H. Schellman and G. Zeller
Version 2.2 February 15, 2003
In this lab you will use data from the NuTeV experiment at Fermilab to measure the Weak Mixing Angle, a fundamental parameter in the Standard Model of Particle Physics.
In 1967 Glashow, Weinberg and Salam proposed a unified theory of weak
and electromagnetic interactions. In this model, electromagnetic interactions
proceed by the exchange of photons and the weak interactions proceed by
the exchange of massive vector bosons. These bosons, the
, 
and 
interact with matter with couplings similar to the
electromagnetic coupling e. The reason these interactions are seen much
less frequently than the electromagnetic interaction is that interaction rates
also depend on the mass of the exchanged particle. The W and
Z are very heavy. This mass effect makes the interaction weak and
short range (the range of the force goes as 1/M).
The Weak Mixing Angle (
) is the central parameter of this unified theory.
It relates the masses of the W and Z bosons and determines their
relative couplings to matter. In this experiment you will count W and
Z mediated neutrino interactions, measure the relative couplings of
W and Z bosons and extract this parameter,
.
The data you will be using comes from the
NuTeV neutrino experiment at
Fermilab. Fermilab is a high energy particle
accelerator which produces neutrinos 
and anti-neutrinos 
with
energies between 10 and 400 GeV. The neutrinos are detected in the NuTeV
neutrino detector which
consists of a target made of a sandwich of iron plates and scintillation
counters and a muon momentum spectrometer made of iron rings wrapped with
toroidal windings.
When a neutrino interacts with a nucleus in the iron, it actually scatters from one of the quarks in a proton or neutron within the iron nucleus; it can interact in one of two ways, either:
 |
Neutral Current (NC) Interaction
Charged Current (CC) Interaction |
The neutrino scattering cartoons are called Feynman diagrams. In the top
diagram, the neutrino emits a which interacts with a d quark. This
first process, called a Neutral Current (NC) interaction, proceeds by the
exchange of the neutral boson. In this interaction, the neutrino scatters
elastically off the quark and does not change its type. The scattered quark
and the remnants of the iron nucleus are detected in the scintillation
counters but the outgoing neutrino is invisible and cannot be detected.
In the bottom diagram, the incoming neutrino emits a
and changes into a 
. The interacts with a d quark and
transforms it into a u quark. In this second process, called a
Charged Current (CC) interaction, the scattered quark is detected in the
scintillation counters (as before) but, since the outgoing particle is a muon,
it can also be detected.
One can thus distinguish between Charged Current (CC) and Neutral Current (NC)
interactions based upon their signatures in the detector (i.e. events with and
without a muon). In addition, one can measure their relative production rates.
These relative rates are simple functions of the Weak Mixing Angle ().
The following is a guide to the particles involved in this laboratory.
)
Muons are charged particles very similar to electrons but with a much larger
mass (
MeV). Muons are minimum ionizing particles. Muons pass through
matter only losing a few MeV of energy for each cm of matter they traverse and
can thus travel long distances even in iron. This property is unique to muons
and makes them very easy to identify in the laboratory.
)Neutrinos are electrically neutral particles which only interact via the weak (and gravitational) interactions - this it what makes them so hard to detect! In 1962, Lederman, Schwartz and Steinberger demonstrated that there were two types of neutrinos: the electron and muon types. Now three are known: the electron, muon, and tau neutrinos.
Muons, electrons, and neutrinos are collectively called "leptons" and have spin 1/2.
The vector bosons introduced earlier mediate the weak interaction. The word "boson" refers to particles with integer spin. "Vector" implies that they have spin 1.
The W boson comes in two charges ( and ) and has a mass of 
GeV.
In this experiment, W bosons are only produced as virtual particles but higher
energy experiments have produced real W bosons both individually and
in pairs.
The Z boson is the neutral partner of the charged W boson and has a
mass of around 
GeV. It has also been observed directly in high energy
proton/anti-proton and electron/positron collisions.
Most of matter is made up of protons and neutrons which are in turn made up of quarks. Quarks interact via the strong, weak, electromagnetic and gravitational interactions. Six types of quarks are known:
The masses given above are approximate as free quarks do not seem to occur in
nature, they always appear as
or qqq bound states with large binding energies.
