Tony Smith's Home Page

There are 3 types ofneutrinos,

one for each of the 3generations of fermions. According to a ParticleData Group ParticleProperties review(revised August 2001) by D. Karlen:

"... The most precise measurements of the number of light neutrino types, Nv, come from studies of Z production in e+e- collisions. The invisible partial width ... is determined by subtracting the measured visible partial widths, corresponding to Z decays into quarks and charged leptons, from the total Z width. The invisible width is assumed to be due to Nv light neutrino species each contributing the neutrino partial width ... as given by the Standard Model. ... The combined result from the four LEP experiments is Nv = 2.984 +/- 0.008 ...".

Solar Neutrinos, VEPNeutrino Oscillatons, and Neutrino OscillationExperiments

As to Neutrino Masses, according to PhysicsNews Update 600 #2, 1 August 2002, by Phil Schewe, James Riordon, andBen Stein: "... The 2dF Galaxy Redshift Survey has scanned250,000 galaxies ... providing a plot of the number of galaxiesversus inter-galaxy distance. Turned into a galactic "powerspectrum," this correlation study can be used to estimate the likelydensity of the constituent species of matter in the universe: baryons(such as protons), cold dark matter (WIMPs), and hot dark matter(neutrinos are the leading candidate). The 2dF work arrives at twobig neutrino conclusions. (1) Neutrinos can account for no more than13% of the matter in the universe and (2) the sum of all the numasses (electron plus mu plus tau) is no more than 2.2 eV. ...".

According to hep-ph/0302191(20 Feb 2003) by Bhattacharyya, Pas, Song, and Weiler "... theWMAP data ... places significant limits on the contribution ofneutrinos to the energy density of the universe ... which translatesinto ...[the sum of all the nu masses being less than] ...0.71 eV (95%C.L.). ...".

For a good introduction to the basic ideas of Neutrino Physics,see the book From Dirac to Neutrino Oscillations, by Tino Ahrens(Kluwer 2000).

However, no book can tell us what is not known, and as of now, alot is unknown about Neutrino Physics, and therefore it is useful toask, about Neutrino Oscillations:

"... Where do we stand ? ..."

According to Gary Feldman ( slide10 of lecture slides for the August 2000 SLAC SummerInstitute ): "...

Solar favored solution(LMA):

Atmospheric favored solution:


All three of these results cannot be correct!...".

Nearly two years later, according to AIP Physics NewsUpdate Number 586 #1, April 24, 2002, by Phil Schewe, JamesRiordon, and Ben Stein: "...

The Solar Neutrino Problem Has Been Closed

The solar neutrino problem has been closed and the ability of neutrinos to change from one type, or "flavor," to another established directly for the first time by the efforts of the Sudbury Neutrino Observatory (SNO) collaboration. ...

Measurements dating back to the 1960's of this neutrino flux were puzzling: only a fraction of the expected number arrived at detectors on Earth. Suspicion naturally fell on the experiments and on the standard solar model (SSM) used to calculate the flux. Soon, however, the neutrinos themselves were implicated. If on their journey to Earth some of the neutrinos (basically solar reactions produce electron-neutrinos exclusively) had changed into muon- or tau-neutrinos, then terrestrial detectors designed only to spot electron neutrinos (e-nu's) would be cheated of their rightful numbers.

SNO scrutinizes a particular reaction in the sun: the decay of boron-8 into beryllium-8 plus a positron and an e-nu. SNO's gigantic apparatus consists of 1000 tons of heavy water (worth $300 million Canadian) held in an acrylic vessel surrounded by a galaxy of phototubes, the whole residing 2 km beneath the Earth's surface in an Ontario mine, the better to filter out distracting background interactions.

Last year SNO reported first results based on reactions in which a solar neutrino enters the detector and either (1) glances off an electron in one of the water molecules (this so-called elastic scattering (ES) is only poorly sensitive to muon and tau neutrinos) or (2) combines with the deuteron to create an electron and two protons, a reaction referred to as a "charged current" (CC) interaction since it is propagated by the charged W boson. The SNO data, when supplemented with ES data from the Super Kamiokande experiment in Japan, provided preliminary evidence a year ago for the neutrino-oscillation solution for the solar neutrino problem. ...

 The new findings update last year's CC and ES data and introduce, for the first time, evidence deriving from a reaction in which the incoming neutrino retains its identity but the deuteron (D) is sundered into a proton and neutron; this is why SNO went to such trouble and expense of using D2O-for the weakly bound neutron inside each D. This interaction, called a neutral-current (NC) reaction because the operative nuclear voltage spreads in the form of a neutral Z boson, is fully egalitarian when it comes to neutrino scattering; unlike last year's ES data, the NC reaction allows e-nu's, mu-nu's, and tau-nu's to scatter on an equal footing.

The upshot: all the nu's from the sun are directly accounted for. The missing nu-e flux shows up as an observable mu-nu and tau-nu flux. This conclusion is established with a statistical surety of 5.3 standard deviations, compared to the less robust 3.3 of a year ago. The measured e-nu flux (in units of one million per cm2 per second) is 1.7 while that for the mu-nu and tau-nu combined is 3.4. (When one includes all the other types of neutrinos, the flux from the sun is billions/cm^2/sec.)

