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    In 1930 physicists knew that they had a serious problem with their understanding of matter and its properties at the subatomic level. Studies of the form of radioactivity known as beta decay had called two of the most fundamental laws of physics into question. But by maintaining a belief in the laws pertaining to the conservation of energy and of linear momentum physicists solved the problem and discovered a new class of particle - the neutrinos.

    Antoine Henri Becquerel (1852 Dec 15 - 1908 Aug 25) discovered radioactivity in 1896 when he left a sample of uranium ore in his desk and it fogged a photographic plate left nearby. In 1899 Ernest Rutherford (1871 Aug 30 - 1937 Oct 19) used the ability of radiation to penetrate thin sheets of material and to cause ionization to divide the radiation into two types - alpha rays and beta rays. The following year Becquerel used the method devised by Joseph John Thomson (1856 Dec 18 - 1940 Aug 30) to study cathode rays and thereby to discover the particle that we call an electron. Thomson had measured the ratio of the particle’s electric charge to its mass and Becquerel found that the particles comprising beta rays have the same ratio, implying that beta rays consist of electrons.

    Rutherford, working with Frederick Soddy (1877 Sep 02 - 1956 Sep 22), improved his techniques enough that in 1901 he showed that the emission of alpha and beta rays coincides with the transmutation of some atoms into atoms of other chemical species. By 1913 Soddy had enough data in hand to hypothesize the radioactive displacement law, which law states that an alpha decay (the emission of a helium nucleus) converts an atom of one element into an atom of the element two places to the left on the Periodic Table of the Elements and that a beta decay (the emission of an electron) converts an atom of one element into an atom of the element one place to the right.

    Exploring those conversions led Lise Meitner (1878 Nov 07 - 1968 Oct 27) and Otto Hahn (1879 Mar 08 - 1968 Jul 28) to reveal, in 1911, their discovery that the electrons emerged from the decays with energies that form a continuous spectrum instead of the discrete spectrum that they had expected. That fact implies that beta decays do not conserve energy. That implication stirred up some controversy among physicists, but further experiments, conducted between 1920 and 1927, confirmed Meitner and Hahn’s results. Other experiments also showed that beta decays appear to violate the law pertaining to conservation of linear momentum.

    In 1930 Wolfgang Pauli (1900 Apr 25 - 1958 Dec 15) wrote a published letter in which he suggested that an atom contains a light, electrically neutral particle, which he called a neutron, in addition to the electrons and protons that physicists already knew about. He suggested that every atom undergoing beta decay emitted one of these particles, in addition to the electron, to carry the missing energy and linear momentum, thereby preserving the validity of the conservation laws. The next year James Chadwick (1891 Oct 20 - 1974 July 24) discovered a heavy, electrically neutral particle, which he called the neutron, thereby coopting Pauli’s name for his postulated particle. In 1933 Enrico Fermi (1901 Sep 29 - 1954 Nov 28), in order to avoid confusion between Pauli’s and Chadwick’s particles, called Pauli’s particle a neutrino, using the Italian diminutive form of neutron, because of the presumed lightness of it (In Italian it also serves as a pun on neutrone, the word for neutron). Fermi used Pauli’s neutrino hypothesis in working out a theory of beta decay that he published in 1934.

    In 1941 Ganchang Wang (1907 May 28 - 1998 Dec 10) suggested using beta capture (inverse beta decay) to detect neutrinos experimentally. Conditions in China did not allow him to conduct the experiment, so an American team did the work. In the 1956 Jul 20 issue of Science Clyde Lorrain Cowan Jr. (1919 Dec 06 - 1974 May 24), Frederick Reines (1918 Mar 16 - 1998 Aug 26), and their team published the results of their experiment.

    Inverse beta decay occurs when a proton absorbs an electron antineutrino and emits a positron to become a neutron. Detecting the event consists of detecting the two 511-kev gamma photons created when the positron annihilates an electron. Confirming the nature of the event consists of detecting the gamma photon that appears five microseconds after the electron-positron annihilation when a nucleus of Cadmium-108 absorbs the neutron.

    Cowan and Reines conducted their experiment at the Savannah River Plant near Aiken, South Carolina, putting their detector eleven meters from the reactor (and twelve meters underground) in order to take full advantage of the presumed flux of 5x1013 neutrinos per square centimeter per second coming from the reactor. The detector consisted of two tanks sandwiched between three scintillator layers containing arrays of photomultiplier tubes. Filled with a solution consisting of 40 kilograms of cadmium chloride dissolved in 200 liters of water, the detectors picked up about three neutrinos per hour. Cowan and Reines had predicted a cross section of 6x10-44 square centimeters and, based on the detection rate, calculated an actual cross section of 6.3x10-44 square centimeters (6.3x10-20 barns).

