Neutrino substructure from flavor oscillations

8/2/2019 Neutrino substructure from flavor oscillations

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Questions On the Usual

Neutrino Flavor Oscillation

Theory*

by John Michael Williams

[email protected]

Wilsonville, Oregon

Copyright 2004, John Michael Williams.All Rights Reserved.

* Some of this paper was posted at CERN as preprintEXT-2002-042, which expanded upon a

poster session given at the SLAC 29th Summer Institute on Particle Physics, "Exploring

Electroweak Symmetry Breaking", at the Stanford Linear Accelerator Center, Stanford,

California, in August 2001.


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Abstract

We question the theoretical cohesion of the usual neutrino oscillation theory.

1. Is it valid to rest a kinematic theory of free particles on the CKM quark mixing matrix, which

explicitly is formulated for bound particles? Our answer is no.

2. Does the usual theory allow a mass hierarchy among neutrino flavors, like that of the charged

leptons? Our answer is no.

3. Can quantum mechanics be applied in the form of a which-path model to justify the usual

theory? Our answer is no.

4. Can a quantum-mechanical wave packet propagation model be useful in the usual theory?

Our answer is no.

5. In the usual one-dimensional theory, is Heisenberg uncertainty in mass state enough to permit a

detectible oscillation in flavor state? Our answer is no.

6. Are quantum properties of the neutrino detector relevant to neutrino oscillation? Our answeris no.

7. Is it valid to apply Heisenberg's uncertainty principle in more than one dimension? Our answer

isyes, and that this implies incoherent oscillations.

8. Can the usual theory predict reasonable mass eigenstate masses? Our answer is, no: the

predicted masses are too great.

9. What are the properties of a physically correct neutrino flavor oscillation theory? Our answer

is that, if based on a CKM analogy, probably it should postulate neutrino substructure.

We conclude by pointing out that neutrino substructure justifies not only a physically correct

PMNS theory, with neutrino mass hierarchy by flavor, but it also justifies alternative theoriessimilar to an atomic theory.

PACS Codes:

12.15.Ff Quark and lepton masses and mixing

13.15.+g Neutrino interactions

14.60.Pq Neutrino mass and mixing


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Nevertheless, I'm still far away from claiming the physical validity of the

equations I have derived here. The reason for this is that I have not

succeeded yet in deriving equations of motion for particles.

-- A. Einstein (1930).

Introduction

It is the postulated free propagation of neutrinos (mass eigenstates) that leads to several

problems in the usual neutrino oscillation theory. Therefore, we concentrate our analysis on the

kinematics of this propagation.

When invoking the velocity v in this paper, we let v represent solely a wave or particle

relationship between time and a geometrical 3-space. We assume only that v implies an

elementary particle or mass eigenstate with an initial creation point and a final annihilation point

which exist in Minkowski 4-space and are well defined after-the-fact of propagation.

We define total energy in terms ofv as E mc= 2

, and momentum as p mv= , in someinertial frame. For massless particles, we require E= and p c= . These are not unusual

definitions, and we never apply them off-shell.

Young's Which-Path Experiment

We first present Young's experiment for photons. However, instead of being concerned with

the superposition pattern, we look at resolution of the distance -- the phase difference -- between

the two slit sources. We pose the question, What is the smallest slit separation detectible by

particle phase as a superposed, as opposed to a single-slit, pattern at some point on the screen?

Young's Experiment for a Photon

Assume a coherent light source such as a laser, and two slits in an opaque screen, or two

spatially separate (Schoen and Beige 2001) coherent light sources. The light, of wavelength ,falls on a distant screen, and an interference pattern appears. The arrangment is shown in Fig. 1:

Figure 1. Young's experiment for a photon.


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The photons originate at sources (slits) separated by distance x i , i for initial interaction.

We set a criterion xf ,fforfinal interaction, for distinguishing one slit versus two by a final

interaction at some given point on the screen.

For simplicity, consider a point directly in front of the upper slit in Fig. 1. We have two

distances rand r to distinguish. We assume somewhat arbitrarily that interference is well-defined if the phase difference xf 4 . Define r r r ; then, the ultimate limit will be

given by Heisenberg's principle in the form,

p r 2 . (1)

This Heisenberg uncertainty should be viewed as a limiting transverse uncertainty causing a

phase uncertainty at the target. We may call this a diffraction (imaging) or an interference

experiment; the distance rwill be the same in either case. In units ofc = 1,

( ) r 2 ; so, r 4 . (2)

We want rin terms of the geometry. By definition, r = + r x ri2 2

, so,

r x r r i= 2

2 2 . (3)

The second term may be ignored, leaving r x ri 2

2 . To get rin terms of the photon

wavelength, by our criterion xf and Eq. (1), we have r ( )16 . Therefore, finally,

r xi 82 . (4)

To see how this formula may be used, call = 610 3. m for visible light. Then, for slit

spacing xi =3

10 m, we find that r 50 m. No identifiable phase of the interference can be

present predictably at much longer distances, so coherence can not be used to reveal it at such

distances, either.

Young's Experiment for a Neutrino

The deBroglie wavelength has been found experimentally to yield accurate results for

interference and diffraction of massive particles, using the paradigm of Young's experiment or,

similarly, of Bragg diffraction. See Sanz et al (2003) for data and literature references. Let us

then replace the slit spacing in Fig. 1 with the diameter of the spatial extent of the initialinteraction creating a neutrino. The question now is whether the diameter of the region of

neutrino creation is enough to project a usable mass eigenstate wavefunction phase difference at

the final interaction point.

For example, a 1 GeV neutrino originating by cosmic muon decay in the Earth's atmosphere

would have a deBroglie wavelength of about 1015 m. For such a neutrino, we might set xi in


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Fig. 1 equal to the 1017 m range of the weak force. Applying Eq. (4), we find that, for a 1 GeV

neutrino created in a weak-force aperture,

r 18

2 10 m. (5)

Alternatively, we might set xi equal to the 1015

m range of the strong force. Then,

r 14

2 10 m. (6)

In the interest of accuracy, it should be mentioned that a better three-dimensional

representation of this neutrino experiment would use a circular aperture of diameter xi , a

Bessel function superposition for the pattern on the screen, and a 2-d criterion; however, the

estimated distance rwould not be changed in any important way.