When neutrinos interact with iron nuclei, their interactions are an incoherent sum of interactions with individual quarks inside the nucleus. Protons are mainly made up of 2 u quarks and 1 d quarks with a additional "sea" of quark-anti-quark pairs and gluons which arise from vacuum fluctuations. Neutrons are the same but with the u and d quarks interchanged. This is a reflection of "isospin symmetry". A typical iron nucleus contains 26 protons and 30 neutrons and is thus almost an equal (isoscalar) mix of u and d quarks. In this laboratory we will ignore scattering off the heavier s quark as its effects are smaller than the statistical precision of the data.
For more information on the elementary particles, see the Particle Data Group web site: http://pdg.lbl.gov/. This site contains databases with full information on elementary particles. Also, the PDG and Fermilab sites both have online explanations of particle physics, the standard model, and experimental procedures. You should go through these sites.
The struck quark in a neutrino interaction is knocked out of the nucleus with
a high energy 
. Here 
denotes the hadronic (quark) energy in the
event and 
denotes the energy of the outgoing neutrino or muon.
The struck quark will appear as a group of particles called hadrons, which
consist of quark/anti-quark pairs or triplets of quarks. These hadrons
interact via the strong interaction and will hit an iron nucleus and cause
secondary interactions, which will then cause tertiary interactions, etc.
The net result is a shower of particles which deposit energy in the iron and
in the scintillation counters. This shower of particles will generally
terminate within a few interaction lengths 
, where is the
typical distance a hadron can travel in iron before interacting. One can thus
estimate the energy transferred to the quark by measuring the energy deposited
in the shower.
The NuTeV detector consists of two parts.
is 90%
where is in GeV.
The tracking chambers are drift chambers, which respond to the position but not the amount of deposited energy. They have an intrinsic resolution of 0.5 mm. In this experiment, you will use them for visualization and to measure the momentum of the scattered muon.
In the Lab E detector, a muon will lose around 200 MeV of energy per scintillation counter. Most of this energy is actually lost in the 4 inches of iron between scintillation counters but it is sampled by the scintillation counters.
The toroid consists of six 55 inch radius iron annuli wound with coils. The magnetic field in the iron is saturated at around 2 T and runs in the azimuthal direction. The current direction is chosen so that the muon is always bent inwards (focussing). Muons experience many small scatters (a process called multiple scattering) which cause them to deviate slightly from the trajectory they would follow in a magnetic field in vacuum. These small random scatters limit the accuracy of momentum determination to 11%.
Please feel free to take a virtual tour of the NuTeV detector.
Neutrinos interact with matter very rarely, of the
neutrinos produced by the Fermilab neutrino beamline every minute, only
10 will interact in the 690 ton NuTeV detector. Because the neutrino has no
electric charge, the detector cannot "see" the incoming neutrino. So how do
we know that a neutrino has interacted?
The following figure shows a typical charged current neutrino interaction in the NuTeV detector.
The neutrino has entered invisibly from the left and interacted at counter 60. A hadronic shower and a muon have been produced; they both show up in the tracking chambers (as X's) and as energy in the scintillation counters (the histogram at the top of the display). In addition to giving us a glimpse of the neutrino interaction, this sample "event display" also provides some calculated information displayed in the right hand corner:
Using these two variables, the LENGTH of an event can therefore be defined as PLACE - CEXIT + 1.
This is the whole name of the game. Recall that in this experiment, neutral and charged current events are distinguished by the number of scintillation counters which fire.
Neutral current events will have just the quark induced shower and will thus only show a signal in a small number of scintillation counters:
 |
A Neutral Current (NC) neutrino interaction in the NuTeV detector. The shower occured between counters 60 and 50 and therefore has a LENGTH of 11 counters. Note that SHEND, the end of the high energy part of the shower is quite close to CEXIT, the last counter to fire. This event is SHORT. |
We will define our NC candidate events as SHORT events, namely those having LENGTHs less than or equal to 20 scintillation counters.
