10 July 2006

# The Cross Section of the Tau-neutrino

### Abstract

The DONuT experiment collected data in 1997 and published first results
in 2000 based on four observed ν~τ~ charged-current (CC) interactions.
The final analysis of the data collected in the experiment is presented
in this paper, based on 3.6 x 10^17^ protons on target using the 800 GeV
Tevatron beam at Fermilab. The number of observed ν~τ~ CC interactions
is 11, in addition to 553 observed ν~μ~ and ν~e~ CC interactions. From
this data we estimate the relative and absolute average ν~τ~ CC cross
sections to be 2×10^-39^ cm^-2^, in agreement with expectations from the
standard model.

## Introduction

The tau neutrino, *ν~ττ~*, was assigned its place in the Standard Model
after its weak partner, the tau lepton (*τ)*, was discovered in
1973[^1]. The observation of identifiable *ν~ττ\ ~*interactions, in a
manner similar to *ν~e\ ~*and *ν~μμ\ ~*interactions, did not immediately
follow. The difficulty of measuring *ν~ττ\ ~*interactions was due to the
relative scarcity of the sources of *ν~ττ\ ~*and having sufficiently
powerful detection methods to unambiguously identify the *τ* lepton
produced in the charged-current interactions. These challenges were
overcome in the observation of four *ν~ττ~* interactions by our group,
the DONuT (Direct Observation of Nu-Tau) collaboration, in 2000[^2],
twenty-seven years after the *τ* was found. Since that report, we have
finished the analysis of the data, more than doubling the total number
of found neutrino interactions of all flavors. We report on the results
of this analysis here, which completes the experiment.

The experiment, equipment and techniques, have been described in detail
elsewhere[^3]^,^[^4]. Here we give a summary, reviewing the essential
parts of the beamline, detectors and analysis methods.

The location of vertices in the emulsion data, tagging leptons and the
subsequent search for secondary vertices, were accomplished with high
efficiency. This allowed a detailed event-by-event analysis with small
and well-known background levels. Further, the large amount of
information in the emulsion/spectrometer system permitted the use of
powerful multivariate methods yielding probabilities of each candidate
event for signal and background.

This report is a summary of the final sample of *ν~ττ~*, *ν~e~*, and
*ν~μμ\ ~*interactions in the emulsion, and from this sample, the
measured *ν~ττ~* cross section is computed. An overview of the detector,
with emphasis on the emulsion is given in Section2. Section 3 details
the important features and limitations of the neutrino interaction
analysis, the secondary vertex analysis, and tau-identification methods
and efficiencies. Section 3 is a survey of the entire interaction data
set, and comparisons with what is expected. The *τ*-containing candidate
events are described in Section 4 together with the all the
charm-containing events that were found. Section 5 details the
systematic uncertainties relevant to the cross section determination,
which is summarized in Section 6.

## Neutrino Beam and Detector

The primary source of tau neutrinos in DONuT was the leptonic decays of
DS mesons produced by 800 GeV protons from the TeVatron at Fermilab. The
protons were dumped into a solid block of tungsten alloy, with a typical
intensity of 8×1012 protons for 20 seconds each minute, or about 20 kW
of power. The DS mesons yield two neutrinos in this decay mode within a
distance of a few millimeters, much less than the interaction length of
six centimeters. Immediately following the beam dump were two dipole
magnets with solid steel poles, providing both absorption of interaction
products and deflection of high-energy muons away from the beam center.
Following the magnets was an additional 18m of passive steel shielding
limited to within 2m of the beamline. Emerging at the end of this shield
were neutrinos and muons. The muons were mostly contained in horizontal
fan-like distributions, on each side of the centerline. The neutrino
beam design is shown in Figure 1.

The target for the neutrinos was 250 kg of nuclear emulsion stacked in
modular fashion along the beamline. The emulsion target was the heart of
DONuT, its capabilities and performance were matched to task of
recognizing neutrino interactions containing tau leptons. The signature
of *ν~ττ\ ~*charged-current (CC) interactions was the decay topology.
Note that 86% of *τ* decays are to a single charged particle (lepton or
hadron) and the typical decay length in the emulsion is only 2 mm. The
emulsion target provided micrometer precision in tracking the products
of the neutrino interactions, resolving the *τ* decay vertex, which was
usually only a kink in the visible track. A total of seven emulsion
modules in the target station were exposed, with a maximum of four
modules in place at any time during the experiment. The DONuT emulsion
modules were the first modern implementation of a design that
interleaves metallic sheets (stainless steel) with emulsion sheets to
blend high mass for interactions with high precision for tau
recognition. Two of the three module types incorporated steel, while one
module employed a third type, which used only emulsion sheets without
steel. The three designs are shown in Figure 2.

The information in small volumes of the emulsion was fully digitized and
incorporated into the analysis in a manner similar to an electronic
detector, though without time information. Integrated into the emulsion
target station were 44 planes of scintillating fiber detectors, used for
reconstruction of the interaction vertex. This vertex information
permitted the scanning and digitization of only a small volume of the
emulsion target, appropriate to the capability of the automatic scanning
machines. The emulsion target station was followed by spectrometer
consisting of a large-aperture dipole magnet and drift chambers. A
lead-glass calorimeter supplemented the emulsion as a way to identify
electrons. Behind the calorimeter, muons were tagged with a steel and
proportional tube system. Lepton identification was important in DONuT,
since an interaction produced with a charmed meson could have a topology
similar the *τ* signal. A *ν~ττ~* interaction does not have a charged
lepton from the primary vertex. The plan of the spectrometer is shown in
Figure 3 with the emulsion target area featured in Figure 4.

## Data Collection and Reduction

### *Triggering and Data Acquisition*

The electronic detectors required a prompt trigger for the digitizing
and readout electronics. A simple and efficient trigger for recording
neutrino interactions required that no charged particles entered the
emulsion from upstream and at least one charged particle emerged from an
emulsion target. This trigger was formed by a series of scintillation
counters consisting of a veto wall upstream of the emulsion target stand
and three hodoscope planes distributed between and downstream of the
emulsion modules. The veto wall consisted of 10 counters and covered a
total area of 140 cm x 152 cm. The dimensions of each counter were 30.5
cm in x, 152 cm in y, and 10 cm in z. For muons, the veto wall
efficiency was determined to be better than 99.9%.

