This week we talked about decay and scattering – applications of relativistic kinematics. And a brief introduction on what information we obtain / derive from decays etc.

Relativistic kinematics

I shall demonstrate the simplest case of these: rest frame decay A->B+C, if they are spin-less, we only have to worry about the linear momentum and energy – thus the final momentum of the two out going particles are deterministic. In A's rest frame, we have 4-momentum conservation:

$$\begin{cases} p_\text{initial} = (E_A, 0, 0, 0) = p_\text{final}\\\\ E_A = m_A = E_B + E_C \\\\ \vec{p}_B + \vec{p}_C = 0 \end{cases}$$

We also have the total 4-momentum invariant conserved as a whole:

$$\begin{split} p_B &= p_A - p_C \\\\ p_B^2 &= p_A^2 + p_C^2 - 2p_A \cdot p_C\\\\ m_B^2 &= m_A^2 + m_C^2 - 2(E_A, 0, 0, 0) \cdot p_C \\\\ &\Rightarrow E_B = \frac{m_A^2 + m_B^2 - m_C^2}{2m_A} \end{split}$$

Another fun derivation is the Compton scattering, which will give a interesting relation between the wavelength change & scatter angel, the outline of the derivation goes like this:

First write down before and after 4 momentums for photon and electron:

$$\begin{cases} p_\gamma = (E, E, 0, 0) \Rightarrow p_\gamma^\prime = (E^\prime, E^\prime \cos{\theta}, E^\prime \sin{\theta},0) \\\\ p_e = (m_e, 0, 0, 0) \Rightarrow p_e^\prime = (\sqrt{m_e^2 + |\vec{p^\prime}|^2}, |\vec{p^\prime}| \cos{\phi}, -|\vec{p^\prime}| \sin{\phi},0) \\\\ \end{cases}$$

Then, you compare total 4-momentum before and after term by term, from:

$$E^\prime\sin{\theta} - |\vec{p}|^2\sin{\phi} = 0$$

you can relate $\sin{\phi}$ to $\theta$ and then go to the $x$ component, use the fact:

$$E^\prime\cos{\theta} + |\vec{p}|^2\cos{\phi} = E$$

to re-express $|\vec{p}|^2$ in terms of $E$ and $\cos{\theta}$. Finally, use the 0-th component to obtain:

$$m_e(E - E^\prime) = E E^\prime (1-\cos{\theta}) \Rightarrow \lambda^\prime = \lambda - \frac{\hbar}{m_e c} (1-\cos{\theta})$$

Decay probability and decay width

For a particle, in its rest-frame, we say that the decay rate is $R$, for a particle can have multiple modes, under the hood: $R = \sum_i R_i$. Since we assume particles don't have 'memory' – so to speak, the decay probability between $t, t+dt$ is constant. Obviously, we get a exponential decay law out of this just like radio active decay we're familiar with. We can associate a mean life time $\tau_i$ for each mode, and because the way math works:

$$R=\sum_i R_i \Rightarrow \frac{1}{\tau} = \sum \frac{1}{\tau_i}$$

We can also define a relative ration to indicate the branching probability of a particular mode over all modes:

$$Br_i = \frac{R_i}{R}$$

An interesting trick comes in as we always want to express things in unit of energy. Notice $[\hbar] = \text{E} \cdot \text{T}$. So we can define a decay width with unit of energy:

$$\Gamma = \hbar \frac{1}{\tau} = \hbar R$$

Of course, branching ratio can also be expressed using this width:

$$Br_i = \frac{\Gamma_i}{\Gamma}$$

Later we will see that this total decay width, for particles with tiny lifetime, corresponds to Full-Width-Half-Maximum in their invariant mass distribution. This is a consequence of the said distribution follows Breit-Wigner distribution.