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On Morphing Neutrinos and Why They Must Have Mass
Eugene Hecht
Citation: The Physics Teacher 41, 164 (2003); doi: 10.1119/1.1557506
View online: http://dx.doi.org/10.1119/1.1557506
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Published by the American Association of Physics Teachers
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On Morphing Neutrinos and
Why They Must Have Mass
Eugene Hecht,
Adelphi University, Garden City, NY
T
his paper explores the recently confirmed hypothesis that neutrinos have mass and that
they spontaneously transform from one type
to another. That immensely important discovery culminates 40 years of experimental research. After
briefly discussing that work, we’ll study the quantum
mechanical explanation of these phenomena elaborating the concepts of particle mixing, and the oscillation
of flavor types. These rather esoteric ideas lead to the
prediction that morphing neutrinos must have mass,
but there’s a much more elegant relativistic argument
that brings us to this same conclusion.
The Solar Neutrino Problem
On Oct. 24, 1995, The New York Times1 carried an
article entitled “Neutrinos Have Mass, Panel Says.”
That panel was a group of researchers at Los Alamos
National Laboratory. The article went on to say that
“physicists reported finding evidence from accelerator
collisions that neutrinos can ‘oscillate,’ or transmute
from one kind to another.2 Such a change implies
that these subatomic particles must have some mass,
however slight.” Alas, the reader was not provided
with any rationale for why “such a change implies”
that neutrinos have mass. What was clear was that the
experiment, though hardly definitive, was a substantial step toward addressing the so-called Solar Neutrino Problem (SNP). This remains an immensely important project occupying laboratories all across the
world; if neutrinos have mass, and it now seems they
do, the standard model, the catechism of particle
physics, must be substantially modified.3 In addition,
164
the cosmological consequences of neutrino mass are
immense. It has been estimated that there are roughly
110 of each type of neutrino per cubic centimeter, distributed throughout the cosmos. Even a tiny neutrino
mass would therefore account for a significant
amount of dark matter and play a profound role in determining the future unfolding of the universe.
The SNP springs from the fact that according to
our best theoretical understanding, the thermonuclear
reactions that power the Sun (primarily the protonproton cycle: p + p → d + e+ + e ) should pour out copious amounts of neutrinos. Yet it’s been known since
the experiments of Raymond Davis Jr., begun in the
late 1960s, that only a fraction of the predicted 1011
neutrinos/cm2 s that “should” arrive at Earth are actually detected.4
A Quantum Mechanical Explanation
A likely solution to this troubling dilemma was apparently first proposed by Pontecorvo5 in 1968. It
was inspired by the highly successful theory of the
neutral K-meson, formulated earlier by Gell-Mann
and Pais (1955).6 The neutral kaon, as it’s often referred to, displays a remarkable decay curve that’s
formed of two distinct exponential segments, as if the
kaon were somehow composed of two different particles, with two very different lifetimes. To explain this
and a variety of other fascinating observations, it was
0
proposed that
—0 both the neutral kaon (K ) and its antiparticle (K ) — which have identical lifetimes and
masses — were mixtures of two distinct but kindred
particles (K01 and K20), which themselves have different
DOI: 10.1119/1.1557506
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lifetimes and masses.7 K01 decays into two pions (K01
→ 2) with a mean lifetime of 8.9 10-11 s, whereas
K02 decays into three pions (K02 → 3) with a mean
lifetime of 5.2 10-8 s. These relatively long lifetimes
indicate that the weak interaction is at work (mean
lifetimes for strong-interaction decays are 10-20 s).
It follows that the
weak interaction “sees” K01 and K02,
—
but not K0 and K0 (which are produced via the strong
interaction). Gell-Mann and Pais suggested, “Since
we should properly reserve the word ‘particle’ for an
object with a unique lifetime, it is the K01 and the K02
—
quanta that are the true ‘particles.’ The K0 and K0
must, strictly speaking, be considered ‘particle
mixtures.’”6
That sort of thing can be understood via quantum
theory, where the state of a particle can consist of a
“superposition” of two or more distinct particle
states.
