The Relationship between Resonance Width and Particle Lifetimes
Alex Olds06/08/2016
Abstract
This Study was done to find and describe the relationship, if any, between Resonance Width ( Γ ) and the lifetimes of relevant particles (τ) .
This was done by utilizing data publicly available from CERN on the resonances of the Z boson, W boson, electron, and muon and comparing them to their known lifetimes, finally deducing the relation, if any. Using the uncertainty principle in form ΔEΔt > (?/2) , it is suggested that there is an inverse relationship between energy uncertainty ( ΔE ) and particle lifetime ( Δt ). When expressed as ΔE = (Γ/2) = (?/(2τ)) , lifetime could theoretically be calculated from resonance width, and vice versa. To test this, known values for Γ for the z-boson and the Higgs Boson were found and inputted into the equation and τ experimentally calculated, then compared to the known value of τ for said particles. This shows that the relationship described above is indeed the relationship between Γ and τ .
Introduction
This study was performed to explain how the resonance width of a given particle relates to said particle’s lifetime, if at all. Knowing this relation would allow to find one or the other by simply knowing either the resonance width or the lifetime. Although this relation is known, this study applied it to the z-boson to find how easy and accurate the relationship is from experimental data and comparing it to known values. This appears to reinforce currently accepted science, without creating any conflicts or gaps in knowledge.
The purpose of this study is to increase the knowledge of the interlacing characteristics of subatomic particles. Knowledge also may be gained on how these characteristics relate.
Procedures
Data was collected from publicly avaiable sources from the ATLAS detector at CERN, on Z -> μ μ events. This data was then plotted as invariant mass vs the Breit-Wigner probability density to form a curve (fig. 1). This curve was then labelled with Γ at half-maximum. From this, the value of Γ can be found. Next, the Uncertainty Principle is taken in the form ΔE = (Γ/2) =( ? /(2τ)) and Γ is substituted with it's value. This gives us a new equation of 1.25 = ( ? /(2τ)).
Results
For this data, Γ= 2.5. We now substitute Γ for 2.5 in ΔE = (Γ/2) =( ħ /(2τ)). This gives us a new equation of 1.25 = ( ħ /(2τ)) and solve for τ. In this case, τ = 2.32847804x10-25 . For reference, the lifetime of the Z-Boson is believed to be roughly 3x10-25 by the particle physics community.
Figures
Discussion and Conclusions
The Resulting lifetime estimation of 2.32847804x10-25 is remarkably accurate, not only is it on the same order of magnitude as the accepted 3x10-25 lifetime, it is far closer than expected, given that the original Γ measurement was an estimation. This result contributes further support for this method, as it does appear that there is, in fact, a definite relationship between the resonance width and the lifetime, as well as showing that this method is a reasonable estimation of that relationship.
Bibliography
Ocariz, J. (2012, October 14). Probability and Statistics for Particle Physicists. KEK-Proceedings, 2013(8), 253-280. doi:10.5170/CERN-2014-001.253