Why do neutrinos with different masses interfere and oscillate? Why are states with different masses but same energy coherent? Overcoming barrier between particle & condensed matter physics

Why do neutrinos with different masses interfere and oscillate? Why are states with different masses but same energy coherent? Overcoming barrier between particle & condensed matter physics

Harry J. Lipkin  Supported in part by U.S. Department of Energy, Office of Nuclear Physics, under contract number DE-AC02-06CH11357. Department of Particle Physics Weizmann Institute of Science, Rehovot 76100, Israel
Physics Division, Argonne National Laboratory Argonne, IL 60439-4815, USA
 
harry.lipkin@weizmann.ac.il
 
Abstract

Neutrino oscillations occur only if it is impossible to determine mass by using conservation laws on measurements of nucleon-lepton system absorbing . No oscillations if detector is mass spectrometer. Beam is split into components with different masses entering different counters. For each event only one counter will click and determine mass. Condensed matter physics needed to describe the detector, show it is not a mass spectrometer and identify which properties of the incident are unobservable. Relativistic quantum field theory can only describe wave function entering detector but not large uncertain momentum transfers to detector nor associated energy-momentum asymmetry. Absorption of incident ’s with different momenta but same energy leaves no trace of initial momentum difference in finite-size detector with effectively infinite mass at rest in laboratory. Undetectable recoil-free momentum is transferred to the detector with negligible energy transfer. The Debye-Waller factor common in X-ray diffraction by crystals gives probability that absorbing ’s with different momenta produce same nucleon-charged-lepton final state. Oscillations in time described in textbooks as interference between states with different energies not observable in realistic experiments. Different energy ’s not coherent because energy can be determined by measurements on initial and final states. Experiments detecting produced by decay observe no electrons even though mass eigenstates produce electrons. Electron amplitude canceled by interference between amplitudes from different mass eigenstates with same energy and different momenta entering massive detector.

I Quantum mechanics missed at particle - condensed matter interface

i.1 A communications gap that deserves serious thought

Consider how a linear combination of waves having different masses are absorbed by a nucleon in a detector. Standard particle physics treatments describe only the coherence properties of the wave function entering the detector. But this is only part of the story. Understanding what happens after the wave function enters the detector is crucial to explain why neutrino oscillations are observed.

Suppose that the neutrino detector is a neutrino mass spectrometer. The beam will be split into components with different masses entering different counters. For each event only one counter will click and determine the mass. There can be no observable oscillations if the detector is a mass spectrometer.

That neutrino oscillations are observed tells us the detector is not a mass spectrometer. Then what is it? What are the properties of the detector that prevent the determination of the mass of the that entered the detector?

Here we need condensed matter physics. The detectors in all neutrino oscillation experiments are condensed matter systems with effectively infinite mass at rest in the laboratory and in thermal equilibrium with its environment. The neutrino is absorbed by a nucleon with the emission of a charged lepton. The nucleon bound in this infinite mass system can absorb two incident neutrino waves with different momenta and produce transitions to the same final state of the detector nucleon. The two momenta are absorbed with the same energy transfer. The momentum difference is absorbed by the infinite mass detector leaving no trace of the momentum difference. The probability of absorbing two waves with different momenta in transitions to the same final state depends upon the relative phase of the two waves. Studying transitions as a function of the position of the detector is studying them as a function of the relative phase. Oscillations are observed when two neutrino waves with different momenta and the same energy are detected at different distances from the neutrino source. This immediately tells us that interference is observable only between neutrino states with the same energy and different momenta. States with different energies will not interfere.

This can be summed up as follows:

  1. No coherence between the amplitudes having different masses if any measurement on the final state of the detector can determine the neutrino mass.

  2. When a two neutrino waves with slightly different momenta are absorbed by a nucleon bound in a large system with effectively infinite mass, there is a probability that the same bound nucleon final state is produced with the momentum difference absorbed by the large system. The probability that this can occur can be calculated by knowing the wave function of the absorbing nucleon. This probability is called the ”Debye-Waller” factor in condensed matter physics.

  3. Interference between absorption of two incident waves with same energy and different momenta leave no observable trace of momentum difference in final state.

This simple physics encounters a communications barrier because:

  • Particle physicists are sure that neutrino oscillations can be described by single-particle quantum field theory and it is not necessary to know any condensed matter physics, They have probably never heard of a Debye-Waller factor and unaware that the presence of interactions with a large system has any relevance to particle physics. They never discuss why neutrino oscillations are observable in real experiments.

