Neutrino oscillations are experimentally observable only as a result of
interference between neutrino states with different masses and THE SAME ENERGY.
All interference effects between neutrino states having different energies are
destroyed by the intereaction between the incident neutrino and the neutrino
detector. Erroneous results are frequently obtained by neglecting the
neutrino-detector interactions.Comment: 6 pages, Two paragraphs of discussion on possible time measurements
  added at en

Lipkin, Harry J.

English

arXiv

Neutrino oscillations are experimentally observable only as a result of interference between neutrino states with different masses and the same energy. All interference effects between neutrino states having different energies are destroyed by the interaction between the incident neutrino and the neutrino detector. Erroneous results are frequently obtained by neglecting the neutrino-detector interactions

Lipkin, H. J.

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 ANL-HEP-TR-02-119            Stodolsky’s Theorem and Neutrino Oscillation Phases—for pedestrians     By  Harry J. Lipkin    Department of Particle Physics Weizmann Institute of Science, Rehovot 76100, Israel   School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel   Argonne National Laboratory, High Energy Physics Division, Argonne, IL 60439 USA     December 9, 2002     *Work supported in part by the U.S. Department of Energy, Division of High Energy Physics, under    Contract W-31-109-ENG-38 and by a grant from US-Israel Bi-National Science Foundation.  Argonne National Laboratory, with facilities in the states of Illinois and Idaho, is  owned by the United States Government and operated by The University of Chicago under the provisions of a contract with the Department of Energy.                DISCLAIMER  This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor The University of Chicago, nor any of their employees or officers, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of document authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof, Argonne National Laboratory, or The University of Chicago.        Available electronically at http://www.doe.gov/bridge   Available for a processing fee to U.S. Department of Energy and its contractors, in paper, from:  U.S. Department of Energy Office of Scientific and Technical Information P.O. Box 62 Oak Ridge, TN  37831-0062 phone: (865) 576-8401 fax: (865) 576-5728 email: reports@adonis.osti.gov   arXiv:hep-ph/0212093 v2   9 Dec 2002Stodolsky’s Theorem and Neutrino Oscillation Phases – forpedestriansHarry J. Lipkin ∗Department of Particle Physics Weizmann Institute of Science, Rehovot 76100, IsraelSchool of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, TelAviv University, Tel Aviv, IsraelHigh Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439-4815, USAlipkin@hep.anl.govAbstractNeutrino oscillations are experimentally observable only as a result of in-terference between neutrino states with different masses and THE SAMEENERGY. All interference effects between neutrino states having differentenergies are destroyed by the intereaction between the incident neutrino andthe neutrino detector. Erroneous results are frequently obtained by neglectingthe neutrino-detector interactions.Stodolsky [1] has given a very simple answer to the confusion that still arises in discussionsof the phase of neutrino oscillations. The relevant literature producing this confusion hasrecently been summarized and clarified [2]. The purpose of this note is to support thisexcellent analysis [2] without engaging in a direct debate against the confusing articles andalso to present a “pedestrian” version of Stodolsky’s work which is hopefully understandableto students and experimentalists.The detection of a neutrino always involves its interaction with a detector that is part ofan environment described by a density matrix in which the energy is diagonal. Unless thisinteraction with the environment is turned off, and no experiment can do this, all relative∗Supported in part by grant from US-Israel Bi-National Science Foundation and by the U.S.Department of Energy, Division of High Energy Physics, Contract W-31-109-ENG-38.1phase information between neutrino states with different energies is destroyed.The question is not whether states of the same momentum and different energies arecoherent, states of the same energy and different momentum or states of the same velocity.There have been many irrelevant arguments about these issues. But states with differentENERGIES ARE NEVER COHERENT in any realistic experiment. States of the sameenergy and different momenta can be coherent, but may not be. This depends upon the waythe measurement is made. But states with different energies can not be coherent.This discussion refers only to neutrino detectors, The usual detector is a nucleon, whichchanges its state after absorbing a neutrino and emitting a charged lepton, and is initiallyeither in an energy eigenstate or in a statistical mixture in thermal equilibriam with itssurroundings. No neutrino detector has ever been prepared in a coherent mixture of energyeigenstates and no such detector has been proposed for future experiments .All arguments about Lorentz invariance are irrelevant. The detector chooses a particularLorentz frame where the detector is at rest and described by a density matrix in which theenergy is diagonal and no interference between states of different energies can be observed.Most treatments do not consider at all the quantum mechanics of the detector. Since thedetector is a quantum system (e.g. a nucleon) which undergoes a transition together withthe neutrino-to-charged-lepton transition, and the initial and final states of the detectorare not measured, the transition probability is the square of the transition matrix elementfor the whole system, averaged over detector initial states and summed over detector finalstates. This immediately kills all interference between neutrino states with different energiesas they are accompanied by different detector states which must have different energiesbecause energy in conserved in the process. The detector states with different energies areorthogonal to one another and all interference terms between them vanish because of thisdetector orthogonality.In this context the “factor-of-two” arguments are seen to be missing an essential point inthe actual neutrino oscillation experiments; namely the role of the detector as a quantum-mechanical system entangled with the neutrino.2The standard textbook neutrino-oscillation wave function, a coherent linear combinationof states with different energies, is not found in real experiments. Thus considerable confu-sion remains even though coherence, interference and dephasing have been extensively dis-cussed and clarified [1,3,4,5,6,7,8,9,10,11,12]. Elementary quantum mechanics and quantumstatistical mechanics tell us that the components of the density matrix describing a neutrinodetector and having different energies are never coherent [1], while neutrino componentswith different masses and different momenta must be coherent to cancel components withthe wrong flavor just outside the neutrino source.. This coherence between source stateshaving the same energy and different momenta can produce coherence between neutrinostates with the same energy and different masses.This physics is illustrated in detail in a toy model [13] for the detection of a neutrino asa transition between an initial state of a neutrino and a detector and a final state of a muonand the same detector. The wave function for the initial state of neutrino and detector isΨi(ν, D) =Nν∑k=1∣∣∣ν(Eν , mk, ~Pk), Di(Ei)〉(0.1)where Nν is the number of neutrino mass states, Eν , mk and ~Pk denote the neutrino energy,mass and momentum and Di(Ei) is the initial state of the detector with energy Ei. If thedetector is a muon detector the final detector state after neutrino absorption isΨf(µ±, D) =Nν∑k=1∣∣∣µ±(Eµ, ~Pµ), D∓kf(E − Eµ)〉(0.2)where Eµ and ~Pµ denote the muon energy and momentum, D∓kf is the final detector stateproduced in the “path k”; i.e. after the absorption of a neutrino with mass mk and emissionof a µ±, and E = Eν + Ei is the total energy which is conserved in the transition.The transition in the detector occurs on a nucleon, whose co-ordinate is denoted by by~X, and involves a charge exchange denoted by the isospin operator I∓ and a momentumtransfer ~Pk − ~Pµ. The detector transition matrix element is therefore given by〈D∓kf∣∣∣ T∓ |Di〉 =〈D∓kf∣∣∣ I∓ei(~Pk−~Pµ)· ~X |Di〉 (0.3)3The overlap between the final detector wave functions after the transitions absorbingneutrinos with masses mk and mj is then〈D∓kf∣∣∣D∓jf〉= 〈Di| ei(~Pj−~Pk)· ~X |Di〉 (0.4)If the quantum fluctuations in the position of the active nucleon in the initial state ofthe detector are small in comparison with the oscillation wave length, h̄/(~Pj − ~Pk),|~Pj − ~Pk|2 · 〈Di| | ~X2| |Di〉  1 (0.5)〈D∓kf∣∣∣D∓jf〉≈ 1 − (1/2) · |~Pj − ~Pk|2 · 〈Di| | ~X2| |Di〉 ≈ 1 (0.6)There is thus effectively a full overlap between the final detector states after absorptionof different mass neutrinos, and a full coherence between the neutrino states with the sameenergy and different momenta.The total energies of the final muon and detector produced after absorption of neutrinoswith different energies are different. These muon-detector states are thus orthogonal to oneanother and there is no coherence between detector states produced by the absorption ofneutrinos with different energies.It is a pleasure to thank Maury Goodman, Yuval Grossman, Boris Kayser, Lev Okun,andLeo Stodolsky for helpful discussions and comments.4REFERENCES[1] Leo Stodolsky, Phys. Rev. D58 (1998) 036006[2] L. B. Okun et al. hep-ph/0211241[3] H.J. Lipkin, Neutrino Oscillations for Chemists, Lecture Notes (unpublished).[4] H.J. Lipkin, Lecture Notes on Neutrino Oscillations for a Quantum Mechanics course(unpublished).[5] H.J. Lipkin, Neutrino Oscillations and MSW for Pedestrians, Lecture Notes (unpub-lished).[6] B. Kayser, Phys. Rev. D 24 (1981) 110.[7] T. Goldman, hep-ph/9604357.[8] H.J. Lipkin, Phys. Lett. B 348 (1995) 604.[9] M.M. Nieto, hep-ph/9509370, Hyperfine Interactions 100 (1996) 193[10] Yuval Grossman and Harry J. Lipkin, Phys. Rev. D55 (1997) 2760[11] A.D. Dolgov, A.Yu. Morozov, L.B. Okun, and M.G. Schepkink, Nucl. Phys. 502 (1997) 3[12] Harry J. Lipkin, hep-ph/9901399, Weizmann Preprint WIS-98/31/Dec-DPP, Tel AvivUniversity preprint TAUP 2537-98, Argonne Preprint ANL-HEP-CP-98-126, available inProceedings of the Europhysics Neutrino Oscillation Workshop (NOW’98) 7-9 September1998. Amsterdam. Published in http://www.nikhef.nl/pub/conferences/now98/[13] H.J. Lipkin, hep-ph/9907551, Phys. Lett. B 477 (2000) 1955

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