Subfield: Modern Physics


Question: 5. Identifying particle processes 

Note: all numerical values in this problem are given in natural units: $c=1$, mass measured in $\mathrm{GeV}=1$, so no unit conversions are necessary, even when dimensions don't seem to align (this allows us to do things like sum momentum and energy without worrying about units).

The 4-momentum of a relativistic particle is the 4-dimensional vector

$$
\vec{P}=\left(E, p_{x}, p_{y}, p_{z}\right),
$$

where $E$ is the particle's energy and $\left(p_{x}, p_{y}, p_{z}\right)$ is the standard 3-momentum (excluding the time component). The total (summed) 4-momentum of a system is always conserved. The components of 4-momentum can be related to the rest mass $m$ of the particle by

$$
E^{2}=m^{2}+p_{x}^{2}+p_{y}^{2}+p_{z}^{2} .
$$

A particle accelerator records the 4-momenta of 2 muon-antimuon pairs produced by some particle processes. Other particles produced by the processes are not measured by the detector. Below are two possible processes that involve muon-antimuon production and down quark-antiquark production. We know that out of the two measured 4-momentum pairs, one corresponds to Process A and the other to Process B. Using the muon-antimuon 4-momenta given below, determine which corresponds to each process. The numbers will be truncated for simplicity and slightly fudged from precise values to account for some small error in the measurement, so don't expect precision for small values (this should not affect the problem). The points will be given for an explanation of how you reached your conclusion; simply guessing the process correctly will result in at most one point. Also note that both processes are valid, you do not need to verify that they could actually happen. A table of (mean) particle rest masses may prove useful:

| Particle | Rest Mass |
| :---: | :---: |
| Higgs | $\sim 125$ |
| Z | $\sim 91$ |
| Top | $\sim 172$ |
| Bottom | $\sim 4.2$ |
| Up | $\sim 0.0024$ |
| Muon | $\sim 0.1$ |
| Neutrino | $\sim 0$ |

Process A: The collision of an up quark-antiquark pair results in the production of one Higgs boson and one $\mathrm{Z}$ boson. The $\mathrm{Z}$ boson then decays into a muon-antimuon pair

Process B: The collision of an up quark-antiquark pair results in the production of a top quark-antiquark pair. The top quark decays into a $\mathrm{W}+$ boson and a bottom quark, while the anti-top decays into a W- boson and an anti-bottom quark. The W- then decays into a muon and an anti-neutrino, while the $\mathrm{W}+$ decays into an anti-muon and neutrino.

Detected muon-antimuon 4-momenta:

х) $(149.0,12.6,-8.8,148.2) ;(163.7,-38.8,67.3,144.1)$

у) $(319.5,-41.3,-24.2,315.8) ;(305.3,-60.1,25.0,298.1)$

Solution:
The sum of the muon-antimuon momenta are:

х) $(312.7,-26.2,58.5,292.3)$

у) $(624.8,101.4,0.8,613.2)$

For the one that corresponds to Process A, this 4-momentum would correspond to the 4momentum of the $\mathrm{Z}$ boson. This summed 4-momentum from Process B should not correspond to any particular particle, as the muon-antimuon pair decays off of two different particles. Applying the relation between the 4-momentum and the rest mass, we find that each 4-momentum would correspond to a $\mathrm{Z}$ with rest mass of:

x) 90.74

у) 63.86

The mass in $\mathrm{x}$ ) is much closer to the true mean $\mathrm{Z}$ mass, so that should be process $\mathrm{A}$, and $\mathrm{y}$ ) is process B.

Final answer: x for A and y for B