  • Condensed matter physicists show no interest in oscillations. This is useless particle physics.

i.2 Nucleon binding prevents determination of momentum transfer

Neutrino oscillations cannot be produced by interference between two states with different masses if measurements on the nucleon-lepton system absorbing the can determine the mass. The nucleon is bound in a quantum-mechanical condensed matter system with well defined energy levels. Neutrino oscillations are observed because the nucleon final state is not a momentum state and the momentum spread prevents determination of the mass. An oscillation wave length cannot be measured unless the size of the detector is much smaller than the wave length of the oscillation to be measured. This localization of the detector nucleon produces a momentum spread which prevents the use of momentum conservation in determining the neutrino mass. There is no energy spread because the transition occurs between discrete energy levels. This momentum spread without energy spread occurs only in the laboratory system.

This uncertainty only in momentum and not in energy in the nucleon-lepton system is not relativistically covariant. Lorentz transformations are useless here because they boost the whole detector. Relativistic quantum field theory can provide a full description of the neutrino wave function incident on a detector. But it cannot describe which coherence properties of the wave function in the laboratory system remain in the nucleon-lepton system after absorption in the detector.

i.3 The absorption by the detector is described by condensed matter physics

The Debye-Waller factor common in X-ray diffraction will be shown below to give the probability that two components of an incident neutrino with a momentum difference can produce the same final quantum state in the detector.

(1)

where denotes the mean square distance of detector nucleon position in the detector from its equilibrium position. The transition leaves no trace of the momentum difference between components of the initial neutrino with different masses.

The physics of neutrino absorption and X-ray diffraction seem to be very different. But the physics is the same for energy and momentum transfer when a single neutrino or photon is scattered by a massive object. Energy is conserved; momentum is not. The interference fringes in X-ray diffraction and neutrino oscillations both arise because no recoil momentum can identify the position of the momentum transfer. These allow two components of the initial photon or neutrino wave functions with the same energy and different momenta or different paths in configuration space to interfere coherently in producing the observed final state.

In an experiment where oscillations are observed the size of the detector denoted by a length must be much smaller than the oscillation wave length.

(2)

i.4 How to solve coherence problems in quantum mechanics

The main question in understanding neutrino oscillations is what prevents the oscillation experiments from being missing mass experiments. The answers are wave-particle duality and coherence.

For a correct description of quantum coherence, consider a two-slit experiment in which an electron passes through a barrier with two slits. The electron is a wave which passes through the screen. the waves coming from the two slits combine and interfere. The interference pattern observed on a screen shows that both parts of an electron wave passed through the slits. The point where the electron will be observed as a particle on a screen cannot be predicted. Only the probability that a single electron will be observed on the screen can be predicted. But this interference requires that no information can be available to determine through which slit the electron passed.

An essential feature of this coherence is wave-particle duality. The passage of the electron through the slits is described as propagation of the electron as a wave. But the detection on the screen requires the appearance of the electron as a particle. Quantum mechanics describes the electron by a wave function that satisfies the Schroedinger equation and expresses both the wave and particle properties.

Neutrino oscillations are created in a two-slit experiment in in momentum space. The neutrino is described by a wave function which contains components with different masses and different momenta. It is detected by a nucleon in a massive detector and emerges as a lepton with definite energy and momentum. We now investigate how two components of the neutrino wave with different momenta can interfere and produce a nucleon and lepton pair with a single momentum. A correct description of this process is obtained by writing down the Schroedinger equation and solving it. But in neutrino detection by a massive detector the Schroedinger equation is so complicated that it is not easily solved. Condensed matter physicists have treated this question many times by finding a small parameter and writing the solution as a power series in the small parameter. In most cases the leading term in the series is sufficient for most purposes. In neutrino oscillations the obvious small parameter is the ratio of the detector size to the oscillation wave length. We sho w below that the leading term in an expansion in this small parameter is adequate.

The application of quantum mechanics to problems of coherence and interference has been confused by the Schroedinger cat paradox. There the cat is alive in a cage and is killed by the arrival of a bullet shot from a gun triggered by the decay of a radioactive nucleus. The suggestion that the cat is in a coherent mixture of life and death is simply wrong. The cat is either alive or dead even if no one opens the cage. The radio active nucleus is not an isolated quantum system. It is coupled to an environment which will change drastically when the nucleus decays, changes its state and emits an , or ray. One does not need to open the cage to see if the cat is alive or dead. The environment always contains the information.