Leaving these results for now, we proceed to our main discussion of the neutrino oscillation

theory.

1. The Usual CKM - PMNS Analogy is Misleading

The Cabibbo-Kobayashi-Moskawa (CKM) matrix seems to represent well the data on flavor

changing weak currents (Falk 2002; Marsiske 2002; Battaglia et al 2003; cf. Abele et al 2002).

In this representation, the flavor state vector of a quark in a hadron is obtained by multiplication

of an initial mass state vector by the CKM matrix. The state vectors are each of three

components (for three generations of quark), and the matrix, V , is defined unitary. For the

CKM matrix defined by

V

V V V

V V V

V V V

ud us ub

cd cs cb

td ts tb

,

for flavor state [ ]q d s bfT

, and for mass state [ ]q m m mm d s bT

, we have,

q qf m

= V . (7)

For the positively charged quarks, VT may be used correspondingly.

Equation (7) represents an interference among the mass state components, each a mass

eigenstate, to define a flavor state vector each component of which is an amplitude of that flavor.Thus, the mass state caused by quark exchange of a Wboson (a charged-current interaction) may

be reexpressed as the resultant flavor state, even though the resulting quark is not in a mass

eigenstate.

Elements of the CKM matrix include mixing angles and phases. How can this work without

kinematics? The answer is simple: Experiments confirming the CKM relationship measure

expectancies, and the expected velocity of the mass eigenstates in a hadron is just zero, in the rest

frame of the hadron. Such experiments have studied K- andB-meson oscillations. So, the


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superposition given by the CKM elements is an expected superposition of quark mass eigenstates

at rest; equivalently, the current of the weak force flows in a hadron at rest. Different quark

velocities during WorZexchange average out to zero, and weak-force distances within a tiny

hadron can't be resolved. There is zero expectancy of motion, so there are no kinematics

involved. Hadrons have unbroken symmetry with regard to quark mass eigenstate momentum.

Furthermore, gluons are believed to account for a large fraction of the hadron rest energy; thisis seen easily by comparing the estimated mass of a u or dquark, in the 10 MeV range, with the

mass of the proton or neutron (3 quarks each), which is in the 1 GeV range. This means that,

within a hadron, a quark could change its mass as it mixed to change its flavor, by conversions

between total hadron rest energy and quark mass. Thus, the fields within a hadron permit flavor

oscillation while at the same time maintaining an approximate mass hierarchy, by flavor, for the

quarks. A freely propagating hadron could maintain a constant mass, as required by energy and

momentum conservation; and, at the same time, its constituent quarks might change mass

individually while still confined.

So, to describe neutrino flavor oscillation, we must assume (a) a neutrino at rest (viz., the

three usual neutrino mass eigenstates at rest), allowing an accurate CKM analogy, based onsymmetry unbroken by momentum; or, (b) a theory which incorporates kinematics of the mixing

mass eigenstates explicitly and accounts for the broken symmetry caused by an expectation of

nonzero neutrino momentum.

The assumption of neutrinos at rest clearly is not applicable, because observable neutrinos

have well defined momenta representing ultrarelativistic velocities.

The usual neutrino oscillation theory postulates a Pontecorvo-Maki-Nakagawa-Sakata

(PMNS) matrix almost exactly analogous to the CKM matrix (e. g., Kayser 2001; McKeown and

Vogel 2004). There is nothing added that accounts for neutrino kinematics, because propagation

distance is folded into the mass state vector and merely alters the mass state phase. Both

vacuum and matter-mediated neutrino oscillations may be expressed as analogies to the CKMapproach (Xing 2002; see also Kayser 2001), although the present paper does not consider matter

oscillations as such.

Clearly, as a CKM hadron propagates (or not), its quark mass eigenstates must superpose,

creating varying quark masses. We assume that the quark mass hierarchy by flavor in a hadron is

maintained by a gluon conversion mechanism; but, regardless, this hierarchy, and the mixing, has

been observed experimentally. As a PMNS neutrino propagates, variation of its mass according

to final interaction flavor would violate physical laws governing kinematics; in case this is not

obvious, it is proven in the answer to the next question below.

Thus, to the extent that CKM and PMNS are analogous, to that extent is the usual neutrino

oscillation theory not credible.

The CKM - PMNS analogy being untenable, the usual theory of neutrino oscillations must

stand on its own, because a neutrino in propagation is so different qualitatively from a quark at

rest in a hadron.


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2. Mass Hierarchy is Precluded by Oscillation

We use the term, "mass hierarchy", to refer to an ordered range of mass values by flavor. We

wish to show that mass of a neutrino must be independent of flavor; or, more generally, that mass

of a particle in free propagation must be the same in its initial and final interactions.

For the present purpose, we assume no difference between an analysis by proper time

evolution and an analysis by distanceL (position) at approximately the speed c, something

typically assumed by proponents of the usual theory (e. g., Giunti 2004).

The usual theory postulates an analogy to Eq. (7) above to explain neutrino flavor oscillations.

This analogy may be seen easily in the equation,

f L m L( ) ( )= V , (8)

in which f is the neutrino flavor state vector, V is the PMNS matrix, and m is the mass state

vector as a function of propagation distanceL. See Kim and Pevsner (1993) for derivations.

All mass eigenstates in the usual one-dimensional theory necessarily undergo just onepropagation; so, in this theory, assuming fixed initial phases, phase ( )L may be factored out in

the mass state as follows:

m L( ) =

m

m

m

L

1

2

3

( ) =

1 1

2 2

3 3

( )|

( )|

( )|

L m

L m

L m

>

>

>

=

A i kL t m

A i kL t m

A i kL t m

1 1 1

2 2 2

3 3 3

exp[ ( )]|

exp[ ( )]|

exp[ ( )]|

>

>

>

=

exp[ ( )]|

exp[ ( )]|

exp[ ( )]|

>

>

>

i L m

i L m

i L m

1 1

2 2

3 3

. (9)

The Aj must be equal for consistency with the unitarity of V ; so, they are left absorbed in the

eigenstates |mj

> in the rightmost expression in Eq. (9), as are other irrelevant constants.