Even the issue of how the neutrino changes from one flavor to another can be addressed by viewing the day-night asymmetry of neutrino flux. When the whole of the earth is between the sun and the detector (night viewing) the oscillation process, which depends on a density of matter through which the nu proceeds, should be speeded up. This type of measurement will also contribute to the eventual study of neutrino mass. An experiment like SNO can measure not mass but the square of the mass difference between nu species. ...".


Here are some excerpts from some of the relevant 2001 and2002 papers, and some other further material:


The SNO CC papernucl-ex/0106015reports experimental results consistent with the Solar favoredsolution:

"... Solar neutrinos from the decay of 8B have been detected atthe Sudbury Neutrino Observatory (SNO) via the charged current(CC) reaction on deuterium and by the elastic scattering (ES) ofelectrons. The CC reaction is sensitive exclusively to nu_e's,while the ES reaction also has a small sensitivity to nu_mu's andnu_tau's. The flux of nu_e's from 8B decay measured by theCC reaction rate is ...[

about 1.75 x 10^6 cm^(-2) s^(-1)

]... Assuming no flavor transformation, the flux inferred fromthe ES reaction rate is ...[ about 2.39 x 10^6cm^(-2) s^(-1) ]... SNO's ES rate measurement is consistent withthe precision measurement by the Super-KamiokandeCollaboration of the 8B flux using the same ES reaction...[ about 2.32 x 10^6 cm^(-2) s^(-1) ]... Comparisonof ...[ the SNO CC nu_e flux ]... to the Super-KamiokandeCollaboration's precision value of ...[ the ES nu_e flux]...yields a 3.3 sigma˙difference, providing evidence thatthere is a non-electron flavor active neutrino component in thesolar flux. The total flux of active 8B neutrinos is thusdetermined to be ...[ about 5.44 x 10^6 cm^(-2) s^(-1) ] close agreement with the predictions of solar models....".


The SNO Day-Night paper nucl-ex/0204009says:

"... The Sudbury Neutrino Observatory (SNO) has measured day andnight solar neutrino energy spectra and rates. For charged currentevents, assuming an undistorted 8B spectrum, the night minus day rateis 14.0% +/- 6.3% +1.5 - 1.4% of the average rate. If the total fluxof active neutrinos is additionally constrained to have no asymmetry,the nu_e asymmetry is found to be 7.0% +/- 4.9% +1.3 - 1.2%.A globalsolar neutrino analysis in terms of matter-enhanced oscillations oftwo active flavors strongly favors the Large Mixing Angle (LMA)solution. ...".


The SNO NC paper nucl-ex/0204008says:

"... Observations of neutral-current neutrino interactions ondeuterium in the Sudbury Neutrino Observatory are reported. Using theneutral current, elastic scattering, and charged current reactionsand assuming the standard 8B shape, the nu_e component of the 8Bsolar flux is

PHI_e = 1.76 + 0.05 - 0.05 (stat.) + 0.09 - 0.09 (syst.) x10^6 cm^(-2) s^(-1)

for a kinetic energy threshold of 5 MeV. The non-nu_e componentis

PHI_mu tau = 3.41 + 0.45 - 0.45 (stat.) + 0.48 - 0.45 (syst.)x 10^6 cm^(-2) s^(-1),

5.3 sigma greater than zero, providing strong evidence for solarnu_e flavor transformation. The total flux measured with the NCreaction is

PHI_NC = 5.09 + 0.44 - 0.043 (stat.) + 0.46 - 0.43(syst.) x 10^6 cm^(-2) s^(-1),

consistent with solar models. ...



According to the 2000Particle DataBook: "... Now, as we have seen, the dM^2 values required toexplain the atmospheric, solar, and LSND oscillations are of threedifferent orders of magnitude. ... Hence, to explain all three of thereported neutrino oscillations, one must introduce a fourth neutrino.Since this neutrino is known to make no contribution to the width ofthe Z0, it must be a neutrino which does not participate in thenormal weak interactions - a "sterile" neutrino. ...".

According to Gary Feldman, the Solarfavored solution disfavors the existence of a sterile neutrino at 95%CL However, Barger, Mafatia, and Whisenant in hep-ex/0106207say: "...We perform a model-independent analysis of solar neutrinoflux rates including the recent charged-current measurement at theSudbury Neutrino Observatory (SNO). We derive a universal sum ruleinvolving SNO and SuperKamiokande rates, and show that the [forthcoming ] SNO neutral-current measurement can only fix thefraction of solar nu_e oscillating to sterile neutrinos if the 8Bneutrino flux normalization is assumed to be known. Imposingadiabatic constraints do give an unique solution for thesterileneutrino fraction, but the uncertainties are large. ... Wediscuss the additional measurements that are needed to determine, ina model-independent way, the oscillation probabilities and thefraction of solar nu_e that may be oscillating to sterile neutrinos.What is needed is a measurement of neutrino-nucleon NC scatteringfor the intermediate-energy neutrinos, which, like the SNO NCmeasurement of high-energy neutrinos, has equal strength for allactive neutrinos. In order to experimentally determine the flux ofintermediate energy neutrinos, a low energy neutrino flux measurementinvolving a neutral currents is needed. ...".

