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| {{More footnotes|date=April 2010}}
| | == primary sector == |
| {{Beyond the Standard Model|expanded=Evidence}}
| |
| '''Neutrino oscillation''' is a [[quantum mechanics|quantum mechanical]] phenomenon predicted by [[Bruno Pontecorvo]]<ref name="Pontecorvo1957">
| |
| {{cite journal
| |
| |author=B. Pontecorvo
| |
| |title=Mesonium and anti-mesonium
| |
| |journal=[[Zh. Eksp. Teor. Fiz.]]
| |
| |volume=33
| |
| |pages=549–551
| |
| |year=1957
| |
| |doi=
| |
| }} reproduced and translated in {{cite journal
| |
| |journal=[[Sov. Phys. JETP]]
| |
| |volume=6
| |
| |pages=429
| |
| |year=1957
| |
| |doi=
| |
| }} and
| |
| {{cite journal
| |
| |author=B. Pontecorvo
| |
| |title=Neutrino Experiments and the Problem of Conservation of Leptonic Charge
| |
| |journal=[[Zh. Eksp. Teor. Fiz.]]
| |
| |volume=53
| |
| |pages=1717
| |
| |year=1967
| |
| |doi=
| |
| }} reproduced and translated in {{cite journal
| |
| |journal=[[Sov. Phys. JETP]]
| |
| |volume=26
| |
| |pages=984
| |
| |year=1968
| |
| |doi=
| |
| |bibcode = 1968JETP...26..984P }}</ref>
| |
| whereby a [[neutrino]] created with a specific [[lepton]] [[flavor (particle physics)|flavor]] ([[electron]], [[muon]] or [[tau lepton|tau]]) can later be [[Quantum measurement|measured]] to have a different flavor. The probability of measuring a particular flavor for a neutrino varies periodically as it propagates. Neutrino oscillation is of [[theoretical physics|theoretical]] and [[experimental physics|experimental]] interest since observation of the phenomenon implies that the neutrino has a non-zero mass, which is not part of the original [[Standard Model]] of [[particle physics]].
| |
|
| |
|
| ==Observations==
| | With these encounters with [http://www.nrcil.net/fancybox/lib/rakuten_LV_133.html ルイヴィトン 財布 モノグラム] Luo Feng [http://www.nrcil.net/fancybox/lib/rakuten_LV_123.html ルイヴィトン パドロック] himself, [http://www.nrcil.net/fancybox/lib/rakuten_LV_14.html メンズ ルイヴィトン] and no pride.<br><br>'the sheer vastness of the universe, the fate of the special, there are adventures, and [http://www.nrcil.net/fancybox/lib/rakuten_LV_111.html バック ルイヴィトン] certainly there are many. really good ...... will combine everything, use good, in [http://www.nrcil.net/fancybox/lib/rakuten_LV_97.html ルイヴィトン中古通販] order to become a masterpiece strong.' Luo Feng illegal channels, 'as long as the earth tens Qianxiu In 'I can officially become the main industry [http://www.nrcil.net/fancybox/lib/rakuten_LV_120.html ルイヴィトンのキーケース] of. By the time I will be able to carry out the laws of perception of a rapid increase. '<br><br>primary sector, real outbreak.<br><br>Feng Luo do now, is to give his foundation is growing fast, [http://www.nrcil.net/fancybox/lib/rakuten_LV_137.html ルイヴィトン ショルダーバッグ] so that when the main Yanfeng sector can achieve strikingly terrible burst of strength!<br><br>......<br><br>that is over half gone.<br>control room<br>meteorite ink asterisk, Feng Luo and Dylan stood in the control room.<br><br>'immediately [http://www.nrcil.net/fancybox/lib/rakuten_LV_73.html ルイヴィトン アクセサリー] began shuttling the universe, into the solar system.'<br><br>'Countdown, 10 ...... 9 ...... 8 ...... 7 ...... 6 ...... 5 ...... 4 ...... 3 ...... 2 ...... 1!' |
| A great deal of evidence for neutrino oscillation has been collected from many sources, over a wide range of neutrino energies and with many different detector technologies.<ref name="G-M Review">
| | 相关的主题文章： |
| {{cite journal
| | <ul> |
| | author=M. C. Gonzalez-Garcia and Michele Maltoni
| | |
| | year=2008
| | <li>[http://www.bmwclassics.co.uk/cgi-bin/gb/guestbook.cgi http://www.bmwclassics.co.uk/cgi-bin/gb/guestbook.cgi]</li> |
| | journal=[[Physics Reports]]
| | |
| | volume=460 | pages=1–129
| | <li>[http://bbs.520.io/forum.php?mod=viewthread&tid=317911&fromuid=80983 http://bbs.520.io/forum.php?mod=viewthread&tid=317911&fromuid=80983]</li> |
| | arxiv=0704.1800
| | |
| | title=Phenomenology with Massive Neutrinos
| | <li>[http://pswa.moo.jp/kaigi/joyful.cgi http://pswa.moo.jp/kaigi/joyful.cgi]</li> |
| | doi=10.1016/j.physrep.2007.12.004
| | |
| |bibcode = 2008PhR...460....1G }}
| | </ul> |
| </ref>
| |
| | |
| ===Solar neutrino oscillation===
| |
| The first experiment that detected the effects of neutrino oscillation was [[Raymond Davis Jr.|Ray Davis's]] [[Homestake Experiment]] in the late 1960s, in which he observed a deficit in the flux of [[Sun|solar]] neutrinos with respect to the prediction of the [[Standard Solar Model]], using a [[chlorine]]-based detector. This gave rise to the [[Solar neutrino problem]]. Many subsequent radiochemical and water [[Cherenkov radiation|Cherenkov]] detectors confirmed the deficit, but neutrino oscillation was not conclusively identified as the source of the deficit until the [[Sudbury Neutrino Observatory]] provided clear evidence of neutrino flavor change in 2001.
| |
| | |
| Solar neutrinos have energies below 20 [[MeV]] and travel one [[astronomical unit]] between the source in the Sun and detector on the Earth. At energies above 5 MeV, solar neutrino oscillation actually takes place in the Sun through a resonance known as the [[Mikheyev–Smirnov–Wolfenstein effect|MSW effect]], a different process from the vacuum oscillation described later in this article.
| |
| | |
| ===Atmospheric neutrino oscillation===
| |
| Large detectors such as [[Irvine-Michigan-Brookhaven (detector)|IMB]], [[Monopole, Astrophysics and Cosmic Ray Observatory|MACRO]], and [[Kamiokande II]] observed a deficit in the ratio of the flux of muon to electron flavor atmospheric neutrinos (see ''[[Muon#Muon decay|muon decay]]''). The [[Super Kamiokande]] experiment provided a very precise measurement of neutrino oscillation in an energy range of hundreds of MeV to a few TeV, and with a baseline of the diameter of the [[Earth]]: the first experimental evidence for atmospheric neutrino oscillations was announced in 1998.
| |
| | |
| ===Reactor neutrino oscillation===
| |
| Many experiments have searched for oscillation of electron [[antimatter|anti]]-neutrinos produced at [[nuclear reactors]]. Such oscillations give the value of the parameter [[PMNS matrix|θ<sub>13</sub>]]. The [[KamLAND]] experiment, started in 2002, has made a high precision observation of reactor neutrino oscillation. Neutrinos produced in nuclear reactors have energies similar to solar neutrinos, a few MeV. The baselines of these experiments have ranged from tens of meters to over 100 km.
| |
| | |
| On 8 March 2012, the [[Daya Bay Reactor Neutrino Experiment|Daya Bay]] team announced a 5.2σ discovery that θ<sub>13</sub>≠0.<ref>http://neutrino.physics.wisc.edu/dayabay/2012-03-08-oscPRL/DYB_EWNP_v5.pdf</ref>
| |
| Two other experiments are currently measuring reactor neutrino oscillation (at the same baseline of a few kilometers) and may eventually confirm the Daya Bay results: [[Double Chooz]] and [[RENO]].
| |
| | |
| ===Beam neutrino oscillation===
| |
| Neutrino beams produced at a [[particle accelerator]] offer the greatest control over the neutrinos being studied. Many experiments have taken place which study the same neutrino oscillations which take place in atmospheric neutrino oscillation, using neutrinos with a few GeV of energy and several hundred km baselines. The [[MINOS]] experiment recently announced that it observes consistency with the results of the [[K2K experiment|K2K]] and [[Super-Kamiokande|Super-K]] experiments.
| |
| | |
| The controversial observation of beam neutrino oscillation at the [[Liquid Scintillator Neutrino Detector|LSND experiment]] in 2006 was tested by [[MiniBooNE]]. Results from MiniBooNE appeared in Spring 2007, and appeared to contradict the findings of the LSND experiment. Results from the [[HARP (Hadron Production Experiment)|HARP-CDP]] group also put the LSND result into doubt.
| |
| | |
| On 31 May 2010, the [[INFN]] and [[CERN]] announced<ref>http://press.web.cern.ch/press/PressReleases/Releases2010/PR08.10E.html</ref> having observed a tau particle in a muon neutrino beam in the [[OPERA experiment|OPERA detector]] located at [[Laboratori Nazionali del Gran Sasso|Gran Sasso]], 730 km away from the neutrino source in [[Geneva]].
| |
| | |
| The currently-running [[T2K experiment]] uses a neutrino beam directed through 295 km of earth, and will measure the parameter [[PMNS matrix|''θ''<sub>13</sub>]]. The experiment uses the Super-K detector. [[NOνA]] is a similar effort. This detector will use the same beam as MINOS and will have a baseline of 810 km.
| |
| | |
| ==Theory==
| |
| | |
| Neutrino oscillation arises from a mixture between the flavor and mass [[eigenstates]] of neutrinos. That is, the three neutrino states that interact with the charged leptons in weak interactions are each a different [[Quantum superposition|superposition]] of the three neutrino states of definite mass. Neutrinos are created in [[Weak interaction|weak]] processes in their flavor eigenstates{{#tag:ref|More formally, the neutrinos are emitted in an [[Quantum entanglement|entangled]] state with the other bodies in the decay or reaction, and the mixed state is properly described by a [[density matrix]]. However, for all practical situations, the other particles in the decay may be well localized in time and space (e.g. to within a nuclear distance), leaving their momentum with a large spread. When these partner states are projected out, the neutrino is left in a state that for all intents and purposes behaves as the simple superposition of mass states described here. See <ref>
| |
| {{cite journal
| |
| |author=Andrew G. Cohen, Sheldon L. Glashow, and Zoltan Ligeti
| |
| |title=Disentangling neutrino oscillations
| |
| |journal=Physics Letters B
| |
| |volume=678 |page=191
| |
| |year=2009
| |
| |doi=10.1016/j.physletb.2009.06.020
| |
| |bibcode = 2009PhLB..678..191C |arxiv = 0810.4602 }}
| |
| </ref> for more information.|group="nb"}}. As a neutrino propagates through space, the quantum mechanical [[Phase factor|phases]] of the three mass states advance at slightly different rates due to the slight differences in the neutrino masses. This results in a changing mixture of mass states as the neutrino travels, but a different mixture of mass states corresponds to a different mixture of flavor states. So a neutrino born as, say, an electron neutrino will be some mixture of electron, mu, and tau neutrino after traveling some distance. Since the quantum mechanical phase advances in a periodic fashion, after some distance the state will nearly return to the original mixture, and the neutrino will be again mostly electron neutrino. The electron flavor content of the neutrino will then continue to oscillate as long as the quantum mechanical state maintains [[Coherence (physics)#Quantum coherence|coherence]]. It is because the mass differences between the neutrinos are small that the [[coherence length]] for neutrino oscillation is so long, making this microscopic quantum effect observable over macroscopic distances.