According to the usual theory, the evolution of the mass state vector as a function of distance

is what is supposed to cause observation of oscillation between flavors. This evolution is

calculated theoretically and not observed. Observation of the neutrino state m L( ) is limited by

quantum theory to the initial and final interaction points.

Unitarity is more than enough to define V in Eq. (8) as invertible. If invertible, then trivially

one also must accept,

V V V1 1= f L m L( ) ( ) , (10)

m L f L( ) ( )= 1V . (11)

Thus, flavor oscillation maps 1-1 to mass oscillation. The flavor amplitudes rotate; the mass

phases rotate. The usual formula for calculating the flavor expectancy is derived from

expressions reducing to Eq. (8), and generally it is written as a probability of transition between

the initial flavor and the flavor at a distanceL above.


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Conservation laws require that the amplitudes of the mass eigenstates not change between

initial and final interactions, as shall be proven next. The usual theory postulates that it is

possible for the initial value m xi( ) to be mixed if flavors oscillated and m xf( ) was mixed; but,

if each mj is constant, so must be the expectancy < >m . And so, by this simple argument,

neutrino flavor oscillations under the usual theory imply there can be no mass hierarchy in the

neutrino sector.

To prove the constancy of mass eigenstate mass during propagation, we first notice that, for

each neutrino, both energy and momentum must be conserved and therefore must not differ

between initial and final interaction points. Of course, this holds no matter which mass

eigenstate is propagated. The proof then is by reductio:

For a free, massive particle, following Einstein we may write,

E m c= 2 E pc m c2 2 2 2= +( ) ( ) , (12)

in whichp represents momentum and m rest mass. For the momentum, we write,

p m v= . (13)

Now here is the reductio proposition: Suppose we allow m to change to m , with 0 < and 1, between the initial and final points of a free propagation. We propose that there exists aspeed w which will allow us to keep both energy and momentum at their original constant

(conserved) values. This just means that if we change m to m in Eqs. (12) and (13), we claimwe can find a unique w such that we must also change v to w in those equations.

(Dis)proof: If we change m as described, then we must change v to v in Eq. (13) to keep

momentum constant. The Lorentz is a function of speed v, so we must change it, too:

=

1

12 2v c

; (14)

solving Eq. (14) for v,

v c c= 2 2 2 , (15)

in which the form, but not the sign, of the square root will matter.

So, from Eq. (15), to conserve momentum in Eq. (13), we need a new speed w defining a new

w such that, ww v= ; or,

w wc c c c

2 2 2 2 2 2 = . (16)

This implies,


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w

c w

v

c v

2

2 2

2

2 2 2

= ( )

, of which we keep wcv

c v=

+ 2 2 2 21( ). (17)

To conserve energy, we must replace with in Eq. (12); this means we need that same

new speed w such that,

w =

1

1

1

12 2 2 2

=

w c v c. (18)

This becomes,

w v c= + 2 2 2 21( ) ; (19)

and, the absurdum comes out if we test consistency by equating w in Eqs. (17) and (19):

cvc v 2 2 2 21+ ( )

= + 2 2 2 21v c( ) , (20)

which can not hold except for speed v c . No approximation was used anywhere above. It

would be irrational to claim that flavor oscillates because of well-defined mass eigenstate mass

differences, but that all eigenstates propagate exactly on the light cone and thus are massless.

A more intuitive demonstration of this problem was described as the " v and v2 paradox" in

Section 4.3 of (Williams 2001), and a simulation demonstrating constancy of mass of each

neutrino is given in Williams (2003).

The conclusion, then, is that, under the usual neutrino oscillation theory, if the masses ofdifferent neutrinos differ by flavor, then the physics of the theory must be bad.

3. Which-Path Mechanics Can't be Applied

Why not attach the neutrino oscillation problem to a different analogy, that of Young's

experiment in photon interference? The Young's experiment which-path paradigm is depicted in

Fig. 1 above, and it works for massive particles. There is an uncertainty as to which path (slit) a

given photon has traversed; thus, the location of the final interaction is given by a superposition

of amplitudes from the two paths possible. By analogy, there should be an uncertainty as to

which mass it was that the neutrino possessed during propagation.

Unfortunately, this analogy can be shown not to work. This is because the analogous neutrinosuperposition is kinematic as well as quantal. To see this, let us consider a typical which-path

paradigm in detail.

A particle is assumed to travel freely, without interaction, on any one of several paths to its

point of final interaction. A path here is a computational device, and the path-based calculations

correctly are exact and classical.


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Wave functions in Young's experiment are meaningless except in context of the initial or final

interaction of a photon. It is axiomatic in quantum mechanics as applied to this problem, that no

measurement may be made, no observation may occur, which would identify which path a

photon traversed. If such identification was made, no interference could occur for that photon.

Thus, the slits may not change the particle kinematics in any way observable at the detector.

Let us look at the energyEand the norm of the momentump on two different paths, and thephase difference, keeping in mind that the hypothetical neutrino mass eigenstate interference

does not postulate any slit geometry between the initial and final interaction. We find that he

existence of more than one interference path entails that theEandp of the particle must be the

same on both paths:

Considermassless particles. The interference of photons is calculated on the assumption that

each path is traversed at speed c, or at the speed of light in the medium of the experiment, and no

discrepancy ever has been reported. If the energy on both paths was not the same, a photon

would shift wavelength depending on which slit it traversed, creating three problems: (a) making

phase calculations on the two paths more complicated (which is not observed); and, (b) making it

possible to tell which slit was traversed, by measuring energy or wavelength in the finalinteraction. In addition, (c) if one path differed in energy from the other, the particles could not

be propagating freely, but must be gaining or losing energy by some interaction between initial

and final points.

The momentum, p c= , of a massless particle is proportional to its energy, E= ; so,

the momentap also must be equal on both paths. For massless particles, equality of energy is

directly equivalent to equality of momentum.