—0
0
In other words, the orthogonal K - and K - states are
composed of a linear combination, a sum and difference, respectively, of K01- and K02- states.8 The decay of
the neutral kaon is mediated by the weak interaction,
and K01 and K02 are associated with eigenstates of the
weak interaction Hamiltonian.
Less formally, suppose we have a beam of particles
of momentum p, each of which can be considered to
be a mixture of two component particles with different masses. These will have different energies, E1 and
E2, and their associated free-particle “matter waves”
will have different frequencies, where E1 = 1 and
E2 = 2 (that’s crucial). The wave functions evolve
via the standard time dependence,9 exp (–i1t) and
exp (–i2t), respectively. When such waves are superimposed, they go in and out of phase, creating a kind
of beat pattern reminiscent of the way energy passes,
or oscillates, between coupled pendulums. At some
point in space and time the waves add, at another they
subtract. By contrast, if the two particle states corresponded to the same mass, and hence the same energy
and frequency, the two matter waves would maintain
a constant relative phase. There would be no beats
and no oscillation of the mixed state.
In an analogous fashion, a newly created K0 traveling freely through
vacuum can spontaneously trans—
form into a K0 (and vice versa) as its constituent K01and K02- states evolve in time via the weak interaction.
Imagine overlapping K01- and K02- wave functions sequentially passing completely in phase (creating a K0),
THE PHYSICS TEACHER ◆ Vol. 41, March 2003
—
and completely out of phase (creating a K0 ), as they
progress through space-time.10 The probability that
an initially pure beam consisting of a specific particle
time, contain a
mixture (resulting in K0) will, at a later—
different particle mixture (resulting in K0) is an oscillatory function of time. That phenomenon, which
0
has
—0 been experimentally confirmed, is known as K –
K oscillation.
The idea of mixing, which might seem counterintuitive when applied to particles, has a powerful classical analog in the analysis of polarized light, whether
it’s via Faraday rotation,11 the response of a linear polarizer,12 or the behavior of circular light in an ordinary birefringent material. For example, one can easily express either right circular (᏾-state) or left circular
(ᏸ-state) polarized light as a linear combination of
two orthogonal linearly polarized (ᏼ-state) light
waves. In vacuum these are coherent, and the state of
the circular light remains unaltered as it propagates.
However, in a properly oriented birefringent medium,
each constituent ᏼ-state travels with a different speed
and wavelength. As they progress, the phases of the
two ᏼ-states evolve differently in space-time. Consequently, although the light may have originally entered as an ᏾-state, it can morph into an ᏸ-state, and
if the journey is long enough, it can oscillate back into
an ᏾-state, and so on.
As for neutrinos, they come in three kinds13 or “flavors”: the electron neutrino (or e-nu, e), the muon
neutrino (or mu-nu, ), and the tau neutrino (or
tau-nu, ). These are the objects that are weakly created along with electrons, muons, and tauons, and so
must be describable by the weak interaction Hamiltonian, and correspond to weak interaction (or flavor)
eigenstates. As these isolated particles sail through
space-time, they are described by an additional
free-particle Hamiltonian whose eigenstates are the
so-called mass eigenstates. But these two behaviors
must be interconnected (by an appropriate transformation). The flavor eigenstates must be linear combinations of the mass eigenstates, and vice versa (recall
the neutral kaons8).
If the neutrino masses are zero, the flavor and mass
eigenstates would be indistinguishable. On the other
hand, if the neutrinos have different nonzero masses
(and therefore different energies and frequencies), and
moreover, if they mix such that each neutrino is a
165
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composite of two or three different-mass components
(having wave functions with phases that evolve in
time), then neutrino oscillations will occur. In other
words, if neutrinos have different masses and, moreover, if the corresponding mass states are not eigenstates of the weak interaction, the neutrinos must be
formed of mixed states.14 That being the case, there
would be spontaneous oscillations between flavor
types as a beam of neutrinos propagated through
space.
Inexplicably, neutrinos have long been the only
massless fermions, and in that regard they’ve been
rather weird little things ever since their inception. It
should certainly be more “aesthetically” pleasing that
they have mass than not. Indeed, one might expect
mass to be a distinguishing characteristic among the
three flavors.
The thermonuclear reactions in the core of the Sun
only generate e-nu’s, and all the early detectors were
appropriately designed to respond to just that flavor.
With this in mind, suppose that neutrinos do oscillate
from one flavor into another; then to the extent that
they have so morphed, there would be fewer e-nu’s at
the Earth-bound detectors — ergo the SNP. There
would also be more mu-nu’s and tau-nu’s present, and
that’s exactly what the Sudbury Neutrino Observatory
in Canada recently reported (at the APS meeting in
April 2002): “The new results are so accurate, the
Sudbury team said, that ‘it is now 99.999 percent
probable that solar neutrinos change type before
reaching Earth.’”15
A Relativistic Explanation
The whole story of how Pauli conceived the neutrino in order to save the concept of conservation of energy, along with the solution16 of the SNP, makes for a
lovely pedagogical account of the way physics works
(which can be proffered both in and out of the classroom). Still, almost any elementary discussion of
neutrino oscillations inevitably brings up the question, “Why does the process of morphing necessitate
that neutrinos have mass?” Trying to answer that
query for a lay audience can be daunting, and even the
most assiduously developed response in terms of particle mixing and beating wave functions is not likely to
be compelling. So, is there a less esoteric answer?
The following is an attempt to address that ques166
tion. Since if neutrinos were massless they would
travel at c, special relativity is a natural place to seek
our straightforward explanation. The total energy (E )
of a particle of mass m is given by E = mc 2, and hence
E/
= mc 2. When m = 0, E/
= 0. Since E is generally
2,
2 and it follows that for
–/c
nonzero, 1/
= 0 = 1v
particles of zero mass, v = c. Particles of zero mass exist only at speed c. Notice that for particles having
some finite mass, the right side of E/
= mc 2 is finite,
and so 1/
cannot equal zero. Such particles must
travel at speeds less than c. How do massless particles
like the photon or graviton behave in time?
Though a complete definition of time is quite beyond our poor powers, let’s at least agree on what we
mean here by the word. Operationally, time is that
which is measured by a clock. Conceptually, time is a
measure of the rate at which events occur, and since an
event is an observable change in a physical system,
time is a measure of the rate at which physical change occurs. Time progresses as change takes place; it is informed by a succession of observably different physical states. Time is activated by change; if NOTHING
changes, time becomes irrelevant, clocks stop, and
equivalently “time stops.” In a universe (or in an isolated portion thereof ) where absolutely nothing happens, time is immeasurable. No clocks tick, no suns
rise, no people age — the so-called “flow” of time is
suspended.
With that in mind, imagine a clock of any sort
(perhaps even a neutral kaon) sailing through space at
a constant speed v. Recall the relativistic equation for
time dilation:
tS
tM = = tS,
2/
2
1
–vc
wherein > 1. Here tS is known as the “proper
time” interval, measured by someone for whom the
events occur at the same location in space — in this
case, someone who is stationary with respect to the
clock. Alternatively, tM is the time interval measured
by someone with respect to whom the clock is moving
at v. For example, suppose a kaon travels a distance l
in the lab from the point of creation to the point of
spontaneous decay. If a technician sees it in motion
for a time interval tM, then l = vtM = vtS, where
tS is the kaon’s proper lifetime as regulated by some
weak-interaction-internal-kaon clockwork.
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By contrast, consider a photon in vacuum — it
travels at speed c; thus = and any finite interval
tS will be perceived (tM) to be infinite. In other
words, someone “at rest” on Earth watching the photon fly by will see its internal clock take infinitely long
between ticks and tocks. He will see the photon’s time
pass infinitely slowly. Or more to the point, he will
see time stop for the photon. When not interacting
with matter, photons are timeless — they can travel for
billions of years (tM) without the passage of any corresponding proper time. And that would imply that a
free photon cannot undergo spontaneous change (as
perceived within its reference frame). Unlike kaons,
left on their own, photons are changeless.