| |
| | |
| On July 19, 2013 the results from the [[T2K experiment]] presented at the [[European Physical Society]] Conference on High Energy Physics in Stockholm, Sweden, confirmed the theory.<ref name="PhysicsNews20130719">[http://www.physnews.com/physics-news/cluster637855326/ "Neutrino shape-shift points to new physics"] ''Physics News'', 19 July 2013.</ref><ref name="BBC20130719">[http://www.bbc.co.uk/news/science-environment-23366318 "Neutrino 'flavour' flip confirmed"] ''BBC News'', 19 July 2013.</ref>
| |
| | |
| ===Pontecorvo–Maki–Nakagawa–Sakata matrix===
| |
| {{Main|Pontecorvo–Maki–Nakagawa–Sakata matrix}}
| |
| The idea of neutrino oscillation was first put forward in 1957 by [[Bruno Pontecorvo]], who proposed that neutrino-antineutrino transitions may occur in analogy with [[Kaon#Neutral kaon mixing|neutral kaon mixing]].<ref name="Pontecorvo1957" /> Although such matter-antimatter oscillation has not been observed, this idea formed the conceptual foundation for the quantitative theory of neutrino flavor oscillation, which was first developed by Maki, Nakagawa, and Sakata in 1962<ref>
| |
| {{cite journal
| |
| |author=Z. Maki, M. Nakagawa, and S. Sakata
| |
| |journal=[[Progress of Theoretical Physics]]
| |
| |volume=28 |page=870
| |
| |year=1962
| |
| |doi=10.1143/PTP.28.870
| |
| |title=Remarks on the Unified Model of Elementary Particles
| |
| |bibcode = 1962PThPh..28..870M }}</ref>
| |
| and further elaborated by Pontecorvo in 1967.<ref>
| |
| {{cite journal
| |
| |author=B. Pontecorvo
| |
| |title=Neutrino Experiments and the Problem of Conservation of Leptonic Charge
| |
| |journal=[[Zh. Eksp. Teor. Fiz.]]
| |
| |volume=53 |page=1717
| |
| |year=1967
| |
| |doi=
| |
| }} reproduced and translated in {{cite journal
| |
| |journal=[[Sov. Phys. JETP]]
| |
| |volume=26
| |
| |page=984
| |
| |year=1968
| |
| |doi=
| |
| |bibcode = 1968JETP...26..984P }}</ref> One year later the solar neutrino deficit was first observed,<ref>
| |
| {{cite journal
| |
| |author=Raymond Davis Jr., Don S. Harmer, and Kenneth C. Hoffman
| |
| |journal=[[Physical Review Letters]]
| |
| |volume=20 |page=1205
| |
| |year=1968
| |
| |doi=10.1103/PhysRevLett.20.1205
| |
| |title=Search for Neutrinos from the Sun
| |
| |bibcode = 1968PhRvL..20.1205D }}</ref> and that was followed by the famous paper of Gribov and Pontecorvo published in 1969 titled "Neutrino astronomy and lepton charge".<ref>
| |
| {{cite journal
| |
| |author=V. Gribov and B. Pontecorvo
| |
| |journal=[[Physics Letters B]]
| |
| |volume=28 |page=493
| |
| |year=1969
| |
| |doi=10.1016/0370-2693(69)90525-5
| |
| |title=Neutrino astronomy and lepton charge
| |
| |bibcode = 1969PhLB...28..493G }}</ref> The idea of neutrino mixing is a natural outcome of gauge theories with massive neutrinos and its structure can be characterized in general.<ref>.
| |
| {{cite journal
| |
| |author=J. Schechter, J.W.F. Valle
| |
| |journal=[[Physical Review D]]
| |
| |volume=22 |page=2227
| |
| |year=1980
| |
| |doi=10.1103/PhysRevD.22.2227
| |
| |title=Neutrino Masses in SU(2) x U(1) Theories
| |
| |bibcode = 1980PhRvD..22.2227S }}</ref>
| |
| In its simplest form it is expressed as a [[unitary transformation]] relating the flavor and mass [[eigenbasis|eigenbases]] can be written
| |
| :<math> \left| \nu_{\alpha} \right\rangle = \sum_{i} U_{\alpha i} \left| \nu_{i} \right\rangle\,</math>
| |
| :<math> \left| \nu_{i} \right\rangle = \sum_{\alpha} U_{\alpha i}^{*} \left| \nu_{\alpha} \right\rangle</math>, | |
| | |
| where
| |
| | |
| * <math> \left| \nu_{\alpha} \right\rangle</math> is a neutrino with definite flavor. α = e (electron), μ (muon) or τ (tauon).
| |
| * <math> \left| \nu_{i} \right\rangle</math> is a neutrino with definite mass <math>m_i</math>, <math>i =</math> 1, 2, 3.
| |
| * The asterisk (<math>{}^*</math>) represents a [[complex conjugate]]. For [[antineutrino]]s, the complex conjugate should be dropped from the second equation, and added to the first.
| |
| | |
| <math> U_{\alpha i} </math> represents the ''Pontecorvo–Maki–Nakagawa–Sakata matrix'' (also called the ''PMNS matrix'', ''lepton mixing matrix'', or sometimes simply the ''MNS matrix''). It is the analogue of the [[Cabibbo–Kobayashi–Maskawa matrix|CKM matrix]] describing the analogous mixing of [[quark]]s. If this matrix were the [[identity matrix]], then the flavor eigenstates would be the same as the mass eigenstates. However, experiment shows that it is not.
| |
| | |
| When the standard three neutrino theory is considered, the matrix is 3×3. If only two neutrinos are considered, a 2×2 matrix is used. If one or more sterile neutrinos are added (see later) it is 4×4 or larger. In the 3×3 form, it is given by:<ref name="pdg">
| |
| {{cite journal
| |
| | author = S. Eidelman et al.
| |
| | year = 2004
| |
| | title = Particle Data Group - The Review of Particle Physics
| |
| | journal = [[Physics Letters B]]
| |
| | volume = 592 | issue = 1
| |
| | doi = 10.1016/j.physletb.2004.06.001
| |
| | url = http://pdg.lbl.gov
| |
| | bibcode=2004PhLB..592....1P
| |
| |arxiv = astro-ph/0406663 }} Chapter 15: ''[http://pdg.lbl.gov/2005/reviews/numixrpp.pdf Neutrino mass, mixing, and flavor change]''. Revised September 2005.</ref>
| |
| | |
| :<math>
| |
| \begin{align}
| |
| U &= \begin{bmatrix}
| |
| U_{e 1} & U_{e 2} & U_{e 3} \\
| |
| U_{\mu 1} & U_{\mu 2} & U_{\mu 3} \\
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| U_{\tau 1} & U_{\tau 2} & U_{\tau 3}
| |
| \end{bmatrix} \\
| |
| &= \begin{bmatrix}
| |
| 1 & 0 & 0 \\
| |
| 0 & c_{23} & s_{23} \\
| |
| 0 & -s_{23} & c_{23}
| |
| \end{bmatrix}
| |
| \begin{bmatrix}
| |
| c_{13} & 0 & s_{13} e^{-i\delta} \\
| |
| 0 & 1 & 0 \\
| |
| -s_{13} e^{i\delta} & 0 & c_{13}
| |
| \end{bmatrix}
| |
| \begin{bmatrix}
| |
| c_{12} & s_{12} & 0 \\
| |
| -s_{12} & c_{12} & 0 \\
| |
| 0 & 0 & 1
| |
| \end{bmatrix}
| |
| \begin{bmatrix}
| |
| e^{i\alpha_1 / 2} & 0 & 0 \\
| |
| 0 & e^{i\alpha_2 / 2} & 0 \\
| |
| 0 & 0 & 1
| |
| \end{bmatrix} \\
| |
| &= \begin{bmatrix}
| |
| c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{-i\delta} \\
| |
| - s_{12} c_{23} - c_{12} s_{23} s_{13} e^{i \delta} & c_{12} c_{23} - s_{12} s_{23} s_{13} e^{i \delta} & s_{23} c_{13}\\
| |
| s_{12} s_{23} - c_{12} c_{23} s_{13} e^{i \delta} & - c_{12} s_{23} - s_{12} c_{23} s_{13} e^{i \delta} & c_{23} c_{13}
| |
| \end{bmatrix}
| |
| \begin{bmatrix}
| |
| e^{i\alpha_1 / 2} & 0 & 0 \\
| |
| 0 & e^{i\alpha_2 / 2} & 0 \\
| |
| 0 & 0 & 1
| |
| \end{bmatrix} \\
| |
| \end{align}
| |
| </math>
| |
| | |
| where ''c''<sub>ij</sub> = cosθ<sub>ij</sub> and ''s''<sub>ij</sub> = sinθ<sub>ij</sub>. The phase factors ''α''<sub>1</sub> and ''α''<sub>2</sub> are physically meaningful only if neutrinos are [[Majorana equation|Majorana particles]] — i.e. if the neutrino is identical to its antineutrino (whether or not they are is unknown) — and do not enter into oscillation phenomena regardless. If [[neutrinoless double beta decay]] occurs, these factors influence its rate. The phase factor ''δ'' is non-zero only if neutrino oscillation violates [[CP symmetry]]. This is expected, but not yet observed experimentally. If experiment shows this 3×3 matrix to be not [[Unitary matrix|unitary]], a [[sterile neutrino]] or some other new physics is required.
| |
| | |
| ===Propagation and interference===
| |
| Since <math> \left| \nu_{i} \right\rangle</math> are mass eigenstates, their propagation can be described by [[plane wave]] solutions of the form
| |
| | |
| :<math> |\nu_{i}(t)\rangle = e^{ -i ( E_{i} t - \vec{p}_{i} \cdot \vec{x}) }|\nu_{i}(0)\rangle,</math> | |
| | |
| where
| |
| | |
| * quantities are expressed in [[natural units]] <math>(c = 1, \hbar = 1)</math>
| |
| * <math>E_{i}</math> is the [[energy]] of the mass-eigenstate <math>i</math>,
| |
| * <math>t</math> is the time from the start of the propagation,
| |
| * <math>\vec{p}_{i}</math> is the three-dimensional [[momentum]],
| |
| * <math>\vec{x}</math> is the current position of the particle relative to its starting position
| |
| | |
| In the [[ultrarelativistic limit]], <math>|\vec{p}_i| = p_i \gg m_i</math>, we can approximate the energy as
| |
| | |
| :<math>E_{i} = \sqrt{p_{i}^2 + m_{i}^2 }\simeq p_{i} + \frac{m_{i}^2}{2 p_{i}} \approx E + \frac{m_{i}^2}{2 E},</math>
| |
| | |
| where E is the total energy of the particle.
| |
| | |
| This limit applies to all practical (currently observed) neutrinos, since their masses are less than 1 eV and their energies are at least 1 MeV, so the [[Lorentz factor]] γ is greater than 10<sup>6</sup> in all cases. Using also ''t ≈ L'', where ''L'' is the distance traveled and also dropping the phase factors, the wavefunction becomes:
| |
| | |
| :<math> |\nu_{i}(L)\rangle = e^{ -i m_{i}^2 L/2E }|\nu_{i}(0)\rangle.</math>
| |
| | |
| Eigenstates with different masses propagate at different speeds. The heavier ones lag behind while the lighter ones pull ahead. Since the mass eigenstates are combinations of flavor eigenstates, this difference in speed causes interference between the corresponding flavor components of each mass eigenstate. Constructive [[interference (wave propagation)|interference]] causes it to be possible to observe a neutrino created with a given flavor to change its flavor during its propagation. The probability that a neutrino originally of flavor α will later be observed as having flavor β is
| |
| | |
| :<math>P_{\alpha\rightarrow\beta}=\left|\left\langle \nu_{\beta}|\nu_{\alpha}(t)\right\rangle \right|^{2}=\left|\sum_{i}U_{\alpha i}^{*}U_{\beta i}e^{ -i m_{i}^2 L/2E }\right|^{2}.</math>
| |
| | |
| This is more conveniently written as
| |
| | |
| :<math>\begin{matrix}P_{\alpha\rightarrow\beta}=\delta_{\alpha\beta} & - & 4{\displaystyle \sum_{i>j}{\rm Re}(U_{\alpha i}^{*}U_{\beta i}U_{\alpha j}U_{\beta j}^{*}})\sin^{2}(\frac{\Delta m_{ij}^{2}L}{4E})\\ & + & {\displaystyle 2\sum_{i>j}{\rm Im}(U_{\alpha i}^{*}U_{\beta i}U_{\alpha j}U_{\beta j}^{*})\sin(}\frac{\Delta m_{ij}^{2}L}{2E}),\end{matrix}</math>
| |
| | |
| where <math>\Delta m_{ij}^{2} \ \equiv m_{i}^2 - m_{j}^2</math>. The phase that is responsible for oscillation is often written as (with ''c'' and <math>\hbar</math> restored)<ref>{{cite arXiv |eprint=hep-ph/0602115 |author1=Minako Honda |author2=Yee Kao |author3=Naotoshi Okamura |author4=Tatsu Takeuchi |title=A Simple Parameterization of Matter Effects on Neutrino Oscillations |class=hep-ph |year=2006}}</ref>
| |
| | |
| :<math> \frac{\Delta m^2\, c^3\, L}{4 \hbar E} = \frac{{\rm GeV}\, {\rm fm}}{4 \hbar c} \times \frac{\Delta m^2}{{\rm eV}^2} \frac{L}{\rm km} \frac{\rm GeV}{E} \approx 1.267 \times \frac{\Delta m^2}{{\rm eV}^2} \frac{L}{\rm km} \frac{\rm GeV}{E},</math>
| |
| | |
| where 1.267 is unitless. In this form, it is convenient to plug in the oscillation parameters since:
| |
| | |
| * The mass differences, Δ''m''<sup>2</sup>, are known to be on the order of {{val|1|e=-4|u=eV<sup>2</sup>}}
| |
| * Oscillation distances, ''L'', in modern experiments are on the order of [[kilometers]]
| |
| * Neutrino energies, ''E'', in modern experiments are typically on order of MeV or GeV.
| |
| | |
| If there is no [[CP-violation]] (δ is zero), then the second sum is zero.
| |
| | |
| ===Two neutrino case===
| |
| The above formula is correct for any number of neutrino generations. Writing it explicitly in terms of mixing angles is extremely cumbersome if there are more than two neutrinos that participate in mixing. Fortunately, there are several cases in which only two neutrinos participate significantly. In this case, it is sufficient to consider the mixing matrix
| |
| | |
| :<math>U = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}.</math> | |
| | |
| Then the probability of a neutrino changing its flavor is
| |
| | |
| :<math>P_{\alpha\rightarrow\beta, \alpha\neq\beta} = \sin^{2}(2\theta)\, \sin^{2} \left(\frac{\Delta m^2 L}{4E}\right)\, \mathrm{(natural\,units)}.</math>
| |
| | |
| Or, using [[SI units]] and the convention introduced above
| |
| | |
| :<math>P_{\alpha\rightarrow\beta, \alpha\neq\beta} = \sin^{2}(2\theta) \, \sin^{2}\left( 1.267 \frac{\Delta m^2 L}{E} \frac{\rm GeV}{\rm eV^{2}\,\rm km}\right).</math>
| |
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| This formula is often appropriate for discussing the transition ν<sub>μ</sub> ↔ ν<sub>τ</sub> in atmospheric mixing, since the electron neutrino plays almost no role in this case. It is also appropriate for the solar case of ν<sub>e</sub> ↔ ν<sub>x</sub>, where ν<sub>x</sub> is a superposition of ν<sub>μ</sub> and ν<sub>τ</sub>. These approximations are possible because the mixing angle θ<sub>13</sub> is very small and because two of the mass states are very close in mass compared to the third.
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| === Classical analogue of neutrino oscillation ===
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| {{multiple image | direction = vertical | width = 200
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| |image1=Coupled.svg
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| |caption1=Spring-coupled pendulums
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| |image2=Schwebungsfall.svg
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| |caption2=Time evolution of the pendulums
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| |image3=Gleichsinnig.svg
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| |caption3=Lower frequency normal mode
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| |image4=Gegensinnig.svg
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| |caption4=Higher frequency normal mode
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| }}
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| The basic physics behind neutrino oscillation can be found in any system of coupled [[harmonic oscillator]]s. A simple example is a system of two [[pendulum]]s connected by a weak spring (a spring with a small [[spring constant]]). The first pendulum is set in motion by the experimenter while the second begins at rest. Over time, the second pendulum begins to swing under the influence of the spring, while the first pendulum's amplitude decreases as it loses energy to the second. Eventually all of the system's energy is transferred to the second pendulum and the first is at rest. The process then reverses. The energy oscillates between the two pendulums repeatedly until it is lost to [[friction]].