In hep-ph/0108191,M.V. Garzelli and C. Giunti take a

Bayesian View Of Solar Neutrino Oscillations.

They say: "... We show that the Bayesian analysis of solarneutrino data does not suffer any problem from the inclusion of thenumerous bins of the CHOOZ and Super-Kamiokande electron energyspectra and allows to reach the same conclusions on the favored typeof neutrino transitions and on the determination of the most favoredvalues of the oscillation parameters in both the Rates and GlobalAnalysis.

Our Bayesian analysis shows that nu_e -> nu_stransitions are strongly disfavored with respect to nu_e ->nu_mu,tau transitions.

... We present also the results of a standard least-squaresanalysis of solar neutrino data and we show that the standardgoodness of fit test is not able to rejects pure nu_e -> nu_stransitions. ...".

For more details about Bayesian Statistics, see, for example,the website of The InternationalSociety for Bayesian Analysis.


 Solar Neutrinos

Using the Standard Solar Model, in which temperature, density, andpressure decrease monotonically from the center of the Sun to itsphotosphere/surface, John N.Bahcall, in his book Neutrino Astrophysics (Cambridge 1989), sayswith respect to his figure 8.1

"... The region in which the pp flux is produced is very similarto the region in which most of the photon luminosity is produced ...0.09 Rs [where Rs is the solar radius] ...the 8B productionis peaked at much smaller radii, 0.05 Rs ...".

A recent detailed description of solar models is given inastro-ph/0010346,by Bahcall, Pinsonneault, and Basu.

In his paper An Introduction to Solar Neutrino Research (hep-ph/9711358),Bahcall says: "... Figure 1...compares the measured and the calculated event rates in the fourpioneering experiments, revealing three discrepancies between theexperimental results and the expectations based upon the combinedstandard model. ...

... The first solar neutrino experiment to be performed was thechlorine radiochemical experiment, which detects electron neutrinosthat are more energetic than 0:81 MeV. After more than 25 years ofthe operation of this experiment, the measured event rate is a factor3.6 less than is predicted by the most detailed theoreticalcalculations ... A SNU is 10^(-36) interactions per target atom persecond. ...

... The second solar neutrino problem results from a comparison ofthe measured event rates in the chlorine experiment and in theJapanese water Cerenkov experiment, Kamiokande. ... The waterexperiment detects higher-energy neutrinos, those with energies above7 MeV ... According to the standard solar model, 8B beta decay is theonly important source of these higher-energy neutrinos. ... we cancalculate the rate in the chlorine experiment that is produced by the8B neutrinos observed in the Kamiokande experiment (above 7 MeV).This partial ( 8B) rate in the chlorine experiment is 3.2 +/- 0.45SNU, which exceeds the total observed chlorine rate of 2.55 +/- 0.25SNU. ... The inference that is most often made from this comparisonis that the energy spectrum of 8B neutrinos is changed from thestandard shape ...

... The results of the gallium experiments, GALLEX and SAGE,constitute the third solar neutrino problem. ... The seemingexclusion of everything but pp neutrinos in the gallium experimentsis the "third" solar neutrino problem. ... ".

In hep-ph/0107022,Raychaudhuri and Sil say: "... Table 3: The ratio of the observedsolar neutrino rates to the corresponding BBP2000 SSM predictionsused in this analysis. ... The Gallium rate is the weighted averageof the Gallex, SAGE, and GNO results.

... SNO will soon start taking data for acalorimetric measurement of neutral current (NC) deuterondisintegration ( nu + d -> nu + p + n ) to which all activeneutrinos contribute equally. ...".

The SNO NC results may be important indetermining whether ( as is indicated by SNO CCresults ) the Sun follows the Standard Solar Model. JohnN. Bahcall, in his book Neutrino Astrophysics (Cambridge 1989),says (at pages 240-242, 405-406):

"... Neutral currents distinguish between explanations of the solar neutrino problem that involve revised astrophysics and explanations that involve neutrino oscillations.
  • For astrophysical explanations, (i.e., nonstandard solar models) the flux of neutrinos is reduced relative to the standard solar model while the energy spectrum remains the same.
  • For explanations involving oscillations, the total flux of neutrinos is the same as in the standard solar model although their flavor distribution is radically different.

Neutral currents are flavor blind. In the standard electroweak theory, neutrinos of different flavors have the same neutral-current interactions. ... The Sudbury solar neutrino collaboration has porposed observing the neutral-current disintegration of deuterium by detecting the gamma rays emitted when the produced neutrons are captured in a doped impurity ...

The cross section for ... the neutral-current reaction ... is independent of the flavor of the incident neutrino. ... the SNO collaboration has had to devise clever methods for ensuring that only a small number of background events can produce neutrons. ... the neutral-current reaction is particularly important for resolving the solar neutrino problem because it gives a flavor-blind measure of the total neutrino flux, providing a test of solar models even if neutrino oscillation occur...".