That brings to mind at least two phenomena that
might at first seem to contradict the above conclusion:
the index of refraction (n = c/v) and pair production.
The speed v defining the index of refraction is the effective speed with which light propagates through a
material medium. It is not the speed of any one photon in the beam — that’s always c. Photons negotiating matter travel in the void between the atoms. The
speed v arises from the ongoing absorption and reemission of the light as it traverses the material medium. Photons only exist at c. As for pair production,
it cannot take place unless there is matter present that
interacts with the photon (thereby conserving momentum and energy). A photon from some distant
star sailing through empty space cannot spontaneously transform into an electron-positron pair.
To finally answer the question at hand, let’s apply
these ideas to a free neutrino traveling through the
void. If the neutrino is massless, it moves at c and so
with = , it must be timeless. A massless neutrino is
changeless; it cannot spontaneously transform. And it
certainly can’t oscillate. On the other hand, a neutrino that morphs, for whatever quantum mechanical
reason, obviously changes. Consequently, it cannot
be timeless and so must travel at less than c. If it travels at less than c, it must have mass. Ergo, morphing
neutrinos have mass.17
References
1. J.N. Wilford, “Neutrinos Have Mass, Panel Says,” The
New York Times, C9 (Oct. 24, 1995). This article
presages a growing popular interest in morphing neutrinos. See for example, G.P. Collins, “SNO Nus Is Good
THE PHYSICS TEACHER ◆ Vol. 41, March 2003
2.
3.
4.
5.
6.
7.
8.
News,” Sci. Am, 18 (Sept. 2001); A. Fisher, “Neutrinos
Weigh In,” Popular Sci., 24 (Sept. 2001); A. MacRobert
and D.Tytell, “Solar-Neutrino Problem Solved,” Sky &
Telescope, 18 (Sept. 2001); R. Kunzig, “The Unbearably
Unstoppable Neutrino,” Discover, 33 (Aug. 2001); P.
Weiss, “Physics Bedrock Cracks, Sun Shines In,” Sci.
News, 388 (June 23, 2001).
The Liquid Scintillating Neutrino Detector (LSND)
was the first accelerator-based experiment to generate
results suggesting neutrino oscillations. The work is
presently being repeated at Fermilab using the MiniBooNE detector, which contains 250,000 gallons of ultra-pure mineral oil.
In the standard model a neutrino is completely polarized; its spin vector is antiparallel to its linear momentum vector. Thus, the neutrino is left-handed, whereas
the antineutrino is right-handed. But these characteristics are relativistically invariant only if the particles travel at c. Otherwise an observer moving faster than the
neutrino in the same direction will see it receding, its
momentum vector reversed, and its handedness
inverted.
The neutrino deficit appears to depend on energy (solar
neutrinos have energies of only up to about 15 MeV).
Davis’s 600-ton (dry cleaning fluid) radiochemical detector in the Homestake mine in South Dakota found
about one-third of the predicted number of events. The
Kamiokande light water Cherenkov experiment (1986)
in Japan recorded about one-half of the anticipated solar neutrino events. Two more recent radiochemical gallium experiments, SAGE and GALLEX, which had
lower energy thresholds, reported values of roughly
70% of the theoretical predictions.
B. Pontecorvo, “Neutrino experiments and the problem
of conservation of leptonic charge,” Sov. Phys. JETP 26,
984–988 (1968).
M. Gell-Mann and A. Pais, Phys. Rev. 97, 1387 (1955).
See, for example, A. Das and T. Ferbel, Introduction to
Nuclear and Particle Physics (Wiley, New York, 1994),
pp. 214–249, or H. Frauenfelder and E.M. Henley,
Subatomic Physics (Prentice Hall, New Jersey, 1991), pp.
242–251.
—
K0 and its antiparticle K0, which have the same mass
and lifetime, are distinguishable by their difference in
strangeness. They are produced in strong-interaction
—
processes such as K+ + n → K0 + p and K– + p → K0 + n.