This is because, in contrast to neutral current events, charged current events will also have a muon going forward in the lab frame and will thus have a lengthened signal due to the muon which penetrates all of the scintillation counters after the interaction until it leaves the detector:
 |
A Charged Current (CC) neutrino interaction in the
NuTeV detector. The interaction occured near PLACE = 46, deposited more than
5 MIPS until SHEND = 33 but the outgoing muon
continued to deposit energy until it left the calorimeter at CEXIT = 1. This event is LONG. |
But we must be careful. In the case where the muon is going backwards in the center of mass frame and the neutrino energy is low, it is possible for the muon to be at large angles in the lab frame and exit out the side of the detector. Such events may be quite short and can be confused with neutral current events. You will be asked to estimate the confusion rate. Below is an example:
 |
A short Charged Current (CC) neutrino interaction in the
NuTeV detector. In this interaction, the muon emerged at a large angle and
left the detector through the side. Note: the LENGTH of this event is PLACE
- CEXIT + 1 = 20 which would classify it as a short "neutral current
candidate", however a visual inspection shows an obvious muon track leaving
the detector. Hence, our LENGTH  20 classification would have
counted this as a NC event, but it's clearly CC. |
One of the features of the weak interactions is that they care about the
helicity of the interacting particles. In particular, the weak vector bosons
can only interact with quarks which have particular helicities (spin
directions relative to their momenta). These helicities are denoted as
left-handed or right-handed. Neutrinos are massless and have only been
observed to occur in a left handed state while anti-neutrinos only occur as
right-handed particles. Quarks can occur in either state.
bosons prefer to interact only with left-handed quarks and
right-handed anti-quarks while bosons have the opposite preference.
Neutrinos can interact with quarks via the exchange of a
in the following ways:
Note that the quarks have changed type but that charge is conserved in the
interaction. Interactions such as
are forbidden by charge conservation. Interactions such as
are forbidden by muon number conservation as the neutrino has muon
number +1 and the 
has muon number -1.
However, anti-neutrinos interact via the exchange of a
:
The cross section for charged current (CC) interactions on a proton or neutron is the sum of the interactions with different quark types:
Where 
is the Fermi weak interaction constant, s is the center of
mass energy of the neutrino and proton, 
and 
represent the fraction
of the proton's momentum carried by quarks of that type.
is the scattering angle in the quark-neutrino center of mass frame. The
reason the 
term appears is that in scattering of a left-handed neutrino
from a left-handed d quark, the total angular momentum is zero and the
interaction is isotropic, while in the scattering of a left-handed neutrino
from a right handed 
quark, the net angular momentum is 1 and the final
state reflects this through the term.
The center of mass scattering angle is related through simple
relativistic kinematics to the fraction of energy transferred from the
neutrino to the quark in the lab frame. The energy transferred to the quark
is measured in the detector and is refered to as 
where "had"
indicates hadronic energy. The fraction of the neutrino energy transferred
to the quark is 
and:
bosons are willing to interact with both quarks and anti-quarks but have
couplings that depend on the charge and helicity of the particle and involve
the Weak Mixing Angle. The cross section is:
In the case of a target with equal number of neutrons and protons, the numbers
of u and d quarks are equal and U=D. The number
of and 
quarks are also equal so 
. For an iron target, there
are actually slightly more D than U quarks but the difference is
small.
If one makes the assumptions: U = D and
, the cross sections for neutral and charged current interactions
at a fixed value of or y can then be related:
At fixed neutrino energy E, the energy transferred to the quark, EHAD, is just y E. This quantity can be easily measured for both neutral current and charged current interactions while y requires a measure of the neutrino energy E. For fixed neutrino energy:
because 
.
The number of events N observed in the experiment will be
where 
is the incoming flux of neutrinos, 
is the density of the
target, L is the length of the target, A is fraction of events
which are detected and 
is the cross section. For EHAD > 20
GeV, the acceptance A is 100% for detecting both charged and neutral
current events. The target density and length are also the same for both
types of events. The flux is different for neutrino and anti-neutrino events.
However, if one measures the number of events as a function of EHAD for
neutrino interactions only, then one can use the previous formula and get:
where 
(r~ 1/2)
Therefore,
So, if we can measure the ratio of neutral current to charged current
interactions, we can extract the Weak Mixing Angle from this relation!
Therefore, you will convert measurements of NC/CC neutrino event ratios
into estimates of the Weak Mixing Angle, 
.
Your data sample consists of around one day's worth of data taken at the NuTeV experiment late in 1996. The reason you do not get the full data sample is because Excel will croak on anything larger. The data comes in one file:
This neutrino data file contains 14 columns of numbers:
