When neutrinos propagate over long distances, the mass eigenstate components
of a flavor eigenstate will become spatially separated due to their different
group velocities. This can happen over terrestrial distance scales if the
neutrino energy is of order MeV and if the neutrino is localized (in a quantum
mechanical sense) to subatomic scales. For example, if the Heisenberg
uncertainty in the neutrino position is below 0.01 Angstrom, neutrino
decoherence can be observed in reactor neutrinos using a large liquid
scintillator detector.Comment: LaTeX, 7 pages, 3 figures; in v1, we had concluded that wave packet
  decoherence may be observable in experiments like Hanohano or LENA. As
  pointed out to us by E. Akhmedov, G. Raffelt, and L. Stodolsky, this
  conclusion is INCORRECT since it was based on incorrect estimates for the
  size of the neutrino wave packets. The paper will be revised to explain this
  problem and to address related question

Kayser, Boris

Kopp, Joachim

English

arXiv

When neutrinos propagate over long distances, the mass eigenstate components of a flavor eigenstate will become spatially separated due to their different group velocities. This can happen over terrestrial distance scales if the neutrino energy is of order MeV and if the neutrino is localized (in a quantum mechanical sense) to subatomic scales. For example, if the Heisenberg uncertainty in the neutrino position is below 10{sup -2} {angstrom}, neutrino decoherence can be observed in reactor neutrinos using a large liquid scintillator detector

UNT Digital Library

arXiv:1005.4081v1  [hep-ph]  21 May 2010FERMILAB-PUB-10-168-TTesting the wave packet approach to neutrino oscillations in future experimentsBoris Kayser∗ and Joachim Kopp†Fermilab, Theoretical Physics Department, PO Box 500, Batavia, IL 60510, USA(Dated: May 25, 2010)When neutrinos propagate over long distances, the mass eigenstate components of a flavoreigenstate will become spatially separated due to their different group velocities. This canhappen over terrestrial distance scales if the neutrino energy is of order MeV and if theneutrino is localized (in a quantum mechanical sense) to subatomic scales. For example, ifthe Heisenberg uncertainty in the neutrino position is below 10−2 Å, neutrino decoherencecan be observed in reactor neutrinos using a large liquid scintillator detector.PACS numbers: 14.60.Pq, 03.65.Yz, 03.65.TaEven though the existence of neutrino oscillations has been unambiguously proven experimen-tally, the theoretical description of this phenomenon is still occasionally disputed, see for exam-ple [1–11]. The reason is that neutrino oscillations, being a space- and time-dependent phenomenon,cannot be fully described in terms of infinitely delocalized energy and momentum eigenstates asis used in most other applications in high energy physics. The signature dependence of neutrinooscillations on the distance between the neutrino source and the detector obviously cannot beobserved unless the source and the detector are localized. The uncertainty principle then impliesthat the source, and consequently the neutrinos that it produces, must be in a superposition ofdifferent momentum states. That is, the neutrino wave function cannot be a plane wave, but mustbe a wave packet [12–22]. For the purpose of this paper, it will be sufficient to consider Gaussianwave packets, but our results will apply also to more general wave packet shapes [22]. We writethe neutrino wave function as〈x|να(t)〉 ∝∑jU∗αj exp[− (x− vjt)24σ2x], (1)where α and j are flavor and mass eigenstate indices, respectively, Uαj are the elements of theleptonic mixing matrix, σx is the width of the wave packet, which depends on the properties of theneutrino source1, and vj is the group velocity corresponding to the jth neutrino mass eigenstate.As the neutrino propagates, the wave packets corresponding to different neutrino mass eigen-states will become separated in space and time due to their different group velocities [12]. After apropagation distance larger than σx/|vj−vk|, the wave packets of the jth and kth mass eigenstateswill have no significant overlap any more and their coherence will be lost. Coherence can be restoredif the detection process is delocalized over distances larger than the wave packet separation [23].If this is not the case, neutrino oscillations will be suppressed. Flavor change is still possible,but it will no longer depend on distance. Detailed calculations show that the να → νβ oscillationprobability for ultra-relativistic neutrinos with an average energy E ≫ mj at long baseline L can∗Email: boris@fnal.gov†Email: jkopp@fnal.gov1 Clearly, the observability of oscillations depends on the localization of the detector in the same way as on that ofthe source. Detector localization is incorporated into the description of oscillations in [18, 19, 22]2be written as [24, 25]P (να → νβ) ≃∑j,kU∗αjUαkU∗βkUβj exp[− 2πi LLoscjk−(LLcohjk)2], (2)with the oscillation lengths Loscjk = 4πĒ/∆m2jk and the coherence lengthsLcohjk =4√2E2|∆m2jk|σx,eff = 7347 km(EMeV)2(7.7× 10−3eV2∆m2jk)(σx,effÅ). (3)Here σx,eff is an effective wave packet width that depends on the spatial delocalization of the sourceand that of the detector, and is dominated by the larger of the two.2Eq. (3) shows that neutrinos from astrophysical sources always arrive at the Earth as completelyincoherent mixtures of mass eigenstates, while observation of decoherence effects in terrestrialexperiments would require a long baseline, low neutrino energy, and small σx,eff .These requirements can be fulfilled for reactor neutrinos observed in a future large liquid scin-tillator detector like Hanohano [26] or LENA [27]. The reactor neutrino event rate peaks atE ∼ 4 MeV, and the distance from a multipurpose detector like Hanohano or LENA to the nearestnuclear power station will be of order L & 100 km to avoid large backgrounds in geo-neutrino,supernova relic neutrino, and proton decay studies.It is more difficult to estimate the wave packet width σx,eff entering in eq. (3). Since it dependson the quantum mechanical localization of the neutrino in space, we have to ask how well we canin principle determine the position of the neutrino production and detection points, given perfectexperimental equipment.For neutrinos emitted from free particles in flight, the spatial uncertainty of the productionprocess will be similar to the mean free path of the parent particle [28].A neutrino emission or detection process in a solid or liquid is localized at least to interatomicdistance scales of σx,eff ∼ O(1 − 10 Å) because the emitting or absorbing atom is continuouslyinteracting with its neighbors. The latter can be viewed as a thermal bath in the sense that theirquantum states at different times are completely uncorrelated. An interaction of a particle witha thermal bath constitutes a measurement in the quantum mechanical sense that localizes theparticle.Another reason why σx,eff cannot be larger than an interatomic distance is that we can inprinciple measure the location of the production and detection vertices to that accuracy in anexperiment. Whenever we detect that a neutrino has been produced or absorbed (for example bydetecting the associated charged lepton), we can in principle quickly bombard the source with ahigh-energy beam of probe particles to determine which nucleus has undergone a transition in theprocess.Finally, a spatial uncertainty much larger than a few Å would imply that all particles par-ticipating in the process would have to be delocalized over many interatomic distances. Imaginean energetic charged particle like an outgoing charged lepton traveling through a solid or liquidmaterial and interacting with the atoms. For concreteness, let us assume that it excites the atomicshells, which subsequently relax by emitting scintillation light. If the energetic particle was delo-calized over many interatomic distances, it would be impossible to tell which atoms is excited. Theresulting scintillation light would then behave as if it came from several atoms simultaneously and2σx,eff depends also on the temporal delocalization of the production and detection processes, which, however, isusually similar to the spatial delocalization.3would therefore exhibit interference patterns. We are not aware of any experimental evidence forsuch interference patterns.For neutrino emission or detection in a solid state crystal, the upper limit on the delocalizationscale can be even tighter than 1–10 Å. To be specific, consider neutrino production in a reactorfuel rod. Once we know the position of the production vertex to within one lattice spacing—whichis possible according to the above arguments—we can make use of our knowledge of the crystalstructure to pin down the exact lattice site where the process must have occurred. In a crystalat zero temperature, the emitting nucleus would be localized at its lattice site to within its ownsize of order few× 10−15 m. At finite temperature T , it will oscillate about this position, with theamplitude of these vibrations being of order [T/mΘ2D]1/2 ∼ 0.01–0.1 Å for Debye temperaturesΘD of few hundred Kelvin and nuclear masses m ∼ 100 GeV. To arrive at this estimate, we haveassumed the nucleus to be bound in a harmonic oscillator potential V = 12mω2x2 with characteristicfrequency ω = ΘD, and the thermal excitation energy to be of order T . We conclude that the spatialuncertainty associated with a neutrino production process in a solid state source is roughly between10−5 Å and 0.1 Å, depending on the temperature. Similar arguments apply to solid state neutrinodetectors.At least for reactor neutrino experiments with liquid scintillator detectors, we can also set alower limit on σx,eff from the fact that the KamLAND experiment has observed neutrino oscillationconsistent with the results from solar neutrino experiments at baselines of O(100 km) [29].3 Thistranslates into the limit σx,eff & 10−3 Å.From the above considerations we conclude that, for reactor neutrino experiments10−3 Å . σx,eff . 101 Å (4)or100 km . Lcoh21 . 1 000 000 km (5)These estimates show that it may well be possible to detect neutrino wave packet decoherenceeffects in terrestrial experiments; this would prove that the wave packet formalism is indeed well-suited to describe neutrino oscillations, and a measurement of the coherence length would provideinteresting information about the quantum mechanics of the production and detection processes.The experimental prospects for detecting neutrino decoherence have been studied previouslyin the context of atmospheric neutrinos [32], short-baseline (O(km)) reactor experiments [33],superbeams [34, 35], and a neutrino factory [33]. Here, we will study the prospects of observingthe predicted wave packet decoherence from eq. (4) by observing reactor neutrinos at very longbaselines of at least a few hundred km in a large liquid scintillator detector like Hanohano orLENA. We have simulated this setup using GLoBES [36, 37], assuming a 5-year run of a LENA-like [27] detector with a fiducial mass of 45 kt and a Gaussian energy resolution of 0.07×(E/MeV−0.8)1/2 MeV. As systematic uncertainties, we include a 3% error on the reactor neutrino flux, a 50%uncertainty on the geo-neutrino flux, a 0.5% energy calibration error, and a 0.5% spectral error,uncorrelated between different energy bins. We study three different detector sites: The Pyhäsalmimine in Finland, the Deep Underground