# Introduction

The existence of neutrinos, weakly interacting particles that travel at
or near the speed of light, was hypothesized in the early 1930's to
explain aspects of beta decay; without a third "invisible" product of
the decay, the conservation of momentum would have been violated. The
neutrino proposed by Pauli had unique properties: it was extremely
light, it had no electric charge associated with it, and it interacted
so weakly it could not be detected. With the inclusion of this new
particle, momentum is conserved in beta decay processes because the
otherwise missing momentum is carried away by the undetected neutrino.
Because a neutrino can traverse a thousand light years of lead with a
very small chance of stopping, existence of these nearly intangible
particles was not verified unambiguously for twenty years.

Currently accepted theories (the "Standard Model") identify three
separate families of lepton: the electron, the muon, and the tau, each
with its associated neutrino (the electron neutrino, muon neutrino, and
tau neutrino). These families are referred to as "flavors".

Neutrinos are produced by natural radioactivity, nuclear reactors,
high-energy accelerators like the Tevatron at Fermilab, and fusion
processes within our sun and other stars. Because neutrinos interact so
weakly that they can pass through vast amounts of matter, they make
useful tools for the study of the interior of our sun and the collapsing
core of supernovae. Experiments designed for astrophysical studies need
large volumes of material to maximize the chance of stopping a neutrino
within their detector. In 1971, an astrophysics experiment (Homestake)
began to see a large discrepancy between the predicted number of
neutrinos incident on the Earth and the number actually detected. While
the results were not conclusive, other experiments built to measure
neutrino flux with different techniques began to see a similar
discrepancy. By 1980, there was considerable interest in the field of
neutrino astronomy.

Beams of neutrinos from accelerators were studied to search for the
"neutrino oscillation effect". The lack of the expected numbers of
neutrinos could be explained if neutrinos of one flavor could transform
into a neutrino of a different flavor. The solar neutrinos could have
oscillated into a flavor to which the detectors were not sensitive. The
results of these experiments have sparked a great deal of activity and
controversy.

This review is divided into six sections. The first section will present
the astrophysical and cosmological results that suggest neutrinos may
have non-zero mass and the mathematical formalisms developed to describe
possible neutrino oscillations. In the second section, a review of
kinematical searches for neutrino masses will provide the most stringent
upper limits on neutrino masses and detail the methods used to define
these limits. The third section will focus on the most recent and
controversial accelerator experiment and compare its result to the body
of existing work. The fourth section will provide an example of a
theoretical prediction based on the existing data set, and the fifth
section will outline how the DZero experiment might go about searching
for non-zero neutrino mass and what its chances are. Finally, the sixth
section will address the question, "What is the best design for a
neutrino experiment?"

Several experiments will be discussed in detail in the text of this
review. For a listing of other neutrino experiments and brief
descriptions, consult Appendix A. A copy of the exam questions prepared
by Professor Gregory R. Snow upon which this review is based is included
as Appendix B. Questions from committee members subsequent to the
initial version of this exam are addressed in Appendix C.

# Indications of Neutrino Mass from Cosmology and Astrophysics

The tentative evidence for massive neutrinos stems primarily from
astrophysical and cosmological studies. Three empirical observations
lead us to suspect that neutrinos may have small but non-zero mass: the
solar neutrino problem, the atmospheric neutrino deficit, and the
missing matter of the universe.

The first observation involves generally accepted theories that model
stellar fusion processes and thus predict the number of neutrinos
produced within stars like our sun. Together, these theories comprise
what is called the Standard Solar Model (SSM). Many aspects of the SSM
have been experimentally verified through analysis of the sun's
electromagnetic spectrum or through helioseismology. If experiment
conflicts with a major prediction of the model, many basic theories must
be reworked. In fact, the SSM predicts many more observable electron
neutrinos from the sun than are detected on Earth; this neutrino deficit
is termed the "solar neutrino problem".

Secondly, cosmic rays incident upon Earth's atmosphere produce neutrinos
as they collide with particles in the upper atmosphere. Cosmic ray
interactions predominantly produce showers of pions and kaons, which
decay according to the following mechanism:

$p\text{or}K \rightarrow \nu_{m} + m$,

which is followed by decay of the muon,

$m \rightarrow e + \nu_{e} + \nu_{m}$.

With knowledge of only these basic processes, the ratio of muon
neutrinos to electron neutrinos should be very close to 2. Detailed
corrections to this number that include rare decay chains do not
substantially change this value. Surprisingly, the measured ratio of
neutrinos $\nu_{m}/\nu_{e}$ from cosmic rays differs from this basic
prediction by 40%. This deficit of atmospheric muon neutrinos is not
reconcilable with the current understanding of the way particles shower.

The third hint that neutrinos may be massive arises from cosmological
considerations. By seeking correlations between the distance to the
furthest stars, the rate at which they are receding, and the age of the
oldest observable stars, cosmologists construct a series of models that
describe the formation of the universe from the initial moments to its
present state. Reasonable constraints (e.g., the oldest stars cannot be
older than the universe) imply limits on the maximum and minimum total
mass of the universe. Luminous matter accounts for only 10% of the mass
required to model the universe in its current observed state, so 90%
must be accounted for by non-luminous matter, or so called "dark
matter". Massive neutrinos are a favored candidate for portions of this
missing matter.

In the following subsections, we consider each of these problems in more
detail. A framework for explaining the source of these effects with
massive neutrinos will be developed.

### The Solar Neutrino Deficit

Table 1 depicts the various nuclear reactions that, according to the
SSM, occur within the sun. Although neutrinos are produced in several
different steps, the following discussion centers around the neutrinos
produced in the initial pp fusion, the Beryllium reaction, and the Boron
reaction. For reactions that produce neutrinos, the resulting neutrino
energy is listed as E. The table also lists the relative frequency of
each reaction as a termination percentage (T%). For instance, of all the
processes that terminate, 15% of them end with a lithium reaction.

![](media/image1.emf){width="4.260416666666667in"
height="4.104166666666667in"}

> **Table 1:** Nuclear processes occurring within the sun \[1\]. Two
> reaction chains are listed; both chains begin with hydrogen nuclei
> (protons) and end with helium production. T% is the termination
> percentage, describing the frequency of the reaction. For reactions
> that create neutrinos, the energy of the neutrino is indicated as E
> (MeV). Not shown is the CNO cycle which is believed to contribute only
> 2% of the sun's luminance.

The SSM predicts a well-defined number of neutrinos incident upon the
Earth in a given period of time. Table 2 lists the neutrino flux
expected at the Earth's distance from the sun, categorized by process.
The uncertainty in the calculation is expressed in parentheses. The most
abundant solar neutrinos are the least energetic and also the most
difficult to detect.

![](media/image2.emf){width="2.3340277777777776in"
height="2.263888888888889in"}

**Table 2:** Neutrino flux is calculated using the Standard

Solar Model for a distance equivalent to the Earth's orbit. \[1\]

Experiments have been designed to detect this flux, most notably the
Homestake mine experiment in South Dakota and Kamiokande in Japan (see
Appendix A for descriptions). Having energy thresholds of 814 keV and
7.5 MeV respectively, these experiments are not capable of detecting the
low energy neutrinos from the pp reaction. The neutrinos from the rare
Boron decay comprise the majority of the solar neutrino data sample.
Figure 1 depicts the energy spectra of the produced neutrinos as
predicted by the SSM. ![](media/image3.emf){width="3.0625in"
height="1.6875in"}

Unfortunately, only half the predicted numbers of $\nu_{e}$ are
detected. One attractive solution involves neutrinos oscillating from
one flavor to another (in this case $\nu_{e} \rightarrow \nu_{m}$) via
the Mikheyev, Smirnov, Wolfenstein effect (MSW). \[3\] A prerequisite
for this effect is a non-zero mass difference between the two flavors of
neutrino, requiring at least one of their masses be non-zero.

The energy of a neutrino within the sun includes its kinetic energy, its
mass energy, and its potential energy. The potential energy results from
weak (charged or neutral current) interactions with the surrounding
protons, neutrons, and electrons \[4\]:

$V(\nu_{e}) = \frac{G_{F}}{\sqrt{2}}\left\lbrack 2N_{e} + \left( - 1 + 4\text{Sin}^{2}q_{W} \right)N_{e} + \left( 1 - 4\text{Sin}^{2}q_{W} \right)N_{P} - N_{N} \right\rbrack$,

where G~F~ is the Fermi coupling constant (a ratio of the mass of the W
and the weak fine structure constant) and N~x~ is the surrounding
density of electrons, protons, and neutrons. θ~W~ is the weak mixing
angle, not to be confused with the mixing angle between neutrinos to be
introduced later. The first term represents W exchange between the
neutrino and electrons. The remaining terms result from Z exchange
between the neutrino and electrons, protons, and neutrons, respectively.
If the sun is electrically neutral overall, the middle two terms cancel
exactly. Also, the number of neutrons in the sun is only one third to
one sixth that of electrons \[4\], so the fourth term can be neglected,
leaving only the first term. The total energy of an electron neutrino
within the sun is then

$E_{\nu_{e}} = \sqrt{p_{\nu_{e}^{2}} + m_{\nu_{e}^{2}}} + \sqrt{2}G_{F}N_{e}$,

where the last term describes the interaction potential energy between
the electron neutrino and the surrounding electrons. The electron
density, N~e~, is expected to have a maximum near the sun's core and
decrease with increasing distance from the center of the sun.

The expression for the energy of some other neutrino,$\nu_{x}$, within
the sun does not include the W exchange term as muon and tau neutrinos
within the sun are not surrounded by a dense medium of their respective
charged leptons. Neglecting the Z-exchange terms as before, the energy
of this neutrino is simply

$E_{\nu_{x}} = \sqrt{p_{\nu_{x}^{2}} + m_{\nu_{x}^{2}}}$.

Now, we search for a form similar to the electron neutrino expression,
so

$$\begin{array}{r}
E_{\nu_{x}} = \sqrt{p_{\nu_{x}^{2}} + m_{\nu_{x}^{2}}} = \sqrt{p_{\nu_{x}^{2}} + m_{\nu_{x}^{2}} + m_{\nu_{e}^{2}} - m_{\nu_{e}^{2}}} \\
 \\
p_{\nu_{x}}\sqrt{1 + \frac{m_{\nu_{e}^{2}}}{p_{\nu_{x}^{2}}} + \frac{m_{\nu_{x}^{2}}}{p_{\nu_{x}^{2}}} - \frac{m_{\nu_{e}^{2}}}{p_{\nu_{x}^{2}}}}\text{.}
\end{array}$$

Using the binomial theorem to approximate the square root, we define
$m_{{\nu_{x}}^{2}} - m_{\nu_{e}^{2}} = {Dm}^{2}$ and

$$\begin{array}{r}
E_{\nu_{x}}»p_{\nu_{x}}\left( 1 + \frac{1}{2}\frac{m_{{\nu_{e}}^{2}}}{p_{{\nu_{x}}^{2}}} + \frac{1}{2}\frac{{Dm}^{2}}{p_{{\nu_{x}}^{2}}} \right) \\
 \\
»p_{\nu_{x}}\left( 1 + \frac{1}{2}\frac{m_{{\nu_{e}}^{2}}}{p_{{\nu_{x}}^{2}}} \right) + p_{\nu_{x}}\left( \frac{1}{2}\frac{{Dm}^{2}}{p_{{\nu_{x}}^{2}}} \right) \\
 \\
»\sqrt{p_{{\nu_{x}}^{2}} + m_{\nu_{e}^{2}}} + \frac{{Dm}^{2}}{2p_{\nu_{x}}}\text{.}
\end{array}$$

If flavor mixing can occur, a non-zero transition probability arises
only when the energies of each flavor‑state are equal. Thus, setting
$E_{\nu_{x}} = E_{\nu_{e}}$, this necessary condition becomes

$\sqrt{p_{\nu_{x}^{2}} + m_{\nu_{e}^{2}}} + \frac{{Dm}^{2}}{2p_{\nu_{x}}} = \sqrt{p_{\nu_{e}^{2}} + m_{\nu_{e}^{2}}} + \sqrt{2}G_{F}N_{e}$.