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| The behavior of this system can be understood by looking at its [[normal modes]] of oscillation. If the two pendulums are identical then one normal mode consists of both pendulums swinging in the same direction with a constant distance between them, while the other consists of the pendulums swinging in opposite (mirror image) directions. These normal modes have (slightly) different frequencies because the second involves the (weak) spring while the first does not. The initial state of the two-pendulum system is a combination of both normal modes. Over time, these normal modes drift out of phase, and this is seen as a transfer of motion from the first pendulum to the second.
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| When the pendulums are not identical the analysis is slightly more complicated. In the small-angle approximation, the [[potential energy]] of a single pendulum system is <math>\tfrac12\tfrac{mg}{L} x^2</math>, where ''g'' is the [[standard gravity]], ''L'' is the length of the pendulum, ''m'' is the mass of the pendulum, and ''x'' is the horizontal displacement of the pendulum. As an isolated system the pendulum is a harmonic oscillator with a frequency of <math>\sqrt{g/L}</math>. The potential energy of a spring is <math>\tfrac12 k x^2</math> where ''k'' is the spring constant and ''x'' is the displacement. With a mass attached it oscillates with a period of <math>\sqrt{k/m}</math>. With two pendulums (labeled ''a'' and ''b'') of equal mass but possibly unequal lengths and connected by a spring, the total potential energy is
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| :<math>V = \frac{m}{2} \left( \frac{g}{L_a} x_a^2 + \frac{g}{L_b} x_b^2 + \frac{k}{m} (x_b - x_a)^2 \right).</math>
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| This is a [[quadratic form]] in x<sub>a</sub> and x<sub>b</sub>, which can also be written as a matrix product:
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| :<math>V = \frac{m}{2} \begin{pmatrix} x_a \ x_b \end{pmatrix} \begin{pmatrix} \tfrac{g}{L_a} + \tfrac{k}{m} & -\tfrac{k}{m} \\ -\tfrac{k}{m} & \tfrac{g}{L_b} + \tfrac{k}{m} \end{pmatrix} \begin{pmatrix} x_a \\ x_b \end{pmatrix}.</math>
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| The 2×2 matrix is real symmetric and so (by the [[spectral theorem]]) it is "[[orthogonal matrix|orthogonally]] [[diagonalizable matrix|diagonalizable]]". That is, there is an angle θ such that if we define
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| :<math>\begin{pmatrix} x_a \\ x_b \end{pmatrix} = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}</math>
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| then
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| :<math>V = \frac{m}{2} \begin{pmatrix} x_1 \ x_2 \end{pmatrix} \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}</math>
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| where λ<sub>1</sub> and λ<sub>2</sub> are the [[eigenvalue]]s of the matrix. The variables x<sub>1</sub> and x<sub>2</sub> describe normal modes which oscillate with frequencies of <math>\sqrt{\lambda_1}</math> and <math>\sqrt{\lambda_2}</math>. When the two pendulums are identical (L<sub>a</sub> = L<sub>b</sub>), θ is 45°.
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| The description of the system in terms of the two pendulums (''a'' and ''b'') is analogous to the flavor basis of neutrinos. These are the parameters that are most easily produced and detected (in the case of neutrinos, by weak interactions involving the [[W boson]]). The description in terms of normal modes is analogous to the mass basis of neutrinos. These modes do not interact with each other when the system is free of outside influence. The angle θ is analogous to the [[Cabibbo angle]] (though that angle applies to quarks rather than neutrinos).
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| When the number of oscillators (particles) is increased to three, the orthogonal matrix can no longer be described by a single angle; instead, three are required ([[Euler angles]]). Furthermore, in the quantum case, the matrices may be [[complex number|complex]]. This requires the introduction of complex phases in addition to the rotation angles, which are associated with [[CP violation]] but do not influence the observable effects of neutrino oscillation.
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| ==Theory, graphically==
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| === Two neutrino probabilities in vacuum ===
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| In the approximation where only two neutrinos participate in the oscillation, the probability of oscillation follows a simple pattern:
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| [[Image:oscillations two neutrino.svg|none|400px]]
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| The blue curve shows the probability of the original neutrino retaining its identity. The red curve shows the probability of conversion to the other neutrino. The maximum probability of conversion is equal to sin<sup>2</sup>2θ. The frequency of the oscillation is controlled by Δm<sup>2</sup>.
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| ===Three neutrino probabilities===
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| If three neutrinos are considered, the probability for each neutrino to appear is somewhat complex. Here are shown the probabilities for each initial flavor, with one plot showing a long range to display the slow "solar" oscillation and the other zoomed in to display the fast "atmospheric" oscillation. The oscillation parameters used here are consistent with current measurements, but since some parameters are still quite uncertain, these graphs are only qualitatively correct in some aspects. These values were used:
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| * sin<sup>2</sup>2θ<sub>13</sub> = 0.10 (Controls the size of the small wiggles.)
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| * sin<sup>2</sup>2θ<sub>23</sub> = 0.97.
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| * sin<sup>2</sup>2θ<sub>12</sub> = 0.861.
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| * δ = 0 (If it is actually large, these probabilities will be somewhat distorted and different for neutrinos and antineutrinos.)
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| * Δm{{su|p=2|b=12}} = {{val|7.59|e=-5|u=eV<sup>2</sup>}}.
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| * Δm{{su|p=2|b=32}} ≈ Δm{{su|p=2|b=13}} = {{val|2.32|e=-3|u=eV<sup>2</sup>}}.
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| * Normal mass hierarchy.
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| {| class="wikitable"
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| |[[Image:oscillations electron long.svg|thumb|none|320px|Electron neutrino oscillations, long range. Here and in the following diagrams black means electron neutrino, blue means muon neutrino and red means tau neutrino.]]
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| |[[Image:oscillations electron short.svg|thumb|none|320px|Electron neutrino oscillations, short range]]
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| |-
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| |[[Image:oscillations muon long.svg|thumb|none|320px|Muon neutrino oscillations, long range]]
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| |[[Image:oscillations muon short.svg|thumb|none|320px|Muon neutrino oscillations, short range]]
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| |-
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| |[[Image:oscillations tau long.svg|thumb|none|320px|Tau neutrino oscillations, long range]]
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| |[[Image:oscillations tau short.svg|thumb|none|320px|Tau neutrino oscillations, short range]]
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| |}
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| ==Observed values of oscillation parameters==
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| * sin<sup>2</sup>(2θ<sub>13</sub>) = {{val|0.092|0.017}}<ref>{{cite journal|author=Daya Bay Collaboration|title=Observation of electron-antineutrino disappearance at Daya Bay|year=2012|journal=Physical Review Letters|volume=108|issue=17|pages=171803|doi=10.1103/PhysRevLett.108.171803|arxiv=1203.1669|bibcode = 2012PhRvL.108q1803A }}</ref>
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| * tan<sup>2</sup>(θ<sub>12</sub>) = {{val|0.457|0.040|-0.029}}. This corresponds to θ<sub>12</sub> ≡ θ<sub>sol</sub> = {{val|34.06|1.16|-0.84|s=°}} ("sol" stands for solar)<ref name="pdg2010">
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| {{cite journal
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| | author=K. Nakamura ''et al''.
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| | year=2010
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| | journal=[[Journal of Physics G]] | |
| | volume=37
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| | pages=1
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| | title=Review of Particle Physics}}
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| </ref> | |
| * sin<sup>2</sup>(2θ<sub>23</sub>) > {{val|0.92}} at 90% confidence level, corresponding to θ<sub>23</sub> ≡ θ<sub>atm</sub> = {{val|45|7.1|s=°}} ("atm" stands for atmospheric)<ref name="pdg2010"/>
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| * Δm{{su|p=2|b=21}} ≡ Δm{{su|p=2|b=sol}} = {{val|7.59|0.20|-0.21|e=-5|u=eV<sup>2</sup>}}<ref name="pdg2010"/>
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| * |Δm{{su|p=2|b=31}}| ≈ |Δm{{su|p=2|b=32}}| ≡ Δm{{su|p=2|b=atm}} = {{val|2.43|0.13|-0.13|e=-3|u=eV<sup>2</sup>}}<ref name="pdg2010"/>
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| * δ, α<sub>1</sub>, α<sub>2</sub>, and the sign of Δm{{su|p=2|b=32}} are currently unknown
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| Solar neutrino experiments combined with [[Kamioka Liquid Scintillator Antineutrino Detector|KamLAND]] have measured the so-called solar parameters Δm{{su|p=2|b=sol}} and sin<sup>2</sup>θ<sub>sol</sub>. Atmospheric neutrino experiments such as [[Super-Kamiokande]] together with the K2K and MINOS long baseline accelerator neutrino experiment have determined the so-called atmospheric parameters Δm{{su|p=2|b=atm}} and sin<sup>2</sup>θ<sub>atm</sub>. The last mixing angle, θ<sub>13</sub>, has been measured by the experiments [[Daya Bay Reactor Neutrino Experiment|Daya Bay]], [[Double Chooz]] and [[RENO]] as sin<sup>2</sup>2θ<sub>13</sub>.
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| For atmospheric neutrinos (where the relevant difference of masses is about {{nowrap|Δm<sup>2</sup> {{=}} {{val|2.4|e=-3|u=eV<sup>2</sup>}}}} and the typical energies are {{val|p=~|1|u=GeV}}), oscillations become visible for neutrinos traveling several hundred km, which means neutrinos that reach the detector from below the horizon.
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| The mixing parameter sin<sup>2</sup>2θ<sub>13</sub> is measured using electron anti-neutrinos from nuclear reactors. The rate of anti-neutrino interactions is measured in detectors sited near the reactors to determine the flux prior to any significant oscillations and then it is measured in far detectors (sited about 2 km from the reactors). The oscillation is observed as an apparent disappearance of electron anti-neutrinos in the far detectors (''i.e.'' the interaction rate at the far site is lower than predicted from the observed rate at the near site).
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| From atmospheric and [[solar neutrino]] oscillation experiments, it is known that two mixing angles of the MNS matrix are large and the third is smaller. This is in sharp contrast to the CKM matrix in which all three angles are small and hierarchically decreasing. Nothing is known about the CP-violating phase of the MNS matrix.
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| If the neutrino mass proves to be of [[Majorana fermion|Majorana]] type (making the neutrino its own antiparticle), it is possible that the MNS matrix has more than one phase.
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| Since experiments observing neutrino oscillation measure the squared mass difference and not absolute mass, one can claim that the lightest neutrino mass is exactly zero, without contradicting observations. This is however regarded as unlikely by theorists.
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| ==Origins of neutrino mass==
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| The question of how neutrino masses arise has not been answered conclusively. In the Standard Model of particle physics, [[fermion]]s only have mass because of interactions with the Higgs field (see ''[[Higgs boson]]''). These interactions involve both left- and right-handed versions of the fermion (see ''[[Chirality (physics)|chirality]]''). However, only left-handed neutrinos have been observed so far.
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| Neutrinos may have another source of mass through the [[Majorana fermion|Majorana mass term]]. This type of mass applies for electrically-neutral particles since otherwise it would allow particles to turn into anti-particles, which would violate conservation of electric charge.
