Kinematics and Feynman Diagrams

Special relativity energy, momentum relations and $\beta$

$E=\gamma mc²$ and $p=\gamma mv$ can be rearranged to write: $\beta=\frac{v}c=\frac{p}E$.

• First write $\gamma=\frac{E}m$ using the fact that $E$ has units $/c²$.

• Write $\gamma=\frac{pc}{mv}$ similarly, using the fact that $p$ has units $/c$.

• Equate these expressions for $\gamma$ and rearrange.

Format for relativistic kinematic questions

Work in the parent particle’s rest frame.

• Compute momentum conservation

• Compute energy conservation

• Apply energy-momentum relationship $E²=m²+p²$

• Find required quantity

If the parent particle has an initial momentum:

• Calculate $\beta=\frac{v}c=\frac{p}E$ and $\gamma=\frac{E}m$ for the parent particle.

• Boost the $(E, pc)$ four-vector for the relevant particle (which the quantity is being found for) using the inverse Lorentz transformation

Assumptions made for COM energy calculations

At very high energies or for massless particles:

• $$W²=|(P_1+P_2)|²\backsim 2P_1P_2$$

• $2P_1P_2=2E_1E_2+2c²\vec p_1\cdot\vec p_2\backsim 2E_1E_2(1-\cos\theta)$ using $E\backsim pc$. This can be applied to two beams colliding, where $\theta$ is the angle at which they collide. For a beam-target collision, this is not necessary.

For a photon emission from quarks within a hadron, which quark is more likely to emit the photon?

Photon couples to electric charge, so the quark with the largest magnitude of charge has a higher probability of emitting the photon.

• Probability of emission is proportional to the square of the quark's charge.

• E.g., for a meson made up of a charm and anti-strange decaying from an excited state, the charm is more likely to emit the photon.

Conserved quantities at vertex of Feynman diagrams

• Electric charge. Note its importance for weak decays, where a W$^\pm$ boson may provide this conservation.

• Baryon number. Baryons carry $B = 1$, anti-baryons carry $B = -1$.

• Lepton number. Leptons carry $L=1$, anti-leptons carry $L=-1$.

• Total energy and momentum of the system (whole diagram).

• Colour charge.

Effect of forces on particles (Feynman diagrams)

• Strong: Couples to quarks only. Connects quarks or produces quark-antiquark pairs. It cannot change the flavour of a quark.

• Weak: Couples to all fermions. Only interaction that can change a quark's flavour (if via W$^\pm$, not via Z). Responsible for decays involving neutrinos and leptons.

• Electromagnetic: Couples to all electrically charged fermions. Usually emitted in decays where an excited particle sheds energy.

Probability of a process occurring

The probability of the process occurring is proportional to $e^{2n}$, where $n$ is the number of vertices in the Feynman diagram for the process.

Rate of interactions of different forces

Via the square root of the coupling constant for each force.

• Strong interactions: fastest, rate given by $\sqrt {\alpha_s}=\sqrt1$, massless gluons.

• Weak interactions: slowest, rate given by $\sqrt{\alpha_{W,Z}}=\sqrt{1/40}$, massive W, Z bosons.

• EM force: square root of coupling constant is $\sqrt{\alpha}=\sqrt{1/137}$, however, as the photon is massless, this increases interaction rate. Speed is intermediate.