Considermassive particles, such as neutrons: If the energies on both paths were not equal, the

neutrons would travel at different speeds, again confounding the phase calculations and maybe

making it possible to distinguish the paths by energy or deBroglie wavelength. In addition, the

assumption of free propagation would be violated, as for massless particles above. So, energymust be the same on each path.

If momentump was not the same on each path, the propagation could not be free; therefore,p

must be the same on each path. Also, ifm or v varied during free propagation in any way,Eand

p would change differentially, making conservation of both impossible during propagation; this

was proven in the answer to question 2 above (cf. Zralek 1998).

An energy or momentum difference by path does not have to be observed in an experiment; it

merely has to be observable in principle. Interference won't be observed unless the experiment

can not distinguish the paths. In general, this would imply that differences should not exceed a

quantum.

So, it appears that, according to Young's paradigm either for massless or massive particles,

particles can interfere in a which-path paradigm only if neither the energy nor the norm of

momentum varies according to path. Conceivably, some other kind of superposition (e. g.,

Garcia et al 2002) might occur, but it would not be a which-path superposition.

Now let us look more closely at the usual oscillating neutrino. The which-path explanation

requires a neutrino to be a particle with some number of mass eigenstates, which are said to


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represent neutrino traversal of divers interfering mass "paths". As above, the interfering

superposition is presumed not to allow the final mass to differ from the initial one.

To describe the which-mass experiment for neutrinos, one might think of allowing the final

interaction always to occur along one geometric path, with the mass eigenstates merely traversing

it at different relative phase shift rates, or speeds. Somehow, a number of mass "slits" are created

in the initial neutrino interaction, and they constrain the masses to be the hypothetical eigenstatevalues, as well as constraining all momenta exactly the same in norm and direction. We accept

this assumption here only in deference to the usual, one dimensional theory.

Notice that, as applied to neutrinos, the which-path paradigm involves a stronger constraint

than in other contexts: Because the neutrino is required to propagate freely, and in addition is

required to be elementary, the superposition must not involve a binding energy, an intermediary

slit, or any other possible interaction: The interference is constrained to be defined by the

neutrino initial interaction alone.

Anyhow, the mass eigenstates are said to have different masses, and this is where the theory is

incompatible with which-path: Consider any two mass eigenstates of mass m1 and m2 : If they

propagate freely, we have

E m c1 1 12

= = =E m c2 2 22 , (21)

because energy may not change by path, and because of energy conservation. Clearly, then, by

energy alone, we have,

2

1

1

2

=m

m. (22)

Because the masses differ, the gammas must differ; therefore, v1 must differ from v2 . This

is why superposition phase varies with distance in the usual theory: | |v v1 2 0 =

| |m m1 2 0 = = 0 ; so, no speed difference means no oscillation.

But, momentum along all paths also must be the same, for reasons immediately above and

given previously. Therefore, we must have,

2 2

1 1

1

2

v

v

m

m= . (23)

Obviously, Eqs. (22) and (23) can not possibly both be true if m m1 2 . The usual neutrino

oscillation theory is self-contradictory in the context of the which-path paradigm. We have notexcepted relativistic particles in this exposition (cf. Zralek 1998), and we have made no

approximation.

So, to avoid self-contradiction, the usual theory, which postulates mass eigenstates, must not

be justified by a which-path paradigm. One way to see this would be that the latter requires

interference of a particle with itself, not of eigenstates of different mass and independent

kinematics in free propagation. Interference as self-interference was postulated by Dirac (v. Kim

2003) and is equivalent to a which-path interpretation of the mechanics. By contrast,


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superposition in the CKM context does not depend on which-mass on a path but on which-mass

at rest (in expectancy) in a hadron: The velocity expectancy of zero means that the quark mass

eigenstate mass differences mean nothing in terms of the interference.

If they exist and obey the laws of physics governing free particle kinematics, the neutrino

mass eigenstates must be degenerate and all have the same mass value.

If the hypothetical mass eigenstates do not obey the laws of physics, it seems meaningless to

invoke them to solve a physics problem.

We recognize that according to the usual theory, the final (superposed) neutrino mass might

vary from trial to trial, and, thus, so would the initial mass. The point here is that whether or not

the phase of the mass state varies to cause flavor oscillation, the paths in the usual theory do not

superpose equal masses. So, which-path can not be applied validly.

4. No Neutrino Propagation by Wave Packet

Recall the factoring out of neutrino mass state phase shown in Eqs. (8) and (9) above,

resulting in the form, A i kL t mj j jexp[ ( )]| > .

In classical electromagnetic (EM) theory, the expression exp[ ( )]i kx t can represent a

propagating EM plane wave. It must be a plane wave, because the expression is in one

dimension and thus ignores all possible variations in the plane orthogonal to the direction of

propagation.

Some oscillation theorists have debated the neutrino wavefunction as though it were some

thing similar to an EM wave, speculating upon differences between plane-wave vs. wavepacket

representations (e. g., Kiers et al 1998; cf. Field 2001). However, an EM wave packet can not be

defined in terms of a single frequency.

So, the first problem with a wavepacket model of propagation of any elementary particle isthat such particles individually never show evidence of any frequency other than the

wavefunction frequency; there never is any envelope detected in the final interaction: All

evidence is that diffraction or interference of massive particles depends solely on the deBroglie

wavelength. Therefore, it seems there can be no neutrino quantum wavefunction envelope

analogous to an EM pulse envelope.

The second problem is the idea advanced by some oscillation theorists that the mass eigenstate

wavefunctions might separate with distance of propagation (e. g., Giunti 2004); this, in the

context of the superposition in the usual theory, implies a wavepacket representation. To avoid

ambiguity, we shall use "nonoverlapped" or "separated" here to refer to any two wavepackets

such that the amplitudes of the wavefunctions are very small on some finite domain between thewavepacket peaks.

Shifting Causes a Subtle Error

The usual theory's derivations depend on exact unitarity of the mass-to-flavor transformation

matrices. If the mass eigenstates can not be treated as longitudinally infinite waves, however,


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the flavor-changing can not be made to work over more than a certain, limited number cycles of

the flavor oscillation.