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| The smallest modification to the Standard Model, which only has left-handed neutrinos, is to allow these left-handed neutrinos to have Majorana masses. The problem with this is that the neutrino masses are surprisingly smaller than the rest of the known particles (at least 500,000 times smaller than the mass of an electron), which, while it does not invalidate the theory, is widely regarded as unsatisfactory as this construction offers no insight into the origin of the neutrino mass scale.
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| The next simplest addition would be to add into the Standard Model right-handed neutrinos that interact with the left-handed neutrinos and the Higgs field in an analogous way to the rest of the fermions. These new neutrinos would interact with the other fermions solely in this way, so are not phenomenologically excluded. The problem of the disparity of the mass scales remains.
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| ===Seesaw mechanism===
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| {{Main|Seesaw mechanism}}
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| The most popular conjectured solution currently is the ''seesaw mechanism'', where right-handed neutrinos with very large Majorana masses are added. If the right-handed neutrinos are very heavy, they induce a very small mass for the left-handed neutrinos, which is proportional to the inverse of the heavy mass.
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| If it is assumed that the neutrinos interact with the Higgs field with approximately the same strengths as the charged fermions do, the heavy mass should be close to the [[GUT scale]]. Note that, in the Standard Model there is just one fundamental mass scale (which can be taken as the scale of {{nowrap|SU(2)<sub>L</sub> × U(1)<sub>Y</sub>}} breaking) and all masses (such as the electron or the mass of the Z boson) have to originate from this one.
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| There are other varieties of seesaw<ref>
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| {{cite journal
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| |author=J. W. F. Valle
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| |year=2006
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| |title=Neutrino physics overview
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| |journal=[[Journal of Physics: Conference Series]]
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| |volume=53 |issue=1 |pages=473
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| |arxiv=hep-ph/0608101
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| |doi=10.1088/1742-6596/53/1/031
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| |bibcode = 2006JPhCS..53..473V }}</ref>
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| and there is currently great interest in the so-called low-scale seesaw schemes, such as the inverse seesaw mechanism.<ref>
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| {{cite journal
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| |author=R.N. Mohapatra and J. W. F. Valle
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| |year=1986
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| |title=Neutrino Mass and Baryon Number Nonconservation in Superstring Models
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| |journal=[[Physical Review D]]
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| |volume=34 |issue=5 |pages=1642
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| |doi=10.1103/PhysRevD.34.1642
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| |bibcode = 1986PhRvD..34.1642M }}</ref>
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| The addition of right-handed neutrinos has the effect of adding new mass scales, unrelated to the mass scale of the Standard Model, hence the observation of heavy right-handed neutrinos would reveal physics beyond the Standard Model. Right-handed neutrinos would help to explain the origin of matter through a mechanism known as [[Leptogenesis (physics)|leptogenesis]].
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| ===Other sources===
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| There are alternative ways to modify the standard model that are similar to the addition of heavy right-handed neutrinos (e.g., the addition of new scalars or fermions in triplet states) and other modifications that are less similar (e.g., neutrino masses from loop effects and/or from suppressed couplings). One example of the last type of models is provided by certain versions supersymmetric extensions of the standard model of fundamental interactions, where [[R parity]] is not a symmetry. There, the exchange of supersymmetric particles such as [[squarks]] and [[sleptons]] can break the lepton number and lead to neutrino masses. These interactions are normally excluded from theories as they come from a class of interactions that lead to unacceptably rapid [[proton decay]] if they are all included. These models have little predictive power and are not able to provide a cold dark matter candidate.
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| ==See also== | |
| * [[MSW effect]]
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| * [[Majoron]]
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| * [[K-meson#Neutral kaon mixing|Neutral kaon mixing]]
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| ==Notes==
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| {{Reflist|group=nb}}
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| * {{cite journal |journal=Reviews of Modern Physics |author1=Gonzalez-Garcia |author2=Nir |volume=75 |doi=10.1103/RevModPhys.75.345 |issue=2 |title=Neutrino Masses and Mixing: Evidence and Implications |pages=345–402 |year=2002|arxiv=hep-ph/0202058|bibcode = 2003RvMP...75..345G }}
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| * {{cite journal |journal=New Journal of Physics |author1=Maltoni |author2=Schwetz |author3=Tortola |author4=Valle |volume=6 |doi=10.1088/1367-2630/6/1/122 |pages=122 |title=Status of global fits to neutrino oscillations |year=2004 |arxiv=hep-ph/0405172 }}
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| * {{cite journal |journal=Physical Review D |author1=Fogli |author2=Lisi |author3=Marrone |author4=Montanino |author5=Palazzo |author6=Rotunno |volume=86 |doi=10.1103/PhysRevD.86.01301 |pages=013012 |title=Global analysis of neutrino masses, mixings, and phases: Entering the era of leptonic CP violation searches |year=2012 |arxiv=arXiv:1205.5254 }}
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| * {{cite journal |journal=Physical Review D |author1=Forero |author2=Tortola |author3=Valle |volume=86 |doi=10.1103/PhysRevD.86.073012 |pages=073012 |title=Global status of neutrino oscillation parameters after Neutrino-2012 |year=2012 |arxiv=arXiv:1205.4018 |bibcode = 2012PhRvD..86g3012F }}
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| ==External links==
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| * Maury Goodman, "[http://neutrinooscillation.org/ The Neutrino Oscillation Industry]" (2006). ''(Provides links to many other neutrino oscillation websites.)''
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| *[http://xstructure.inr.ac.ru/x-bin/revtheme3.py?level=3&index1=-155642&skip=0 Review Articles on arxiv.org]
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| {{Use dmy dates|date=September 2010}}
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| {{DEFAULTSORT:Neutrino Oscillations}}
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| [[Category:Leptons]]
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| [[Category:Standard Model]]
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| [[Category:Electroweak theory]]
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| [[Category:Physics beyond the Standard Model]]
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