Neutrino Cross Section

    Ever since the opening decades of the Twentieth Century, when Ernest Rutherford and his team began the practice of using collisions to study atomic structure, physicists have taken the cross section of a particle as one of the particle’s more important fundamental properties. Over the following decades physicists discovered that particles have several kinds of cross section, one for scattering other particles and one for interacting with other particles. For convenience in calculations they defined the basic unit of cross section as the barn (as in the broad side of) to correspond to 10-24 square centimeter. They chose that particular number because it more or less matches the typical nuclear radius of about 10-12 centimeter.

    Reines and Cowan based their original detection of neutrinos on the reaction of antineutrino plus proton yielding neutron plus positron. This interaction is related by crossing symmetry to the decay of the neutron, the simplest example of beta decay. In fact physicists sometimes refer to it as "inverse beta decay". So it would appear that the study of beta decay could give physicists an approach to determining the cross-section for the neutrino interaction.

    Enrico Fermi’s Golden Rule provides the basis for calculating the cross-section for the interaction. If we can find a matrix element for the weak interaction in one of these reactions it should be comparable to the corresponding matrix element for the other interaction. Fermi’s Golden Rule enables us to calculate interaction cross sections from the square of the amplitude of a process (derived from the associated Feynman diagrams) and the density of final states.

    Applying Fermi’s theory of beta decay in certain special cases, some physicists estimated the coupling strength of the weak interaction, the interaction responsible for creating or annihilating neutrinos. By that means they estimated 10-43 cm2 = 10-47 m2 = 10-19 barn for the neutrino cross section in the beta decay range of energy. That stands at about 20 orders of magnitude less than the cross-section for nuclear processes that occur at low energy. For example, U-235 has a cross section that lies between 2 and 3 barns for fission induced by thermal neutrons.

    With this nominal cross section, we can make some estimates of rates at which neutrino scattering interactions occur. Multiplying the cross section by the nucleon density in a material gives a number of interactions that occur per meter of material, and the reciprocal of that number gives us an estimate of the neutrinos’ mean free path. For water, with a density of 1000 kg/m3, and lead, with a density of 11,400 kg/m3, we can estimate the mean free path for a neutrino from the following mean free path equations.

(Eq’ns 1)

To calculate the mean free path in those equation we have divided the volume per nucleon by the cross section of the neutrino interaction. Because we want only an approximate value and because the masses of the proton and the neutron do not differ significantly, we use the same mass mn for both nucleons.

    We can gain a better perspective on these distances by converting them to lightyears (one lightyear = 9.46x1015 meters). We get, respectively, 17.97 lightyears and 1.59 lightyears. So this estimate of the neutrinos’ mean free path comes out to more than a light year of lead.

    This cross section can also give us an estimate of the number of events which we can expect to observe in a given size of detector. We calculate the rate at which the detector absorbs neutrinos, in particles per second, through the product

(Eq’n 2)

in which phi represents the neutrino flux through the detector (in particles per square meter per second), sigma represents the interaction cross section (in square meters), en represents the density of nucleons in the detector (in particles per cubic meter), and vee represents the volume of the detector.

    Physicists have also discovered that the nominal neutrino cross section for an interaction with a nucleon increases with the energy carried by the neutrino. At energies far above the range of energies carried by neutrinos from radioactive decay, from 20 to 200 Gev (billions of electron-volts) the description of the neutrino cross section forms a sloping straight line.

Neutrino Quantum Number

    In 1953 Emil Jan Konopinski (1911 Dec 25 - 1990 May 26) and H.M. Mahmoud (no data found) published "The Universal Fermi Interaction" in Volume 92 of The Physical Review and in that paper postulated a new conservation law. Taking the hypothesis that the same form of interaction acts among any set of spin-1/2 particles, they applied the interaction law found for beta decay to processes involving muons (discovered in 1936). They had to address an ambiguity that arises from the various ways in which we can take the correspondence between the particles of muon decay and beta decay. They resolved the ambiguity by arguing we get a correct description of a unique correspondence if muon decay ejects two like neutrinos.

    Konopinski and Mahmoud argued that physicists had previously made unjustifiably broad interpretations of the Universal Fermi Interaction. They asserted that we should only expect to detect processes in which two normal particles (vs antiparticles) get annihilated and two get created. In that case we must treat the positive muon as the normal particle (if we take the neutron, proton and electron as normal particles) in order to avoid the expectation, contrary to experimental facts, that muon capture by a proton may yield electrons. They then concluded that any muon decay ejects two like neutrinos. Thus they presented the precursor of the law of conservation of lepton number.

