Abstract

The purpose of this investigation was to graphically analyze the Z boson (Z 0 ) as detected by the Large Hadron Collider detector to determine the decay rate and lifetime of the Z 0 .  We analyzed graphs of the decay of the Z boson to μμ and ee, and found its lifetime to be very short. The Heisenberg Uncertainty Principle was used to determine the lifetime of the Z boson and explain how the conservation laws remained conserved.

Introduction

A Z-Boson is a force carrier that is made up of either one positron or one electron or two muons of opposite charge. When the neutral Z boson decays, it produces either a quark and the matching flavor of antiquark or a lepton and the matching flavor or anti-lepton.  It must obey the conservation laws to maintain neutrality (Bohm & Harshman, 2000).

The lifetime of a particle (denoted by τ) is governed by the decay rate (Γ), which is the probability per unit time that a given particle will disintegrate. For decays less than or equal to 10 -24 seconds, such as the Z 0 , it is impossible to measure the exact lifetime (Abe et al., 2000).  The decay rate is measured using the full width of the particle’s mass in centimeters.   It should be noted that the decay rate (Γ) in s -1 is the same physical quantity as the natural decay width (Γ) in MeV.  Though they have different SI units, they are related by the reduced Planck’s Constant (h-hat), which is a physical constant in quantum mechanics and is equal to: 6.582 x 10 -22 MeV’s (Arthurs & Goodman, 1988).

1. Decay rate: τ = 1/Γ

2. Decay width: τ = h-hat/Γ

The relationship between the lifetime and the width of a decaying particle state can be explained by the Heisenberg Uncertainty Principle, which characterizes small and short-lived products, such as the Z boson, that are produced in high energy collisions in accelerators.  The uncertainty principle in the form below suggests that for particles with extremely short lifetimes, there will be significant uncertainty in the measured energy, where the particle lifetime (τ) is equal to the change in time (t).

3. Heisenberg Uncertainty Principle: ΔE = Γ/2 = h-hat/2τ

4. Heisenberg Uncertainty Principle: Δt ≥ h-hat/(4πΔE)

Thus it was hypothesized that the Heisenberg Uncertainty principle could be used to predict the lifetime of the Z 0 , and explain the discrepancy between its decay and lifetime (Maggiore, 1993).  In the above equations, the principle is rewritten so that energy uncertainty (ΔE) can be determined and the lifetime can be implied from it.

Procedures

To determine the lifetime τ  and the decay rate Γ of Z 0 Boson, the Z→μμ and Z→ee data from the Large Hadron Collider was analyzed to find the width distribution of z signature. Both data were found under the Exploration tab. The μ+μ- mass for Z→μμ data and the e+e- mass for Z→ee data were plotted. For the Z→μμ plot, the width of the peak was determined at bin width of 0.45. For the Z→ee plot, the width was analyzed at bin width of 0.45.

  Using the Heisenberg Uncertainty Principle, Γ and τ  were found using the equation ΔE= Γ/2= h-hat /2τ by replacing the ΔE(change in energy) with the width distribution and using 6.582×10 −16 eV*s for h-hat (reduced Planck’s Constant), the Γ and τ  could be solved. The analyzed Γ and τ  numbers were then compared with the actual values.

Results

Z→μμ (Figure 1)
Using the mass plot, the width distribution was between 90.9-91.4 GeV/c 2 , where the actual mass, 91 GeV, lied. This width was then used to get the lifetime and decay rate.

ΔE= 2/ Γ= h-hat /2τ

ΔE = 91.4-90.9 = 0.5 GeV = 5×10⁸eV
5×10⁸eV = Γ/2 = (6.58211928×10^-16 eV*s)/2
Γ= 84.0 MeV, τ= 6.582×10^-25 sec

Z→ee (Figure 2)
The width of the plot was found to be between 91.4-91.8. The Uncertainty Principle was used again for the lifetime and decay rate.

ΔE= 2/ Γ= h-hat /2τ

ΔE = 91.8-91.4 = 0.4 GeV = 4×10⁸eV
4×10⁸eV=Γ/2=(6.58211928×10^-16 eV*s)/2
Γ=84.0 MeV, τ=8.228×10^-25 sec

Discussions & Conclusions

Through the data from the Large Hadron Collider, the decay rate and lifetime numbers of Z→μμ and Z→ee data were determined. According to the obtained plots, the decay rate and lifetime of Z→μμ were 84.0 MeV and 6.582×10^-25 s, respectively. Also the decay rate and lifetime of Z→ee were 83.9 MeV and 8.228×10^-25s, respectively. The validity of the calculated data was confirmed when they it was found to be approximately close to the accepted lifetime of 1×10^-25s (Abrams et al., 1989).

According to the calculated data, the Z particle has a very short lifetime and almost immediately decays into a low energy particle; conservation of energy seems to be violated.  According to the Heisenberg Uncertainty Principle, if the time of a process is exceedingly short, then the uncertainty in energy can be very large.  Thus these high-energy force carrier particles can exist if they are short lived.  Thus energy is conserved because the energy of the initial and final decaying Z 0 is equal.  The Z 0 exists for such a short time that it cannot be observed (Collins, Soper, & Sterman, 1985).

Bibliography

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