Unitarity is equivalent to rotation, but not distortion, of the group represented. If the neutrino

mass eigenstate wavefunctions in the usual theory are treated as wavepackets, then their

individual envelopes will shift during propagation in some significant way before superposition

in the final interaction.

Such a shift necessarily would change the mixing contribution of each mass eigenstate

wavefunction relative to the others. Then, wherever in the mutual phase space the final

interaction point was located, the relative amplitudes would be stretched (not rotated) to values

different from what they were in the initial interaction. Any change in wavepacket superposition

overlap must represent a change in the norm of all values in at least one column of the mixing

matrix V , relative to the other columns. This must change the determinant of V , causing loss

of unitarity.

Separation Causes a Big Error

Separation of the mass eigenstate wave packets in this variant of the oscillation theory usually

is invoked in a cosmological context. However, this merely makes obvious the above error in

allowing them to shift:

Let us assume that a nonoverlapped mass eigenstate could interact, and ignore flavor and other

implications about V . If the mass eigenstate wave packets separated spatially along the

direction of the neutrino propagator, one mass eigenstate would precede the others by some

distance of nonoverlap, L . As the propagation distanceL increased, so would L increase

proportionally. But, probability of interaction of a particle increases with distance traversed,

other things being equal. Thus, the probability of an interaction always would be greater, and

would continually increase, for mass eigenstates with greater speeds than others. Theprobability that the final neutrino mass would be that of the fastest eigenstate then must increase

withL, relative to the probability that it would be another.

So, if nonoverlapped mass eigenstates could interact, the mass expectancy of the neutrino

could change with distance of propagation, contradicting conservation laws, as above. Any value

of L greater than zero, however tiny, implies a calculated(not measured) violation of

kinematic conservation laws, so any such value is forbidden in the theory. In fact, the same

argument might be applied to any shift at all, whether there remained an overlap or not.

Even disregarding small violations of conservation laws, let us allow that mass eigenstates

modified to be wavepackets can interact in their volume of overlap, so that all mass eigenstates

contributed amplitude equally in this volume. Then, because the volume of overlappedamplitude always is a subset of the total volume of a wavefunction, the interaction rate must

decline as the overlap does.

So, finally, if we allow individual neutrinos to be described by wavepackets, we imply at best

that the neutrino cross-section goes to 0 as distance of propagation increases. This is behavior

not predicted of any other elementary particle. Maybe new physics, but more likely an error.


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Separation of mass eigenstate wavepackets merely pulls apart the mixing matrix, rendering

flavor undetermined and conservation laws violated.

A wavepacket model of single elementary particles can not account for the momentum in the

packet envelope: Because all experiments show only the wavefunction deBroglie wavelength,

appending wavepacket properties to a particle wavefunction requires that the wavepacket be

treated as a probability amplitude envelope somehow unrelated to the momentum-equivalentamplitude of the position wavefunction itself.

Wavepackets are a classical electromagnetic concept, and the present author thinks they

should not be introduced into quantum theory without careful forethought. A particle

wavefunction is defined only by the particle interactions, and the wavefunction does not exist

during propagation the way a classical sonic or EM pulse does. Quantum theory requires that a

particle not experience any interaction during free propagation; therefore, it seems meaningless

to claim that the deBroglie parameter of a single,elementary particle should be associated during

propagation with a wave or a classical wavepacket of some kind.

Wavepackets of single neutrinos or mass eigenstates are unphysical and may imply violations

of unitarity if they are fit into the usual oscillation theory.

5. No 1-d Oscillation by Mass Uncertainty

Let us accept for now that the propagation is one-dimensional, as in the usual theory, and in

effect is along a line joining the neutrino initial and final interaction points. Is it possible that

the quantum superposition postulated in the usual neutrino oscillation theory might be derived

from application of the Heisenberg uncertainty principle?

No, this doesn't seem to work. The proof here is that the typical m2 fits by oscillation

theorists can not correspond to any reasonable oscillation lengthL created by uncertainty:

The rest mass of any freely propagating particle may be expressed in terms of its momentum

as,

m p v= ; or, (24)

m p v2 2 2= ( ) . (25)

If we accept the usual theory's approach, we may ignore v because it is so close to c that it

won't matter. This seems to make sense because v is so close to c for an ultrarelativistic

neutrino, that there is not much change possible. So, we set dv 0 ; and, differentiating Eq. (25)

and solving for the momentum differential,

d mp dp

v( )

( )

2

2

2=

dp

d m v

p=

( )( )2 2

2

. (26)

Using dp of Eq. (26) as the p in the Heisenberg limit x p = 2 , writing the differentials

as differences, and solving the result for x ,


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J. M. Williams 2004-04-22 v. 1.0 1

x LH =p

m v 2 2( )=

m

m v2

, (27)

substituting p mv= , the subscript identifying this as a Heisenberg-specific propagation interval.

This uncertainty in mass thus in principle might prevent accurate measurement of distance, asclaimed, relaxing the requirement of energy conservation during propagation. Indeed, the

uncertainty in mass in Eq. (27) would permit an "oscillation" distance LH such that no

measurement could demonstrate a change in momentum within that distance.

But, a typical solar neutrino studied for oscillations has an energy of 15 MeV; from E mc= 2

and a neutrino rest energy mc2 of, say, 1 eV, = 15 106 . Calling v c for present purposes,

and choosing a typical neutrino oscillation theory value of m2

=3

10 ( )eV c2 2 ,

LH

34

10 3 10

10 16 10 15 10

8

3 19 6

( )

( . )( )

11

10 m, (28)

which is quite microscopic. A detector composed of atoms wouldn't notice it.

Therefore, the complementary uncertainty inx because of the precision inp implied by a

typical m2 would seem beyond present measurement capability, if velocity doesn't matter.

From an equivalent perspective, Eq. (28) shows that the hypothetical mass differences among

eigenstates should be measurable with great precision without preventing observation of

oscillations, under assumptions of the usual theory.

6. Quantum Properties of the Detector Don't Matter

Disappearance of Solar neutrinos was observed by Davis (e. g., Bahcall and Davis 1976),

using a chemical experiment in an event observation interval of months. In this experiment,

neither energy nor momentum was measured; distance is not measured within meters.

