Hadrons decay thanks to the strong interaction, in this process they get lighter, so how do lighter hadrons decay, for example $\pi^+$ or $\pi^0$

💼 Case: the decay of $\pi^0$ and $\pi^+$

Decays via the electromagnetic force into photons

Decays via the weak force into, for example, an antimuon and a muon-neutrino

Electromagnetic force

🔦 Photon properties:

• Force carrier of the electromagnetic force

• Massless spin-1 boson that couples to the electric charge by does not carry one itself

• A photon can interact as follows

  

  where $f$ are fermions

• A photon ca be considered virtual if it exists within $\Delta E \Delta t\ge \hbar$ in that case its energy can be greater than the available energy

• The strength of coupling is given by: $\alpha_\text{em}=\frac{e^2}{4\pi\epsilon_0 \hbar c} \approx \frac{1}{137}$ where $e$ is the electric charge

Other properties of electromagnetism:

• Quarks carry fractional charges (eg $\frac 23 e$ for up quark) this reduces the coupling to the fraction squared time $\alpha _\text{em}$ (eg $(\frac 23 )\alpha _\text{em}$ for the up quark)

• The photon is massless and does not carry electric charge itself so the e.m. interaction is $\infin$

• The scattering of a particle on another is described by the Coulomb potential

  $$ V(r)=-\alpha_\text{em} \frac{\hbar c}{r} $$

Weak interaction

👻 $W$ bosons properties

• Force carrier of the weak interaction

• Spin-$1$ bosons with $\pm 1$ electric charge for $W^\pm$ respectively

• $W$ bosons can interact as follows

  

  here it is a $\mu$ muon $\nu$ neutrino interaction ie $W$ boson coupling to leptons

• Mass $80 \, \rm GeV$ so it has range $R \approx c \Delta t =\frac{\hbar}{m_W c}$

• Since it has a short range it takes a lot time for a particle to decay via this force

• It conserves the individual lepton number

• They do not conserve quark number to allow for $\beta$ decay for example

⚽ Goal: lets take a closer look at quark couplings with

If we look at the 💃 Example: Kaon decay $K^{-}\to \mu+\overline \nu_\mu$

Here an $s$ quark changed into a $u$ quark via $W$ boson interactions, showing that they can change the flavour inter-family

We can include this type of interactions to the other using the quark mixing hypothesis by Cabbibo

💫 Cabbibo hypothesis: $d$ and $s$ quarks participate in the weak interaction via the following

$$ \begin{aligned} d'&=d \cos \theta_C+ s \sin \theta_C \\ s'&=-d \sin \theta_C+s\cos \theta_C

\end{aligned} $$

where $\theta_C$ is known as the Cabbibo angle and: