Discovering the $W$ and $Z$ bosons was quite hard due to their weight. It thus required a lot of mass from energy for them to become “real”. This second (which I don't know if it examinable) goes through a history of particle collider physics experiments to discover these particles
The accelerator emitted $400\,\rm GeV$ protons to a stationary target
⚽ Goal: find out if the proton has enough energy to create a real $Z$ or $W$ boson
Lets start by calculating the centre of mass energy $\sqrt{s}$
$$ \begin{aligned} S&=(q_\text p+q_\text t)^2=q^2 _\text p+ q^2 _\text t+2 q _\text pq \text t \\ &=m^2 \text p+m^2 \text t+2(E\text p m\text t- \vec p\text p\cdot \vec 0) \end{aligned} $$
where the subscript $_\text p,\ _\text t$ stands for proton, target
Now we assume the masses $m_\text p$ and $m_\text t$ are negligible compared to $E_\text p$ which thus gives
$$ \sqrt{s}\simeq \sqrt{2 E_\text p m_\text t} =\sqrt{800\,\rm G eV^2}\simeq 30\,{\rm GeV}< m_Z\simeq 90\,{\rm GeV} $$
💎 Conclusion: this accelerator didn't produce particles with high enough energy for detection
This accelerator produced a proton and anti-proton with energy of $270\, \rm GeV$ and made them collide
💼 Case: $540\,\rm GeV$ is a lot of energy however not all is converted into $Z$ and $W$ bosons
We can draw the Feynman diagram of the allowed interactions as follows
💎 Conclusion: quarks of $p$ and $\overline p$ only store a fraction of the energy thus we need $\sqrt{s}\gg m_{W,Z}$
The lifetime of $W$ and $Z$ bosons is around $\sim 3\cdot 10^{-25}\,\rm s$ or about $1\,\rm attometer$ of decay length. This is much smaller than the $\sim 1\,\rm cm$ detector resolution, we thus need to consider the products
💼 Case: lets find the decay products of $Z$ bosons