PHYS10362 — 2nd Assignment: Z0 boson
March 2025
The Z0 boson is an elementary particle that mediates the weak nuclear interaction. It is studied at particle physics facilities such as Fermilab and CERN where beams of particles collide and their products are analysed.
In this assignment, you are tasked with deducing the mass and lifetime of the Z0 boson by studying data from electron-positron collisions resulting in electron-positron products i.e. e-e+ → Z0 → e-e+ events. If possible, you should also compute the uncertainty of these values. Your work should produce some graphics to support your analysis.
Theory
The Feynman diagram for the process we are interested in is shown below in figure 1.
Figure 1: Lowest order Feynman diagram depicting a Z0 boson resonance in e-e+ annihilation. The boson then decays to e+ e- . Time increases on the horizontal axis, anti-fermion lines point backwards.
When beams of electrons and positrons collide head-on, the rate, R, at which e-e+ pairs are produced through this interaction is given by:
R = σ · L (1)
where L is the instantaneous luminosity and is determined by the properties of the beam and im- portantly for this assignment, σ is the cross-section of the interaction and has dimensions of area. Typically σ is expressed in units of a barn(b) where 1b = 10-28 m2 (100 fm2 ). The underlying physics of the interaction is encoded in the cross-section e.g. its magnitude depends on the coupling strength of the weak interaction and the mass and lifetime of the Z0 boson.
The likelihood that the Z0 boson produces a given final state is proportional to a quantity called the partial width. For instance the partial width for Z0 → e-e+ , which we will denote by Γee, is 83.91 MeV. Whilst the partial width for Z0 decaying into hadrons is 1744 MeV; a reflection of the fact that a Z0 → hadrons event is significantly more likely than a Z0 → e-e+ event.
The cross-section for e-e+ → Z0 → e-e+ is given by
(2)
where σ is in natural units of GeV-2 . In natural units ℏ = c = 1. Here the mass of the Z0 boson is mZ and its width is ΓZ which is related to the lifetime of the Z0 boson, τZ by:
(3)
and in natural units ΓZ has units of energy.
E is the centre-of-mass energy of the colliding beams.
Project description
An experiment has taken place to determine the mass and width of the Z0 boson by measuring the cross-section at different centre-of-mass energies. Two detectors were used intermittently over different energy ranges. The data can be found in z boson data 1 .csv and z boson data 2 .csv. Unfortunately, there were some faults that led to invalid data points which you must filter out.
You are tasked with obtaining mZ and ΓZ from this data by fitting to equation 2.
Do not attempt to turn this into a linear problem and do not try and fit the parameters independently. This will significantly over-complicate the problem and will almost certainly return the wrong result.
Previous studies suggest that mZ ~ 90 GeV/c2 and ΓZ ~ 3 GeV. To convert equation 2 from natural units, you should use
(4)
where mb are mili-barns.
Your program should:
• Read in, validate and combine both data files.
• Perform a minimised χ2 fit by simultaneously varying mZ and ΓZ .
• Calculate both mZ and ΓZ to four significant figures in GeV/c2 and GeV respectively.
• Calculate the reduced χ2 to 3 decimal places.
• Calculate τZ to 3 significant figures in seconds.
• Produce a useful plot of your result.
• Ideally, you should also find the uncertainties on mZ , ΓZ and τZ to the appropriate precision.
With regards to style, in addition to what was asked for in the previous assignment, we expect your code to:
• Have a useful file check that halts the code neatly if there is an issue.
• Read in data using inbuilt functions; do not ask the user to input the file names.
• Use inbuilt functions to perform the minimisation.
• Be versatile and applicable to data with similar validation issues.
• To make any plots by attaching axes attributes to figure objects.
• Save any plots as a .png file.
• Achieve a linter score of at least 9.80/10.00 (maximum penalty of 10 marks: a score of 8.33/10.00 corresponds to a deduction of 1.57 marks and -2.40/10.00 corresponds to a deduction of 10.00 marks).
Additional marks are available for extra features. You do not need to include them all to get full marks for this aspect. Can you display extra information in these plots? Can you format these plots nicely? Can it be applied to systems with different decay products? Could it be applied to different files with different validation issues? Can you make the initial guess on mZ and ΓZ general? Could it work without knowing the initial value for Γee?
An inbuilt function is anything that comes with the Spyder release (5.4.3) on the university PCs e.g. numpy. You can use any packages that come with Spyder but do NOT use ones that you have installed separately on your laptop. The pylint score we will use is that obtained on the university PCs i.e. pylint version 2.16.2 with Spyder 5.4.3 and Python 3.11.7. You should check your linter score on the university PCs and that your code runs there without any issues before submitting it.
More detail on how the mark is split can be found in the illustrative rubric on BlackBoard.