Two planes of scintillating fibers, T1 and T2, were located downstream
of the second and fourth target modules, respectively. Each plane was 70
cm x 70 cm in area and segmented into eight (T1) or nine (T2) 10 cm
bundles. A third scintillator hodoscope, T3, was located downstream of
the target/SFT box. It was composed of eight counters, each 10 cm x 80
cm and 5 mm thick.

Two triggers were used during the course of the experiment. The design
goal of the trigger system was to keep data acquisition live time at
greater than 85%, which would correspond to a trigger rate of 6 Hz: The
main trigger (Trigger A) required: (1) hits in T1, T2 and T3 consistent
with ≥ 2 charged tracks; (2) track angles \> 250 mrad; and (3) no hits
in the veto wall. The detector elements for Trigger A are shown in Fig.
4. Trigger A was the sole physics trigger for the first 53% of the
recorded data. The fact that it required more than one charged particle
compromised the efficiency for triggering on single multiplicity
neutrino interactions. This compromise was necessary since the trigger
rate for a single track was very high due to background processes
initiated by through-going muons from the dump. Because of the limited
speed of the SFT readout system, discussed above, the live-time of the
experiment would have dropped far below the design goal. The measured
average rate for trigger A was 4.5 Hz corresponding to a live-time of
90%. During the final 6 weeks of data taking (47% of the recorded data),
a second trigger (Trigger B) was implemented in order to include single
track interactions that were lost in Trigger A. This trigger used the
MLUs to require a proper 1-track pattern and, in addition, required at
least one minimum ionizing track in the electromagnetic calorimeter.
With the addition of Trigger B, the trigger rate increased to 5:5 Hz and
the live-time decreased to 87%.

The efficiency of the triggers for neutrino interactions was calculated
using simulated events with actual geometries and measured efficiencies
for each counter. It was estimated that the efficiency was 98% for
triggering on charged-current neutrino interactions of electron-- and
muon--neutrinos, 84% for neutral-current interactions, and 97% for nt
interactions.

The architecture of the DAQ was based on the Fermilab DART product
\[7\], using VME-based microprocessors to control the transport of data
from the VME buffers to a host computer. The host computer served as
both the data monitor and as the data logger to tape (Exabyte 3500). The
average event size was 100 kB; with a throughput of 10 MB per beam cycle
of one minute.

### *Filtering, Stripping and Scanning*

A total of 6.6×10^6^ triggers for 3.6×10^17^ protons on target were
recorded onto tape. However, from calculations, only about 10^3^
neutrino interactions were expected for this proton exposure. This
implied that the great majority of the triggers were background
processes satisfying the simple trigger requirements of Section 3.3.
Data from the electronic detectors were used to extract the neutrino
interaction candidates in a two-step process.

First, data from the SFT and from the drift chambers were used to
reconstruct tracks and to search for a vertex near one of the emulsion
targets. This filter reduced the number of events by a factor of 300.

In the second step, the filtered triggers were examined individually by
a physicist using graphical display software. This stage rejected events
originating from particle showers produced by high-energy muons and
checked for errors in reconstruction and other pathologies. About 90% of
the events were rejected quickly and with high confidence. This visual
scanning reduced the data by another factor of 20, yielding 868
interaction candidates.

The estimated total efficiency for retaining a tau neutrino interaction
vertex with the electronic detectors was 75%.

### *Neutrino Event Sample*

The result of the filtering and scanning selection was the neutrino
interaction data sample. This sample of 868 events were highly likely to
be interactions from (all flavors) of neutrinos with the interaction
vertex located within the fiducial volume of the emulsion. In this
sample, we report on the complete analysis of 552 events with the
neutrino interaction vertex *located* in the emulsion. In the remainder
of this paper, we will refer to these 552 as the "located" events.
Although locating the vertex in the emulsion was attempted for each of
the 870 events, some of these events are difficult to find due to
factors discussed below in Sec. 3.1. The ν~μ~ CC events, however, are
identifiable using only the electronic spectrometer information. There
are 400(??) events in the sample of 868, which have an identified muon
track (40%??). This compares well to the 38%(??) rate found for the
sample of 552 located in the emulsion. No further analysis was done for
the events in the interaction sample that are *not* located.

## Overview of Data Analysis

## Event Reconstruction

Event reconstruction is a three step process. First, the information
from the electronic detectors is used to fit charged tracks and
reconstruct the vertex from the neutrino interaction. Next, this vertex
and its estimated errors are used to determine the location and size of
the volume in the emulsion that is subsequently scanned. Finally, once
the emulsion information is digitized, a second round of track fitting
and vertex fitting was performed. The electronic detectors were needed
to predict a vertex position with a precision of about 1mm transverse
and 10mm along the neutrino beam direction. This volume size was
well-matched to the capabilities of the emulsion scanning machines used
at Nagoya University.

The scintillating fiber tracker, which was interleaved between the four
emulsion modules, was the principal detector for making the initial
vertex prediction. A complete reconstruction of the neutrino vertex,
with all tracks unambiguously fit spatially, was made for only about 30%
of the final sample. These were usually low-multiplicity vertices
located near the downstream end of an emulsion module. Each module
corresponded to 2.5 to 3 radiation lengths and 0.2 interaction lengths
so secondary interactions giving rise to more charged particles was a
common occurrence. In the majority of the events, tracks could be easily
constructed in a single view (two-dimensional tracks) but not in space
without ambiguous solutions. Nevertheless, this information was usually
sufficient for predicting volumes used in event location in the
emulsion. Once the vertex was located using the emulsion information,
all spatial ambiguity of the neutrino interaction tracks was resolved.