By conservation of momentum, $p_{\nu_{x}} = p_{\nu_{e}}$, so the square
root terms cancel. Finally, because the neutrino is highly relativistic,
the total energy is a good approximation for the momentum. The condition
for flavor oscillation within the sun becomes

Because the Fermi energy of Eqn. 1 is quite small, neutrinos may undergo
a flavor change only if their energy is high compared to the mass
difference making the left side of the equation similarly small. Thus,
the MSW formalism predicts a lack of "high-energy" solar neutrinos
compared to the SSM while leaving the prediction for "low-energy"
neutrinos unchanged. The precise energy value required to satisfy Eqn. 1
must be determined by experiment. For high-energy neutrinos, the
probability of a flavor change is not unity even if Eqn. 1 is satisfied;
discussion of mixing angle is reserved for the next section.

To explain solar neutrino data, neutrinos must either be massive and
allow mixing between flavors, or aspects of the SSM must be reworked.
Theorists have offered some explanations other than the MSW solution
that do not require massive neutrinos (modifications to the current
solar convection scheme) so the evidence for neutrino mass from solar
data is not yet conclusive.

### The Atmospheric Neutrino Deficit

Very high-energy particles (including photons) originating from outside
our solar system can collide with atoms in our upper atmosphere and
generate neutrinos. Predicting the number of muon and electron neutrinos
that are generated by these "cosmic rays" is possible; however, the
theoretical predictions of the total flux of both
$\nu_{m}\text{and}\nu_{e}$ vary by as much as 30%, depending upon which
models are used. Because the absolute fluxes are so difficult to
calculate, a prediction of the muon neutrino flux relative to the
electron neutrino flux is generated. Predictions of the ratio of
neutrino fluxes agree to within 5% of one another and are therefore
suitable to compare to the ratio measured by experiment.

In atmospheric neutrino experiments, neutrinos are detected from all
directions, not just from above. This observation indicates that
neutrinos generated in the atmosphere on the far side of the Earth
propagate \~10,000 km through the planet to reach the detectors. If one
considers only "upward-going" neutrinos and "downward-going" neutrinos,
one finds that the ratio $\nu_{m}/\nu_{e}$ is anomalously low for
upward-going neutrinos but is in agreement with theoretical predictions
for downward-going neutrinos. Note that these measurements were made by
the Kamiokande experiment, which is only sensitive to neutrinos with an
energy above 5 MeV \[6\]; there is no evidence that neutrinos of lower
energy exhibit this anomalous behavior. Furthermore, not all experiments
have confirmed this high-energy muon neutrino deficit. Figure 2 displays
the ratio$\nu_{m}/\nu_{e}$ determined by experiment divided by the
predicted ratio of$\nu_{m}/\nu_{e}$. This so-called "ratio of ratios"
has an expected value of unity. ![](media/image4.emf){width="2.0625in"
height="1.525in"}

The deficit observed by Kamiokande can be explained if the$\nu_{m}$
oscillate to one of the other two flavors of neutrinos. The following
derivation does not require the neutrino to travel through a dense
medium as the MSW calculation did, so the following effect is referred
to as a "vacuum oscillation". If neutrinos are massive and can mix, the
flavor eigenstates result from a linear combination of mass eigenstates.
In matrix form, this relationship is expressed as

$\begin{pmatrix}
\nu_{e} \\
\nu_{m} \\
\nu_{t}
\end{pmatrix} = \boxed{M}\begin{pmatrix}
\nu_{1} \\
\nu_{2} \\
\nu_{3}
\end{pmatrix}$, where

$$\boxed{M} = \begin{pmatrix}
\text{Cos}q_{em}\text{Cos}q_{et} & \text{Sin}q_{em}\text{Cos}q_{et} & \text{Sin}q_{et} \\
 - \text{Sin}q_{em}\text{Cos}q_{\text{mt}} - \text{Cos}q_{em}\text{Sin}q_{et}\text{Sin}q_{\text{mt}} & \text{Cos}q_{em}\text{Cos}q_{\text{mt}} - \text{Sin}q_{em}\text{Sin}q_{et}\text{Sin}q_{\text{mt}} & \text{Sin}q_{\text{mt}}\text{Cos}q_{et} \\
\text{Sin}q_{em}\text{Sin}q_{\text{mt}} - \text{Cos}q_{em}\text{Cos}q_{\text{mt}}\text{Sin}q_{et} & - \text{Cos}q_{em}\text{Sin}q_{\text{mt}} - \text{Sin}q_{em}\text{Cos}q_{\text{mt}}\text{Sin}q_{et} & \text{Cos}q_{et}\text{Cos}q_{\text{mt}}
\end{pmatrix}$$

and θ~xy~ are the *mixing angles* between flavors*,* to be determined by
experiment. This unitary transformation matrix$\boxed{M}$ is similar in
form to rotation matrices in classical mechanics. To simplify the
calculation, assume that muon neutrinos created in the atmosphere
oscillate only into tau neutrinos and that oscillations between electron
neutrinos and tau neutrinos can be neglected (let θ~eμ~ = θ~eτ~ = 0 in
the matrix). Then, the matrix simplifies to

$\boxed{M} = \begin{pmatrix}
1 & 0 & 0 \\
0 & \text{Cos}q_{\text{mt}} & \text{Sin}q_{\text{mt}} \\
0 & - \text{Sin}q_{\text{mt}} & \text{Cos}q_{\text{mt}}
\end{pmatrix}$.

Explicitly, the muon neutrino eigenstate expressed in terms of neutrino
mass states is

$\nu_{m} = \left( \text{Cos}q_{\text{mt}}\nu_{2} + \text{Sin}q_{\text{mt}}\nu_{3} \right) = \begin{pmatrix}
0 \\
\text{Cos}q_{\text{mt}} \\
\text{Sin}q_{\text{mt}}
\end{pmatrix}$

Consider an atmospheric muon neutrino created at time t=0. At some later
time, the mixture of mass states has evolved into

$\nu(t) = \begin{pmatrix}
0 \\
\text{Cos}q_{\text{mt}}e^{- {iE}_{2}t} \\
\text{Sin}q_{\text{mt}}e^{- {iE}_{3}t}
\end{pmatrix}$,

where E~2~ and E~3~ are the energies associated with mass-states 2 and 3
given the initial momentum. The probability of the neutrino oscillating
into a tau neutrino is

$P\left( \nu_{m} \rightarrow \nu_{t} \right) = \left| \left\langle \nu_{t} \middle| \nu(t) \right\rangle \right|^{2} = \left| \text{Cos}q_{\text{mt}}\text{Sin}q_{\text{mt}}\left( e^{- {iE}_{2}t} - e^{- {iE}_{3}t} \right) \right|^{2}$.

Because the mass of the neutrino is small compared to its momentum, we
may approximate the energy as

$E_{j} = \sqrt{p^{2} + m_{j^{2}}} = p\sqrt{1 + \frac{m_{j^{2}}}{p^{2}}}»p\left( 1 + \frac{1}{2}\frac{m_{j^{2}}}{p^{2}} \right)$,

where j indicates a particular mass state (either ν~1,~ ν~2,~ or ν~3~),
and we have again approximated using the binomial expansion. The
probability becomes

$P\left( \nu_{m} \rightarrow \nu_{t} \right) = \left| \text{Cos}q_{\text{mt}}\text{Sin}q_{\text{mt}}e^{- ipt}\left( e^{- i\frac{m_{2^{2}}}{2p}t} - e^{- i\frac{m_{3^{2}}}{2p}t} \right) \right|^{2}$.

Using

$$\begin{array}{r}
\left| \frac{1}{2}\left( e^{- ix} - e^{- iy} \right) \right|^{2} = \frac{1}{4}\left( 2 - \left( e^{i(y - x)} + e^{- i(y - x)} \right) \right) = \frac{1}{4}\left( 2 - 2\text{Cos}(y - x) \right) \\
 \\
\frac{1 - \text{Cos}(y - x)}{2} = \text{Sin}^{2}\left( \frac{y - x}{2} \right)
\end{array}$$

the probability becomes

$P\left( \nu_{m} \rightarrow \nu_{t} \right) = 4\text{Cos}^{2}q_{\text{mt}}\text{Sin}^{2}q_{\text{mt}}\text{Sin}^{2}\left( \frac{m_{3^{2}} - m_{2^{2}}}{4p}t \right) = \text{Sin}^{2}(2q_{\text{mt}})\text{Sin}^{2}\left( \frac{m_{3^{2}} - m_{2^{2}}}{4p}t \right)$.

Because the neutrino is very relativistic, the "flight-time" is
simply-related to the distance travelled,$t»L/c$and the momentum is
simply-related to the energy,$p»E/c$, thus

$P\left( \nu_{m} \rightarrow \nu_{t} \right) = \text{Sin}^{2}(2q_{\text{mt}})\text{Sin}^{2}\left( \frac{1}{4}{Dm}^{2}\frac{L}{E} \right)$.

This equation describes the probability of a muon neutrino oscillating
into a tau neutrino if we may assume that the energy is relativistic and
neglect other flavor oscillations. The values of Sin^2^ (2θ~μτ~) and
∆m^2^ are physical constants to be determined by examination of data at
different values of $L/E$. The oscillation length $L_{\text{osc}}$, the
distance required for the neutrino oscillation probability to oscillate
from zero to its maximum and back to zero, is found by

$$\begin{array}{r}
\text{Sin}^{2}\left( p\frac{L}{L_{\text{osc}}} \right) = \text{Sin}^{2}\left( \frac{1}{4}{Dm}^{2}\frac{L}{E} \right) \\
 \\
 \Rightarrow L_{\text{osc}} = \frac{4pE}{{Dm}^{2}}\text{.}
\end{array}$$

In the literature, the probability equation is often expressed with
explicit units for the mass, distance, and energy:

Here, the numerical factor of 1.27 results from the unit conversion.
Note that this general result does not depend upon the flavor of the
target neutrino; our mixing matrix,$\boxed{M}$, would have simplified
similarly for any one oscillation pair. While this derivation explicitly
assumed mu to tau, the result would be the same for mu neutrinos to
electron neutrinos, hence the subscript μτ is omitted from the mixing
angle.

This derivation has been criticized in the literature for considering
only one type of mixing and neglecting the other two. In a three‑flavor
mixing analysis, equal and opposite phases may cancel, depending on the
mixing angles, yielding very different results for the same values of
∆m^2^ and θ. \[7\] Because there is no *a priori* reason to neglect two
of three mixings, the standard two‑flavor analysis presented here may be
flawed. \[8\]

To explain the observed atmospheric neutrino mixing, we require that for
$L»\text{20}\text{km}$ the transition probability be nearly zero (our
downward neutrinos) and for $L»\text{10},\text{000}\text{km}$ the
probability be appreciable. We find that for atmospheric neutrinos of 6
GeV energy or higher, the best fit to *all* Kamiokande data is
${Dm}^{2}»\text{10}^{- 2}\text{eV}^{2}$, with an upper limit of around
${Dm}^{2} \leq \text{10}^{- 1}\text{eV}^{2}$. Despite the apparent
popularity of representing the entire data set as a single point, data
analysis should proceed in a different manner.