We now explore in detail the implications of an unobserved recoil momenta using a model-independent description of the detector nucleon absorbing the neutrino. It applies not only for the case where the detector nucleon is bound in a crystal[2, 3, 4, 5] but also for amorphous solid, liquid and gas detectors where the detector is confined to a region of linear dimension satisfying (2).

Ii Detailed analysis of effects of observed momentum transfer

ii.1 Coherence in absorption of two states with different masses

Consider the amplitudes for absorption of two neutrino states with slightly different masses. If energy and momentum are exactly conserved in the nucleon-lepton system, the two final nucleon-lepton states are orthogonal. There is then no interference and there are no oscillations. There is interference only if no measurement on the final state can determine the momentum of the entering neutrino. To have interference there needs to be a momentum transfer to the whole detector which is just enough to cover up the change denoted by in the momentum of the entering neutrino.

The detector is a large many-body system at rest in the laboratory with a discrete energy level spectrum. Consider the transition matrix element for the case where the detector makes a transition between an initial detector eigenstate denoted by to a final state denoted by which differs from the initial state by the absorption of a with momentum , a nucleon charge change and the emission of a charged lepton. The transition between the same initial detector eigenstate to the same final state by the absorption of a with momentum is denoted by

(3)

where denotes the distance between the position of the nucleon and the center of the detector. For a small change

(4)

where we have taken the leading term in the expansion of the small parameter . and denotes the length of the detector. We see that a change by an amount in the neutrino momentum will not be detected by measuring the transition as long as the size of the detector satisfies (2) and is much smaller than the oscillation wave length. If absorption of two neutrino states with slightly different momenta can produce the same change from energy level A to energy level B there is coherence. One only sees that there was a transition from A to B. One cannot know which neutrino mass produced the transition. The momentum difference is taken up by the whole detector.

The derivation of (4) is independent of the model for the detector. It applies not only for the case where the detector nucleon is bound in a crystal[2, 3, 5] but also for amorphous solid, liquid and gas detectors where the detector is confined to a region of linear dimension satisfying (2).

ii.2 Application to detection of ’s produced in decay

We now apply eqs.(3 - 4) to treat and explain the observation that no electrons are produced in the detector. We consider two neutrino mass eigenstates, denoted by and with momenta and and include the transition.

(5)
(6)

where and denote the momenta of the initial nucleon state, the final electron and the incident neutrino and is the interaction producing the weak transition, Using eq (4) and neglecting the small parameter gives

(7)

Where we note that the ratio of the weak transition matrix elements is equal to the ratio of the elements of the flavor-mass mixing matrix denoted by and and neglect the dependence of the weak transition matrix element on the small momentum difference .

Consider an incident neutrino which is a linear combination of the two mass eigenstatates

(8)
(9)

The probability that an incident neutrino is absorbed with electron emission is seen to vanish if is just the right mixture of mass eigenstates to be orthogonal to the electron neutrino state . This explains the failure to observe electrons in the detection of neutrinos from decays.

Iii Oscillations can arise only if mass is unobservable

iii.1 Momentum and energy conservation violations in the lepton-nucleon system

The wave function of a emitted in a weak decay is a linear combination of states containing different masses, energies and momenta. The is observed in a detector by an interaction which changes the charge of a nucleon and emits a charged lepton.

(10)

oscillations and the failure to observe electrons in decay[1] can occur only if there is coherence and interference between components of the neutrino wave function with different neutrino masses. If oscillations are observed something must prevent knowing the neutrino mass The neutrino mass is measurable if the lepton energy and the energy and momentum changes in the detector nucleon are all observable. They are all observable if:

  1. The neutrino energy is equal to the sum of the change in detector nucleon energy and the lepton energy

  2. The neutrino momentum is equal to the sum of the change in detector nucleon momentum plus the lepton momentum

The mass is determined in this “missing mass” experiment. If the mass can be determined by measurements on the initial and final states there can be no interference and no oscillations.

But neutrino oscillations are observed. What is wrong with this argument?