## Reconstruction of Emulsion Information

1.  **Event Location**

Using the information from the SFT, the approximate location of the
neutrino interaction vertices were reconstructed and used to locate
these events within the volume of the emulsion target. The typical
volume that was digitized for event location was 5 mm × 5 mm × 15 mm.
Because the emulsion target was constructed by stacking emulsion plates
that must be disassembled for development, a method for precisely
realigning the plates was employed. A large number of background tracks
recorded in the read-out volume (high energy muons from the dump) were
used. It was these tracks that were used to precisely align the emulsion
layers. The complete tracks were built layer by layer. Each track
recognized in an emulsion layer (micro-track) was examined to see if it
had a connectable micro-track in the adjacent emulsion layers. The
parameters of interest are the distance between the emulsion layers (L),
the relative shifts in transverse direction (x,y) and the shrinkage of
the emulsion layers. The measured ∆θ  was also affected by emulsion
distortions. From the angular and position displacement distributions
the above parameters can be determined.

Once a predicted emulsion volume in the target was scanned and aligned,
track pairs were examined to if they formed a vertex. To select these
tracks, the following criteria were applied: (1) Tracks must start
within the volume and have no connectable micro-tracks in the two
adjacent upstream emulsion layers. This requirement rejects the
penetrating muon tracks. (2) Tracks must be constructed from at least
three micro-tracks and have a good chi-square fit. These requirements
reduce the number of low momentum tracks. (3) The remaining tracks were
tested for vertex topology. At least three (two?) tracks were required
to be associated where all impact parameters at the best vertex position
were less than 4 μm. After these vertex requirements were imposed, only
a few vertex candidates remained. To confirm a vertex candidate, (i) the
emulsion plates near the vertex point were studied using a manually
controlled microscope to check for consistency of the neutrino
interaction hypothesis (*i.e.* neutral particle interaction) (ii) the
emulsion track information was compared with the hits in the SFT to
verify all tracks were associated with the same event.

## Momentum from Coulomb Scattering

Although the thickness of the emulsion modules was a disadvantage in
reconstructing the vertices from the spectrometer data, this depth was
also used to the benefit of the experiment. The very high spatial
precision of the tracking along with an adequate sampling rate was used
to calculate the momentum of tracks extracted from the visible
scattering between emulsion plates. In order to gain maximum
sensitivity, the emulsion data was subjected to calibration procedure to
remove local geometrical distortions in the emulsion layers using the
always-present penetrating muon tracks. Details in this procedure are
given in \[Ref NIM Emul paper\].

The upper limit of the momentum measured using scattering was limited by
the number of samples, the angle of the track, the quality of the
emulsion data and the type of emulsion module. The typical upper limit
(1-σ) was 25 GeV/c. A comparison of track momenta measured with both the
emulsion and spectrometer is shown in \[Fig\].

## Secondary Vertex Analysis

For the located events, the emulsion was digitized again for a volume
optimized around the position of the vertex. This volume was smaller,
2.5 mm × 2. 5mm × 12 mm. The track reconstruction algorithm was the same
as that used for vertex location. The tau decay search was divided into
three distinct categories distinguished by topology: (1) one-prong
decays where the tau passed through at least one emulsion layer (Long
decay search), (2) one-prong decays where the only the daughter was
recorded in emulsion (Short decay search) and (3) three-prong or trident
decays where the tau passed through at least one layer (trident search
). Details of the Long and Short decay search have been previously
published \[Ref NIM-emul\].

The Long decay search imposed the following criteria:

1)  The parent track length was \< 10 mm.

2)  The impact parameter of the parent to the primary vertex

    1.  was \< 5 μm if there are at least two segments

    2.  was \< (5 + 0.01×δ*z*) μm if there was one segment, where δ*z* ≡
        distance from parent segment to vertex

3)  The minimum distance between extrapolated parent and daughter tracks

    1.  was \< 5 μm if there are at least two segments

    2.  was \< 5 + 0.01×δ*z*) μm if there was only one parent segment

4)  The kink angle was \> 4 times the angular measurement error OR the
    impact parameter of the daughter to the primary vertex is \> 4 times
    the error in the position

5)  The digitzed parent and daughter tracks may or may not coalesce :

    1.  If the parent and daughter tracks are distinct the parent must
        be at least one segment and the daughter at least three segments

    2.  If the parent and daughter tracks are not distinct in the
        digitized data then part assigned to the parent track must be at
        least four segments and the daughter part at least four
        segments.

> Candidate tracks passing the above criteria were checked by a
> physicist using a microscope to ensure that the kink track could not
> be associated with emulsion tracks upstream of the vertex and was not
> due to alignment problems, *e*^+^*e*^-^ pairs or other pathologies.
>
> The Short decay search required the following criteria:

1.  The daughter track was at least three segments,

2.  The impact parameter to the primary vertex was \< 200 μm.

Again, the candidate daughter track was checked visually to insure that
it could not be connected upstream of the vertex. The above criteria are
topological and were intended to find real kinks and tridents in the
data. This was the first step to extract tracks of interest, which were
either actual decays or hadronic interactions in the modules. The
criteria for separating decays, using criteria derived from kinematic
considerations, is given in Section 4.5, below.

## Lepton Tagging

If an electron or muon was associated with the primary vertex of the
neutrino interaction then the interaction was rejected from the tau
analysis, even if a secondary vertex were found. Lepton identification
was achieved with the electronic spectrometer for both electrons and
muons and the emulsion information was employed for electron
identification. The muons were found by using proportional tube planes
interleaved with steel for absorbing hadrons. A muon tag was assigned to
a track if there were at least four of the possible six prop tube hits.
The per tube efficiency for muons was measured to be 0.96, and the
geometrical efficiency of the muon ID system was estimated by Monte
Carlo to be 0.76, yielding an overall efficiency of 0.73.

The electron analysis was less straightforward since it involved several
systems. Since the emulsion modules are 2 to 3 radiation lengths, most
events containing electrons will exhibit showers in the scintillating
fiber tracker and calorimeter. These two systems are used to find the
most likely initial energy of the electron from an algorithm using both
energy (pulse height) and geometrical shower development. Emulsion
information can be used by searching for electron-positron track pairs
within 20μm of the track under investigation. This method was effective
for vertices located in the upstream part of each module. The efficiency
for electron tagging using the spectrometer was estimated using Monte
Carlo to be 0.80. The electron tagging efficiency using emulsion data
varied with depth from 0.86 for tracks passing through at least 2 *X*~0~
to 0.0 for tracks at the last (downstream plate) with an integrated
efficiency of 0.66.