Similar disappearance is observed in the Super-Kamiokande water Cerenkov detector (Fukuda

et al 2001), or in the Sudbury deuterium-water Cerenkov detector (Ahmad et al 2002), with

observation intervals of milliseconds and energy or momentum measurement within 20%, and

simultaneous distance measurement within centimeters. Different chemical apparatus such as

SAGE (Abdurashitov et al 1999) and GALLEX (Altmann 1998) all confirm about the same

disappearance (oscillation) rate under similar conditions but under diverse constraints on the

observation. There is no quantum-mechanical commonality among these detectors.

Observation of oscillations therefore is unrelated to quantal properties of the detector.

These detectors are just analogous to observation screens in Young's experiment. Although

expressed in terms of quantum mechanics, oscillations are not being detected by anything

contributing a quantum-mechanical effect.


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J. M. Williams 2004-04-22 v. 1.0 1

7. Oscillation in 2-d Is Incoherent

Let us recall that the usual neutrino oscillation theory postulates that a neutrino may be

described during propagation as a superposition of some number, here taken to be three, of mass

eigenstates. Each such eigenstate usually is viewed as a massive, independently propagating

particle such that its position wavefunction shifts in phase relative to the others as a function ofneutrino proper time (propagation distance).

Measurement of the flavor constitutes a measurement of the final relative mass phases under

the usual theory of superposition; therefore, the mass state may be subject to Heisenberg's

uncertainty principle. The Heisenberg uncertainty refers to measurement, not to the determinism

used in wavefunction calculations. For this reason, the calculations in the section above on

"Young's Which-Path Experiment" may be interpreted as implying any arbitrarily large spatial or

temporal incoherence of the particle(s) at the point of creation. The coherence is irrelevant to

the uncertainty.

The point of this is made rather easily: The usual oscillation theory fails in two dimensions

except at microscopic propagation distances.

Recall that in the Fig. 1 representation for Young's experiment we have one transverse

dimension (of uncertainty) and one dimension in the direction of propagation. There is Lorentz

contraction in the direction of propagation but none in the second dimension. So, for a freely

propagating particle, the transverse uncertainty in positionx, x , will grow proportionally with

propagation distance. Photons, and massive particles at speed v just below c, fit the same Eq.

(4).

Call the propagation lengthL; set xf = 4 ; using Eq. (4) define r xx if 82 ; then,

x

x Lr

fxf

x

2

24 8L

xi. (29)

For 1 GeV atmospheric neutrinos and coherence by weak force range as in Eq. (5), would

be about 1015 m,L would be about 10

7 m, and,

x 30

10

4

10

8 10

7

34 10

9 m. (30)

For 10 MeV Solar neutrinos and coherence by weak force range, would be about 1013 m,L

about 1011 m, and,

x 26

10

4

10

8 10

11

34 10

17 m. (31)

So, the uncertainty in position of a mass eigenstate implied by a creation region the size of the

range of the weak force is two orders of magnitude greater than the diameter of the Earth for 1

GeV atmospheric neutrinos, and it is enormously greater for Solar neutrinos. The position


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J. M. Williams 2004-04-22 v. 1.0 1

uncertainty here may be interpreted as an uncertainty in the phase of a mass eigenstate during

neutrino propagation, which phase is used to superpose it with the others at the final interaction

point.

If we constrain superposition to be on a line passing through the Earth, so that the final

interaction occurs somewhere on Earth, the phase differences among three such mass eigenstates

will have to be essentially indeterminate and totally random in terms of mass or flavor. So, witha superposition in just two spatial dimensions, neutrino flavor oscillation phase will not be

observable because of the Heisenberg uncertainty alone.

Disappearance of flavor in general still will be observed, because incoherent oscillations and

three neutrino flavors imply loss of 2/3 of all neutrinos by a flavor-sensitive detector. This

happens approximately to fit all presently known data on neutrino oscillations.

The situation is much better if we drop the weak current derivation of the usual PMNS theory

and assume that, somehow, the size of the initial hadron is what defines the initial phases of the

neutrino mass eigenstates. Perhaps this might be rationalized by saying that neutrino mass

eigenstate initial phases are determined at creation by a strong interaction. We then have, as in

Eq. (29) above,

For 1 GeV atmospheric neutrinos and coherence by hadron size,

x 30

10

4

10

8 10

7

30 10

5 m. (32)

For 10 MeV Solar neutrinos and coherence by hadron size,

x 26

10

4

10

8 10

11

30 10

13 m. (33)

So, interestingly, if we avoid the usual theory and merely postulate any sort of otherwise

unconstrained flavor oscillation, we arrive at the conclusion that oscillation phase will not be

observable if phase coherence results from a weak interaction.

However, oscillation phase may be observable for ~GeV neutrinos if we postulate that

coherence results from a strong interaction -- or, more generally, from some interaction with

spatial extent substantially greater than the range of the weak force.

Therefore, if oscillation phase is observable, the usual theory is disproven. If oscillation

phase is not observable, the usual theory might be tenable, but the mixing angles and mass

differences inferred from it can not be meaningful.

8. The Usual Theory Predicts Masses Too High

Because the usual theory yields only mass (squared) differences, we integrate to obtain the

masses.

Atmospheric muon neutrinos, which we shall emphasize here, are found to disappear when

studied by an electron- and muon-flavor detector; it seems fairly certain that they are oscillating


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J. M. Williams 2004-04-22 v. 1.0 1

to a flavor not detected, such as tauon. The majority of recent experiments have applied the

usual theory to define an oscillation parameter phase space on mass-eigenstate m2 and V

mixing angle sin2 . The m2 for the large mixing angle (LMA) solution currently preferred

for atmospheric oscillations is within an order of magnitude or so of102 (eV c2 2) (e. g.,

Toshito 2002) . But, this estimate seems inconsistent with the size of the Earth:Consider an atmospheric muon neutrino of 1 GeV energy. Its deBroglie wavelength would

be = h p = hc E mc[ ( ) ]2 2 2 1 2 . Suppose the neutrino mass to be 100 eV c2 or less. Then,

with negligible error, hc E 1512 10. m.