The corresponding two linear orthonormal combinations of weak interaction eigenstates are |K0 =
—
1
1
{|K20 + |K01} and |K0 = {|K20 – | K01}. The
2
2
167
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9.
10.
11.
12.
etcetera...
13.
factor 1/2 is for normalization of the wave functions.
The weak interaction does not conserve—
strangeness and
the initially orthogonal states |K0 and |K0 will not remain orthogonal in time.
D.J. Griffiths, Introduction to Quantum Mechanics
(Prentice Hall, New Jersey, 1995), pp. 20–24 and
44–48.
0
0
The K1- and K2-states actually decay, so there ought to
be a multiplicative exponential term present in the
phase that depends on the lifetime of each state. Since
this will not be the case with neutrinos, we’ll neglect it
here. See, for example, W.B. Rolnick, The Fundamental
Particles and Their Interactions (Addison Wesley, Reading, MA, 1994), pp. 219–224.
W.C Haxton and B.R. Holstein, “Neutrino physics,”
Am. J. Phys. 68, 15–32 (Jan. 2000).
The New Physics, edited by P. Davies (Cambridge University Press, New York, 1989), p. 380.
Each lepton comes in both particle and antiparticle varieties, making a total of six electrically charged and six
neutral leptons. These form three — electron, muon,
and tau — particle-neutrino couplets: e and e; and
; and and . Lepton number (L) is conserved in
all presently attainable processes involving any members of this family. Each of the three subsets of leptons
(electron-type, muon-type, and tau-type) was also
thought to conserve its own lepton number (Le, L,
L). This cannot, however, be strictly true if morphing
takes place.
14. For a more mathematically detailed discussion of neutrino mixing, see D.H. Perkins, “Neutrino Oscillations,” in Critical Problems in Physics, edited by V.L.
Fitch, D.R. Marlow, and M.A.E. Dementi (Princeton
University Press, Princeton NJ, 1997), pp. 201–219.
Also see W.B. Rolnick, The Fundamental Particles and
Their Interactions (Addison Wesley, Reading MA,
1994), pp. 375–378, and Kane, Modern Elementary
Particle Physics (Addison Wesley, Reading MA, 1993),
pp. 303–309.
15. E. Lane, “Study: Neutrino Has Mass,” Newsday, D3
(April 30, 2002).
16. “New SNO Data Resolves Solar Neutrino Problem,”
APS newsletter 2(6) 3 (June 2002).
17. The converse, massive neutrinos morph, is not necessarily true.
PACS codes: 03.65, 03.30, 14.00
Eugene Hecht is the author of a number of books, including two on the American ceramic artist George Ohr, and
six on physics. Among the latter are the introductory texts
Physics: Algebra Trig. and Physics: Calculus, both published by Brooks/Cole, and Optics, published by AddisonWesley. His main interests are the history of ideas and the
elucidation of the basic concepts of physics. He spends
most of his time teaching, studying physics, and training
for his third degree black belt in Tae Kwan Do.
Department of Physics, Adelphi University, Garden
City, NY 11530; [email protected]
etcetera... Editor
Albert A. Bartlett, Department of Physics,
University of Colorado, Boulder, CO 80309-0390
Relativity and Clocks
“Clocks accurate to one part in 1017 — a millisecond in three million years — will be
easily thrown out of whack by two relativistic effects. First there is time dilation:
moving clocks run slow. ‘A frequency shift of 10–17 corresponds to a time dilation
due to walking speed,’ Gibble said. The other confounder is gravity. The stronger
its pull, the slower time passes. Clocks at the top of Mount Everest pull ahead of
those at sea level by about 30 microseconds a year. ‘We already have to correct
for this effect when we compare clocks on different floors of our building,’ Sullivan
said. Raising a clock 10 centimeters will change its rate by one part in 1017.”1
1. W. Wayt Gibbs, “Ultimate clocks,” Sci. Am. 287, 86–93 (Sept. 2002).
168
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