## Tau / Charm Recognition Analysis

1.  **Topology and Kinematics**

There were three analyses used to extract the tau signal. First, a
relatively simple set of selection criteria was applied to the entire
data set of 550 \[??\] events. This type of analysis was reported by us
in a previous publication \[Ref fop\]. The second method used five
parameters measured from the data (four parameters for 3-body decays)
and compared the values to the expected 5-dimensional (or 4-dim)
distributions from a Monte Carlo. A probability for each hypothesis (i.e
tau, charm or background) was obtained from this multivariate analysis.
A third analysis using artificial neural networks, trained on Monte
Carlo, was used to extract tau events as well as classify the type of
interaction (i.e. ν*~e~*, ν~μ~, ν~τ,~, or neutral current).

The charged-current interactions of ν~τ~ produce a τ lepton that
typically decays within 2mm of its origin. Thus, the topological
signature for tau events is a track from the primary vertex that gives a
secondary vertex at a short distance consistent with the kinematics of
the decay. As noted above, there must be no other lepton from the
primary vertex. The second background of significance was hadronic
interactions that appear in the emulsion data as a kinked track or a
three-body decay. The following sets of criteria was derived from Monte
Carlo studies to efficiently extract the tau signal from with minimal
background. It is a modified version of the selection criteria of \[Ref
fop\]. For Long decays the cuts were:

-   The parent angle w.r.t. the neutrino direction \< 0.2 radians.

-   The daughter angle w.r.t. the neutrino direction \< 0.3 radians.

-   The kink angle (for one-prong candidates) \< 0.25 radians.

-   The impact parameter to the primary vertex \< 500 μm.

-   The transverse momentum of the decay was \> 250 MeV/*c* for hadrons
    and electrons and \> 100 MeV/*c* for muons.

-   The daughter momentum \> 1 GeV/*c*.

-   The sum of the impact parameters of the daughters in three-body
    decays was \< 600 μm.

In the case of trident secondary vertices, at least one of the secondary
tracks must pass all of the above requirements.

For Short decays the cuts were the same as above except that the kink
angle cannot be defined since the parent angle is unknown. Here we
replaced the kink angle by the minimum kink angle, which was defined as
the angle between the daughter and a line passing through the primary
vertex and the daughter track projected to the emulsion boundary.

The above criteria

1.  **Mutivariate Analysis**

    1.  *Introduction*

Since the tau neutrino charged-current interactions are the main
interest of this paper, an additional, more detailed analysis is
performed on the tau and charm events. This analysis is based on
Bayes\'s equation (cite) and provides relative probabilities for the
following possibilities: a tau event, a charm event, and a hadronic
event. A tau event is a tau neutrino charged-current interaction which
produces a tau lepton. This tau lepton decays to one or three charged
particles. A charm event is a muon or electron neutrino interaction
which produces a charged charm particle. This charm particle decays to
one or three charged particles, and the primary lepton is not
identified. A hadronic event is an electron or muon neutrino interaction
which produces a hadron in the final state. The hadron interacts and
produces either one or three charged particles, and the primary lepton
is not identified. This analysis is also applied to the charged charm
candidates, but not the neutral charm candidates since they are not
background to the tau neutrino charged-current interaction.

Each tau candidate must either be a tau decay or a background event. The
relative probabilities for each possibility are calculated using
Bayes\'s equation (cite). This analysis provides relative probabilities
for each tau and charm candidate, which are used to define a set of tau
neutrino interactions for the cross section measurement. Bayes\'s
equation is

$$P\left( \left\{ x \right\}|i \right) = \frac{\Gamma_{i}\Pi\left( \left\{ x \right\}|i \right)}{\Gamma_{i}\Pi\left( \left\{ x \right\}|i \right) + \Sigma_{\text{bkg}}\Gamma_{\text{bkg}}\Pi\left( \left\{ x \right\}|\text{bkg} \right)}$$

$$$$

where {*x*} is a set of the event\'s parameters. For kink events, this
includes five parameters, the production angle, the polar angle, the
decay length, the kink angle, and the daughter momentum, and for trident
events this includes four parameters, the production angle, the polar
angle, the decay length, and the sum of the daughters\' impact
parameters. These parameters are described in below. Eqn. \_\_\_
provides the relative probability of the event being a tau event, given
the set of parameters and is used to calculate the relative
probabilities for each possibility for each tau and charm candidate.

Γ~i~ is the prior probability of the event being an *i* type event. The
prior probability is the likelihood of each event type. The expected
number of tau neutrino interactions from the event data set is
calculated and used for the prior probability. Similarly, the charm and
hadronic prior probabilities are the expected number of charm and
hadronic background events in the data set. Π({*x*}\|*i*) is the
probability density function evaluated at a set of measured parameters,
{*x*}, of event type *i*, which is calculated using the fraction of
simulated events residing in a region of parameter space centered on the
specific values of the event\'s parameters. The sum, Σ~bkg~ Γ ~bkg~ Π(
{x}\|bkg), is over all hypotheses except the *i*th hypothesis.

2.  *Parameters*

The analysis used the following parameters, θ, the polar angle
asymmetry, ∆φ, the decay length of the parent track, *L*, the kink
angle, α, the daughter momentum, *P~d~*, and the sum of the daughter
impact parameters, Σ*IP*. The Monte Carlo was used to simulate these
parameters for tau neutrino interactions and muon and electron neutrino
interactions which resulted in either a charm which decayed to one or
three charged particles or a non-charmed hadron which interacted and
produced one or three charged particles. The charm and interaction
events were the main potential backgrounds to the signal events. The
analysis provided relative probabilities for each of these possibilities
for each tau and charm candidate. Figures \_\_\_\_ illustrate the
parameters. The simulated distributions of each parameter were
normalized using the weights provided for each event in the Monte Carlo;
these weights were a product of the probability of generation and
interaction

figure -The kink event parameters

figure -- The Delta Phi parameter

figure- The trident event parameters

The production angle θ is the angle between the neutrino direction and
the direction of the parent,

$$\theta = \text{cos}^{- 1}\left( {\hat{P}}_{p} \cdot {\hat{P}}_{\nu} \right)$$

where *P~p~* is a unit vector in the direction of the parent track, and
*P~ν~* is a unit vector in the direction of the neutrino. The neutrino
direction was calculated from the center of the beam dump to the
interaction point; this assumes the neutrino was produced in the center
of the beam dump. The maximum uncertainty for this assumption was 1.5
mrad, which occurred when the neutrino was produced at the outer edge of
the beam dump. Since the tracks undergo multiple scattering, the
direction of the parent track was measured from the first three emulsion
segments on the emulsion track. This parameter provided significant
distinction between the tau and interaction events. The production of
tau leptons in high energy tau neutrino interactions was mainly in the
forward direction; therefore, the tau distribution is peaked near zero.
The distributions of θ for simulated tau, charm, and interaction events
are shown is figure \_\_\_.

∆φ is the polar angle asymmetry, which is calculated using

$$\Delta\varphi = \text{tan}^{- 1}\left( \frac{\Sigma_{i \neq \tau}\text{sin}\Phi_{i}}{\Sigma_{i \neq \tau}\text{cos}\Phi_{i}} \right) - \Phi_{\text{parent}}$$

where

$\Phi_{i} = \text{tan}^{- 1}\left( \frac{\text{tan}\alpha_{y}}{\text{tan}\alpha_{x}} \right)$
for $\alpha_{y} \geq 0$

and

$\Phi_{i} = \text{tan}^{- 1}\left( \frac{\text{tan}\alpha_{y}}{\text{tan}\alpha_{x}} \right) + \pi$
for $\alpha_{y} \leq 0$.

where α~y~ and α~x~ are the production angles of the parent track in the
Y-Z and X-Z planes respectively. Using this definition of ∆φ, the tau
distribution is peaked at π. The polar angle imbalance, ∆φ, is the
measure of the momentum balance between the parent and all other
particles produced in the interaction. The particles which produced
background kinks were not constrained to this momentum balance since the
kinks were not caused by the lepton from the neutrino interaction. The
polar angle provided significant distinction between the charm and
interaction backgrounds and the tau events. The ∆φ distributions for
simulated tau, charm, and interaction events are shown in figure
\_\_\_\_.

The decay length, *L*, is the length of the parent particle, defined as
the distance from the neutrino interaction position to the point of the
decay or interaction, as measured in the emulsion data. This parameter
provided distinction between the interaction backgrounds and the tau
events. The tau and charm distributions are similar; they are
exponentials and are strongly peaked at zero. The decay length
distributions are shown for simulated tau, charm, and interaction events
in figure\_\_\_\_\_\_

The daughter angle, α is the angle between the direction of the parent
track and the direction of the daughter. This was calculated in a manner
similar to the production angle, using the last three segments of the
parent track and the first three segments of the daughter track. This
parameter was only used for the kink events. The kink angle provided
significant distinction between the tau and interaction events. Large
kinks were more probable for interaction events than for tau events
because there were many more low momentum hadrons. The distributions of
this parameter for simulated tau, charm, and interaction events are
shown in figure \_\_\_\_.

The daughter momentum, *P~d~*, was measured either using the deflection
of the track in the magnet or using the emulsion data to measure the
momentum from multiple scattering. This parameter was only used for the
kink events. The uncertainty in the multiple scattering measurement of
momentum was due to a limited number of track segments in the emulsion.
The tracks with low momentum scattered more; thus, the momentum
measurement from multiple scattering was better for lower energy tracks.
The lower momentum tracks often did not reach the drift chambers, and,
therefore, had no spectrometer momentum measurement. The daughter
momentum provided a distinction between the tau events and the
interaction backgrounds. The distributions of the daughter angle for
simulated tau, charm, and interaction events are shown in figure
\_\_\_\_.

Σ*IP* is the sum of the impact parameters of the daughter particles,
which was used only for the trident events. The impact parameter was
calculated using the daughter angle and the length of the tau lepton
track: $\text{IP} = L\text{sin }\alpha$ . The sum of the impact
parameters was closely related to the kink angles of the daughters and
the length of the tau track. Since the tau track was generally shorter
than the primary track from a hadronic interaction, the tau distribution
peaked at lower values. The distributions for the tau, charm, and
interaction events are shown in figure \_\_\_\_.

3.  *Prior Probabilities*

The analysis is performed on four types of candidates, the tau kink, the
tau trident, the charm kink, and the charm trident candidates. Since
each of these events have different characteristics, each hypothesis,
tau, charm, and interaction, will have a separate prior probability for
each event type, tau kink, tau trident, charm kink, and charm trident.

The prior probability, Γ*~i~*, of an event is the number of expected
events of each type divided by the total number of events. The prior
probabilities are calculated using the expected number of each event
type and the probability that the event type will pass the event
requirements.

4.  *Probability Densities*

The probability density is a measure of how likely it is that a
candidate is a tau, charm, or hadronic event based on its parameters.
The neutrinos were generated using the NUGEN \\cite{NUGEN} neutrino
generator package. The neutrino-nucleon interactions were simulated
using the LEPTO event generator \\cite{LEPTO}, which provides a list of
simulated tau leptons from tau neutrino charged-current interactions and
charm particles from electron or muon neutrino charged-current
interactions. The hadronic interactions were simulated using the GEISHA
hadronic interaction simulator \\cite{GEISHA}. The particles were
propagated through the detector using the GEANT detector simulation
\\cite{GEANT}. An interval is defined for each parameter centered on the
value of the event\'s parameters, which are listed in Sec. 5.2.2.

The probability density evaluated at {*x*} is:

$$\Pi\left( \left\{ x \right\}|i \right) = \frac{W_{\Delta v}}{W_{\text{total}}\Delta v}$$

where Π({*x}\|i*) is the probability density function evaluated at a set
of measured parameters, {*x*}, *W*~∆*v*~ is the weight of all the
simulated events which fall within this volume of parameter space, ∆*v*,
and *W~total~* is the total weight of all simulated events.

5.  *Prior Probabilities*

Table \_\_\_\_ summarizes the prior probabilities for all tau and
charged charm candidates for both kink and trident events. The candidate
column refers to the type of candidate, and the material column refers
to where the decay occurs.

INSERT TABLE

6.  *The Results of the Multivariate Analysis*

Table \_\_\_ lists the results, which are relative probabilities of each
hypothesis for each of the tau and charged charm candidates. These were
calculated using Bayes's equation, and the prior probabilities and
probability densities listed above. According to this analysis one of
the tau candidates is more likely a hadronic interaction than a tau
neutrino charged-current interaction.