To observe oscillations the position of the detector must be known with an error much less that the oscillation wave length. Heisenberg position-momentum uncertainty prevents the neutrino momentum from being known with sufficient precision to determine the neutrino mass. The interaction between the neutrino and a detector with effectively infinite mass allows a finite unobserved momentum to be transferred to the detector without energy transfer. This is the same physics as the recoilless momentum transfer in photon scattering by a crystal in X-ray crystallography. Thus:

  1. The neutrino energy IS equal to the sum of the change in detector nucleon energy and the lepton energy

  2. The neutrino momentum IS NOT equal to the sum of the change in detector nucleon momentum plus the lepton momentum

    • The nucleon is not free but bound in a large system with effectively infinite mass.

    • The system can absorb momentum without energy transfer.

    • This “missing momentum” prevents the determination of the neutrino mass

  3. Neutrino absorption is not a “missing mass” experiment.

iii.2 Text books are misleading. No connection with real experiments

Neither text books nor relativistic quantum field theory tell us what is unobservable. Some energy or momentum must be unobservable to produce oscillations.

Text books tell us that a at rest with definite flavor is a coherent mixture of energy eigenstates. Interference between these states produces oscillations in time between different flavors. The ’s oscillate as coherent mixtures of states with different energies. Text books don’t tell us how such an oscillation is created or observed in any real experiment.

  1. No experiment has ever seen a at rest

    • Neutrinos observed in experiments have no unique rest frame

    • Components of wave function with different masses have different rest frames.

  2. ’s with different energies cannot interfere because their energy is measurable.

  3. Detectors in experiments observing oscillations do not measure time and destroy all interference between states with different energies

What is observable depends on the quantum mechanics of the detector.

iii.3 No oscillations in a “missing mass” experiment

The original Lederman-Schwartz-Steinberger experiment[1] found that the neutrinos emitted in a decay produced only muons and no electrons. Experiments now show that at least two neutrino mass eigenstates are emitted in decay and that at least one of them can produce an electron in a neutrino detector. The experimentally observed absence of electrons can be explained only if the electron amplitudes received at the detector from different neutrino mass eigenstates are coherent and exactly cancel.

The neutrinos are linear combinations of mass eigenstates with different masses, different energies and different momenta. The detector must know that the relative phases of relevant amplitudes will cancel the production of an electron. This can only be understood by investigating the quantum mechanics of the detector.

A missing mass experiment was not performed.

iii.4 X-Ray and Mössbauer physics needed to understand interference in detectrion

  1. The detector has a definite position in the laboratory system for all times and has effectively infinite mass.

  2. Energy in the laboratory system is conserved; momentum conservation is violated as in X-ray diffraction by crystals and the Mössbauer effect[2, 3].

    • In the Mössbauer effect a photon is scattered by an atom in a crystal. Energy in the laboratory system is conserved. A missing recoil momentum is absorbed by the crystal with negligible energy loss.

    • In experiments the is absorbed by a nucleon in a detector. Energy in the laboratory system is conserved. A missing recoil momentum is absorbed by the detector with negligible energy loss.

    • The same quantum state of the crystal or detector is produced by transitions with different momentum transfers and the same energy transfer.

    • No measurement on the final state can determine momentum of the initial photon or .

  3. The detector can absorb ’s with a small momentum difference and the same energy transfer and produce the same final state.

  4. Momentum difference and mass difference between two states is not observable.

Iv Basic quantum mechanics of coherence in detection

iv.1 Quantum mechanics of the detector

In neutrino oscillation experiments the neutrino is absorbed by a detector which is a complex interacting many-body system at rest in the laboratory. It is described in quantum mechanics by a Hamiltonian. Diagonalizing this Hamiltonian gives a discrete energy spectrum. Two neutrinos with different masses incident on the detector can be absorbed coherently only if they produce exactly the same final state wave function of the many-body system. Since the energy spectrum is discrete, only neutrinos with the same energy can be absorbed coherently and produce oscillations. Coherence and interference arise when neutrino eigenstates with different masses and momenta but the same energy are absorbed by a detector and produce the same transition from a given set of detector energy eigenstates. How this occurs is treated explicitly below.

iv.2 Difference between space and time measurements

In neutrino experiments the distance between source and detector can be measured with arbitrary precision. The transit time between source and detector cannot. The neutrino arrives at the detector as a finite wave packet over an appreciable time interval. The detection can occur at any time that a portion of the wave is still in the detector. This gives an uncertainty in transit time which cannot be shortened. Furthermore, if the neutrino wave packet contains components with different masses, the centers of the two wave packets arrive at the detector at different times. This time difference has produced a controversy about a factor of two.

The proper way to treat this problem is to note that oscillations can occur only if the time intervals for the two wave packets still overlap almost completely and interference can occur. The time of an individual neutrino arrival can be measured. The probability for emission of a charged lepton with a definite flavor depends on the relative magnitudes and phases of the contributing mass eigenstates at this time. The length of the wave packet in time is sufficiently short so that the relative phases and the flavor of the emitted charged lepton do not change appreciably with the exact time of observation. However the relative phase does change with changes on the position of the detector. This phase change with distance produces neutrino oscillations.

It is well known that at long times like the times of arrival from a supernova the wave packets separate and there is no interference and no oscillations. We consider here intermediate times which are long enough to produce oscillations and short enough so that there is nearly complete overlap between the wave packets of neutrinos with different eigenstates and different velocities.

This asymmetry between time and distance is essential for understanding neutrino oscillations and is not easily treated by relativistic quantum field theory.

iv.3 Constraints on the detector nucleon wave function

Neutrino oscillations can be observed only if the detecting nucleon is confined for all times to a region of space in the laboratory system much smaller than the oscillation wave length.

The probability of finding the detector nucleon outside the detector must vanish for all times. The state of the detector nucleon in quantum mechanics is described by a wave function or density matrix which gives a time-independent vanishing probability for finding the nucleon outside the detector. The density matrix describing the detector nucleon must have coherence and interference between components with different momenta at each energy which cancel out the probability of finding the nucleon outside the detector.

iv.4 Implications of space-time condition on the detector

This space-time condition on the detector nucleon wave function is crucial for a description of oscillations, missed in theoretical investigations, e.g. [6, 7]. and not included in formulations based on quantum field theory. Only components of the incident wave function with the same energy and different momenta are coherently absorbed, produce the same transitions between two detector nucleon eigenstates and interfere to create the observed oscillations. Since the momenta producing these transitions are not observable absorption is a which-path experiment in momentum space. Oscillations in configuration space are produced by interference between final states with same energy and different momenta. The experimental observation of oscillations shows that the wave function entering the detector contains coherent linear combinations of states with same energy, different momenta and definite relative phases[8].

  1. Components of an incident neutrino with the same energy and different momenta can produce coherent transition amplitudes between two detector nucleon states that both have a vanishing probability of finding the nucleon outside the detector.

  2. The momentum of the neutrino that produced the transition in the detector is not observable.

  3. Neutrino detection is a “two-slit” or “which-path” experiment in momentum space.

iv.5 A “which path” experiment with Dicke superradiance

Absorption of two states with different momenta gives the same detector transition. Dicke superradiance[9] arises when several initial states produce same final state. The transition matrix element depends upon the relative phases of these amplitudes. The state with maximum constructive interference is called “superradiant”. The states orthogonal to the superradiant state are called “subradiant” The relative phase between states having different momenta changes with the distance between source and detector. This phase change gives transitions between superradiant and subradiant states and produces the observed oscillations.

Neutrino detection is a “which-path” or “two-slit” experiment in momentum space. Dicke superradiance explains oscillations with distance. Momentum conservation is violated in the nucleon-lepton system by absorption on a nucleon in the detector. The detector absorbs the missing momentum - like crystal in the Mössbauer effect. Absorption of two states with different momenta produce the same detector transition.

That the error in the neutrino momentum is sufficiently large to allow neutrino oscillations is easily seen. The size of the detector must be much smaller than the wave length of the oscillation in space in order for oscillations to be observable. This implies that the difference in momenta of the interfering neutrino waves is much smaller that the spread in the momentum of the detector.

iv.6 Energy-momentum asymmetry crucial to understanding oscillations

An energy-momentum asymmetry not treated in covariant treatments arises from the asymmetry between space and time in the detector nucleon wave function. The probability for finding nucleon outside detector spatial region vanishes for all times. The detector nucleon wave function must then vanish in space outside detector for all times.

  • Components of the wave function at each energy must cancel outside the detector

  • Components with the same energy and different momenta can be coherent

  • Interference between states with different energies and same momentum cannot vanish outside detector

  • Absorption of ’s with different momentum and same energy can be coherent.