Science and Engineering Laboratory (DUSEL) in SouthDakota, USA, and a site in Hawaii. Unless otherwise noted, we take into account the neutrino fluxfrom all nuclear power stations in the world [38], as well as the geo-neutrino background, which3 The possibility cannot be ruled out that KamLAND is affected by neutrino decoherence at a subdominant level.In this case, a reanalysis would be required, leading to modifications of the global fit of the solar oscillationparameters. The experiment we propose in this paper would be able to determine if such a reanalysis is indeednecessary. Decoherence effects in KamLAND have been studied for example in [30, 31].42 4 6 8 10100101102103E @MeVDNumberofΝeventsLENA  Pyhäsalmigeo-Νreactor-ΝΣx = 0.1Þ2 4 6 8 10100101102103E @MeVDNumberofΝeventsLENA  PyhäsalmiΣx = 0.01Þ2 4 6 8 10100101102103E @MeVDNumberofΝeventsLENA  PyhäsalmiΣx = 0.001ÞFigure 1: The neutrino spectrum in a 45 kt (fiducial) liquid scintillator detector located in the Pyhäsalmimine in Finland for different values of the wave packet width σx,eff . The red curve is the geo-neutrinobackground, while the black dots are the reactor neutrino signal.dominates the event rates below E . 3.3 MeV.4 The reactor neutrino spectrum we are using isbased on [29, 39], while the geo-neutrino spectrum has been kindly provided to us in machine-readable form by the author of ref. [40]. The cross sections for neutrino detection in inverse betadecay are from ref. [41].In fig. 1, we show the effect of decoherence on the neutrino spectrum expected at the Pyhäsalmimine in Finland. For large σx,eff , where Lcoh ≫ L, an oscillation pattern is visible, even thoughit is smeared due to the overlap of signals from different reactors at various baselines. For smallσx,eff oscillations are completely smoothed out by decoherence.To quantify this observation and determine the discovery potential for decoherence effects, wehave simulated the expected event spectrum for different values of σx,eff , and have then performeda fit assuming no decoherence. The resulting χ2 is plotted in fig. 2. We find that prospects forobserving decoherence are best at the Pyhäsalmi site due to the proximity of only few nuclearreactors. This keeps washout of oscillations due to the superposition of many reactor spectra atmany different baselines small. On the other hand, there is no nearby reactor that would make thedetector blind to events from distant reactors that could carry information about decoherence. Inthe DUSEL scenario, there are no nearby reactors either, but washout due to the larger numberof nuclear reactors in the eastern United States is a problem. In Hawaii, on the other hand, thenearest nuclear reactors are thousands of kilometers away, so that the number of events is too smallfor the presence or absence of an oscillation pattern to be seen. Moreover, at such long baselines,consecutive oscillation maxima are very close in energy, so the detector resolution becomes an issue.Thus we conclude that the optimal detector site for a decoherence measurement is one that has afew (but not too many) nuclear reactors at baselines of several 100 km.Fig. 3 shows that this condition is fulfilled in many locations around the world. In the toppanel, we have mapped the locations of nuclear power stations [38] (as of 2000), while the bottompanel shows the significance at which wave packet decoherence can be detected if the wave packetwidth is 0.005 Å, close to the lower end of the allowed range from eq. (4). We have checked thata larger values σx,eff = 0.01 can only be detected in few places in Japan, while even larger valuescannot be detected anywhere in the world with the experimental setup we have simulated. Notethat for each grid point in fig. 3, we have simulated only the 32 closest nuclear reactors to speed4 Note that the data on reactor sites available to us dates from 2000, so some recently commissioned stations maynot be included. To partly compensate for this, we assume all stations that were under construction in 2000 to beoperational at full power by now.50.001 0.01 0.1 1 10024681012Σx @ÞDΧ21Σ2Σ3Σ45 kt HfiducialL TASDDUSELHawaiiPyhäsalmiSignal: 30 closest reactorsBG: Geo-ΝGLoBES 2010Figure 2: Discovery reach for neutrino decoherence in a 45 kt (fiducial) detector located at DUSEL, Hawaii,or Pyhäsalmi. Only the Pyhäsalmi site allows for good sensitivity to decoherence effects.up the numerical computations.In conclusion, we have presented theoretical arguments showing that neutrino wave packets in areactor experiment should have a width between 0.001 Å and 10 Å. The lower limit is based on theobservation of oscillations at KamLAND, while the upper limit has been estimated by consideringthe localization of typical neutrino production and detection processes. These numbers imply thatwave packet decoherence due to different group velocities of different neutrino mass eigenstatescan occur over distances of 102–106 km for O(few