INSERT TABLE

2.  **Analyses using Neutral Networks**

The technique of Artificial Neural Networks (ANN) was used in the DONUT
data analysis for the cases where event classification was needed. The
technique is illustrated in Fig. A1. The network consists of several
layers of neurons: the input layer, the output layer and several hidden
layers. The weights w~ij~ represent the strength of the connection ij.
The input to the network is the vector **x** = (x~1~,x~2~, ..., x~n~),
with components the variables {x~i~, i=1,n} and the output is the vector
**y** = (y~1~,y~2~, ..., y~n~). The training of the network is performed
by presenting the ANN with pairs of vectors (x,y) and adjusting the
weights w~ij~ in such a way as to minimize a cost function. The training
set can be either experimental information or Monte Carlo. In the latter
case special attention must be taken to make sure that the Monte Carlo
describes the data satisfactorily.

In DONUT ANNs were used in the following analyses: (i) Selection of
spectrometer events to scanned and subsequently reconstructed. For this
previously selected by eye events were used as the training set. Using
this technique a second selection became possible without visual scan.
(ii) Classification of neutrino interactions into CC and NC and
subsequently into muon neutrino and electron neutrino interactions. In
this case the training set was consisting of Monte Carlo events and
special effort was made to make sure that Monte Carlo described well the
data. (iii) Selection of tau neutrino interactions and separation from
interactions imitating taus, namely, charm production by muon and or
electron neutrinos and background from hadron scattering. In this case
each event was characterized by its probability to belong to each of the
three categories (tau, charm, hadron scattering) was determined. The
training set was provided from Monte Carlo.

> Details of how ANN analysis was implemented will be given in sections
> \*\*\*\*.

## Survey of Data

## Expected Composition

The expected number of interactions, along with the type of interaction
was predicted by using known and measured quantities in a Monte Carlo.
One must define interaction "type" by the way an event was classified in
DONuT. Charged-current interactions of ν~μ~, ν*~e~* and ν~τ~ were
defined by recognition of the lepton at the primary vertex. However, the
NC type of event was defined as all events not recognized as CC events.
This class of event included not only genuine NC events, but also those
CC events where the lepton was not identified. Also,
![](media/image1.pct)interactions could be distinguished only for
![](media/image2.pct)CC interactions where the sign of the muon was
determined in the magnetic spectrometer. Table (??) shows the expected
fraction and number of events of each of the four interaction types
measured in the experiment. Note that although the prompt (from charm
decays) and non-prompt (from π and *Κ* decays) components are separated
in the calculation, they are not directly distinguished in the analysis.

+----------+----------+-----------+------------+----------+----------+
|          | ν*~e~*   | ν*~μ~* CC | ν*~μ~* CC  | ν*~τ~*   | NC       |
|          | CC       | prompt    |            | CC       |          |
|          |          |           | non-prompt |          |          |
+----------+----------+-----------+------------+----------+----------+
| fraction | 0.238    | 0.199     | 0.159      | 0.018    | 0.382    |
+----------+----------+-----------+------------+----------+----------+
| fraction | 132      | 110       | 88         | 10       | 212      |
| ×        |          |           |            |          |          |
|          |          |           |            |          |          |
| 553      |          |           |            |          |          |
+----------+----------+-----------+------------+----------+----------+
| data     | 143      | 210       |            | 9        | 191      |
+----------+----------+-----------+------------+----------+----------+
| di       | 11       | 12        |            | -1       | -21      |
| fference |          |           |            |          |          |
+----------+----------+-----------+------------+----------+----------+

## ν~μ~ CC events

The identification of muons using the spectrometer was straightforward
and was efficient, so this category of interactions was considered the
most reliable. The muons were selected by requiring at least four hits
recorded in the proportional tube muon system out of a possible maximum
of six. The efficiency of this tag for ν~μ~ CC events was estimated from
Monte Carlo to be 0.75±0.03. The number of ν~μ~ CC events found in the
data was 210 events, which gives the fraction of the total located
events as 0.38±0.03. The expected fraction of the muon sample was highly
affected by the number of muon neutrinos that originated from decays of
π^±^ and K^±^ in the primary beam dump. Although there is only a small
(10^-3^) probability that they decay with *p* \> 1 GeV/*c*, the number
of estimated ν~μ~ s from π and K decays (which we call "non-prompt") is
comparable to the ν~μ~ s from decays of charm mesons (termed "prompt"
sources). The value of this non-prompt fraction of ν~μ~ events can be
estimated by direct, Monte Carlo calculation, or by extraction from the
data. In the data, very few ν*~e~* interactions are likely to be from
non-prompt sources. The number of ν~*e*~ CC events should be very nearly
the same as that from the prompt part of the ν~μ~ flux. Also, momentum
spectrum of muons from ν~μ~ CC interactions will have two different
components. The different methods yield consistent values for the ratio
of prompt muon neutrinos to the sum of prompt and non-prompt ν~μ~ . A
FLUKA-based Monte Carlo result was used for the determining the
non-prompt fraction as it produced spectra and yields most consistent
with the observations. The ratio of prompt to total ν~μ~ CC interactions
was estimated to be 0.57±0.07.

The ratio of the number of neutrino interactions with μ^+^ to the number
with μ^-^ should be computable from the cross sections of neutrinos and
anti-neutrinos as well as the detector efficiencies and acceptances. The
naïve estimate of 0.5 for this ratio was modified by Monte Carlo
computations to 0.63. This ratio from the 553 located events was
0.67±0.08.

There are three events in the located sample that have two identified
muons. One event has muons of opposite sign with one from the primary
interaction vertex and the other from a secondary, decay vertex. This is
event is identified as a ν~μ~ CC interaction producing a charmed meson.
The other two di-muon events are same-sign tracks, which probably
indicates that one of the tracks, a charged π, decays in-flight.

The momentum spectra for the muons from the located ν~μ~ CC events are
presented in Figure 5.

## ν~e~ CC events

## Electron Identification Methods

The ν*~e~* interactions comprised the second largest set of events,
after the ν~μ~ events. Both the ν*~e~* and the ν~μ~ events had the
decays of charm particles in the dump as their primary source. However,
only 5%(??) of the located ν*~e~* events originated from non-prompt
sources. The identification of electrons, essential to classifying
ν*~e~* CC events, was not as straightforward as muon identification. In
DONuT, the primary method was to use the scintillating fiber system to
record shower development in conjunction with energy in the lead glass
calorimeter. Since each emulsion module was 2 to 3 radiation lengths in
thickness, the total material in the path of an electron could be as
large as 9 radiation lengths. Thus, the target mass, interleaved with
fine-grained detectors, serves as a calorimeter in itself. The lead
glass, 7 m away, served to record the electromagnetic energy escaping
from the target/fiber system and was primary used to supplement the data
for interactions in the third and fourth (last) emulsion target modules.
For electrons that traversed at least two radiation lengths in the
emulsion modules, the emulsion data was used to search for *e*^+^*e*^-^
tracks (from bremstrahlung photons) within 50μm along the length of each
track attached to the primary vertex. Monte Carlo studies were used to
estimate the electron tagging efficiency for the neutrino energy
spectrum expected from charm decays. It was found to be 0.80. It was not
very sensitive to possible systematic shifts in the spectrum from
variation in the production models since the method is most effective
for high-energy electrons (*E* \> 20 GeV, efficiency 0.88). The mean
expected electron energy form CC interactions is 35 GeV(??).

## Electron Energy Estimates and ν~e~ CC Spectrum

The target/fiber system, used for electron ID, was also used to estimate
the electron (or gamma) energy. Since the scintillating fiber system
response was calibrated to minimum ionizing particles, the total pulse
height in a shower could be summed for each station providing a direct
measure of energy. The energy estimates at each station were input
variables for an algorithm to compute electron energy from shower
development. The calorimeter information was added for showers that
penetrated less than six radiation lengths of emulsion (approximate
shower maximum). This energy resolution (δ*E*/*E*) from Monte Carlo
events was 30%.

Since the beamline could not be configured for transport of electrons,
electron ID relied heavily on Monte Carlo simulation. A selection of
probable electrons from interactions in the last module, analyzed for
momentum in the spectrometer and energy in the calorimeter, showed that
the calorimeter calibration was consistent with the muon calibration
method.

## Nu-tau Signal

The methods described in Sections 4.3, 4.4 and 4.5 applied to the 553
located events yielded the results for the neutrino interactions with an
identified τ lepton. The list of events with a tau candidate is listed
in Table (XX).

<table style="width:100%;">
<colgroup>
<col style="width: 12%" />
<col style="width: 9%" />
<col style="width: 9%" />
<col style="width: 9%" />
<col style="width: 11%" />
<col style="width: 11%" />
<col style="width: 9%" />
<col style="width: 9%" />
<col style="width: 9%" />
<col style="width: 8%" />
</colgroup>
<tbody>
<tr class="odd">
<td>Event</td>
<td><p>Decay length</p>
<p>(mm)</p></td>
<td><p>Decay</p>
<p>Angle</p></td>
<td><p>IP</p>
<p>(μm)</p></td>
<td><p><em>p</em></p>
<p>dau.</p>
<p>(GeV/<em>c</em>)</p></td>
<td><p><em>p<sub>T</sub></em></p>
<p>(GeV/<em>c</em>)</p></td>
<td><em>P</em>(τ)</td>
<td><em>P</em>(<em>c</em>)</td>
<td><em>P</em>(int)</td>
<td></td>
</tr>
<tr class="even">
<td>3024 30175</td>
<td>4.47</td>
<td>0.093</td>
<td>416</td>
<td>5.2</td>
<td>0.48</td>
<td>0.64</td>
<td>0.36</td>
<td>0.00</td>
<td></td>
</tr>
<tr class="odd">
<td>3039 01910</td>
<td>0.28</td>
<td>0.089</td>
<td>24</td>
<td>4.6</td>
<td>0.41</td>
<td>0.96</td>
<td>0.04</td>
<td>0.00</td>
<td></td>
</tr>
<tr class="even">
<td>3140 22143</td>
<td>4.83</td>
<td>0.012</td>
<td>60</td>
<td>22.2</td>
<td>0.26</td>
<td>0.97</td>
<td>0.03</td>
<td>0.00</td>
<td></td>
</tr>
<tr class="odd">
<td>3333 17665</td>
<td>0.66</td>
<td>0.011</td>
<td>8</td>
<td>53</td>
<td>0.69</td>
<td>0.99</td>
<td>0.01</td>
<td>0.00</td>
<td></td>
</tr>
<tr class="even">
<td>3024 18706</td>
<td>1.70</td>
<td>0.014</td>
<td>23</td>
<td>50</td>
<td>0.70</td>
<td>1.00</td>
<td>0.00</td>
<td>0.00</td>
<td></td>
</tr>
<tr class="odd">
<td>3139 22722</td>
<td>0.53*</td>
<td>0.022*</td>
<td>12</td>
<td>15.8</td>
<td>0.35</td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td>3296 18816</td>
<td>0.80</td>
<td>0.054</td>
<td>38</td>
<td>5.0</td>
<td>0.27</td>
<td>0.71</td>
<td>0.29</td>
<td>0.00</td>
<td></td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td>0.190</td>
<td>148</td>
<td>1.3</td>
<td>0.25</td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td></td>
<td></td>
<td>0.130</td>
<td>112</td>
<td>1.9</td>
<td>0.25</td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td>3334 19920</td>
<td>8.88</td>
<td>0.017</td>
<td>147</td>
<td>11.6</td>
<td>0.20</td>
<td>1.00</td>
<td>0.00</td>
<td>0.00</td>
<td></td>
</tr>
<tr class="even">
<td></td>
<td></td>
<td>0.011</td>
<td>98</td>
<td>15.7</td>
<td>0.17</td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td>0.11</td>
<td>94</td>
<td>3.2</td>
<td>0.04</td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td>3250 01713</td>
<td>0.83</td>
<td>0.133</td>
<td>110</td>
<td>1.3</td>
<td>0.17</td>
<td>0.71</td>
<td>0.03</td>
<td>0.26</td>
<td></td>
</tr>
<tr class="odd">
<td></td>
<td></td>
<td>0.192</td>
<td>161</td>
<td>2.4</td>
<td>0.46</td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="even">
<td></td>
<td></td>
<td>0.442</td>
<td>354</td>
<td>0.5</td>
<td>0.21</td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
</tbody>
</table>

## Systematic Uncertainties

## Protons on Target

The number of 800 GeV protons that struck the dump were measured by
devices that integrate charge collected from secondary emission from a
foil. These monitors were calibrated with a beta source before the
experiment began. Several times during the course of the run, these
devices were calibrated against coil pickups and other monitors
installed in the accelerator extraction complex. These checks showed
that these primary beam monitors were consistent within 5% at
intensities of 5×10^12^ to 1×10^13^ protons per spill. Losses in the
beamline were negligible (\~10^-5^ ) and no other corrections were
applied.