The ratio of all muon neutrinos to all electron neutrinos is not useful
for an oscillation search, the important parameter is $L/E$. Each
experiment should bin its data in terms of $L/E$, but most newer
experiments lack the statistics to do so at this time.
![](media/image5.emf){width="1.85in"
height="1.85in"}![](media/image6.emf){width="1.8972222222222221in"
height="1.8972222222222221in"}

A plot of Eqn. 2 yields a sinusoidal function with a maximum of unity, a
minimum determined by the mixing angle, and a period determined by ∆m^2^
(Figure 4a). Experiments are not capable of measuring the neutrino ratio
with infinitely fine resolution in $L/E$, so Eqn. 2 should not be used
to perform a fit. The *average* of Eqn. 2 over a given interval in $L/E$
must fit the observed data. This interval should match the bin size of
the data.![](media/image7.emf){width="2.6770833333333335in"
height="2.6770833333333335in"}![](media/image8.emf){width="2.6770833333333335in"
height="2.6770833333333335in"}

### The fit to the multi-GeV Kamiokande data using this modified Eqn. 2 yields the preferred values

### $$\begin{array}{r}
{Dm}^{2} = 0\text{.}\text{010} \pm 0\text{.}\text{0024} \\
q = 0\text{.}\text{66} \pm 0\text{.}\text{24},
\end{array}$$

where a bin size of 0.7 \*$L/E$ was used to produce a smooth function
(Figure 4b). The steep drop between the first and second points drives
the ∆m^2^ fit; beyond the second point, the solution quickly approaches
its average value, which is determined by the mixing angle θ. To reduce
the uncertainty of the fit with Eqn. 2, several more points are required
along the first quarter-cycle of the of the solution. The very large
uncertainty on the value obtained for theta from the fit underlines the
need for more data in the low $L/E$ range. Note that several experiments
found no disagreement with the predicted ratio of neutrinos. Fitting all
the data at once is difficult, so theorists tend to exclude some data
sets from consideration when they make their predictions.

Some measurements of the neutrino ratio $\nu_{m}/\nu_{e}$ disagree
significantly with theoretical predictions that do not incorporate
neutrino mixing. Vacuum oscillations of massive muon neutrinos to
another flavor could explain this deficit, but the current data samples
make comparisons from one experiment to another difficult. Fits to
existing data sets yield large uncertainties that cannot decrease until
more data has accumulated.

###  The Dark Matter Problem

According to the generally accepted Big Bang hypothesis, a very large
number of neutrinos must have been produced during the birth of the
universe. While some interacted during the early moments of expansion,
most remained as a uniform sea of particles. The neutrinos are now
believed to have "cooled" to a very low energy and experience a
negligible number of interactions. Calculations estimate a particle
density of 113 "relic neutrinos" of each flavor per cubic centimeter. In
addition to this number, cosmic neutrinos are created from type II
supernovae, which produce $\text{10}^{\text{58}}$ neutrinos of \~10 MeV
each. \[10\] About 10 supernovae occur each second in the observable
universe. Finally, stars generate a small percentage of their power in
the form of neutrinos during the normal course of their lives. While
neutrinos interact very weakly, the universe is very densely populated
by them. If neutrinos have mass, no matter how small, they will
collectively have a large effect on cosmological calculations.

A typical cosmological model of the early universe invokes inflation, an
accelerated expansion, in the first moments. The model explains the
measured size of the universe and includes some details of galaxy
formation, but it requires the density of the universe to be precisely
that density that allows the universe to expand forever at a rate that
asymptotically approaches zero. If the density were greater than the
*critical density*, the universe would eventually collapse in upon
itself; the theory of inflation would also be invalid, leaving us with
no way to explain the universe as it is observed.

Tests of the rate of expansion, as described by the Hubble constant,
indicate that the density is very close to the critical value. To reach
critical density, most of the matter in the universe (90%) must be
non-luminous; the stars we observe are not massive enough. The
constraint of explaining the observable universe allows one to make
educated guesses about the possible masses of neutrinos.

If all of the non-visible matter in the universe consisted of massive
neutrinos, critical density would be reached if the mass of the three
known neutrino flavors (just one of each) summed to 30 eV. \[11\] This
value provides a hard upper limit on $\sum_{}^{}m_{\nu}$. This initial
upper limit is further modified by dividing dark matter into a hot, high
velocity component and cold, low velocity component. Hot dark matter
should comprise approximately 30% of the total dark matter mass. \[12\]
Identifying neutrinos as the hot component of the missing matter, the
constraint on neutrino masses becomes

$0 \leq \sum_{\alpha = e,m,t}^{}m_{\nu_{\alpha}}\text{10}\text{eV}$.

Cosmologists are anxious to determine whether the universe is open or
closed. Recent measurements have not yet been able to make the
determination, so there is a great deal of interest in learning more
about dark matter to refine models. Neutrinos are the only candidate
particles for hot dark matter known to exist in nature; all other
candidates are hypothetical. \[13\]

### Summary

Two formalisms, the vacuum oscillation formula and the MSW effect, have
been introduced to explain inconsistencies between astrophysical
theories and experiment. While the MSW effect is negligible except for
extremely dense media, the vacuum oscillation formula is used
extensively to make predictions for accelerator experiments.

The preceding three arguments suggest that neutrinos may be massive and
consequently mix between flavors. The 40% deviation of the number of
observed solar neutrinos from theoretical predictions holds limited
weight due the model-dependence of initial flux calculations.
Atmospheric neutrino measurements are not consistent with one another;
some experiments deviate significantly from predictions, while others
observe no deviation. In addition, massive neutrinos would satisfy the
need to identify some portion of the missing dark matter required by
most cosmological theories, but there is no compelling cosmological
reason neutrinos *must* be massive.

Astrophysical phenomena are not manipulable systems where inputs can be
varied to fully study the outputs, as is the case with most terrestrial
experiments. Instead, experimentalists must rely heavily on their models
of solar activity and cosmic ray showers, introducing a great deal of
uncertainty into the predicted results. The determination of neutrino
masses and mixing angles (if any) will most likely result from
terrestrial experiments and not from additional astrophysics searches.

# Kinematic Searches for Neutrino Mass

The most direct searches for neutrino masses study the decay of
particles at rest. By conservation of momentum, the vector sum of all
product momenta must be zero. Furthermore, the total energy (including
mass energy) of the final states must equal the initial total energy.
Because neutrinos are so difficult to detect, one measures the energy of
the "visible" products of a decay and calculates the missing energy
required to balance the initial conditions. This calculation specifies
the mass of the non-visible particle, assuming the energy, direction,
and identities of the other final states are determined.

Consider a simple two-body decay. A particle of mass M at rest decays
into two particles with masses and momenta m~1~, p~1~ and m~2~, p~2­~,
respectively (Figure 5).

Because there are only two products of this decay, the momentum of
particle 1 must be equal in magnitude to particle 2's momentum and
opposite in direction. The final energy must always be equal to the
initial energy. Even if particle 2 is not detected, its mass and energy
are still completely determined by a measurement of particle 1. For
instance the mass of particle 2 is expressed as

In a decay with three or more products, the calculation is not so
simple. The rigid requirement of equal and opposite momenta in the
two-body decay no longer applies. The momentum of the third particle, if
very massive, is impossible to measure accurately because the momentum
is smaller than the resolution power of detection techniques. In a
three-body decay, a study of the momentum *spectrum* of the fast-moving
visible product will be required.

The following is a list of preferred reactions used to search for the
neutrino masses of the three standard model flavors. In each case, the
experimental method setting the most stringent mass limit is described.

### Tritium β decay

In this decay,
$_{1}^{3}H \rightarrow_{2}^{3}\text{He} + e^{-} + {\overline{\nu}}_{e}$,
a Tritium atom with one proton and two neutrons converts to a Helium
isotope (2 protons 1 neutron), emitting a beta particle (an electron)
and an electron anti-neutrino. The massive nucleus remains essentially
at rest, making a momentum measurement impossible.
![](media/image11.emf){width="3.397222222222222in"
height="2.2597222222222224in"}

A study of the electron momentum spectrum should reveal the presence or
absence of neutrino mass. If the neutrino has m^2 ^= 0, the electron
momenta of many decays should have a distribution that decreases to zero
frequency at a value predicted by a two-body decay of tritium into only
a helium isotope and an electron. (If the neutrino has zero mass and
vanishingly small momentum, then the electron must have all the momentum
required to balance the He^3^ recoil.) If instead, the neutrino has
non-zero m^2^, the electron energy spectrum must approach zero frequency
at some upper limit prior to the zero-mass two-body prediction.

Instead, experimenters find that the tail consistently exhibits an
*excess* of events (Figure 6), indicating a negative value for $m^{2}$
and a seemingly imaginary mass. While most experiments have
uncertainties that make their m^2^ result consistent with zero, some
results indicate a significantly negative value.

One explanation of this non-physical result is that a molecular ion
HeT^+^ forms during the course of the experiment. While models of this
ion do not indicate an excitational mode that would explain the observed
excess of events, no confirming experimental data on this ion's
excitational spectrum exist. An experiment to investigate the orbital
spectrum of the helium-tritium molecular ion would require a detector
sensitive to very-ultraviolet photons but such devices have not yet been
developed. \[15\]

The current limits on the mass of the electron neutrino have been set by
tritium β decay experiments. Because the center of the probability
distribution of m^2^ has a nominally negative value, the upper limit of
3.5 eV is lower than it would be had the central value been zero. It is
perhaps better to quote a higher limit as the negative value may
indicate a poor understanding of systematic biases in the method or
contamination of the data sample. The method used to determine the mass
limit is discussed further at the end of this section.

### Pion Decay

Assamagan et al. have designed a small unnamed experiment to search for
muon neutrino mass at the Paul Scherrer Institute in Switzerland. The
simple experiment involves a 590 MeV proton beam incident on a graphite
target, producing copious π^+^ mesons. A fraction of these pions come to
rest within the target near the surface. When these "surface pions"
decay via $p^{+} \rightarrow m^{+} + u_{m}$, some of the generated muons
will have the correct momentum vector to enter the experimental
apparatus.

The apparatus consists of a momentum selector designed to accept a 1%
wide momentum band centered on 29.75 MeV/c. This band kinematically
selects only those muons that likely originated from the decay of a
surface pion. The momentum selector also acts as a charge identifier.
After passing through several collimators, the muon beam strikes a
simple magnetic spectrometer consisting of a homogenous magnetic field
and a silicon micro-strip detector.

The experimenters invoke the CPT theorem (they assume the mass of the
$p^{-}$ is equal to the mass of the $p^{+}$) and use the most recent
data for the mass of the $p^{-}$ in their calculation. Because there are
two different values of pion mass from two recent analyses, they perform
their calculation twice, yielding

$m_{{\nu_{m}}^{2}}\text{=-}0\text{.}\text{148} \pm 0\text{.}\text{024}M\text{eV}^{2}/c^{2}$

and

$m_{{\nu_{m}}^{2}}\text{=-}0\text{.}\text{022} \pm 0\text{.}\text{023}M\text{eV}^{2}/c^{2}\text{.}$

Because the first value is negative by 6.2 standard deviations, they
choose to exclude the pion mass that results in value ( a ). The second
neutrino mass value ( b ) is quoted as the final result. They calculate
a pion mass that corresponds to a value for massless neutrinos and quote
it as a lower limit on the true pion mass. The magnitude of this change
is slightly larger than the one sigma magnitude of the quoted error on
the "original" pion mass. \[16\]

### Tau Decay

Measurements of tau decay into pions and a neutrino have been performed
by the ARGUS collaboration, the OPAL collaboration, and more recently
ALEPH, which set the current upper limit on the mass of the tau
neutrino. Unlike the other experiments described in this document,
ARGUS, OPAL, and ALEPH are collider detectors and were not designed for
the specific purpose of neutrino physics. This subsection will focus on
ALEPH, which was installed in the LEP e^-^e^+^ beam-line at CERN to
study Z boson decays.

Similar to many collider detectors, ALEPH is composed of many
sub-detectors arranged in concentric cylinders that surround a beam
pipe. The innermost sub-detectors effectively track the momentum and
charge of product particles as they emerge from the beam pipe. Charge is
indicated by the curvature of the trajectory within a 1.5 Tesla magnetic
field. \[17\] The surrounding calorimeter measures particle energies and
the outermost layer detects muon position and momentum.