This crucial energy-momentum constraint is valid only in the laboratory frame. Covariant treatments and relativistic quantum field theory cannot explain this constraint.

V Why different approaches give the same answer

Consider a simplified two-component state with two components having momenta and with energies and and squared masses and . Changes produced by a small change in the squared neutrino mass satisfy the relation

(11)

When these neutrinos travel through a distance in a time their phase is given by:

(12)

Realistic experiments detect the at a known and definite distance from the source. The time of detection of the is not measured. The transit time of the center of the wave packet traversing the distance can be estimated using the group velocity of the neutrino wave packet

(13)

The relative phase between the two components observed at point is

(14)

If the relative phase depends only on the squared mass difference between mass eigenstates. It is independent of the particular chosen values of and as long as these satisfy the relation (11) We note that if and we are comparing two components with the same energy. Treatments where the two have different energies choose and also have . Thus choosing states with the same energy and different momenta gives the same result as choosing states with the same momentum and different energies.

This explains the apparent miracle that different treatments of oscillations always give the same answer for the relation between the oscillation wave length and the mass squared difference between mass eigenstates. Quantum fluctuations in the transit time of the wave packet do not affect this result since neither approach uses the fluctuating value of .

Vi Conclusions

Neutrino scillations cannot occur if the momenta of all other particles participating in the reaction are known and momentum and energy are conserved. Heisenberg uncertainty allows interference between neutrino mass eigenstates with different momentum transfers out of the nucleon-lepton system. Interference observed in experiments arises from momentum transfers to a large detector, analogous to the momentum transfer to a whole crystal in coherent diffraction of X-rays and the Mössbauer effect. The wave function of the nucleon absorbing the neutrino in an oscillation experiment is confined to a region whose scale is much smaller than oscillation wave length. The condition that this wave function must vanish for all times outside this region in the laboratory system is not properly included in covariant treatments. It shows that the oscillations are produced by interference between neutrino states with different masses and momenta but the same energy. A complete description of the decay process must include the interaction with the environment and violation of momentum conservation.

Vii Acknowledgement

The theoretical analysis in this paper was motivated by discussions with Paul Kienle at a very early stage of the experiment in trying to understand whether the effect was real or just an experimental error. It is a pleasure to thank him for calling my attention to this problem at the Yukawa Institute for Theoretical Physics at Kyoto University, where this work was initiated during the YKIS2006 on “New Frontiers on QCD”. Discussions on possible experiments with Fritz Bosch, Walter Henning, Yuri Litvinov and Andrei Ivanov are also gratefully acknowledged. The author also acknowledges further discussions on neutrino oscillations as “which path” experiments with Evgeny Akhmedov, Eyal Buks, Avraham Gal, Terry Goldman, Maury Goodman, Yuval Grossman, Moty Heiblum, Yoseph Imry, Boris Kayser, Lev Okun, Gilad Perez, Murray Peshkin, David Sprinzak, Ady Stern, Leo Stodolsky and Lincoln Wolfenstein.

References

  • [1] G. Danby et al, Physical Review Letters 9 (1962) p.36
  • [2] Harry J.Lipkin, Quantum Mechanics, North-Holland Publishing Co. Amsterdam (1973) pp.33-110
  • [3] Harry J. Lipkin, in Proceedings of the 10th Conference on the Intersections of Particle and Nuclear Physics, San Diego, California, 26-31 May 2009. Published in AIP Conf.Proc.1182:44,2009.
  • [4] Harry J. Lipkin,Phys.Lett. B642 (2006) 366,hep-ph/0505141
  • [5] A.D. Dolgov et al, Nucl.Phys. B729 (2005) 79, hep-ph/0505251
  • [6] Evgeny Kh. Akhmedov and Alexei Yu. Smirnov, hep-ph/0905.1903 and references therein
  • [7] Andrew G. Cohen, Sheldon L. Glashow, Zoltan Ligeti, hep-ph/0810. 4602
  • [8] Harry J. Lipkin, hep-ph/0801.1465 and hep-ph/0805.0435
  • [9] R. H. Dicke, Phys. Rev. 93, 99 (1954)
  • [10] Yuval Grossman and Harry J. Lipkin, Phys.Rev. D55 (1997) 2760, hep-ph/9607201
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