    In the late 1950's Raymond Davis Jr (1914 Oct 14 - 2006 May 31) and Don S Harmer (no data found) asked whether neutrinos have an antiparticle (other neutral particles, such as the neutral pion and the photon, are their own antiparticles). Cowan and Reines used the reaction

(Reaction 1)

in which an antineutrino merges with a proton and yields a neutron and a positron. The crossed reaction,

(Reaction 2)

must also occur and do so at a rate approximately equal to that of Reaction 1, given equal densities of neutrons and protons along with equal neutrino fluxes. Davis and Harmer looked for signs of the analogous reaction,

(Reaction 3)

in which an antineutrino replaces the neutrino of Reaction 2. They found that spraying neutron-rich material with antineutrinos did not yield any emitted electrons and thus inferred that neutrinos and antineutrinos exist as distinct particles and not as different aspects of one kind of particle.

    The mathematical description of Konopinski and Mahmoud’s rule involves assigning a lepton number to each kind of particle. Negative muons, electrons, and neutrinos get L=+1 and positive muons, positrons, and antineutrinos get L=-1. The other particles - protons, neutrons, and photons - get L=0. We then state the rule by saying that the sum of the lepton numbers on one side of the reaction must equal the sum of the lepton numbers on the other side of the reaction.

    Up until 1962 physicists assumed the existence of only one kind of neutrino. Given what physicists knew of the nature of matter in the first half of the Twentieth Century, that was a reasonable assumption. Up to midcentury no particle that physicists had discovered had shown alternative forms, aside from the antimatter counterpart. Then physicists noticed a glitch in the law of conservation of lepton number.

Neutrino Flavors

    In 1962 Leon M. Lederman (1922 Jul 15 - ?), Melvin Schwartz (1932 Nov 02 - 2006 Aug 28) and Jack Steinberger (1921 May 25 - ?) and their team showed that more than one type of neutrino exists, doing so by first detecting interactions of the muon neutrino (whose existence someone had already hypothesized with the name neutretto, another Italian diminutive like the one we see in words like libretto. We might also use neutrello, using the Italian diminutive found in Monticello). The issue of more than one kind of neutrino arose because physicists noticed that a certain decay does not occur: they noticed that a negative muon does not decay into an electron and a photon. Particle physicists had, by 1960, devised a simple rule of thumb to guide them in their researches: Richard Feynman, according to the belief popular among physicists, had declared that whatever the laws of nature do not explicitly forbid, they must mandate. Thus the non-existence of a direct decay of a muon into an electron and a photon necessitates the existence of whatever prevents that decay from happening.

    The nature of the whatever lies implicit in the fact that muons do decay into electrons. But in those decays a neutrino and an antineutrino accompany the electron. To explain why one decay happens and the other can’t physicists conceived the idea of mu-ness: they assumed that neutrinos exist in two flavors, each flavor represented by a different lepton number. Thus we can describe the decay of a negative muon,

(Reaction 4)

as involving an electron antineutrino and a muon neutrino. In that reaction the muon number equals plus one on both sides of the arrow and the electron number adds up to zero.

    In 1962, then, Lederman and his collaborators put that proposition to the test by using 1014 antineutrinos emanating from decays of negative pions. Striking protons, those antineutrinos could conceivably produce two reactions,

(Reactions 5)

When the experimenters looked at their data, they found 29 instances of the first reaction (which conforms to conservation of muon number) and no instances of the second reaction (which does not conform to the conservation law).

    When the third type of lepton, the tau, was discovered in 1975 at the Stanford Linear Accelerator Center, physicists expected that it, too, would have an associated neutrino (the tau neutrino). First evidence for this third neutrino type came from the observation of missing energy and momentum in tau decays analogous to the beta decay leading to the discovery of the neutrino. The first detection of tau neutrino interactions was announced in summer of 2000 by the DONUT collaboration at Fermilab, making the tau neutrino the latest particle of the Standard Model that physicists have observed directly; its existence had already been inferred from both theoretical consistency and experimental data from the Large Electron–Positron Collider.

The Solar Neutrino Problem

    Starting in the late 1960s, several experiments found that the number of electron neutrinos arriving from the Sun lies between one third and one half the number predicted by the Standard Solar Model. This discrepancy, which became known as the solar neutrino problem, remained unresolved for some thirty years. Physicists resolved it through postulating the existence of neutrino oscillation and mass.