From a sample of 76 $e^{+}e^{-} \rightarrow t^{+}t^{-}$ events, ALEPH
selects events with at least one tau fragmenting into five daughters,
each with an energy greater than 100 MeV. If one or more of the
daughters is identified as an electron or photon, the event is rejected.
\[18\] Several other cuts are made to ensure the daughters are correctly
identified as pions. The remaining 23 events are classified as
$t \rightarrow 5p$. ALEPH also identified two events from the tau-tau
sample as $t \rightarrow 5p + p^{0}$. These two events are included in
the mass analysis.

The invariant mass of the hadrons from the events is used as the input
for a *maximum likelihood function*. The function describes the
probability of obtaining the observed distribution of pions for various
possible neutrino masses. The functional form includes selection
efficiency, energy-resolution effects, and the theoretical distribution
function of the decay mode. This method does not yield a central value
directly because input parameters in the non-physical region do not
contribute to the likelihood. Instead, this method directly calculates
an upper limit on the probable neutrino mass. ALEPH finds the value of
the tau neutrino mass is 95% likely to be below 23.8 MeV. \[18\] While
one might expect the actual value for the tau neutrino mass to be less
than 100 eV or zero, ALEPH has set the best empirical limit to date.
Fixed target experiments dedicated to the search for the mass of the tau
neutrino should be able to set much more stringent limits.

### Neutrinoless Double Beta Decay

An arbitrary element of atomic number Z and atomic mass A, can undergo a
double beta decay according to the following mechanism:

$_{Z}^{A}X \rightarrow_{Z + 2}^{A}X + e^{-} + e^{-} + \nu_{e} + \nu_{e}$,

which is simply two beta decays (as discussed for tritium) occurring
simultaneously. One extension of the standard model hypothesizes that it
may be possible to emit two beta particles without emitting neutrinos.
Such a neutrinoless decay requires that neutrinos be their own
anti-particles. \[11\] This violation of lepton number is not allowed
for SM (Dirac) neutrinos, but is described in some super-symmetric
models with Majorana neutrinos.

In the Majorana decay, an electron is emitted from a neutron via
$n \rightarrow p + e^{-} + {\overline{\nu}}_{e}$ and simultaneously, the
neutrino is absorbed by a neighboring neutron via
$n + \nu_{e} \rightarrow p + e^{-}$. This mechanism requires
$\nu_{e}º{\overline{\nu}}_{e}$, hence the statement that Majorana
neutrinos violate lepton number conservation by 2. An alternative to the
Majorana scenario is if there exists at least one right-handed partner
to the known (left-handed) neutrino flavors. A $\nu$ might then
oscillate into a $\nu_{R}$, similar to the vacuum oscillations discussed
above. (Because the right-handed neutrinos do not experience weak
interactions as the three known neutrinos do, they are sometimes
referred to as *sterile neutrinos*.) This extension to the standard
model may also permit neutrinoless double beta decay.

Experiments searching for a signal in this channel favor the element
Germanium. For an ordinary beta decay with two single decays
coincidentally close together in time the two electron energy spectrum
should decrease to zero at the highest energy, as for tritium decay. For
the neutrinoless case with Majoron exchange, the electrons are
kinematically forced to have the maximum energy, resulting in a single
line at the high end.

This technique should provide an order of magnitude greater mass
sensitivity \[19\] than the tritium experiments current limit, but only
if neutrinos are, in fact, their own anti-particles. If neutrinos are
not their own anti-particles, the data sample will be entirely composed
of mock double beta decays. Thus far, not a single neutrinoless double
beta decay has been observed.

### Summary

Despite the strong hints from astrophysics and cosmology, kinematic
searches have yet to produce definitive proof of non-zero neutrino mass.
Disturbingly, the electron and muon neutrino results consistently favor
a negative value for m^2^. While this anomalous result would not be
statistically surprising for a single experiment, especially for very
small or identically zero neutrino mass, it is not expected to occur in
all experiments.

The methods used to determine the most stringent mass limits of the
three flavors of neutrino have been summarized. Because some of the best
fits to neutrino masses are consistent with zero or non-physical
imaginary values, the masses have not been determined. The usual
Bayesian approach to setting upper limits on masses must be modified to
accommodate a negative value of the mass fits, as summarized in Figure 7
(c.f. Phys. Rev. D, Review of Particle Physics, Sec. 3). Recall that for
the tau neutrino mass the limit was calculated directly with a maximum
likelihood function, and it was not possible to find a central value for
the mass.

![](media/image12.emf){width="1.6576388888888889in"
height="1.14375in"}**Figure 7:** The central value of the mass fit is
negative. The black

arrow indicates a point where 90% of the distribution is to the left.

Because some of the distribution is in a non-physical region, the 90%
confidence limit is found by considering only the portion of the

distribution in the physical region. The 90% mass limit then corresponds
to the border between the black and white regions near

the right corner of the figure.

Table 3 lists the most stringent mass limits to date from the
experiments described in this section. The electron and muon neutrino
values define a 90% confidence limit, but the tau neutrino limit is at
95% confidence. While the electron neutrino mass is well constrained,
the muon and tau neutrino mass values are expected to lie much lower
than the current limits.

# Neutrino Oscillation Experiments

As discussed in the astrophysics section, the probability of observing a
flavor oscillation depends upon the mixing angle, the mass difference,
and the parameter $L/E$. The best opportunities for a transition occur
when the distance travelled by the neutrino is long and the neutrino
energy is small. Presently, several experiments are searching for vacuum
oscillations of various neutrinos, and more experiments are in the
planning stages. (See Appendix A for a sample set of neutrino
experiments.) The typical scheme for current experiments is to search
for $\nu_{m} \rightarrow \nu_{e}$ because the former are so easy to
create and the latter are relatively easy to detect. These searches are
not sensitive to the absolute value of neutrino masses but only to the
mass difference between flavors.
![](media/image15.emf){width="2.4270833333333335in"
height="2.384027777777778in"}

In the past, there have been announcements that neutrino oscillations
had been verified, but all these claims have been disproven. On January
31, 1995 a *New York Times* front page story claimed that evidence for
neutrino oscillations was observed at Los Alamos National Laboratory.
The initial data have since been published in Physics Review Letters
\[20\] and the extraordinary result has generated many papers from
theoreticians. Because a great deal of controversy surrounds this
result, the experiment merits further discussion here.

The Los Alamos Scintillating Neutrino Detector (LSND) consists of 185
tons of baby oil and liquid scintillator surrounded by 1220
photomultipliers. The detector searches for the interaction
${\overline{\nu}}_{e} + p \rightarrow e^{+} + n$ produced by a beam of
${\overline{\nu}}_{m}$ implying
${\overline{\nu}}_{m} \rightarrow {\overline{\nu}}_{e}$ oscillations.
The full process is described by the
reactions![](media/image16.emf){width="2.4298611111111112in"
height="2.2944444444444443in"}

$\begin{array}{r}
{\overline{\nu}}_{e} + p \rightarrow e^{+} + n \\
n + p \rightarrow \text{deuteron} + g\text{.}
\end{array}$

The detector triggers on a positron and the characteristic 2.2 MeV gamma
ray in coincidence. The positron energy is recorded only after a 2.2 MeV
photon is detected. From the incoming neutrino flux one expects only a
few events due to background electron neutrinos, as modeled in a Monte
Carlo event generator.

A 780 MeV proton beam incident on a water target produces mainly
${\overline{\nu}}_{m}$ along with some $\nu_{u}$ and $\nu_{e}$
contamination. \[20\] The detector is located approximately 30 meters
from the neutrino source. LSND computes a "likelihood ratio" R to
characterize how often a 2.2 MeV photon might accidentally be detected
at the same time as a positron. R is determined using laser calibration
events and cosmic ray neutron events.

The result of their experiment over the past three years is depicted in
Figure 8. LSND finds a significant event peak around 40 MeV. Consistency
checks such as cutting the fiducial region in half do not affect the
fraction of excess events observed.

The previous LSND results were first published in 1995, but the data
suffered from low statistics and concentration of events in the front of
the detector. Most of these indications pointed to a poor understanding
of background sources. While the data shown in Figure 8 have not yet
been published in a refereed journal, the paper indicates a much better
set of event selection criteria has been found. In Figure 9, the best
fits to the published LSND results are shown with the astrophysics
predictions. The latest predictions (only slightly different) are shown
in Figure 10. ![](media/image17.emf){width="3.290277777777778in"
height="2.2916666666666665in"}

In Figure 9, the preferred values from the best fit to the data are
plotted as a point and the uncertainty in the parameters defines the
oval regions about these points. The fit parameters generated with the
Kamiokande data in an earlier section lie within the oval marked
"Atmospheric Neutrinos." For LSND, the data could be described by
several different sets of ∆m^2^ and θ. Their ovals do not overlap those
of the astrophysics experiments, with the exception of the fit with the
largest uncertainty.![](media/image18.emf){width="3.095833333333333in"
height="3.036111111111111in"}

Recall that the fit to the Kamiokande data indicated a ∆m^2^ value of
0.01 GeV. The lowest point in $L/E$ was 5 km/GeV and had no deviation;
the deviation from the massless neutrino prediction began near 10
km/GeV. For LSND, the deviation (when converted to the same units)
begins much sooner, around 0.75 km/GeV. The characteristic length of the
neutrino oscillations begins 10 times sooner for LSND than has been
observed by Kamiokande and other experiments. Other accelerator
experiments have found no deviation from the massless neutrino
prediction.

Experiments with a null result (no evidence for neutrino oscillations)
define excluded regions of ∆m^2^ ­­Sin^2^ (2θ) *parameter space.* The
curves for many selected neutrino oscillation experiments are plotted in
Figure 11. A comparison to Figure 10 reveals that the most favored
points of the LSND fit fall within some of the excluded regions. Several
experiments had a 90% chance of detecting LSND's claimed oscillations
but did not.

### Summary

While the details of the neutrino oscillation searches differ, the
underlying physics is the same. A muon neutrino beam is created with a
high-energy proton beam incident upon a target and a detector searches
for the appearance of a neutrino of different flavor some distance away.
The results are used to infer two parameters: the mixing angle θ,
characterizing the amplitude of the probability of a transition, and the
mass difference, ∆m^2^, characterizing the distance required for flavor
oscillation.

Several experiments are searching for better limits in the
$\nu_{m} \rightarrow \nu_{t}$ sector, while others are collecting more
data in the $\nu_{m} \rightarrow \nu_{e}$ channel with lower energy
thresholds than previous work or faster rates due to higher acceptance.
The results of LSND have sparked intense interest in the μ-e channel
because they are the only accelerator experiment to have found a signal.
Accelerator experiments are not able to determine the absolute value of
neutrino masses, only the mass differences.

# Models for Existing Data

Working within the assumption that there are only three flavors of
neutrino and excluding only LSND's data (currently the most
controversial result), we can assemble the data from all experiments to
formulate certain constraints on neutrino masses and mixing angles.
Gathering all data together narrows the search space, yielding the best
chance to detect physics beyond the Standard Model. The other method
used to derive limits assumes the existence of currently unknown
neutrino flavors or excludes major portions of the worldwide data set
from the calculation. Table 4 has been assembled \[15\] with the first
method: only the three known flavors of neutrino exist, but they are
linear combinations of massive eigenstates, making them subject to
vacuum oscillations and the MSW effect. Also, the neutrino mass states
are assumed to be *hierarchical*: m~1~ \<\< m~2~ \<\< m~3~, similar to
the quark mass structure. The masses and mixing angles calculated using
other schemes yield slightly different results, but not significantly
so. Not surprisingly, all theoretical models have a common theme: the
masses and mixing angles of neutrinos lie just out of reach of the
ability of current experiments to find them.

![](media/image19.emf)

> **Table 4:** Predicted masses and mixing angles θ~ij~ calculated from
> experimental data. The calculation requires input assumptions:
>
> 1 - neutrino masses are assumed to be hierarchical ... m1 \<\< m2 \<\<
> m3
>
> 2 - negative m^2^ values are excluded from the input data
>
> These values should be considered to be an [example]{.underline} of a
> set that satisfies most of the data. Other models will predict
> different masses and angles.

# Searching for Neutrino Mass with the DZero Detector

DZero was designed to study high-mass states (such as the top quark and
the W and Z bosons) and high-p~t~ phenomena. \[21\] The Tevatron is a
proton-antiproton collider operating with a center of mass energy of
1800 GeV. At this energy the constituents of the proton and anitproton
act as nearly-free particles. These constituents, called partons, share
the momentum of the entire particle between them. It not possible to
determine the exact fraction of the total momentum a particular parton
has at the moment of collision, therefore the momentum of the products
in the laboratory frame of reference is not necessarily the momentum in
the center-of-mass frame of the collision. To extract information
from$p\overline{p}$ collisions, DZero considers only that fraction of
the momentum that is transverse to the beam-pipe. The transverse
momentum approaches the true momentum when the products emerge from the
beam-pipe at a 90° angle.

To determine DZero's ability to search for neutrino mass, we ignore the
difference between momentum and transverse momentum; by assuming
particles decay from rest in the center of the DZero detector, we make
the physics calculation as simple as possible. The tau neutrino mass
limit was set by a similar collider detector, so we briefly consider the
method of ALEPH first: study the decay $t \rightarrow \nu_{t} + 5p$.

Because the tau lepton decays into hadrons so quickly, DZero has a
difficult time identifying tau events; the lack of a central magnetic
field makes it impossible to resolve the closely spaced pion energy
deposits from one another. The decay of the Z into the other two lepton
flavors cannot determine neutrino masses because the muon escapes the
detector before it decays and the electron does not decay at all.
Studying the decay of Z into leptons will not allow DZero to determine
neutrino masses.

Now consider the decay of the W, specifically, the decay
$W^{+} \rightarrow m^{+} + \nu_{m}$. The W decays to one lepton and one
neutrino 32.4% of the time. Of these W-to-lepton events there is an
equal chance for each flavor, meaning that 10.8% of all W events result
in a muon and a muon neutrino; the signature for this decay is a lone
muon. We assume that the detector is *perfect*, always measuring energy
correctly and with unlimited precision, we also assume the W decays from
rest in the center of the calorimeter. From Eqn. 3 in the section on
kinematics, the energy of the muon for the two body decay
$W \rightarrow m + \nu_{m}$ is

$E_{m} = \frac{M_{W^{2}} + m_{m^{2}} - m_{\nu_{m}^{2}}}{2M_{W}}$.

Entering Particle Data Book values for the mass of the W (80.33 GeV),
the mass of the muon (0.105 GeV), and assuming a zero mass for the
neutrino, the expected muon energy is 40.1651 GeV. Recalculating with
the neutrino mass set at the current upper limit (0.000160 GeV) we find
the answer to be identical to the massless case to at least 10 figures.
The large mass of the W in the numerator completely suppresses the
effect of adding a non-zero neutrino mass to the equation. In fact,
setting the *muon* mass to zero changes the expected energy only one
ten-thousandth of a percent! The DZero detector is not sensitive to the
mass of the *muon* in W decays, so it is certainly not sensitive to the
mass of the muon neutrino. Because the masses in the electron sector are
smaller, the sensitivity vanishes for electron neutrinos as well. If
DZero could identify tau leptons with precision, the difference between
m^2^=0 and m^2^=160keV is still \~4 keV, below the resolution capability
of the calorimeter and more than 30 times smaller than the current
uncertainty on the mass of the W. We must conclude that the W decay mode
is not an effective measure of neutrino mass.

Even when simplifying the physics and assuming the detector is ideal,
DZero in its current form cannot search for neutrino mass. It is
impossible to search for massive tau neutrinos at DZero as has been done
at ALEPH, because there is no sample of$Z \rightarrow t^{+}t^{-}$
events. These events must be included in the data set, but without a
central magnetic field they are indistinguishable from quark-quark dijet
events. The mass of the W makes an analysis of the W two-body decay
insensitive to small neutrino masses. The future upgrade of the detector
includes plans for a central magnetic field, so the next incarnation of
DZero will have a much greater ability to search for massive tau
neutrinos.

# Designing a Neutrino Oscillation Experiment

Given infinite resources, several experiments could be designed to
gather neutrino data and definitively confirm neutrino oscillations. The
exercise of designing an impractical experiment can be a valuable
experience because it forces the architect to decide what information is
lacking in the body of existing knowledge and how to gather the
information in the most complete way. In this section, an experimental
design is presented without the restrictions of practicality imposed.
While fanciful and even comical, the advantages of the unlimited
experiment will be incorporated into a more viable design, and the
disadvantages will be addressed to prevent their inclusion.

Because the topic of discussion in this section is hypothetical and
"free-wheeling," a more informal tone has been adopted, if only for this
section.

### Idea 1: "Pie in the Sky"

Build a proton accelerator ring around the Earth. Once per day or so,
the Earth will rotate into a favorable position. Make the beam of
protons strike a very dense target, producing a beam of primarily muon
neutrinos that leaves the Earth tangent to the surface. The newly formed
neutrino beam travels outward into space where a neutrino detector (a
Voyager-type spacecraft) is slowly drifting toward the edge of the solar
system. The orbit should be designed to wrap around Neptune and return
to the Earth.

While the plan is admittedly just a dream, it is not an impossible one.
Fermilab sends a muon neutrino beam into space every time it "dumps" the
proton beam. A neutrino detector would be in the vicinity of Pluto now,
had Voyager II been filled with a finely segmented drift chamber similar
to the one used by NOMAD.

This implausible idea has several advantages that make this brief
presentation worthwhile:

1.  The extremely long baseline ensures adequate time for neutrinos to
    undergo oscillations.

2.  The distance travelled by the neutrinos slowly varies with time as
    the detector travels, yielding a new point in $L/E$ with every burst
    of neutrinos. This wide range in distances maximizes the detector's
    sensitivity to vacuum oscillations.

3.  The detector searches for the MSW effect for several weeks each year
    because the neutrinos must pass through the sun to reach the
    detector.

Disadvantages:

1.  The spacecraft is subject to a large influx of cosmic rays.

2.  The extreme cold of space prevents the use of most conventional
    neutrino-sensitive media.

3.  The sensitivity of a neutrino detector precludes the use of standard
    atomic batteries (a tritium β source) to power the equipment aboard
    the vessel.

4.  The detector cannot be serviced in the event of break-down.

While whimsical, the main idea at the core of this discussion could be
used to good effect. Perhaps the key to observing neutrino oscillations
lies not in building bigger and better detectors, but in the more clever
implementation of a simple detector.

### Idea 2: The Submersible

The only practical places to build neutrino experiments are underground
or underwater to avoid cosmic ray contamination. The goal of this
section is to present a viable and innovative detector design under the
constraint of limited funding. The design should maximize physics
exploration while minimizing the need for new technology or heroic
construction efforts. To keep the advantages of the previous scheme, we
turn our attention now to the deep sea.

I propose sinking a mobile experiment in the ocean. This is not to
suggest the use of seawater as a Cerenkov medium like the NESTOR
experiment \[22\], but merely as shielding. Because the detector will be
suspended in water, a rigid and expensive tank will not be required.
Instead, Kevlar sheets could be stretched over a titanium framework to
contain a scintillating liquid as is used by Borexino \[23\]. The
experiment will not suffer from the tremendous pressure because the
interior pressure will match the ambient pressure on the exterior. This
design makes the experiment easy to move because the scintillating
liquid can be pumped out before the framework is brought aboard a
transport ship.

Trigger control can originate from a ship/platform on the ocean surface
or from shore. The only equipment needed in the detector are the
photomultipliers that line the interior of the soft tank and the
fiber-optics needed to read them out.

At depths of more than two kilometers, the ocean is very calm and cool.
\[22\] No sea life lives at these great depths, so the detector will
rest unmolested.

Advantages:

1.  **Mobility:** The detector can be placed in the Pacific Ocean near
    California, on the order of three thousand kilometers from Fermilab,
    the source of a neutrino beam. After a number years of data-taking,
    the detector can be placed in Lake Michigan, on the order of 50
    kilometers away. The Gulf of Mexico lies at an intermediate
    distance. When the Large Hadron Collider at CERN begins operation,
    the detector can be placed in the Mediterranean Sea close to NESTOR.
    A survey is required before moving the submersible to a new site,
    but the placement is more flexible than the requirement of an
    existing mine shaft or planning new construction.

2.  **Low Energy Threshold:** The liquid scintillator proposed by the
    Borexino collaboration has a threshold energy of 0.25 MeV. Like that
    used in radiochemical experiments, this medium is sensitive to solar
    pp neutrinos but the data is analyzed in real-time, allowing energy
    and direction measurements.

```{=html}
<!-- -->
```
1.  **Excellent Cosmic Ray Shielding:** The NESTOR experiment will lie
    four kilometers beneath the ocean surface. This is twice the
    shielding material of underground experiments.\
    \
    ![](media/image20.emf){width="3.7402777777777776in"
    height="4.480555555555555in"}

To the exterior of the tank must be attached the following equipment
(Figure 12):

-   simple valves to protect the detector from pressure changes

-   corrosion resistant read-out electronics

-   data cables, power feed, and main hoist line

-   external photomultipliers to assist with beam location

-   backup batteries, a signal beacon, and an Ultra Low Frequency (ULF)\
    communications antenna for limited remote control\
    in the event the cables snap

Power, data storage, and control can be provided from shore via an
umbilical cable. Ideally the experiment could find a semi-permanent base
near an oil drilling platform at sea to minimize the length of cables
required. The NESTOR experiment proposes a seven mile cable system
\[22\], sufficient for most sites.

Any particle collider can provide the muon neutrino beam. Proper
placement of the experiment within the beam could be accomplished by
mounting photomultipliers on the *outside* of the detector to look for
the beam from its Cerenkov effect in the surrounding seawater. \[22\]
The proposed MINOS experiment firing a$\nu_{m}$ beam from Fermilab to
Soudan, Minnesota, expects the beam to spread to \~ 200 meters by the
time it reaches the detector 780 km away. \[24\] A similar beam spread
would indicate a width of about 800 meters offshore in the Pacific with
a corresponding drop in flux. These numbers describe the width of the
"good part" of the beam, where the neutrino distribution is nearly
uniform. The beam could be made tighter with existing technology, but
the beam width is limited by the precision with which it can be aimed.

### Summary

Using the brainstorming exercise of unlimited funding, a viable detector
design has been formulated. The design incorporates the desirable
qualities of mobility, low energy thresholds, and excellent cosmic ray
shielding. The detector is more simple in design and therefore less
expensive than most accelerator experiments in terms of capital outlay.
The mobility makes the detector flexible enough to provide data in the
most interesting sectors of ${Dm}^{2} - \text{Sin}^{2}q$ space without
foreseeable end.

# Review Summary

Three observations lead theorists to suppose that neutrinos may have
non-zero mass: the solar neutrino problem, the atmospheric neutrino
deficit, and the "missing mass" of the universe. Efforts are underway to
better quantify the observed deviations of solar and atmospheric
neutrinos from theoretical predictions, but current results are not yet
precise enough to confirm neutrino oscillation formalisms. The two most
widely accredited mechanisms for neutrino oscillations, the MSW effect
and the vacuum oscillation formalism, were presented in full detail.

Experiments searching for kinematic consequences of neutrino mass have
set the most stringent upper limits on the mass of the electron
neutrino, the muon neutrino, and the tau neutrino. While most of the
results have been consistent with zero mass, these experiments all find
the square of the neutrino mass to be negative. The non-physical results
of these experiments indicate the presence of a poorly understood
systematic effect that makes the results suspect. The upper limit set
for the electron neutrino mass is of the order of 1 eV but the limits on
the muon and tau neutrino masses are far less constrained.

Neutrino oscillations may occur with a short characteristic length or a
long characteristic length, depending upon the neutrino energy and the
mass difference between flavors. Experiments with several different
distance scales search for oscillations of muon neutrinos into each of
the two other known flavors. Several experiments that were built to
study astrophysical phenomena are now studying neutrinos produced by
accelerators as well. The only experiment claiming to have detected a
statistically significant signal, LSND, is still studying its
systematics because the signal lies in a region of parameter space that
has been excluded by several other experiments.

A proposal for a new type of neutrino experiment was presented. The
unique design does not require the development of new technology to
implement and the materials required for the construction of the
detector are inexpensive. The design has the advantage of mobility,
giving it the capability to search for neutrino oscillations at any
distance scale from tens of miles to the diameter of the Earth. The
detector can migrate to share a neutrino beam from an accelerator by
positioning itself behind existing experiments. By selecting only
materials found in published proposals for other detector designs, the
submersible detector has been made viable in principle.

This review of neutrino physics is as comprehensive as possible, written
to introduce all aspects of the topic in reasonable detail. The reader
should now be adequately prepared to peruse most journal articles
germane to this rapidly developing field.

# References

\[1\] Bahcall, John N., "Neutrinos from the Sun: An Astronomical
Puzzle," Mercury, pp. 53‑63, Vol. XIX, Number 2, March/April 1990.

\[2\] Winter, K., "Neutrino Properties and Interactions," Invited talk
presented at the 1995 International Symposium on Lepton-Photon
Interactions, Beijing.

\[3\] Wolfenstein, L., Phys. Rev. D **17**, 2369 (1978); S.P. Mikheyev
and A. Yu. Smirnov, Yad. Fiz. **42**, 1441 (1985), \[Sov. J. Nucl. Phys.
**42**, 913 (1985)\].

\[4\] Kaysar, Boris, "Lectures on the Physics of Massive Neutrinos,"
Proceedings of the Fourth BCSPIN School in Physics, Puri, India.
January, 1993.

\[5\] Zuber, K., "Neutrino Oscillations and Dark Matter," Proceedings
*Aspects of Dark Matter in Astro- and Particle Physics (Dark 96)*,
Heidelberg Sep. 1996, Germany.

\[6\] Hirata, et. Al, The Kamiokande-II Collaboration, "Observation of a
Small Atmospheric $\nu_{m}/\nu_{e}$ Ratio in Kamiokande," Phys. Lett. B
[280]{.underline}, 146 (1992).

\[7\] Conforto, G., et. al., "A complete solution to neutrino mixing,"
hep-ph/9606226, 4 Jun 1996.

\[8\] Ahluwalia, D. V., "Notes on the Kinematic Structure of the
Three-Flavor Neutrino Oscillation Framework," hep-ph/9612471 v2 27 Dec
1996.

\[9\] Kang, Kyungsik, Jihn E. Kim, Pyungwon Ko, "A simple modification
of the maximal mixing scenario for three light neutrinos,"
hep‑ph/9503436, March, 1995. Submitted to Z. Phys. C.

\[10\] Renton, Peter, [ElectroWeak Interactions]{.underline}, An
Introduction to the Physics of Quarks and Leptons, Cambridge University
Press, 1990.

\[11\] Morrison, Douglas R.O., "Review of Neutrino Masses,"
CERN-PPE/95-47, 5 April, 1995. Lectures at Conf. on Current topics in
Astrofundamental Physics, Erice, Italy, Sep 4-16, 1994. Published in
School of Astrophysics 1994:93-117 (QB461:A7:1994).

\[12\] Valle, José W.F., "Schemes for Neutrino Mass and Mixing,"
hep-ph/9410380 26 Oct 1994.

\[13\] Minakata, Hisakazu, "Three-Flavor Analysis of Neutrino Mixing
with and without Mass Hierarchy," hep-ph/9612259, 5 Dec 1996. Presented
at Inauguration Conference of the Asia Pacific Center for Theoretical
Physics (APCTP), Seoul, Korea, 4-10 Jun 1996 and at Institute for
Nuclear Theory Summer Program on Nucleosynthesis in the Big Bang, Stars
and Supernovae, Seattle, WA, 24 Jun - 30 Aug 1996.

\[14\] Belesev, A.I., et. al., "Results of the Troitsk experiment on the
search for the electron antineutrino rest mass in tritium beta-decay,"
Phys Lett. **B350**, (1995), 263-272, and Weinheimer, Ch., et. al.,
(Troitsk), "Improved limit on the electron-antineutrino rest mass from
tritium β decay," Phys. Lett. **B300,** 210, 1993. Also, V. M. Lobashev,
in Proc. of Neutrino 96, ed. K. Enqvist K. Huitu and J. Maalampi ,, (
World Scientific , Singapore, 1996) p.0.

\[15\] Adelberger, et. Al., The N1 Working Group, "Kinematical Probes of
Neutrino Mass," NSF-PT-95-01 (Snowmass '94). Talk given at Summer Study
on High Energy Physics: Particle and Nuclear Astrophysics and Cosmology
in the Next Millennium, Snowmass, CO, Jun 29 - Jun 14, 1994. Published
in Snowmass Summer Study 1994:195-214 (QB461:D2:1995).

\[16\] Assamagan, K., et. al., "Measurement of the muon momentum in pion
decay at rest using a surface muon beam," Phys. Lett. **B335**, 231,
1994.

\[17\] Decamp, D., et al., The ALEPH Collaboration., Nucl. Instr. Meth.
A 294 (1990) 121.

\[18\] D. Buskulic et al., ALEPH, "An upper limit for the τ neutrino
mass from $t \rightarrow 5p(p^{0})\nu_{t}$decays," Phys. Lett. **B349**,
585 (1995).

\[19\] Kim, C.W., "Neutrino Physics: Fundamentals of Neutrino
Oscillations," hep-ph/9607391, 22 Jul 1996.

\[20\] Athanassopoulos, C., et. al., "Evidence for
${\overline{\nu}}_{m} \rightarrow {\overline{\nu}}_{e}$ oscillations
from the LSND Experiment at LAMPF," nucl-ex/9605003, 9 May 1996.

\[21\] Abachi, S., The DZero Collaboration, Nuclear Instruments and
Methods in Physics Research A **338,** 185-253, 1994.

\[22\] Capone, A., "NESTOR: a neutrino Astroparticle Physics Laboratory
for the Mediterranean, the physics perspective," and Bottai, S.,
"Present Status of the NESTOR Project," talks given at the ICRC \'95
Conference.

\[23\] Baltz, A.J., and J. Weneser, "IMPLICATION OF GALLIUM RESULTS ON
THE POSSIBILITY OF OBSERVING DAY - NIGHT MATTER OSCILLATIONS AT SNO,
SUPER­KAMIO­KANDE AND BOREXINO," Phys. Rev. **D50** 5971-5979, 1994.

\[24\] Ables, E., et al. The MINOS Collaboration. "P-875: A
Long-baseline Neutrino Oscillation Experiment at Fermilab." February
1995.

\[25\] Cherry, M.L., et al., "THE HOMESTAKE LARGE AREA SCINTILLATION
DETECTOR AND COSMIC RAY TELESCOPE," Proceedings, Solar neutrinos and
neutrino astronomy, Lead, SD, 1984.

\[26\] Anselmann, P., et al., "First results from the ^51^Cr neutrino
source experiment with the GALLEX detector," Phys. Lett. **B342**,
(1995) 440-450.

\[27\] Abdurashitov, J.N., et al., "Results from SAGE (The
Russian-American Gallium solar neutrino Experiment)," Phys. Lett.
**B328** (1994) 234-248.

\[28\] Kajita, T., Kamiokande, "Recent Results from Kamiokande on Solar
and Atmospheric Neutrinos," presented at 26th International Conference
on High Energy Physics (ICHEP 92), Dallas, TX, 6-12 Aug 1992.

\[29\] Bahcall, John N., and Eligio Lisi, "Tests of Electron Flavor
Conservation with the Sudbury Neutrino Observatory,"
Phys.Rev.D54:5417-5433,1996.

\[30\] Ayres, David S., SOUDAN-2, "INITIAL DATA FROM THE SOUDAN 2
EXPERIMENT," Proceedings of the 25th International Conference on High
Energy Physics, Singapore, 1990.

\[31\] Allison, W.W.M., "PROPOSAL FOR A LONG BASELINE NEUTRINO
OSCILLATION EXPERIMENT FROM FERMILAB TO SOUDAN: UPDATE TO P822,"\
FERMILAB-PROPOSAL-P-822-UPD, Mar 1994.

\[32\] Cline, David B., "STUDY OF RARE SIGNALS WITH THE ICARUS
DETECTOR," talk given at International Workshop on Future Prospects of
Baryon Instability Search in p decay and\
n \-\--\> anti-n Oscillation Experiments, Oak Ridge, TN, 28-30 Mar 1996.

\[33\] Resvanis, L.K., et al., NESTOR Collaboration, "NESTOR: A NEUTRINO
PARTICLE ASTROPHYSICS UNDERWATER LABORATORY FOR THE MEDITERRANEAN,"\
Nucl. Phys. Proc. Suppl. 35 294-300, 1994.

\[34\] Gaisser, Thomas K., Francis Halzen, and Todor Stanev, "Particle
Astrophysics with High Energy Neutrinos," Phys. Rept. **258** 173-236,
1995.

\[35\] Arik, E., et al., CHORUS Collaboration, "STATUS REPORT ON CHORUS
EXPERIMENT AT CERN," Nucl. Instrum. Methods A360 (1995) 254-257.

\[36\] 1996-7 Funding Request for Fermilab E803/COSMOS Short-Baseline
Neutrino Oscillations to $\nu_{t}$, January 8, 1996.

\[37\] Tanimoto, Morimitsu, "STRUCTURES OF NEUTRINO FLAVOR MIXING MATRIX
AND NEUTRINO OSCILLATIONS AT CHORUS AND NOMAD," Phys. Rev. **D53**
6632-6637, 1996

\[38\] Ellis, John, "Outlook On Neutrino Physics," invited talk,
Neutrino 96, Helsinki, June 1996

\[39\] Arzarello, F. Ciralli, et al., "HELLAZ: a high rate solar
neutrino detector with neutrino energy determination. CERN-LAA/94-19,
4/7/94.

# Appendix A

Description of Selected Neutrino Experiments

## Astrophysics Experiments

### I. Counting Experiments

This class of experiment involves simple counting of numbers of neutrino
events. These experiments are unable to determine the neutrino's
incident energy or direction because the event is detected weeks after
the fact.

A.  The Homestake experiment\
    A shaft in the Homestake gold mine in Lead, South Dakota houses a
    tank filled with $C_{2}\text{Cl}_{4}$ (perchloro­ethylene), a
    cleaning fluid. \[25\] The depth of the mine shields the experiment
    from background cosmic rays. Neutrinos freely pass through the
    ground, interacting with the $C_{2}\text{Cl}_{4}$ fluid via\
    $\nu_{e} + \text{Cl} \rightarrow e^{-} + \text{Ar}E_{\text{thr}} = \text{814}\text{keV}$.\
    The fluid is cycled through an extraction apparatus that pulls
    approximately 20 Ar atoms out per month. Approximately one Ar atom
    will decay radioactively in a month and the extraction technique is
    estimated to be 90% efficient, so the total number of Ar atoms
    generated by neutrino events per month is around 22.\
    \
    While the experiment is unable to determine the angle or energy of
    incident neutrinos, it is the longest running neutrino experiment in
    the world, taking data for the last 30 years (counting since 1967).
    Its strength is that it is sensitive to very low energy (0.8 MeV)
    neutrinos, including solar Be neutrinos in addition to the Boron
    neutrinos.

```{=html}
<!-- -->
```
B.  Similar Experiments:\
    GALLEX (near Rome, Italy) and SAGE (Baksan Valley, Caucasus)\
    \
    Under the shielding of Italy\'s highest mountain outside the alps,
    the Gran Sasso d\'Italia in the Abruzzo region 150 km east of Rome,
    the Gallium-based neutrino experiments GALLEX and SAGE, search for\
    $\nu_{e} + \text{Ga} \rightarrow \text{Ge} + e^{-},E_{\text{thr}} = \text{233}\text{keV}$.\
    GALLEX uses GaCL in solution \[26\], while SAGE uses metallic
    gallium. \[27\] Because gallium melts at \~ 86 °F, the metal is
    easily kept in liquid form and the produced germanium can be
    chemically extracted every few weeks.\
    \
    These are two of the lowest energy-threshold experiments currently
    running or planned; they will be sensitive to solar pp neutrinos.
    GALLEX has taken data since 1986 and SAGE has been in operation
    since 1990. Both experiments have qualitatively confirmed results
    from Homestake and Kamiokande. These experiments, like Homestake,
    are unable to determine incident angle or energy of neutrinos and
    are insensitive to muon neutrinos.