"... Do we really understand the solar interior wellenough to predict neutrino fluxes accurately? ..."

is a question asked by Carroll and Ostlie in their book AnIntroduction to Modern Astrophysics (Addison-Wesley 1996).

The SNO papersnucl-ex/016015and nucl-ex/0204008report experimental resultswhich, along with those of Super-Kamiokande,are consistent with the existence of neutrinooscillations and with the Standard Solar Model.

However, it is also possible that Non-Standard Solar Modelscoupled with Non-Standard (or No) Neutrino Oscillations may also beconsistent with experimental results.

Therfore, it may be useful to describe some possible StandardSolar Model modifications with respect to InternalSolar Temperatures and Nuclear ReactionRates.

Internal Solar Temperatures and theStandard Solar model.

Carroll and Ostlie say "... if the temperature of the solarinterior was approximately 10% lower than theory predicts, the numberof solar neutrinos emitted from the core would come into agreementwith observation. ...". Their figure 11.4


shows how temperature (in units of 10^6 degees K, or about 200 eV)varies from the center of the Sun to its photosphere/surface.

I have added a blue line at about 10x 10^6 degrees K, or about 2 keV, to suggest the possibility that

the model of B. V.Vasiliev,

in which the interior of the Sun (as well as other celestialbodies) has the structure of "... an ion lattice in a metal. Thislattice is deformed by gravity and then the electron gas adapts itsdensity to this deformation. ... the electrons form the degeneratedFermi gas. ... the action of gravity on matter is compensated by theelectric force and the gradient of pressure is absent. ... thedensity of a substance inside a star is constant. ...", may solve thesolar neutrino problem.

About helioseismological observations of the Sun, Carrolland Ostlie say: "... pressure provides the restoring force for... nonradial oscillations ... called p-modes. .... gravity is thesource of the restoring force for another class of nonradialoscillations called g-modes ... The oscillations observed in the Sunfall into two general categories:

... p- modes are concentrated below the photosphere within theSun's convection zone ... g-modes are found ... below the convectionzone. ... a definite identification of the g-modes has not been made,and even the existence of the most prominent g-mode is doubted bysome astronomers. Its 100-minute period is exactly one-ninth of asolar day, raising the suspicion that it is an obserational artifactproduced by the Sun's influence on Earth's atmosphere. ...

... it appears that the convection zone ... occupies roughly theouter 30% [by radius] of the Sun's interior. ... it isstrongly suspected that the p-modes are diven by tapping into theturbulent energy of the convection zone itself, where the p-modes areconfined. ...".

William J. Kaufmann III, in his book Universe (fourth edition,Freeman 1994) shows in Figure 18-24

the observed rotation rates of the solar interior, saying: "...the surface rotation pattern, which varies from 23 days at theequator to 35 days near the poles, persists throughout the Sun'sconvective envelope. The Sun's radiative core seems to rotate likea rigid body. ...".


Nuclear Reaction Rates and the StandardSolar Model

are discussed by Fukasakuand Fujita, who have shown that "...the solar neutrino data(KAMIOKANDE experiment) can be well described by the [StandardSolar Model] with careful employment of nuclear data ... The mainpoint is that the sinple-minded product ansatz of Coulomb plusnuclear parts should have a few percent uncertainties which inducethe large reduction of the neutrino flux from 8B. Also, if theelectron capture of 7Be inside the sun is suppressed, then the GALLEXexperiment can be understood by the SSM calculation."

Fukasaku andFujita note that Darand Shaviv have made "... new calculation ... which shows verysimilar results ..."



VEP, qVEP,and ssVEP Models of NeutrinoOscillation

IgorKulikov has shown that the GravitationalEquivalence Principle can be violated in quantum field theories atfinite temperature.

In the D4-D5-E6-E7-E8 VoDou Physicsmodel, all 3 flavors of neutrinoare massless at tree level. ( The tree-level masslesscharacter of the neutrino means that its ComptonRadius Vortex is probably effectively aslarge as our universe. ) Therefore, at tree level, theD4-D5-E6-E7-E8 VoDou Physics model hasno neutrino mixing due to the Kobayashi-Maskawa mass-related matrix,and so has no tree-level neutrino mixing by the MSW mechanism.

However, the tree-level massless neutrinos of the D4-D5-E6-E7-E8VoDou Physics model could be mixed by Violation of thegravitational Equivalence Principle, in two ways:

As of June 2001, VEP is consistentwith Solar Neutrino Experiments but may not be consistent withhigh-energy Atmospheric Neutrino Experiments,

while qVEP for massless neutrinos isconsistent with Solar Neutrino Experiments and may also be consistentwith Atmospheric Neutrino Experiments.




Mureika, inhep-ph/9612391, says, about 3-flavor neutrino oscillationsinduced by VEP: "... Neutrino oscillations induced by aflavor-dependent violation of the Einstein Equivalence Principle(VEP) have been recently considered as a suitable explanation ofthe solar electron-neutrino deficiency. ...

