected additionally to the pump laser, we observed not only the SR delay time shortened but also its angular distribution sharpened significantly. Figure 21 shows SR angular distribution with (right) and without (left) the Stokes laser. In this case, the fwhm angular width became from Δθ no 27 mrad to Δθ st 1.0 mrad; the ratio is about 27. It can be concluded that injection of Stokes laser restricts SR to a single mode and increases the initial coherence. The details can be found in reference [6]. Figure 21: SR angular distribution with (right) and without (left) the Stokes laser. Note the changes both in the vertical and horizontal scales. The small dots (red) are the fit results to the experimental data (black) with a 2-dimensional gauss function. The solid lines show the projection of the fits onto the x-plane. The data were taken at n e = m 3. References [1] From Wikipedia, the free encyclopedia; [2] A. Fukumi et al., Progr. Theor. Exp. Phys. 2012, 04D002 and arxiv v1 [3] D.N. Dinh, et al., Phys. Lett. B719, 154 (2012) and arxiv v1[hep-ph]. [4] M. Yoshimura, N. Sasao, and M. Tanaka, Phys. Rev. A86, (2012) and arxiv v1 [quant-ph]. [5] S.Kuma, Y.Miyamoto, K.Nakajima, A.Fukumi, K.Kawaguchi, I.Nakano, N.Sasao, M.Tanaka, J.Tang, T.Taniguchi, S.Uetake, T.Wakabayashi, A.Yoshimi, M.Yoshimura, J.Chem.Phys. 138, (1-6) (2013) [6] Chiaki Ohae, et. al., Submitted to Journal of the Physical Society of Japan (JPSJ). 11

12 4 Appendix 4.1 More on Neutrino Mixing We begin with the following formula given in the text: ψ(x) =cosθ ν 1 e i(e 1t p 1 x) +sinθ ν 2 e i(e 2t p 2 x) (11) In the expression, the time t is given by t = x/ v, where v is the average velocity of two neutrinos with mass m 1 and m 2 expressed by v = p 1 + p 2. (12) E 2 + E 2 The important quantity when calculating the probability P ab is the phase shift δφ =(p 1 p 2 )x (E 1 E 2 )t (13) between the two neutrino states. Putting Eq.12 into Eq.13, we obtain (for x = L) δφ =(p 1 p 2 )L E2 1 E2 2 L = m2 1 m2 2 L Δm 2 L 21 p 1 + p 2 p 1 + p 2 2E (14) where the relation Ei 2 = m2 i + p2 i is used. Then Eq.11 may be rewritten as ψ(l) =e i φ { ( ) ( ) cos θ ν a cos θ ν b sin θ e iδφ/2 +sinθ ν a sin θ + ν b cos θ e iδφ/2} Applying ν b from the left and squaring it, we finally obtain ν b ψ(l) = e i φ sin(2θ) eiδφ/2 e iδφ/2 2 P ab = ν b ψ(x = L) 2 ) =sin 2 (2θ)sin (Δm 2 2 L 21 (15) 4E This is the basic oscillation formula. In reality there exists 3 neutrino flavors, and thus it is necessary to extend the above formula. The 3-flavor mixing matrix is given by c 13 0 s 13 e iδ c 12 s 12 0 V = 0 c 23 s s 12 c s 23 c 23 s 13 e +iδ 0 c c 12 c 13 c 13 s 12 e iδ s 13 = c 23 s 12 e iδ c 12 s 13 s 23 c 12 c 23 e iδ s 12 s 13 s 23 c 13 s 23 (16) s 12 s 23 e iδ c 12 c 23 s 13 e iδ c 23 s 12 s 13 c 12 s 23 c 13 c 23 The neutrino parameters measured by experiments are summarized in Table 1. Table 1: Summary of Neutrino Mixing Parameters mass or mass squared difference mixing angle Δm (±0.26) [ev 2 ] sin 2 θ (±0.018) Δm (±0.12) [ev 2 ] sin 2 θ (±0.008) m ν < 2.3 [ev](*) sin 2 θ (±0.003) m < (??) (**) δ CP unknown 12