    It began with John Norris Bahcall (1934 Dec 30 - 2005 Aug 17) using a model of the solar interior to calculate the flux of solar neutrino flux reaching Earth’s surface. He calculated that flux as about 5 x 106 neutrinos per square centimeter per second. For convenience physicists us the Solar Neutrino Unit (SNU), which equals one detection per second per every 1036 target atoms. Bahcall’s theoretical flux then becomes 7.9±2.6 SNU. In what physicists call the "solar neutrino problem", only about a third to one half this many, 2.1±0.3 SNU (measured over 18 years as of 1990), were measured in early experiments. Current experiments at the Sudbury Neutrino Observatory suggest that neutrino oscillation transmutes some of the solar electron neutrinos into muon and tau neutrinos. When the experimenters take that postulate into account, the flux comes out in close agreement with the Bahcall estimate.

    On the experimental side, while Bahcall tweaked his model of solar fusion, neutrino detection proceeded in the Homestake gold mine, near Deadwood, South Dakota. In their detector Raymond Davis Jr, and Don S. Harmer used 100,000 gallons of perchloroethylene to detect the massless grains of matter that we call neutrinos (about 1/4 of the chlorine used in the detector is the desired Cl-37, which absorbs a neutrino into one of its neutrons, which then spits out an electron and becomes a proton, thereby turning the chlorine into Ar-37). Davis started in 1955 to detect neutrinos from proton-proton fusion. But proton-proton neutrinos carry too little energy to trigger a conversion of chlorine to argon, the easiest method of neutrino detection available at the time. In 1958, though, theorists calculated that 15% of helium nuclei in the sun’s core would undergo the Helium-3-Helium-4 fusion that then absorbs a proton and produces Beryllium-8, some of which would decay into Boron-8 plus a 14-Mev neutrino. Theory eventually predicted that Davis should get one and a half neutrino detections per day on average, but Davis was getting one half of a detection per day, the roughly two SNU noted above. By 1988 Kamiokande II confirmed the result by finding the light from electron-neutrino collisions.

    In creating the Standard Model of particle physics physicists had assumed that neutrinos have no mass, flying at the speed of light like photons, and that they cannot change flavor. Those assumptions made the solar neutrino problem extremely vexing: it seemed to have no solution. However, if neutrinos had mass, they could change flavor, or oscillate between flavors. Such oscillation might solve the problem.



    Bruno Pontecorvo (1913 Aug 22 - 1993 Sep 24) provided a practical method for investigating neutrino oscillations. In 1957 he used an analogy with kaon oscillations and over the subsequent 10 years he developed the mathematical formalism and the modern formulation of vacuum oscillations. In 1985 Stanislav Mikheyev and Alexei Smirnov (expanding on 1978 work done by Lincoln Wolfenstein) noted that flavor oscillations can change when neutrinos propagate through matter. This effect, which physicists call the Mikheyev–Smirnov–Wolfenstein effect (MSW effect), is important to understand because many neutrinos emitted by fusion in the Sun pass through the dense matter in the solar core (where essentially all solar fusion takes place) on their way to detectors on Earth.

    Mikheyev, Smirnov, and Wolfenstein found in their calculations that the presence of electrons in matter changes the energy levels of the propagation eigenstates of the neutrinos. That effect comes about due to certain weak-force interactions; specifically, charged current coherent forward scattering of the electron neutrinos. Coherent forward scattering gives us an effect analogous to the electromagnetic process that leads to the refractive index of light in a material medium differing from that in vacuum. That means that neutrinos propagating in matter have an effective mass that differs from that of neutrinos propagating in vacuum. Because neutrino oscillations depend upon the squared mass difference of the neutrinos, neutrino oscillations in matter may differ from those in vacuum. If we have antineutrinos, the conceptual point remains the same but the effective charge to which the weak interaction couples (called weak isospin) has the opposite algebraic sign.

    The MSW effect becomes important at the very high electron densities occurring in the part of the Sun where electron neutrinos originate. The high-energy neutrinos detected, for example, in SNO (Sudbury Neutrino Observatory) and in Super-Kamiokande, originate as the higher mass eigenstate in matter and remain as such as the density of solar material through which they pass changes. (When neutrinos go through the MSW resonance the neutrinos have the maximal probability to change their nature, but it happens that this probability is negligibly small – physicists sometimes call this effect propagation in the adiabatic regime). Thus, the neutrinos of high energy leaving the sun exist in a vacuum propagation eigenstate that has a reduced overlap with the electron neutrino detected through charged current reactions in the detectors. The transition between the low energy regime (in which the MSW effect is negligible) and the high energy regime (in which the oscillation probability is determined by matter effects) lies in the region of about 2 MeV for the solar neutrinos.

    For high-energy solar neutrinos the MSW effect plays an important role and leads physicists to the expectation that Pee = sin2θ, where θ=34° is the solar mixing angle and Pee represents the probability of an electron neutrino coming into the detector as an electron neutrino. Experimenters at SNO measured the flux of solar electron neutrinos to be ~34% of the total neutrino flux (the electron neutrino flux measured via the charged current reaction, and the total flux via the neutral current reaction). The SNO results agree well with the expectations of theory, thereby resolving the solar neutrino problem. Earlier, Kamiokande and Super-Kamiokande measured a mixture of charged current and neutral current reactions, that also support the occurrence of the MSW effect with a similar suppression, but with less confidence.