### II. Real-Time Experiments (Cerenkov and Ionization Experiments)

A.  The Kamiokande experiment\
    Located in the Kamioka mine in Japan, this detector consists of a
    cylindrical water tank holding 4500 tons of water. \[28\] The
    perimeter is surrounded with phototubes. An incident neutrino may
    strike an electon and transfer some of its momentum to it, knocking
    it free of the $H_{2}O$ molecule with some of the neutrino's
    momentum. The elastic reaction is\
    \
    $\nu_{x} + e^{-} \rightarrow \nu_{x} + e^{-}$.\
    \
    The resulting particles are detected by their emitted Cerenkov
    radiation (and accompanying particle shower in the case of the
    electron). A charged particle emits Cerenkov radiation when its
    speed exceeds the speed of light in a refractive medium such as
    water. This velocity limit sets the energy detection threshold of
    water Cerenkov experiments at \~7.5 MeV.\
    \
    The timing of the phototube signals determines the track's position
    within the tank. The total charge accumulated by the phototubes
    determines the energy of the electron or muon, which is used to
    calculate the energy of the original neutrino. The products of muon
    neutrinos can be distinguished from electron neutrinos from the
    presence (electron) or absence (muon) of showering radiation in
    addition to the Cerenkov track. Collaborators believe their
    identification to be accurate to within 2%. \[6\]\
    \
    Kamiokande's successor sits only 200 m southwest of the original
    experiment. The cylindrical SuperKamiokande stands on one end, 42 m
    in height and 39 m in diameter. The event rate scales with volume,
    so Super-K expects approximately 22 times the counting rate of
    Kamiokande. (Note: To avoid contamination from non-neutrino events,
    the detectors only consider interactions that occur some distance
    away from the walls. While the Super-K is about 10 times larger than
    the original, the *fiducial volume* is 22 times larger.) Super-K
    should begin data-taking later in 1997.\
    \
    There are plans in the works for a neutrino beam to travel from KEK
    (the Japanese center for high energy physics research) to the
    Kamioka site. SuperKamiokande will then have a data set free of
    astrophysical model-dependence. The expected sensitivity is ∆m^2^ \>
    10^‑4^ and Sin^2^ θ \> 0.1.\
    \
    Similar experiment: IMB

B.  Sudbury Neutrino Observatory (SNO, Creighton mine near Sudbury,
    Canada)\
    Rather than use ordinary water, SNO uses deuterium-water, or heavy
    water, as a medium. (A deuteron is a hydrogen atom with an extra
    neutron in the nucleus, just as tritium has two extra neutrons.) In
    addition to the elastic scattering observed by Kamiokande, SNO is
    also sensitive to the two reactions\
    \
    $\nu_{e} + d \rightarrow p + p + e^{-}$\
    and\
    $\nu_{x} + d \rightarrow p + n + \nu_{x}$,\
    \
    which are expected to occur at a rate of 20-30 events per day for
    each process, compared to 2-3 events per day for the elastic
    process. \[29\] All processes have an energy threshold close to
    5 MeV.

C.  Borexino\
    Very similar to both of the preceding Cerenkov experiments, Borexino
    will use florescent and wavelength-shifting chemicals instead of
    water as a detection medium. This will lower their expected energy
    threshold to 0.250 MeV. \[23\] They expect 20-60 events per day from
    solar ^7^Be neutrinos alone.

D.  Soudan II\
    This experiment lies ½ mile beneath Soudan, Minnesota. This detector
    is a tightly packed honeycomb array of drift tubes with steel as the
    absorbing medium, with dimensions of 5.4m high by 8m across by 15m
    long. \[30\] There is a possibility of sending a neutrino beam from
    Fermilab (730 km) to the detector. This accelerator-phase of the
    experiment will be dubbed MINOS, but it is still in the planning
    phase. MINOS will have a near-beam detector located at Fermilab in
    addition to the Soudan site. \[31\]

E.  ICARUS\
    This detector will use 600 tons of liquid Argon as an ionizing
    medium, searching for the reactions\
    $\nu_{x} + e^{-} \rightarrow \nu_{x} + e^{-}$ (5 MeV threshold)\
    from any neutrino, and\
    $\nu_{e} + \text{Ar} \rightarrow K \ast + e^{-}$.\
    They hope to add additional 600-ton modules in the future. \[32\]

F.  NESTOR\
    This experiment will use seawater as a Cerenkov medium. Large towers
    mounted with photomultipliers will be built on the floor of the
    Mediterranean sea, some 3800 meters below the surface. \[33\] The
    experiment can be easily expanded by building more towers. The
    proposal for this experiment was used to determine the viability of
    the Submersible outlined the design section of this review.\
    \
    Similar Experiments: DUMOND (deep water), BAIKAL (lake), AMANDA
    (Antarctic ice) \[34\]

## Accelerator Experiments

A.  CHORUS (Switzerland)\
    This experiment searches for the oscillations of muon neutrinos into
    tau neutrinos. They search for the creation of tau leptons via\
    \
    $\nu_{t} + \text{nucleon} \rightarrow t + \text{anything}$\
    followed by\
    $t^{-} \rightarrow m^{-} + \nu_{m} + \nu_{t}$.\
    \
    Like all accelerator-based neutrino oscillation experiments, the
    neutrino beam is created by sending high-energy protons into a
    target, creating pions and kaons which decay into muons and muon
    neutrinos. A dense barrier stops the muons, leaving only the
    neutrinos. At CERN, the protons have 450 GeV of energy and the
    barrier is a 410 m thickness of iron and soil. \[35\]\
    \
    The detector from front to back consists of

-   a nuclear emulsion target backed by a scintillating fiber (sci-fi)
    tracker to detect tau leptons,

-   a hexagonal magnet with sci-fi in front and behind to determine the
    momentum of charged particles,

-   a fiber tracking region without magnetic field,

-   an iron and sci-fi calorimeter measuring the energy and position and
    also stopping all particles but muons,

-   and a muon spectrometer, consisting of streamer tubes and drift
    chambers interspaced with magnets.\
    \
    The most important element of the detector is the nuclear emulsion
    target (many layers of special photographic film) designed to detect
    the short-lived tau. Even at very high energies, the tau travels
    only 100 micrometers before it decays. Chorus will take data for two
    years and then pause to develop the emulsion and reconstruct the
    tracks originating from tau leptons. A kink in the track through the
    emulsion will mark the decay of a tau. The remainder of the detector
    will be used to track the daughters of the tau to simplify their
    search through the emulsion for the parent particle's position.\
    \
    Similar experiment: COSMOS (Fermilab) starting 2001 and running for
    four years. \[36\]

C.  NOMAD\
    \
    The sister experiment to CHORUS is housed in the same hall, uses the
    same beam, and searches for the same phenomenon with minor
    differences in technique. \[37\] NOMAD will search for tau decay
    within their innovative drift chamber. The chamber walls are
    composed of Aramid fiber honeycomb to decrease the chances of
    bremsstrahlung (literally, \"braking radiation\", due to
    interactions with nuclei) that would artificially kink particle
    tracks.

# Appendix B

Comprehensive Examination for John Krane

Prepared by

Prof. G. Snow

Dept. of Physics and Astronomy

University of Nebraska

Approved by Krane Supervisory Committee in December, 1996.

NEUTRINO MASSES AND NEUTRINO OSCILLATIONS

THE QUESTION:

Your task is to review the status of what we know about neutrino masses
and neutrino oscillations. Our current view of neutrinos is that there
are 3 species of neutrinos associated with the charged leptons
(electron, muon, tau), that they are likely to be massless, and that
they do not \"mix\", i.e. that lepton number is absolutely conserved.
However, a major area of experimental HEP research focuses on measuring
neutrino masses and searching for neutrino oscillations.

You will want to address all of the following sub-questions and any
others you feel make your paper a complete review.

NEUTRINO MASS:

Review the status of our knowledge of the masses of the 3 neutrinos;
describe the decay processes studied in each case, the experiments and
techniques which have led to the most stringent upper limits on the
mass. What limits the sensitivity in each case, and why are the upper
mass limits so different for the 3 species? At DZERO, we routinely
produce millions of electron-, muon-, and tau-neutrinos. Imagine using
the DZERO detector to measure neutrino mass \-- how might we go about
making a mass measurement for one or more of the species? Could DZERO
make a neutrino mass measurement which is competitive with the
experiments above? Why or why not (be quantitative)? What would be the
astrophysical consequences of determining that one or more of the
species has non-zero mass? How would this enter into the dark matter
problem? A few years ago, an experiment claimed to measure a non-zero
mass for one of the neutrinos \-- what is the status of this claim?

NEUTRINO OSCILLATIONS:

Review the formalism used to describe transitions of one neutrino
species into another; what would such a transition imply about neutrino
mass? Review the present status of neutrino oscillation searches \--
what techniques have been used in the searches, which experiments have
produced the most stringent limits and how are these limits expressed
quantitatively? Both at Fermilab and CERN, major short- and
long-baseline oscillation programs are underway or proposed. What is the
status of current efforts and proposals, what improved sensitivity to
oscillations can we expect if the proposed efforts go forward, what is
the timescale? If the world had to choose to carry out only one of the
proposed long-baseline projects, which would be the better investment
physics-wise and why? Given infinite DoE/NSF funds and an infinite
number of graduate students, how would you design the world\'s best
neutrino oscillation experiment?

# Appendix C

The initial version of this review generated several comments and
corrections from members of the examination committee. Typographical
corrections and mathematical clarifications have been incorporated into
the document where required, for instance, the original derivation of
Eqn. 2 suffered from a conceptual blunder that has been corrected; I
start with the full unitary mixing matrix rather than a simplified one.
Some requests for additional information are answered in this section
rather than the body of the work to preserve the conceptual flow of the
original document. Included here are selected comments from committee
members with responses.

*1) Page 1, line 7: for pedagogical effect, define \"small.\"*

> Less than or equal to 50%. I do not think an explicit calculation is
> necessary, Bahcall and other authors use the stock phrase as an
> indicator of the magnitude of difficulty faced by experimenters in the
> field of neutrino physics. This "light-year of lead" statement is a
> favorite of so many authors of popular articles that the introductory
> remarks would not be complete without it.

*2) Page 5, bottom line: Which aspects should be reworked? What are some
of the explanations other than MSW? Wasn\'t there a PRL recently on
this?*

> The neutrinos detected by most terrestrial experiments originate from
> Boron decay, representing only 2% of the reactions within the sun.
> While this percentage can be computed accurately with the help of
> "reasonable" assumptions, this value may be quite inaccurate should
> the assumptions prove incorrect. These assumptions include the
> abundance of elements at the time of formation of the sun and the
> nature of the Be→B reaction. This reaction is poorly measured (cross
> section results vary by as much as 100%) and has only been measured at
> energies higher than that within the sun (117 keV compared to 1-20 keV
> in the sun). There are uncertainties involved with the extrapolation
> to solar energies and uncertainties in the effects of plasma
> screening. \[11\] The solar opacity affects the luminosity and
> therefore influences the estimates of reaction chain ratios. \[1\] For
> these reasons, many authors complain that solar neutrino theorists
> habitually underestimate the uncertainties in their predictions.

*3) Page 10 middle. I am not sure that it follows that if cap omega is
less than 1 that the standard inflation scenario is invalid.*

> I think you mean "...if the density of the universe were *greater than
> 1...*" The standard inflation scenario suggests that Ω is no larger
> than unity. It is common practice (in the articles I have referenced)
> to use Ω=1 as a limiting case.

*4) Page 12, top: Why does low velocity preclude detection? Neutrons are
detected this way all the time.*

> It does not preclude the detection of the presence of a particle, only
> the knowledge of its velocity. For Troitsk, the momentum resolution
> for electrons was of the order of 3-4 eV. For tritium decay, the
> *maximum* recoil momentum of the Helium atom is of the same order.
> Thus, the momentum of the most massive product could not be measured
> accurately even if the same measurement technique applied for Helium
> as for electrons (it does not). The offending sentence has been
> modified for clarity on this point.

*5) Page 12. What is the standard name for fig.6? I wouldn\'t call -39
(+/-) 34 \"significantly negative!\" Last sentence of next-to-last
paragraph: very infrared?.*

> Figure 6 is a "Linearized Beta Spectrum". Troitsk has subtracted their
> estimated background from the measured spectrum and taken the cube
> root. To first order, this should make the spectrum a straight line,
> second-order effects include any excitations of the HeT^+^ ion.
>
> While the Troitsk result is nearly consistent with zero neutrino mass,
> a Los Alamos experiment found -147±68±41 eV (in 1991) and Lawrence
> Livermore Laboratories found -130±20±15 eV.
>
> The vibrational-rotational spectrum of the HeT^+^ ion lies in the IR
> region, however, excited electrons should produce photons with
> wavelengths less than 100 nm---very ultra-violet (but I did correct
> the spelling).

*6) Page 13, Pion section: The author seems taken aback that there might
be experiments without names.*

> Indeed, in the field of high-energy physics, a name for the detector
> and collaboration is almost mandatory. With more than three names on
> the author list, a reference will only contain the first name and "et
> al." There is much more recognition (scientific and political) in a
> collaboration name. If not for detector and collaboration names, one
> might find people changing their names to something like "John
> AAA-Readme".

*7) Page 18, middle: three flavors can be assumed from SLAC results.*

> I quote here the recent LEP result (Warsaw '96) which indicates
> 2.989±0.012 Standard Model neutrinos (weakly interacting neutrinos),
> however, this result does not exclude the existence of postulated
> super-symmetric partners to the neutrinos.
>
> A new family of [right-handed]{.underline} neutrinos would not couple
> to the W or Z boson and therefore would not experience the weak force.
> Searches for the super-symmetric partner to the W (often denoted W~R~)
> place an upper limit on the mass of these "sterile" neutrinos at
> several hundred GeV; the DZero experiment has set the most stringent
> upper limit on the mass of a sterile neutrino. Other than a brief
> mention in the section on double-beta decay (and now in the cosmology
> section), discussion of super-symmetric partners to neutrinos is
> beyond the scope of this work.

*8) I think idea 2 won\'t work with thin-glass-envelope PMTs. One would
have to have thick mu-metal-shell specials made.*

> The NESTOR collaboration proposes to use a standard PMT protected with
> a thick glass spherical housing. Both the PMTs and the housings are
> off-the-shelf equipment; oceanographers have used similar designs for
> the past 50 years. Titanium spheres will protect the required
> front-end electronics. For a full discussion, see reference \[22\], by
> S. Bottai.

*9) Page 12, perhaps some introductory material on four vectors should
precede the final equation or a reference.*

> This formula (my Eqn. 3) is in the Review of Particle Properties
> published by Physical Review D ("the PRD manual"). It is also a
> typical problem in particle physics texts to derive Eqn. 3; Griffiths,
> the textbook used at UNL, is no exception.

*10) Page 12, the tritium beta decay experimental setup should be
explained, otherwise the reader has no way to evaluate systematic
effects.*

> Their experiment is [really]{.underline} complicated and I did not
> want to get into it (this is to say, authors of journal articles
> complain that it is complicated). I'll beg out of this and refer you
> to reference \[14\].

*11) Page 14, Fourth paragraph. What IS the theoretical expectation for
the tau neutrino mass - from where does it come?*

> The statement merely indicates that 23.8 MeV is not much of a limit by
> comparison to the electron and muon sectors. The sentence has been
> changed to reflect this. Fixed target experiments such as CHORUS and
> COSMOS should be able to set more stringent limits once their results
> become available.

*12) Page 16, why would LSND expect to find a 40 MeV peak for positron
energy if the neutrino had mass. Perhaps some kinematics are required
here?*

> The muon anti-neutrinos produced from pions at rest have a calculated
> energy spectrum with a maximum energy of 52.8 MeV. The shape of the
> spectrum is very similar to the curve of the beta decay spectrum of
> Figure 6. At the same time, the detector has an energy threshold,
> below which a progressively smaller percentage of signal events is
> detected. The solid and dashed curves in Figure 8a are the result of
> both effects: a detector efficiency that increases toward unity with
> increasing positron energy and a positron energy spectrum that
> decreases to zero near 53 MeV in correspondence to the spectrum of
> ${\overline{\nu}}_{m}$ that produced it (via the conceptual chain
> ${\overline{\nu}}_{m} \Rightarrow {\overline{\nu}}_{e} \Rightarrow e^{+}$).
> In Figure 8b, the effect is exacerbated with the likelihood cut that
> has an effect similar to the detector efficiency. Once last
> complication: there is background noise due to
>
> $\nu_{e} + C \rightarrow e^{-} + N$.
>
> Electrons produced by the background have an energy \< 36 MeV. The sum
> of these effects explains the shape of the dashed curve (the
> background curve).
>
> Assuming the points are the result of neutrino oscillations with
> several possibilities for ∆m^2^ and θ, the information in the text can
> be used to check LSND's calculation. Given that there is an apparent
> peak near 40 MeV and the distance travelled by the neutrinos is \~30
> meters, I use Eqn 2 to find a value of ∆m^2^ that will make the square
> of the sine a maximum. That is:
>
> $$\begin{array}{r}
> 1\text{.}\text{27}{Dm}^{2}\frac{L}{E} = (2k - 1)\frac{p}{2},k = 1,2,3,\text{.}\text{.}\text{.} \\
>  \\
>  \Rightarrow {Dm}^{2}»1\text{.}7\text{eV},5\text{.}2\text{eV},8\text{.}7\text{eV},\text{.}\text{.}\text{.}
> \end{array}$$
>
> The ∆m^2^ results of this "Mickey Mouse" calculation lie close to or
> within the acceptable regions from LSND's more rigorous calculations
> (Figure 10). For sin^2^ θ, you need to know the % of flux that you
> have accumulated.

*13) Page 19, This is entirely optional: given the expected Run II
tracking resolution can D0 contribute to the tau mass limit in Run II?*

> ALEPH has a better energy resolution in their EM calorimeter than does
> DZERO, most likely because the e^‑^e^+^ interaction has no underlying
> event to deposit extraneous energy in the detector. This energy
> resolution disadvantage at DZERO will not play a large role because
> the EM cal will be used to distinguish π^±^s, π^0^s and photons, the
> bulk of the analysis will rely on the central tracking system for
> momentum measurement. DZero's central magnetic field will be 33.3%
> stronger than ALEPH's and should provide a comparable or better
> momentum resolution as a result. The one difference between the
> experiments is the initial p~t~ of the intermediate Z boson at a
> hadron collider. Because the colliding partons may have some non-zero
> transverse momentum, the subsequent particles will have a small
> transverse momentum imbalance that will result in a more difficult
> measurement. DZERO will provide a stronger limit than ALEPH if we
> obtain a "clean" event sample and can measure the initial p~t~ well
> enough that the momentum resolution does not suffer. (By clean, I mean
> single interactions.)
>
> ALEPH reports their mass (momentum) resolution for 5π events was 15
> MeV (350 MeV).

*14) Page 22 - just for fun - how much would John\'s Submersible cost?
Shall we call it UNO Underwater Neutrino Observatory?*

This question is left as an exercise to the reader. I love the name
"UNO"!

*15) Page 27 - I seem to recall that the Homestake experiment saw
oscillations of the neutrino count with time anti-correlated with sun
spots(?) Are these deemed significant?*

> The Homestake experiment shut down for 18 months in 1984-1986.
> Previous to the shut down, the counting rate fluctuated with the
> inverse of the sunspot cycle by seven standard deviations. After the
> shutdown, the rate has been stable and independent of sunspot cycle.
> Kamiokande has been in operation since 1987 and has never seen a
> variation corresponding to sunspots. It could be that something
> unusual happened after 1987, but it is more likely that Homestake had
> difficulty with their initial setup. See reference \[11\] for
> additional discussion.

16. *You state on page 10 that 30 eV is a hard upper limit. While I
    believe this upper limit, it is not hard. For example, in the case
    of neutrino decay, it is completely relaxed. Since all that is
    needed for neutrinos to decay is lepton number violation, and since
    neutrino oscillations need lepton number violation as well, neutrino
    decay is not hard to imagine. Diagrams for neutrino decay tend to
    put the lifetimes as long, but this is extremely model dependent.*

> I neglected consideration of unstable neutrinos. Big Bang
> neucleosynthesis limits metastable neutrino masses to lie outside an
> intermediate range. A metastable tau neutrino would have mass
>
> $m_{\nu_{t}} < 0\text{.}\text{95}\text{MeV}\text{or}\text{32}\text{MeV} < m_{\nu_{t}}$
>
> I am not convinced the mass is completely unconstrained in the
> unstable case. \[38\] In any event, I tried to make it clear that if
> you assume only standard model neutrinos with the exception that they
> have mass, you get the limit of 30 eV for the total mass of the three.

*17) I would disagree with your assertion on page 9 about L/E. It is
most useful for experiments to report what they measure. In this case it
is the mu/e ratio, visible energy and zenith angle. They measure neither
L nor E-nu. At low energy, where all of the contained events are, the
zenith angle distribution is barely distorted by big L/E effects. With
an exponentially falling Energy spectrum, the energy distribution
provides a poor lever arm for L/E physics. I believe it is important to
keep in mind the L/E_nu aspect of the physics, but not in the manner you
suggest.*

> This point is well taken. Because the scattered electrons are very
> relativistic, there is a nearly exact correlation between the angle of
> incidence and the distance an atmospherically generated neutrino has
> travelled, however, this does not exclude cosmic neutrinos from the
> data sample. Also, the difference between the energy of the scattered
> electron and the incident neutrino is more randomized than I first
> thought.
>
> When an electron from is "kicked out" of an atom, its kinetic energy
> is decreased by the energy required to escape from the atomic
> potential. For the complex atoms, this potential energy subtraction
> can vary greatly depending on the orbital from which the electron
> originates.
>
> (A new experiment designed to measure E~υ~ to within 5% will use He
> gas as the ionization medium. The simple atomic potential of helium
> does not unduly complicate the eυ~e~ cross section (as do nuclear
> wave-function overlap integrals \[38\] ), so the neutrino energy
> should be calculable with electro-weak theory. The low (100 keV)
> energy threshold will allow HELLAZ to collect and distinguish
> neutrinos originating from the beryllium, boron, pp, and pep
> reactions.)
>
> Nonetheless, it would be useful if experiments presented their data in
> terms of L and E (or theta and E), perhaps as ordered pairs or in a
> three-dimensional plot. The effects of atomic potentials in elastic
> collisions can be simulated and useful information can be extracted.
> The mini-analysis presented on page 9 was not intended to model every
> effect, only the "first-order" effect. As it stands, the mini-analysis
> result lies within the published result region.