MeV) neutrinos. In higher energy experimentsor in experiments in which neutrinos are produced by decay in flight of some parent particle, thewave packet width can be much larger, while in a hypothetical experiment in which O(few MeV)neutrinos are emitted and detected by nuclei bound in solid state lattices at very low temperature,the localization can be of O(few × 10−5) Å, corresponding to a coherence length of order 1 km.We have then performed detailed numerical simulation to show that wave packet decoherence isobservable in a future large liquid scintillator detector like Hanohano or LENA, but only if thecoherence length is at the lower end of the expected range and if the detector site is far (fewhundred kilometers), but not too far (. 1 000 km) from the nearest nuclear power station.The authors are indebted to Walter Potzel and Michael Wurm for very useful discussions onlarge liquid scintillator detectors. Fermilab is operated by Fermi Research Alliance, LLC underContract No. DE-AC02-07CH11359 with the US Department of Energy.[1] L. Stodolsky, Phys. Rev. D58, 036006 (1998), hep-ph/9802387.[2] H. J. Lipkin, Phys.Lett. B642, 366 (2006), hep-ph/0505141.[3] E. K. Akhmedov, J. Kopp, and M. Lindner, JHEP 05, 005 (2008), 0802.2513.[4] S. M. Bilenky, F. von Feilitzsch, and W. Potzel, J. Phys. G35, 095003 (2008), 0803.0527.[5] E. K. Akhmedov, J. Kopp, and M. Lindner (2008), 0803.1424.[6] A. N. Ivanov, R. Reda, and P. Kienle (2008), arXiv:0801.2121 [nucl-th].[7] C. Giunti (2008), arXiv:0801.4639 [hep-ph].[8] H. Kienert, J. Kopp, M. Lindner, and A. Merle (2008), 0808.2389.6Σx,efftrue= 0.00590%3ΣFigure 3: Top: A map of all nuclear reactors running or under construction as of 2000 (red dots), togetherwith the proposed detector sites at Hawaii, Pyhäsalmi, and DUSEL (blue dots). Data on nuclear reactors isfrom [38]. Bottom: Discovery reach for wave packet decoherence with σx,eff = 0.005 Å at different geographiclocations. Note that, due to limited resolution, the plot hides the fact that very close to any nuclear reactor,the sensitivity greatly decreases because the detector is blinded by short-baseline neutrinos in that case.[9] A. G. Cohen, S. L. Glashow, and Z. Ligeti (2008), 0810.4602.[10] B. Keister and W. Polyzou, Phys.Scripta 81, 055102 (2010), arXiv:0908.1404.[11] R. G. H. Robertson (2010), 1004.1847.[12] S. Nussinov, Phys. Lett. B63, 201 (1976).[13] B. Kayser, Phys. Rev. D24, 110 (1981).[14] C. Giunti, C. W. Kim, and U. W. Lee, Phys. Rev. D44, 3635 (1991).[15] K. Kiers, S. Nussinov, and N. Weiss, Phys. Rev. D53, 537 (1996), hep-ph/9506271.[16] W. Grimus and P. Stockinger, Phys. Rev. D54, 3414 (1996), hep-ph/9603430.[17] W. Grimus, P. Stockinger, and S. Mohanty, Phys. Rev. D59, 013011 (1999), hep-ph/9807442.[18] M. Beuthe, Phys. Rept. 375, 105 (2003), hep-ph/0109119.[19] C. Giunti, JHEP 11, 017 (2002), hep-ph/0205014.[20] C. Giunti and C. W. Kim, Fundamentals of Neutrino Physics and Astrophysics (Oxford UniversityPress, Oxford, UK, 2007).[21] E. K. Akhmedov and A. Y. Smirnov (2009), 0905.1903.7[22] E. K. Akhmedov and J. Kopp, JHEP 04, 008 (2010), 1001.4815.[23] K. Kiers and N. Weiss, Phys. Rev. D57, 3091 (1998), hep-ph/9710289.[24] C. Giunti, C. W. Kim, and U. W. Lee, Phys. Lett. B274, 87 (1992).[25] M. Beuthe, Phys. Rev. D66, 013003 (2002), hep-ph/0202068.[26] J. G. Learned, S. T. Dye, and S. Pakvasa (2008), 0810.4975.[27] M. Wurm et al. (2010), 1004.3474.[28] F. Torres and M. Guzzo, Braz.J.Phys. 37, 1273 (2007).[29] K. Eguchi et al. (KamLAND), Phys. Rev. Lett. 90, 021802 (2003), hep-ex/0212021.[30] T. Araki et al. (KamLAND), Phys. Rev. Lett. 94, 081801 (2005), hep-ex/0406035.[31] G. Fogli, E. Lisi, A. Marrone, D. Montanino, and A. Palazzo, Phys.Rev. D76, 033006 (2007),arXiv:0704.2568.[32] E. Lisi, A. Marrone, and D. Montanino, Phys. Rev. Lett. 85, 1166 (2000), hep-ph/0002053.[33] M. Blennow, T. Ohlsson, and W. Winter, JHEP 0506, 049 (2005), hep-ph/0502147.[34] N. C. Ribeiro et al., Phys. Rev. D77, 073007 (2008), 0712.4314.[35] A. Sakharov, N. Mavromatos, A. Meregaglia, A. Rubbia, and S. Sarkar (2009), 0903.4985.[36] P. Huber, M. Lindner, and W. Winter, Comput. Phys. Commun. 167, 195 (2005), hep-ph/0407333,URL http://www.mpi-hd.mpg.de/~globes.[37] P. Huber, J. Kopp, M. Lindner, M. Rolinec, and W. Winter, Comput. Phys. Commun. 177, 432 (2007),hep-ph/0701187, URL http://www.mpi-hd.mpg.de/~globes.[38] The United Nations Environment Programme, Nuclear power stations of the world (1999), URLhttp://www.grid.unep.ch/GRID_search_details.php?dataid=GNV181.[39] H. Murayama and A. Pierce, Phys. Rev. D65, 013012 (2002), hep-ph/0012075.[40] S. Enomoto, Ph.D. thesis, Tohoku University (2005), available fromhttp://kamland.stanford.edu/GeoNeutrinos/GeoNuResult/.[41] P. Vogel and J. F. Beacom, Phys. Rev. D60, 053003 (1999), hep-ph/9903554.

Testing the wave packet approach to neutrino oscillations in future experiments

Abstract

Similar works

Full text

Available Versions

UNT Digital Library