    For low-energy solar neutrinos, on the other hand, the matter effect is negligible and the formalism of oscillations in vacuum remains valid. The size of the source, the core of the sun, is significantly larger than the neutrinos’ oscillation length, therefore, averaging over the oscillation factor, one obtains Pee = 1 (sin22θ)/2. For the same value of the solar mixing angle (θ=34°) this corresponds to an electron neutrino survival probability of Pee 60%. This calculation is consistent with the experimental observations of low energy solar neutrinos by the Homestake experiment (the first experiment to reveal the solar neutrino problem), followed by GALLEX, GNO, and SAGE (experiments that use gallium as a neutrino detector), and, more recently, the Borexino experiment. These experiments provided further evidence for the MSW effect.

    Those results gain further support from the reactor experiment KamLAND. The Kamioka Liquid Scintillator Antineutrino Detector (KamLAND) is an experiment at the Kamioka Observatory, an underground neutrino observatory near Toyama, Japan. It was built to detect electron antineutrinos. Situated in the old Kamiokande cavity in a horizontal mine drift in the Japanese Alps, the site is surrounded by 53 Japanese commercial power reactors. KamLAND detects the electron antineutrinos that come from the decays of radioactive fission products in the reactors’ nuclear fuel. Like the intensity of light from a distant star, the isotropically emitted flux follows the inverse-square rule, decreasing as 1/R2 for increasing distance R from the reactor. The experiment is sensitive to the estimated 25% of antineutrinos from nuclear reactors that exceed the threshold energy of 1.8 MeV and thus produce a signal in the detector.

    Oscillation of the neutrinos makes them oscillate into flavors that an experiment may not be able to detect. That oscillation leads to a dimming in the intensity of the detected electron antineutrinos. KamLAND is at a flux weighted average distance of roughly 180 km from the reactors which makes the experiment sensitive to the neutrino mixing associated with the large mixing angle (LMA) solution to the solar neutrino problem.

    KamLAND detects electron antineutrinos via the inverse beta decay reaction (antineutrino plus a proton yields a positron plus a neutron) which has a 1.8 MeV antineutrino energy threshold. The prompt scintillation light from the emitted positron gives an estimate of the incident antineutrino energy, Eν = Eprompt + <En> + 0.8 MeV, where Eprompt represents the prompt event energy including the positron’s kinetic energy and the electron-positron annihilation energy. The quantity <En> represents the average neutron recoil energy, which is only a few tens of keV. The neutron gets captured by a hydrogen atom about 200 microseconds later, emitting a characteristic 2.2 MeV gamma ray. This delayed coincidence signature provides a very powerful tool for distinguishing antineutrinos from backgrounds produced by other particles.

    The researchers at KamLAND started data taking in January 2002 and, with only 145 days of data, they reported its first results. Without neutrino oscillation, the experimenters expected to see 86.8±5.6 events, with 2.8 background events. However, they observed only 54 events. The KamLAND researchers confirmed this result with a 515 day data sample, when they expected to detect 365.2±23.7 events in the absence of oscillation, while they actually detected 258 events (with 17.8±7.3 background events). This establishes antineutrino disappearance, and thus oscillation, at the 99.998% significance level.

    To describe the mechanism of oscillation, we say that flavor determines how the neutrino interacts with matter and the assumed mass determines how the neutrino propagates through space. With each mass we associate a wave and all three kinds of wave are present in any emitted neutrino. Interference of the waves produces different mixtures of the three kinds of neutrino (think of beats), so that we have an average mixture that determines the probabilities of detecting one or another kind of neutrino. Using ratios of electron:muon:tauon, we have the ratio emitted from a neutron decay as 1:0:0, which evolves into 5:2:2. A complete pion decay gives us 1:2:0, which evolves into 1:1:1, the ratio we receive from the sun. Incomplete pion decay, in which the muon does not decay, gives us the ratio 0:1:0, which evolves into 4:7:7. Note that the final state always gives us the same ratio of muon neutrinos to tau neutrinos.

    We thus have two trios of quantum mechanical eigenstates: three flavors (electron, muon, and tau) and three mass states. A pure state of one trio consists of a mixture of the other trio’s states; thus, for example, a pure electron neutrino state consists of a mix of all three mass states. Because the Schrödinger waves associated with the mass states have different frequencies, the waves interfere, so that as they propagate they oscillate among the flavor states. This is similar to kaon oscillations.

    To represent the mismatch between the two sets of eigenstates we use the following equations:

(Eq’ns 3)

in which the subscript ay refers to the flavor eigenstates of the neutrino í and the subscript eye refers to the mass eigenstates, the kets representing the eigenstates themselves in Dirac’s bra-ket notation. The asterisk in the first of those equations tells us, as usual, to use the complex conjugate of Uai, the neutrino mixing matrix. If we had set up those equations to represent the states of antineutrinos, then the complex conjugate would be taken in the second equation and not in the first.

Propagation and Interference as the Source of Oscillation:

    Given a mass eigenstate |νi>, we can describe the evolution of the state with a plane-wave solution of Schrödinger’s Equation,

(Eq’n 4)

in which Ei represents the total energy of the eigenstate, t represents the time elapsed since the start of the propagation, pi represents the linear momentum associated with the eigenstate, and x represents the distance that the neutrino moves in the time t. We know that the mass of a neutrino must correspond to an energy much less than one electron-volt and that the total energies of the neutrinos that physicists deal with take values on the order of one Mev or greater, so we have an ultra-relativistic situation (one in which the Lorentz factor exceeds one million). In that situation we can approximate the total energy of the eigenstate as

(Eq’n 5)

Noting that x ct, we make the appropriate substitution from that equation into Equation 4, which yields a description of the evolved eigenstate as

(Eq’n 6)

    To calculate the probability that a neutrino of a given flavor transforms into one of a different flavor as it traverses a given distance, we use Dirac’s version of Born’s theorem. For the probability of a quantum system going from an a-state to a b-state we use the Dirac bracket

(Eq’n 7)

If we begin with an a-flavor neutrino and let it travel a distance x to a detector, the probability that the detector will find that is has become a b-flavor neutrino conforms to the statement that

(Eq’n 8)

In the second equality in that equation we have used Equation 6 to replace the flavor eigenfunctions with the corresponding arrays of mass eigenfunctions. We must make that substitution because the flavor eigenfunctions don’t describe the neutrino’s propagation. The mass eigenfunctions, based on three different masses, describe propagation of the associated waves at three different speeds. The interference of the three waves then creates beats, which correspond to the different flavors.

    Note that our use of the Dirac bracket formalism gives us an extension of Werner Heisenberg’s matrix-mechanics formulation of the quantum theory, as distinct from Erwin Schrödinger’s wave-mechanics version.

Neutrino Helicity

    Contrary to the expectation underlying the theory of neutrino oscillations, a neutrino flies at the speed of light, so its helicity has the same value for all observers. Before 1957 physicists believed that neutrinos were evenly divided between left-handed and right-handed, like photons. But physicists discovered that all neutrinos are left-handed and all antineutrinos are right-handed.

    In 1929 Hermann Weyl responded to the publication of Dirac’s equation by working out a simple theory of massless particles of spin-1/2. In that theory the particles all have a fixed helicity. Initially dismissed as unrealistic, the idea was validated in 1957 by an experiment that showed neutrinos have a fixed helicity.

    In their paper "Helicity of Neutrinos", (received 1957 Dec 11 and published Feb 1958), Maurice Goldhaber (1911 Apr 18 – 2011 May 11), Lee Grodzins (1926 ?? – ?), Andrew W. Sunyar (no data found) described an experiment that confirmed that neutrinos are not their own antiparticles (like photons), but exist as separate particles and antiparticles. By combining analysis of circular polarization and resonant scattering of ã rays following orbital electron capture in Europium-152 they inferred the helicity of the neutrino. Assuming the most plausible spin-parity assignment for this isomer compatible with its decay scheme, they found that the neutrino is "left-handed", that its spin vector points in the direction opposite the direction pointed by its linear momentum vector (negative helicity). The Goldhaber, Grodzins and Sunyar experiment stands as one of the most brilliant experiments of the twentieth century, one that has become a cornerstone of the physics of weak interactions and of particle physics in general.

    Helicity denotes an important property of particles, neutrinos in particular. It relates the orientation of the particle’s spin and the direction of its linear momentum. The relative orientations of spin and linear momentum for neutrinos and antineutrinos is apparently fixed and intrinsic to the particles. The neutrino spins counterclockwise, antineutrinos spin clockwise as seen from behind.

    For neutrinos the spin vector always points opposite the direction of the linear momentum vector and physicists refer to this as "left-handed", whereas the antineutrinos always have "right-handed" helicity. This evokes the picture of the right-hand rule for determining the orientation of the angular momentum vector of a system. In this convention, we use the curled fingers of the right hand to indicate the sense of the spin or orbital motion and then say that the thumb points in the direction of the defined angular momentum. We use the linear momentum of the particle to define a preferred direction in space. If you curl the fingers of your right hand to show the sense of the "spin" of the antineutrino, your thumb then points in this linear momentum direction. Thus physicists call it a "right-handed" particle. For the neutrino you would use the curled fingers of your left hand to get your thumb to point in the direction of the linear momentum.

    This "left-handed" vs "right-handed" characterization does not apply to particles that carry mass, such as electrons. If an electron has its spin vector oriented to the right and it travels to the right, then we classify it as right-handed. But from the reference frame of someone who travels to the right faster than the electron does, its velocity vector would point to the left while the orientation of its spin vector would remain unchanged. Thus the electron would appear as a left-handed particle to any observer occupying that reference frame.

    For neutrinos, however, which travel at the speed of light, nobody can accelerate to a greater speed and thereby change the neutrinos’ "handedness". We say that the neutrinos have "intrinsic parity", classifying all of them as left-handed. This causes the weak interactions which emit neutrinos or antineutrinos to violate the law pertaining to conservation of parity.

    As noted above, technically the property which physicists call left-handed and right-handed goes by the name "helicity". Mathematically we define the helicity of a particle as the ratio of the z-component of the particle’s spin vector to the magnitude of the spin. Using this definition, we say that the right-handed antineutrino has a helicity of +1 and that the left-handed neutrino has a helicity of -1.

    But how can anyone measure the helicity of the most elusive particles known to science? Neutrinos almost never interact with matter when they traverse it, which fact implies great difficulty in measuring their properties, such as spin. In fact physicists detect particles by observing their interactions with gross matter, such as by observing the small condensation trails that a charged particle leaves in a Wilson cloud chamber. So how did Goldhaber and his collaborators measure the spin and linear momentum of neutrinos?

    They designed and conducted an experiment using Europium-152, a radioactive isotope with some useful characteristics. It has zero total angular momentum (J=0: no net spin) and it decays rapidly (i.e. it has a short half-life), transmuting into excited Samarium-152 by capturing one of its own inner electrons, thereby emitting a neutrino in one direction, and a Samarium nucleus in the other.

    Excited Samarium-152 has a total spin J=1 (in units of Planck's constant), and also has a very useful property: it decays into its fundamental state – its ground state – by emitting a 960 keV photon so quickly (in 0.07 picoseconds) that it has insufficient time to change its direction of motion due to thermal effects, even in a solid sample of Europium at room temperature. The photon, a particle with one unit of spin, has the spin either aligned or anti-aligned with its own direction of motion, and, due to conservation of angular momentum, aligned with the original spin of the excited Samarium. If the photon emerges along the direction of motion of the excited Samarium, the direction of its spin will align with the direction of the spin of the neutrino emitted in the original decay of Europium and it will have the same sign. This fact means that once Goldhaber and his team measured the photon spin properties, they could then infer the neutrino's spin properties.

    So we have a nucleus of Eu-152 (spin 0) absorbing one of its electrons (spin ½), emitting a neutrino (spin ½), and becoming an excited nucleus of Sm-152 (spin 1). The Samarium then spits out a photon (spin 1) whose spin vector points in the direction opposite that of the spin vector of the emitted neutrino. Detecting that photon then allows the experimenter to determine the helicity of the neutrino.

    The experiment must thus detect only those photons emitted by the excited Samarium. But those photons do not necessarily possess all the energy that comes from the de-excitation of the Samarium: emission of the photon causes the Samarium atom to recoil in accordance with Newton’s third law of motion. Thus the Samarium atom keeps to itself some of the energy, thereby denying the emitted photon the full 960 kev released in the de-excitation. As a consequence the photon, when it encounters another atom of Samarium, will not have enough energy to pump it to the excited state. The exception comes about if the photon gets emitted in the same direction in which the Samarium atom travels: thus the photon gets kicked forward and thereby gets doppler-shifted to higher energy. So only those photons will have enough energy to do resonant scattering with a Samarium nucleus.

    Goldhaber’s team achieved their measurement by exploiting the fact that photons with energy of about one MeV have different probabilities of interacting with the atoms in a sample of iron, depending upon whether the magnetization of the iron points parallel or antiparallel to the photons’ direction of propagation. That fact emerges from the electrons in the iron having a greater probability of absorbing photons if their spins point antiparallel to the spin of the photons. Thus Goldhaber’s team measured gamma photon polarization with polarized iron.