The monitors were digitized and recorded at the experiment, and gated by
the triggering electronics. The total of 3.54×10^17^ protons were
recorded during the live-time of the experiment. A systematic
uncertainty of 5% was assigned to the value of the total protons on the
dump.

## Neutrino Production from 800 GeV Protons

The tau-neutrino cross section on nucleons can be measured relative to a
known process, such as the muon-neutrino interaction cross section
(relative measurement), or directly, given the flux of tau-neutrinos
produced in the dump (absolute measurement). The cross sections from
both methods are calculated in Section 8, and rely on knowing the flux
at the emulsion targets as a function of energy. The estimate of the
neutrino flux, absolute or relative to each other, required an estimate
of the production of all significant sources of neutrinos from proton
interactions in the dump. These sources of ν~τ~ , as well as ν~μ~ and
ν*~e~* , are primarily charmed mesons. Their cross sections and
branching ratios, extracted from the literature, were incorporated into
the analysis. For the absolute measurement, the tau-neutrino flux is
estimated from the following terms:

1.  the production of *D~S~*, *D*^±^, Λ~c~ and *B*^±^ hadrons from 800
    GeV proton interactions

2.  the decay form factor for each process

3.  the decay branching fractions of τ→ ν~τ~ *X*

4.  the acceptance of produced tau-neutrinos passing through an emulsion
    module

Terms (1), (2) and (3) are either direct measurements taken from the
published literature, or are estimates from other measurements
extrapolated to the relevant energy of this experiment. The statistical
and systematic uncertainties are substantial. Terms (1) and (2)
constituted the major source of uncertainty in the absolute cross
section. Table A gives a summary the measurements used in this
calculation.

  ---------------------------------------- ------------------------------
  Process                                  Value

  σ(*pN* → *D*^±^*X*)                      11.3±2.2

  σ(*pN* → *D*^0^*X*)                      27.4±2.6

  σ(*pN* → *D~S~X*)                        5.2±0.7

  σ(*pN* → *Λ~c~X*)                        5.4±2.1

  σ~tot~(*pW*)                             1650

  α                                        0.99±0.03

  *n*                                      7.7±1.4

  *b*                                      0.83±0.11
  ---------------------------------------- ------------------------------

**Table A.** Values of physical processes used in the analysis

to compute neutrino cross sections. The charm cross section

in a material of atomic number, *A*, is assumed to be proportional

to *A*^α^. The charm production differential cross section is assumed

to be proportional to (1-\|*x~F~*\|)*^n^* exp(-*bp~T~^2^*). All cross
sections are

in millibarns.

In Table A, the values for α, *n* and *b* were used in the simulated
production of *all* charm states that give rise to neutrinos.

The effect of varying the values listed in Table A on the number of
neutrino interactions in the emulsion was estimated by changing each
value by one standard deviation in the Monte Carlo. (Also listed in
Table A)

## Electronic Efficiencies

## Analysis Efficiencies

## Filters and Scanning

The process of selecting triggers as neutrino interaction candidates was
a sequence of two operations: (1) an encoded filter applied to all
triggered events and (2) a scan of the remaining events by physicists
using graphical displays. In (1) triggers were selected for an emulsion
search by requiring that a vertex reconstructed from tracks be within an
emulsion target. In addition, the event trigger timing between two
counters was required to be within 10 ns, and showering from nearby muon
interactions was required to be absent. The filter (1) reduced the
triggers by a factor of 300 and from Monte Carlo studies was found to
have efficiencies for keeping interactions to be 0.98 (for CC events)
and 0.96 (for NC events).

The second reduction (2) was needed to eliminate common pathologies
allowed by the filter (1). It reduced the data by a factor of 20, which
kept the amount of emulsion data for scanning to a manageable level. The
efficiency of (2) was found to be 0.865±0.066, and the error is from
systematic effects.

## Location

The efficiency for locating the primary vertex in the emulsion was given
directly as the ratio of the number events found and the number of
events tried. This ratio is 553/868 or 0.637±0.035.

## Decay Search

The efficiency of the tau decay search was determined in two ways: (1)
Monte Carlo estimates only, and (2) using secondary (hadronic)
interactions found in the emulsion data during the decay search. The
topological efficiencies, from restriction in decay length, and angles
was derived directly from simulations of neutrino interactions. Other
factors included emulsion data efficiency (both physical and software),
and algorithm efficiency (Short and Long decay searches) using emulsion
data.

The results obtained for method (1) are presented in Table (xx). The
estimate for the overall systematic uncertainties in these efficiencies
is 5% (??).

  ----------------------- ----------------------- -----------------------
                          ![](media/image3.pct)   ![](media/image4.pct)

  1-prong (hadronic       0.52                    0.54
  decay)                                          

  1-prong (electronic     0.61                    0.68
  decay)                                          

  1-prong (muonic decay)  0.63                    0.69

  3-prong decay           0.74                    0.80

  Σ                       0.59                    0.63
  ----------------------- ----------------------- -----------------------

Method (2) ignored the differences in the topology of the interaction
sample and the decay sample and the decay efficiency is simply the ratio
of the number of interactions found to the number expected.

## Nu-tau Cross Section

## Conclusions

[^1]: Perl et al.

[^2]: First paper, Phys. Lett. B

[^3]: NIM Emulsion paper.

[^4]: NIM Spectrometer paper.