The diameter of the Earth is about 107 m; so, in the view of the usual theory, the number of

cyclesNfrom the atmosphere on one side of the Earth to a detector on the other would be,

N = 101

10 8 107 7 21

E

hccycles. (34)

Note thatNis proportional to energy. The mass eigenstates are forbidden to differ observably

in energy, so Eq. (34) applies to any mass eigenstate. Let us assume that the measured

disappearance is because of a phase shift of, say, 1 4 cycle between some two mass eigenstates.

Then, the cycle count ratio for these mass eigenstates must be,

N N

N

+ = + +

1 1

1 4

8 1021

N

N

N

N

3 10

19 . (35)

Recall that the Lorentz gamma may be defined in terms of a ratio of total energy to rest energy

as, = E m . From Eq. (34), a phase shift of 1/4 cycle would be equivalent to an energy shift of

about the ratio on the right of Eq. (35). Because mass is proportional to energy by the Lorentzrelation, from Eq. (35), the equivalent mass ratio also must be the same. So,

m m 19

3 10 . (36)

Therefore, for two particles, or mass eigenstates, of energy about 1 GeV, and differing in mass

by the ratio given in Eq. (36), the phase of their deBroglie-defined wavefunctions will have

shifted by 1/4 cycle by the time they have propagated a distance about equal to the diameter of

the Earth. A greater mass ratio will of course entail a proportionately greater phase shift.

The calculation above allows errors inEand without much changing the result. Under the

usual theory, the mass of a neutrino is not definite because of the mixing; nevertheless, one mayassume that the mass can be indefinite only within the range of masses of the mass eigenstates;

and, if so, the calculation in Eq. (36) must hold for the neutrino within, say a couple of orders of

magnitude.

The ratio in Eq. (36), calculated independently of the usual theory, seems utterly incompatible

with m2 estimates from the theory. For example, suppose a 10

1 eV c2 neutrino with one

mass eigenstate 0.9 of the mass of the other: Then m2 2 10 3 ( )eV c2 2 , close to the range


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J. M. Williams 2004-04-22 v. 1.0 1

of the usual data fits. Then, m 005. eV c2 ; and, from Eq. (36), we get a completely

incredible m 5 1017 eV c2 . We know the electron neutrino must be under 20 eV c2 from

the SN1987a supernova data; it probably is below 1 eV c2 (Bilenky et al 2002). If m 1 eV c2

in Eq. (36), then m 19

10 eV c2 ; and, m

2 is many orders of magnitude below the estimates

of the usual theory.

The only straightforward way to resolve these discrepancies would seem to be to increase the

cycle count in the numerator of Eq. (35). If, for example, we let the disappearance be caused by

a phase difference N of1018 full cycles, then 10 8 1018 21 2N N 10 , implying

m m 2

10 , which could agree at least approximately with the m2 of the usual theory fits,

for a neutrino mass somewhat less than 1 eV c2 .

But, this would make the oscillation cycle length about 10 107 18 11

= m. This means that

the observed atmospheric loss of neutrinos never could be anything but a constant, incoherent,

distance-independent effect.

Considering the above, it seems unreasonable to insist on the relatively large m2 values

usually claimed to fit neutrino data. With such values, essentially incoherent flavor oscillations

must be occurring, making the estimated values of the mixing-matrix parameters of the usual

theory very unrealistic.

Furthermore, independent of the usual theory, if oscillations are occurring, the above

calculations imply that there must be an approximate energyEand distanceL independence in

the long-time averaged data for the atmospheric or Solar data. Failure to observe approximate

energy independence would indicate presence of a phenomenon other than, or in addition to, the

usual neutrino oscillations.

The K2K data (Ahn et al 2001), which showed apparently distance-independent disappearance

about the same as in the Super-K atmospheric data, therefore may be interpreted to support the

calculation in Eq. (36). Energy independence is less consistent with the data, but it is not our

purpose here to apologize for the usual theory.

9. Neutrino Substructure For a CKM - PMNS Analogy

All current neutrino oscillation data, except for those of LSND (Aguilar 2001), are

disappearance data, consistent with loss of about 2/3 of the total neutrino flux, when the

detector is flavor-specific. When the detector is flavor indifferent, as for the SNO neutral

current results (Ahmad et al 2002), no oscillation is evident. This is consistent withincoherent oscillations, except possibly for LSND.

In the section above on "Oscillation can be Coherent only in 1-d Propagation", we find that

if we ignore contradictions in the derivations, the usual theory can predict coherent

oscillations only if the region (aperture) of the initial interaction creating the mass eigenstates

exceeds the range of the weak force by several orders of magnitude.


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J. M. Williams 2004-04-22 v. 1.0 2

So, let's assume that a neutrino has substructure, is not point-like, and is bigger than a

hadron ( > 1510 m) in some sense. Perhaps an elementary particle (as opposed to the

wavefunction associated with its interaction points) can have spatial extent; perhaps not.

Maybe this is an issue of terminology, only. Then, we can postulate three mass eigenstates

analogous to those of the CKM approach outlined above, and allow these mass eigenstates to

be at rest (in expectancy) in the inertial frame of the neutrino during propagation. Allquestions answered negatively above suddenly become affirmative for the usual PMNS theory

reformulated this way. Furthermore, the oscillations no longer need be incoherent, and the

current curve fits of the usual theory may be rationalized and reused unchanged in the context

of neutrino substructure.

Conclusion

Neutrino substructure opens up a new subleptonic world. Accepting a substructure

hypothesis permits rationalization of the usual PMNS oscillation theory as one possible path for

future theory development.

Analogies also may be built upon hadronic or atomic theory. Such an analogy was presented

in Williams (2002): Briefly, an oscillating structure with flavor features called quirks (for

obvious CKM-relevant reasons) was postulated to expand in a short proper-time interval after

neutrino creation. With fewer free parameters than the usual theory, all present neutrino

oscillation data could be accounted for, including the otherwise outlying LSND result.

Yet other approaches based on substructure obviously are feasible. We suggest revising the

current Standard Model now, because oscillations already invalidate it. A new model possibly

should banish the neutrino sector to complexity and replace it with the elements of which the

neutral leptons may be composed.

Acknowledgements

The author thanks Steven Yellin for insightful discussions which clarified understanding of

several points in the present paper. The author's first glimpse into this kinematic side of neutrino

flavor resulted from discussions with Michael E. Peskin.

References

Abdurashitov, et al, 1999. "Measurement of the Solar Neutrino Capture Rate by SAGE and

Implications for Neutrino Oscillations in Vacuum". Physical Review Letters, 83(23), 4686

- 4689.

Abele, H., et al, 2002. "Is the Unitarity of the Quark-Mixing CKM Matrix Violated in Neutron

Beta-Decay?", Physical Review Letters, 88(21), 211801-1 - 211801-4.

Aguilar, A., et al, 2001. "Evidence for Neutrino Oscillations from the Observation ofnubar_e

Appearance in a nubar_mu Beam",arXiv, hep-ex/0104049.


21/22

J. M. Williams 2004-04-22 v. 1.0 2

Ahmad, Q. R., et al, 2002. "Direct Evidence for Neutrino Flavor Transformation from Neutral-

Current Interactions in the Sudbury Neutrino Observatory", Physical Review Letters, 89(1),

011301-1 - 011301-6.

Ahn, S. H., et al, 2001. "Detection of Accelerator-Produced Neutrinos at a Distance of 250 km",

arXiv, hep-ex/0103001.

Altmann, M., 1998. "GALLEX Solar Neutrino Observations: Results from the Total Data Set"

(1998), Talk at the 33rd Rencontres de Morionde "Electroweak Interactions and Unified

Theories" (Les Arcs, March 14 - 21). At http://www.e15.physik.tu-

muenchen.de/gallex/moriond98.ps.

Bahcall, J. N. and Davis, R. Jr., 1976. "Solar neutrinos: A Scientific Puzzle", Science, 191, 264

- 267.

Battiglia, M., Buras, A. J., Gambino, P., Stocchi, A. (Eds.), 2003. "The CKM Matrix and the

Unitary Triangle",arXiv, hep-ph/0304132.

Beuthe, M., 2001. "Oscillations of neutrinos and mesons in quantum field theory",arXiv, hep-

ph/0109119.

Bilenky, S. M., Giunti, C., Grifols, J. A., and Masso, E., 2002. "Absolute Values of Neutrino

Masses: Status and Prospects",arXiv, hep-ph/0211462.

Einstein, A., 1930 "Unified Field Theory based on Riemannian Metrics and distant Parallelism",

Math. Annal. 102, pp. 685 - 697. Translated by A. Unzicker; slightly paraphrased.

Falk, A. F., 2002. "Flavor Physics and the CKM Matrix: An Overview",arXiv, hep-ph/0201094.

Field, J. H., 2001. "The Description of Neutrino and Muon Oscillations by Interfering

Amplitudes of Classical Space-Time Paths",arXiv, hep-ph/0110064.

Fukuda,et al

., 2001. "Solar 8B andhep

Neutrino Measurements from 1258 Days of Super-Kamiokande Data",arXiv, hep-ex/0103032.

Garcia, N., Saveliev, I. G., and Sharonov, M., 2002. "Time-Resolved diffraction and

interference: Young's interference with photons of different energy as revealed by time

resolution", Philosphical Transactions of the Royal Society of London A , 360, 1039 - 1059.

At http://www.pubs.royalsoc.ac.uk/

phil_trans_phys_content/news/interference.html.

Giunti, C., 2004. "Theory of Neutrino Oscillations",arXiv, hep-ph/0401244.

Kayser, B., 2001. "Neutrino Mass, Mixing, and Oscillation", Proceedings of the Theoretical

Advanced Study Institute 2000 (Boulder, Colorado, June 2000),arXiv, hep-ph/0104147.

Kiers, K., Nussinov, S., and Weiss, N., 1995. "Coherence Effects in Neutrino Oscillations",

arXiv, hep-ph/9506271.

Kim, C. W. and Pevsner, A., 1993. Neutrinos in Physics and Astrophysics, (Chur, Switzerland:

Harwood Academic Publishers).

Kim, Y-H., 2003. "Two-Photon Interference Without Bunching Two Photons",arXiv, quant-

ph/0304030.


22/22

J. M. Williams 2004-04-22 v. 1.0 2

Marsiske, H., 2002. "Measurements of CKM Elements and the Unitary Triangle",arXiv, hep-

ph/0209349.

McKeown, R. D. and Vogel, P., 2004. "Neutrino Masses and Oscillations: Triumphs and

Challenges",arXiv, hep-ph/0402025.

Sanz, A. S., Borondo, F., and Bastiaans, M. J., 2003. "On the loss of coherence in double-slitdiffraction experiments",arXiv, quant-ph/0310095.

Schoen, C. and Beige, A., 2001. "An Analysis of a Two-Atom Double-Slit Experiment Based on

Environment-Induced Measurements",arXiv, quant-ph/0104076.

Toshito, T., 2002. "Super-Kamiokande atmospheric neutrino results", Talk at 36th Rencontres de

Moriond Electroweak Interactions and Unified Theories, March 2001,arXiv, hep-

ex/0105023.

Williams, J. M., 2001. "Asymmetric Collision of Concepts: Why Eigenstates Alone are Not

Enough for Neutrino Flavor Oscillations". arXiv, physics/0007078.

Williams, J. M., 2002. "Toward a Kinematically Correct Theory of Neutrino FlavorOscillations". CERN preprint CERN-EXT-2002-063.

Williams, J. M., 2003. "Some Problems with Neutrino Flavor Oscillation Theory". CERN

preprint CERN-EXT-2002-042. Revised in 2003.

Xing, Z., 2002. "Wolfenstein-like Parametrization of the Neutrino Mixing Matrix",arXiv, hep-

ph/0211465.

Zralek, M., 1998. "From Kaons to Neutrinos: Quantum Mechanics of Particle Oscillations",

arXiv, hep-ph/9810543.

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Neutrino substructure from flavor oscillations