    Thus the experiment proceeded as follows: The experimenters placed a sample of Europium-152, with a half-life of about 9 hours, into a niche cut into a block of magnetized iron. A conical lead shield prevents gamma photons emerging from the iron block from reaching the photo-multiplier detector directly from the Europium source. A ring of Samarium oxide (Sm2O3) serves as a scatterer, where photons can do resonance absorption and get re-emitted in random directions. The ring lies below the lead shield and has at its center an instrument to detect the secondary radiation from de-excitation of the Samarium undergoing resonant scattering (a photo-multiplier will do). The actual detector consists of a sodium iodide scintillating crystal, which records the photon signal by converting a gamma photon into a minuscule flash of visible light. The photo-multiplier tube then detects the visible light. The experimental procedure then consisted of counting the ticks emitted by the photo-multiplier tube, reversing the magnetization of the iron, and then counting again. The orientation of the iron’s magnetization that yields the more counts thus reveals the helicity of the neutrinos.

    Another relevant experiment displays even greater elegance. Physicists know that pions all have spin zero and they decay into muons and their associated neutrinos in the following reactions:

(Reactions 6)

By detecting the muons emerging from those reactions and measuring their spins, the experimenters can thus determine, via conservation of angular momentum, the spins of the neutrinos.

Real/Virtual Neutrino Collisions

    The fact that helicity determines the fundamental nature of the neutrino tells us that neutrinos cannot have mass, as required by the theory of neutrino oscillations. If a neutrino has mass, it must travel slower than the speed of light. That fact necessitates that a particle that exists as a neutrino in some inertial frames exists as an antineutrino in other frames. That possibility enables the possibility of violating the law of conservation of lepton number.

    Rather than discard a conservation law, we discard the idea that neutrinos have mass. We conceive the neutrino as a quantized vibration of the weak-force field, like the photon shaken off a changing electromagnetic field. Thus we assert that neutrinos must fly at the speed of light. But the inference from experiment that neutrinos oscillate among their three flavors (electron, mu, and tau) leads physicists to treat the oscillation as a kind of decay that requires the neutrinos to possess some kind of an internal clock counting half lives. That presumed fact seems to necessitate that neutrinos not fly at the speed of light. How, then, might a particle for which time does not elapse undergo change?

    The fundamental nature of the vacuum may answer that question. We conceive the quantum vacuum as a seething froth of virtual particles. For example, under the right circumstances a photon carrying more that 1.022 Mev can vanish and realize an electron-positron pair in consequence. In essence the photon has transferred its properties to a virtual electron-positron pair, making the pair real while the photon becomes virtual. Assume that, in like manner, neutrinos collide with virtual neutrinos in the quantum vacuum and exchange properties. An electron neutrino might collide with a virtual mu neutrino and hand off its energy and spin, thereby making the mu neutrino come real while the electron neutrino becomes virtual. Mean free path and velocity are equivalent to an internal clock counting half lives.

    Neutrinos originate in the sun as electron neutrinos, but within eight minutes oscillate into equal numbers of electron, mu, and tau neutrinos. This is not a decay process. It is a neutrino changing identity, not splitting into other particles. Thus, there is no half life to be affected by time dilation. Instead of decay, oscillation occurs by neutrinos exchanging identities with virtual neutrinos in the quantum vacuum.

Do Neutrinos Fly Faster Than Light?

    In 2011 researchers in Europe found hints in their data that certain flocks of neutrinos going from CERN (on the Swiss-French border near Geneva) to the Gran Sasso National Laboratory in L’Aquila in Italy did so at a speed slightly greater than that at which a pulse of light would have made the same journey. Physicists working on the Italian OPERA experiment from 2009 to 2011 reported the slightly superluminal neutrinos. Using GPS, they determined the distance traveled as about 730 km, determined to within 20 cm (which adds an uncertainty of less than a nanosecond to the travel time of 2.44 milliseconds). The neutrinos’ flight time, calculated from the data, took 60 nanoseconds less than the speed-of-light time. The neutrinos came from protons blasted into graphite at CERN.

    But if the neutrinos had flown faster than light, they would have shed energy, in a manner analogous to the production of Cherenkov radiation (in which electrons traveling faster than the speed of light in water shed blue photons to lose energy), by generating a trail of electrons and positrons, which were not detected in the experiment. A loss of energy in the neutrinos was not detected. The error may come from General Relativistic effects on the atomic clocks used to time the journey, gravity pulling harder on the Swiss clock than on the Italian clock. Further, supernova 1987a popped at a distance of 168,000 lightyears, so if neutrinos actually fly at about 25 parts per million faster than light, the neutrinos (detected in Japan, the United States of America, and Russia) would have arrived about four years earlier than the light from the supernova, instead of almost simultaneously with it as detected.

    As it turns out, the OPERA experimenters traced the missing 60 nanoseconds to a faulty electrical connection in their detector. Most physicists suspected that something of that sort would turn up and the incident illustrates the importance of checking all possible sources of error in experiments, usually by someone else repeating the experiment on different equipment.


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