Weak Mixing Angle Lab Decription

Measurement of the Weak Mixing Angle

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 tex2html_wrap_in line1171, tex2html_wrap_inline1173 and tex2html_wrap_inline1175 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 (tex2html_wrap_inline1183) 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, tex2html_wrap_inline1183.

The data you will be using comes from the NuTeV neutrino experiment at Fermilab. Fermilab is a high energy particle accelerator which produces neutrinos tex2html_wrap_in line1197 and anti-neutrinos tex2html_wrap_inline1199 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 displaymath1201

Charged Current (CC) Interaction displaymath1201

The neutrino scattering cartoons are called Feynman diagrams. In the top diagram, the neutrino emits a tex2html_wrap_inline1175 which interacts with a d quark. This first process, called a Neutral Current (NC) interaction, proceeds by the exchange of the neutral tex2html_wrap_inline1175 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 tex2html_wrap_inline1171 and changes into a tex2html_wrap_inline1211. The tex2html_wrap_inline1171 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 (tex2html_wrap_inline1183).

The Cast of Characters

The following is a guide to the particles involved in this laboratory.

Muons (tex2html_wrap_inline1229)

Muons are charged particles very similar to electrons but with a much larger mass (tex2html_wrap_inline1225 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.

The Muon Neutrino (tex2html_wrap_inline1197)

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 (W, Z)

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 (tex2html_wrap_inline1171 and tex2html_wrap_inline1173) and has a mass of tex2html_wrap_inline1239GeV. 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 tex2html_wrap_inline1247 GeV. It has also been observed directly in high energy proton/anti-proton and electron/positron collisions.

Quarks (u,d,s,c,b,t)

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 tex2html_wrap_inline1265 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.

What is Going On When a Neutrino Interacts?

The struck quark in a neutrino interaction is knocked out of the nucleus with a high energy tex2html_wrap_inline1293. Here tex2html_wrap_inline1295 denotes the hadronic (quark) energy in the event and tex2html_wrap_inline1297 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 tex2html_wrap_inline1301, where tex2html_wrap_inline1301 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.

Experimental Signatures in the NuTeV Detector

The NuTeV Detector

The NuTeV detector consists of two parts.

Please feel free to take a virtual tour of the NuTeV detector.

Neutrino Signature in Matter

Neutrinos interact with matter very rarely, of the tex2html_wrap_inline1308 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?

What This All Looks Like

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:

Telling Neutral Current and Charged Current Events Apart

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 tex2html_wrap_inline169 20 classification would have counted this as a NC event, but it's clearly CC.

Theoretical Predictions for Production Rates

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. tex2html_wrap_inline1171 bosons prefer to interact only with left-handed quarks and right-handed anti-quarks while tex2html_wrap_inline1173 bosons have the opposite preference.

Neutrinos can interact with quarks via the exchange of a tex2html_wrap_inline1171 in the following ways:



Note that the quarks have changed type but that charge is conserved in the interaction. Interactions such as tex2html_wrap_inline1334 are forbidden by charge conservation. Interactions such as tex2html_wrap_inline1336 are forbidden by muon number conservation as the neutrino has muon number +1 and the tex2html_wrap_inline1338 has muon number -1.

However, anti-neutrinos interact via the exchange of a tex2html_wrap_inline1173:



The cross section for charged current (CC) interactions on a proton or neutron is the sum of the interactions with different quark types:



Where tex2html_wrap_inline1350 is the Fermi weak interaction constant, s is the center of mass energy of the neutrino and proton, tex2html_wrap_inline1354 and tex2html_wrap_inline1356 represent the fraction of the proton's momentum carried by quarks of that type. tex2html_wrap_inline1358 is the scattering angle in the quark-neutrino center of mass frame. The reason the tex2html_wrap_inline1360 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 tex2html_wrap_inline1364 quark, the net angular momentum is 1 and the final state reflects this through the tex2html_wrap_inline1360 term.

The center of mass scattering angle tex2html_wrap_inline1358 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 tex2html_wrap_inline1370 where "had" indicates hadronic energy. The fraction of the neutrino energy transferred to the quark is tex2html_wrap_inline1374 and:


tex2html_wrap_inline1175 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 tex2html_wrap_inline1175 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 tex2html_wrap_inline1364 and tex2html_wrap_inline1394 quarks are also equal so tex2html_wrap_inline1396. 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 tex2html_wrap_inline1396, the cross sections for neutral and charged current interactions at a fixed value of tex2html_wrap_inline1358 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 tex2html_wrap_inline1424.

The number of events N observed in the experiment will be


where tex2html_wrap_inline1430 is the incoming flux of neutrinos, tex2html_wrap_inline1432 is the density of the target, L is the length of the target, A is fraction of events which are detected and tex2html_wrap_inline1438 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:



displaymath1450(r~ 1/2)


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, tex2html_wrap_inline1456.

The Data Sample

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:

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