At heart, the MSW and VEP mechanisms are the same. Bothrely on the existence of two distinct eigenstates of the neutrinos.In the former [ MSW ] case, there are mass andelectroweak eigenstates ... For the latter [ VEP ], the mass eigenstate is replaced with a gravitational one ... asthe VEP neutrinos are considered massless ...

.... The MSW mechanism is ...[ more often ]... seriouslyconsidered ... due to the basic underlying hypothesis of VEP,which requires a generation-dependent violation of theEinstein Equivalence Principle (i.e. flavor-dependent coupling tothe background gravitational potential ) in the neutrino sector,and tends to give most physicists a headache. ...

We investigate the differences which arise by consideringthree-flavor VEP neutrinos oscillating against fixed backgroundpotentials, and against the radially-dependent solar potential. Thiscan help determine the sensitivity of the gravitationally-inducedoscillations to both constancy and size (order of magnitude) of Phi....

... While most papers on VEP tend to agree on the form of theoscillation mechanism, there is some debate as to which backgroundpotential is at work. Most works have assumed that neutrinos feelthe potentialdifference of the sun from their creation points, to their exit onthe surface, which ranges between [about] (-7,-2) x10^(-6). Alternatively, ...[Halprin,Leung, and Pantaleone suggest]... that the neutrinos feelinstead the constant background potential of the Local Supercluster(Great Attractor), which in the neighborhood of the solar system is |PHIsc | = 3 x 10^(-5). This overpowers the surface potential of theSun by almost a factor of 10. Results from the COBE data suggest thatthe dominant local background potential is very close to this value.Furthermore, ... one should be careful to consider the overallpotential contribution of the Universe. ... The contributors of the...[ local background ]... potentials are, respectively, theEarth, Sun, Milky Way, Local Supercluster, the Universe, and anarbitrary constant C. ... The constant C comes from an arbitrarycosmological model ... "we cannot 'step outside the universe' to[measure the actural value of C] ... For this work, we willassume the value of C = 10^(-3). ...".

A more recent summary of the status of VEP neutrino oscillationis given by C.N. Leung, in hep-ph/0002073: "... Suppose the neutrinosviolate the principle of equivalence. A consequence will be thatdifferent neutrino types (i.e., different neutrino gravitationaleigenstates) will couple to gravity with a slightly differentstrength. Suppose the neutrino weak interaction eigenstates are notthe same as their gravitational interaction eigenstates, but arelinear superpositions of them. Then, as a neutrino of definitef;avour (e.g., nu_mu) travels through a gravitational field, thegravitational components of the flavour neutrino will developdifferent dynamical phases, which will result in [ VEP ]neutrino flavour oscillations. In other words, neutrino oscillationexperiments provide a laboratory to test the equivalence principlefor neutrinos ... It can be concluded ... that

The dependence of VEP oscillations on Phi(r) is a consequence ofviolating the principle of equivalence. It is also a source ofuncertainty. ... our local value of | Phi | spans from 6 x 10^(-10)for the Earth's gravitational potential to about 3 x 10^(-5) for thegravitational potential due to our local supercluster. Its value maybe even bigger if the contributions from more distant sources can beestimated. Since Phi appears to be dominated by distant sources, itsvalue varies little over a typical neutrino path length forearthbound experiments and may thus be treated as a constant. ... Incontrast to the case of flavour oscillations induced by a neutrinomass difference where the oscillation length increases with theneutrino's energy, VEP oscillations are characterized by anoscillation length that decreases with increasing neutrino energy... We now confront VEP ... with ... atmospheric and solarneutrino experiments. Positive evidence for nu_mu -> nu_tautransitions has been established by the Super-Kamiokande (SK)Collaboration ... analyses that included ... upward-going muonevents, which corresponded to atmospheric neutrinos with higherenergies: E up to about 10^3 GeV, found that VEP and VLI oscillationswere not compatible with the data. In particular, Fogliet al, in hep-ph/9904248, did a very thorough analysis of the VEP... mechanisms with the SK data. They allowed a power-law energydependence for the oscillation length: lambda proportional to E^(-n), and found that, at 90% C.L., n = 0.9 +/- 0.4, which identified themass mixing mechanism as the mechanism for the observed nu_mu-> nu_tau transitions. VEP ... oscillations, which correspond to n= 1, are clearly incompatible with the data. ... As constrainingas the SK atmospheric neutrino data are, they only tell us about VEP... in the nu_mu -> nu_tau sector. To study flavour transitionsinvolving nu_e, we turn to solar neutrino experiments. ... usingthe improved Standard Solar Model predictions ... and treating the 8Bneutrino flux as a free parameter, arecent study [ by Gago, Nunokawa and Zukanovich Funchal ],hep-ph/9909250, finds that long-wavelength VEPoscillations can ... account for the solar neutrino deficit....".

Gago, Nunokawaand Zukanovich Funchal, in hep-ph/9909250, say: "... We havereexamined the possibility of explaining the solar neutrino problemthrough long-wavelength neutrino oscillations induced by a tinybreakdown of the weak equivalence principle of general relativity. Wefound that such gravitationally induced oscillations can provide aviable solution to the solar neutrino problem. ...".


Adunas,Rodriguez-Milla, and Ahluwalia, in gr-qc/0006021, say: "... ifthe inertial and gravitational masses are considered as operationallydistinct objects, quantum gravity carries an inherent quantum inducedviolation of the equivalence principle (qVEP) ... This view hasthe further support in that the very operational procedures thatdefine inertial and gravitational masses are profoundly different,and it is more so in the quantum context. The essential physical ideais that if any one of the two quantities A and B carries an intrinsicquantum uncertainty Delta, then from an operational point of view theequality of A and B cannot be claimed beyond the fractional accuracyDelta / ( (A + B) / 2 ). We shall consider this general statement inthe context of the equality of the inertial and gravitational massesto establish a quantum-induced violation of the equivalence principle(qVEP) ... We now immediately note that the very quantum constructthat defines the flavor eigenstates does not allow them to carry adefinite mass. Therefore, within the stated framework, the equalityof the inertial and gravitational masses looses any operationalmeaning beyond the flavor-dependent fractional accuracy ... We nowexplore how the above obtained qVEP can be studied in some realisticexperimental settings.

VEP,qVEP , ssVEP, andExperiments:

As stated by Adunas,Rodriguez-Milla, and Ahluwalia, in gr-qc/0006021, in terms ofenergy power-law E^n dependence,

In light of that, as of June 2001:

VEP is consistent with Solar Neutrino Experiments but may not beconsistent with high-energy Atmospheric Neutrino Experiments. Inparticular, the analysis of high-energy (10-10^3 GeV) upwardthrough-going muon UPmu events at SuperKamiokande of Fogliet al, in hep-ph/9904248, concludes that the VEP model (redline)

is inconsistent with the UPmu data (note thatFogli et al use a sign convention for the energy depenedence powerlaw that is the opposite of that used by Adunas et al, so that n = 1on the Fogli et al figure corresponds to n = -1 in the Adunas et alpaper).

Fogli et al say that the MSW neutrino oscillation mechanism (redline)

is consistent with the UPmu data (note that Fogliet al use a sign convention for the energy depenedence power law thatis the opposite of that used by Adunas et al, so that n = -1 on theFogli et al figure corresponds to n = 1 in the Adunas et alpaper).

Fogli et al also dislikes the fit of the red line for n = 0 andqVEP

to the UPmu data, but in this case I do not agree with Fogli etal. Their MSW straight line does not follow the general curve of theUPmu data as well as the qVEP curve. Although the qVEP curve issomewhat below the UPmu data, while the MSW line is less below it, Ithink that the better shape-fit of the qVEP curve is important. Inother words, I think that the fancy statistical best fit techniquesused by Fogli et al are, in this case, misleading. Perhaps more dataand/or better theoretical understanding of the qVEP model mightinidicate that qVEP is indeed consistent with the SuperKamiokandeUPmu data.


As if the situation were not complicated enough, there is also asuperstring-motivated VEP model, ssVEP, described by Halprinand Leung, in hep-ph/9707407, where they say: "... neutrinooscillations that results from a model of equivalenceprinciple violation suggested recently by Damour and Polyakov asa plausible consequence of string theory ... will take placethrough interaction with a long range scalar field of gravitationalorigin even if the neutrinos are degenerate in mass. The energydependence of the oscillation length is identical to that in theconventional mass mixing mechanism. ... We therefore concludethere is the distinct possibility that the solar neutrino deficit maybe telling us about a nonuniversal scalar gravitational interactionrather than the existence of a neutrino mass difference. Thisoscillation mechanism is phenomenologically distinguished from theconventional mass mixing mechanism by providing a rationale for thepossibility that effective neutrino mass differences pertinent tosolar neutrinos are small while true neutrino masses are orders ofmagnitude larger - with degeneracy protected by a family symmetry....".

As Fogli et al,in hep-ph/9904248, say: "... An alternative, string-inspired VEPmechanism, leading to an energy exponent n = -1 rather than n = 1,has been recently considered in [ Halprinand Leung, in hep-ph/9707407 ]. Its phenomenology would beindistinguishable from the standard case, as far as nu_mu ->nu_tau oscillations are involved. ...". (note thatFogli et al use a sign convention for the energy depenedence powerlaw that is the opposite of that used by Adunas et al, so that n = -1as used by Fogli et al corresponds to the MSW mechanism).

Since, as Fogli et al say, the SuperKamiokande data cannotdistinguish between MSW and ssVEP, some other experiments will beneeded to try to do so. Halprin and Leung suggest experiments to tryto make direct measurements of neutrino masses and to see if thereexists neutrinoless double beta decay.



My skepticism about the conclusion of Fogliet al, in hep-ph/9904248, that the upward through-going muonsUPmu at SuperKamiokande exclude any energy power-law other than thatof the MSW mechanism (and particularly their exclusion of VEP andqVEP), is based in part on the paper hep-ex/9806038entitled Measurement of the flux and zenith-angle distribution ofupward through-going muons in Kamiokande II+III, which papercontains the figure

in which the upper (solid) line corresponds to expectation of nullNeutrino Oscillation, while the lower (dashed) line corresponds toexpectation of nu_mu <-> nu_tau Neutrino Oscillation. To my(statistically unsophisticated) eye, the two lines don't look thatmuch different.


Further, I am not even sure that we now have a full understandingof muon physics in the relevant energy range of about 10 - 10^3 GeV.For example, consider thepaper hep-ph/0106199, Attempts at Explaining the NuTeV Observation ofDi-Muon Events, by Dedes, Dreiner, and Richardson, in which theysay: "... The NuTeV Collaboration has observed an excess in theirdi-muon channel ... The NuTeV Collaboration has searched forlong-lived neutral particles ... with mass ...[ at least ]...2.2 GeV and small interaction rates with ordinary matter. They lookfor the decay of the neutral particles in a detector which is 1.4 kmdownstream from the production point. They observe 3 mu-mu eventswhere they only expect to see a background of 0.069 +/- 0.010 events.The probability that this is a fluctuation of this specific channelis about 8 x 10^(-5), which corresponds to about 4.6 sigma Theprobability for a fluctuation of this magnitude into any of thedi-lepton channels is about 3 sigma ...".

I am not going to try to explain the NuTeV results, just tosay that I don't fully understand them, and I am not sure thatanybody else really does, either.


Neutrino Oscillation Experiments:



Experimentalsearches for 3-flavor NeutrinoOscillations have been conducted by the KARMEN experiment at theneutron spallation facility ISIS of the Rutherford AppletonLaboratory. 800 MeV protons hit a beam-stop and produce neutrons andpions. Negative pions are absorbed in the heavy target nuclei,whereas positive pions decay to produce muons plus mu-neutrinos, andthe muons decay to produce positrons plus mu-antineutrinos pluse-neutrinos. January 1998 KARMEN results show no evidence ofoscillation.


According to Gary Feldman( slide10 of lecture slides for the August 2000 SLAC Summer Institute )the LSND experimentshows positive results for "...

nu(mu)bar -> nu(e)bar oscillation with dm^2 > 0.2(eV/c^2)^2 ...".

LSND andMiniBOONE are discussed in the paper AN EXPERIMENTALIST'S VIEW OFNEUTRINO OSCILLATIONS by Antonella de Santo, hep-ex/0106089,which paper says, in part:

"... The KARMEN detector ...[is]... located 17:5 mdownstream the neutrino source (about half the LSND baseline) ... theKARMEN2 result is compatible with the no-oscillation hypothesis ...the skeptical reader may find the LSND and KARMEN2 results quitecontradictory. However it has to be stressed that the two experimentshave sensitivities which peak at slightly different values ofdelta_m^2. Moreover, before any definitive conclusion can be drawnfrom the various results, a combined analysis of all experimentaldata should be performed. This has been tried for LSND and KARMEN,but only on subset of the collected data. In conclusion it is fair tosay that no definitive answer has yet been given to the LSND puzzleand that more experimental data is needed. ... The MiniBOONEexperiment, which is now being built at Fermilab, will address theLSND effect by searching for nu_mu -> nu_e oscillations in thesame region of parameter space. ... If oscillations occur withparameters compatible with LSND solutions, after one year of runningMiniBOONE should observe an excess of about 1500 nu_e events abovebackground expectations, with a significance of about 8 to 10 sigma.On the other hand, in case of null result, after two years of runningMiniBOONE should be able to completely exclude the entire LSNDallowed region at 90% C.L. ... While both the solar and theatmospheric neutrino deficit have been confirmed by more than oneexperiment, using different experimental techniques, no otherexperiment has confirmed LSND observations. For this reason, givenalso the enormous impact that accepting the LSND result would have onour understanding of sub-nuclear phenomena, with the introduction ofa light sterile neutrino, there is a tendency in the scientificcommunity to adopt a "conservative" attitude and to exclude LSND frommost phenomenogical models and fits to neutrino oscillationparameters, until more solid experimental data become available....".


According to an article by Phillip F.Schewe and Ben Stein in PhysicsNew Update for 5 June 1998, the Bulletin of Physics News ofThe American Institute of Physics:

"NEUTRINO OSCILLATION HAS BEEN DEMONSTRATED at theSuper-Kamiokande lab in Japan to a higher degree of certaintythan in previous experiments. Neutrinos, weakly interactingelementary particles only detected for the first time in 1956, arethought by some theorists to reside in a kind of schizoid existence;that is, a neutrino would regularly transform (or oscillate) amongseveral alternative neutrino states, each having a slightly differentmass. Such a theory would help to explain the apparent shortfall ofneutrinos coming from the Sun. The oscillation proposition has beentested using four neutrino sources: the Sun, Earth's atmosphere,reactors, and particle accelerators. Some tests find tentative butambiguous evidence for oscillation. Today, at the Neutrino '98conference in Takayama, Japan, theSuper-Kamiokande collaboration (comprising 100 scientists from 23institutions in Japan and the US) is announcing the most exactingevidence yet for neutrino oscillation. They study neutrinos made whencosmic rays from outer space strike the upper atmosphere. Someneutrinos, those made overhead above Japan, travel about 20 km or sobefore entering the underground detector. Other neutrinos, those madein the atmosphere on the far side of the globe, have a travel path of12,700 km into the detector. In either case, they create, among otherthings, a high energy electron or muon, which in turn emits atelltale cone of light (Cerenkov radiation) observed by an array ofthousands of photodetectors mounted in a tank filled with pure water.Sorting events by electron neutrino or muon neutrino, by high energyor lower energy, and by zenith angle (overhead approach or throughthe Earth), statistical evidence for oscillation becomes evident. A1-GeV muon neutrino seems to oscillate every few hundred miles. Fouryears ago, the same group, using a smaller detector, reportedpreliminary evidence on the basis of 200 events (PhysicsToday, Oct 1994). The new report is based on several thousands ofevents, and provides an approximate mass difference (the test cannotrender any neutrino species' mass directly) of about 0.07 eV. Becausethey are so numerous in the universe, neutrinos, with even a smallmass, might play an important role in the formation of galaxies. (See,and Physics News,NeutrinoOscillation)"


With respect to the Super-Kamiokande demonstration of neutrinooscillation, the New York Times gave it major coverage in a Friday, 5June 1998, article by Malcolm W. Browne entitled

Elusive Subatomic Particle Has Mass, ScientistsReport

The article stated "... The Super-Kamiokande experiments suggestthat the difference between the masses of muon nutrinos and othertypes of neutrinos is only about 0.07 electron volts (a measure ofparticle mass). This does not yield a value of the masses themselves,only of the difference between those of muon neutrinos and othertypes. Although the mass of the neutrino of any flavor must be small,Totsuka [Dr. Yoji Totsuka, leader of the coalition and directorof the Kamioka Neutrino Observatory] said, it may be severalelectron-volts, and if so, the overall gravitational effect on theuniverse would perhaps be significant. ..."

The article did nto explicitly discuss theVEP neutrino oscillation model in which neutrinos can oscillate andstill remain massless.

It is interesting to note that

there was no actual measurement of neutrino mass, andthat the mass difference that might be associated in some models withthe observed oscillation was only 0.07 electron volts, far smallerthan the "several electon-volts" that might "perhaps" have a"significant" "overall gravitational effect on theuniverse".

Why was the headline and coverage so sensationalistic aboutneutrino mass? The next day, Saturday, 6 June 1998, the New YorkTimes had another major article by Malcolm W. Browne with thisheadline:

Finances Worry Neutrino Researchers

According to David Kestenbaum's article in Science(281 11 Sep 1998 pp. 1594 - 1595), "... Super-Kamiokande'smeasurement pins down ... the difference in mass between[electron-type e-neutrinos and muon-type mu-neutrinos] and soindicates only that one of them must have a mass of at least 0.07electron volts ... earlier experiments had set an upper limit,showing that the electron neutrino, for instance, must have a massless than 1/30,000 of the electron's. ...". Since the electron massis about 0.511 MeV = 511 keV = 511,000 eV, the upper limit of thee-neutrino mass is 511,000 / 30,000 = 17 eV.

The Super-Kamiokande results are consistent with the e-neutrinobeing massless and with the mu-neutrino having a mass of 0.07 eV, andalso consistent with the VEP neutrino oscillationmodel in which neutrinos can oscillate and still remainmassless.


In hep-ex/0107009,Final Results from the Palo Verde Neutrino OscillationExperiment, Boehm et. al. say: "... In the experiment, thenubar_e interaction rate has been measured at a distance of 750and 890 m from the reactors of the Palo Verde Nuclear GeneratingStation ... There is no evidence for neutrino oscillation and themode nubar_e -> nubar_x was excluded at 90% CL for delta_m^2> 1.1 x 10^(-3) eV^2at full mixing, and sin^2( 2 theta ) > 0.17at large delta_m^2 ... Our measurements, along with those reported byChooz and Super-Kamiokande, excludes two family nu_mu -> nu_emixing as being responsible for the atmospheric neutrino anomalyreported by Kamiokande ... recent data from Super-Kamiokande favorthe nu_mu -> nu_tau oscillation channel over nu_mu -> nu_e...".


In hep-ex/0106102,the NOMAD Collaboration says: "... The NOMAD experiment wasdesigned in 1991 to search for nu_tau appearance from neutrinooscillations in the CERN wide-band neutrino beam produced by the450 GeV proton synchrotron. ... the experiment is sensitiveto differences of mass squares delta_m^2 > 1 eV^2 ... Theexperiment was motivated by theoretical arguments suggesting that expects a nu_mu mass of about 3 x 10^(-3) eV and a nu_tau mass ofabout 1 eV, or higher. Furthermore, in analogy with quark mixing,neutrino mixing angles were expected to be small. ... The analysisof the full NOMAD data sample gives no evidence for nu_tauappearance. ...".




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