The cross section formula contains a factor of Lorentz-invariant phase space (LIPS) for the final state. What does this factor represent physically?
AThe density of available final states consistent with energy-momentum conservation — more available states means higher probability of the transition
BThe Lorentz contraction of the target particle
CThe quantum interference between different final states
DThe normalization of the initial-state wave functions
Phase space is the volume of momentum space accessible to the final-state particles, weighted by the relativistic density of states and subject to the constraint of total energy-momentum conservation. A process with more available final states (more phase space) has a higher rate, even if |M|^2 is the same. This is why heavy particles have more decay channels and higher total decay rates than light particles — more final states become kinematically accessible as the mass increases. Phase space also explains why three-body decays are typically slower than two-body decays: the phase space integration is more restrictive.
Question 2 Multiple Choice
Two processes have the same |M|^2 but different final-state multiplicities. Process A produces 2 final particles; process B produces 4. Which has the larger phase space, and why?
AProcess B, because more particles always means more phase space
BProcess A, because each additional particle introduces a factor of (2pi)^{-3} and a mass-shell delta function, which restricts the available phase space despite adding more degrees of freedom
CThey have equal phase space because |M|^2 is the same
DIt depends entirely on the masses of the final-state particles
The comparison depends on the kinematics — specifically, the masses of the final-state particles relative to the total available energy. Each additional particle adds three momentum components but also adds an on-shell constraint and a factor of 1/(2E_i). If the total energy is much larger than the sum of final-state masses, adding particles can increase the phase space volume. If the energy is barely enough to produce all particles, phase space is severely restricted. There is no universal rule — the specific masses and total energy determine the outcome.
Question 3 True / False
The lifetime of an unstable particle is the inverse of its total decay rate: tau = 1/Gamma_total, where Gamma_total is the sum of partial decay rates to all kinematically allowed channels.
TTrue
FFalse
Answer: True
Each decay channel i has a partial decay rate Gamma_i computed from |M_i|^2 integrated over phase space. The total decay rate is Gamma_total = sum of all Gamma_i. The lifetime is tau = hbar/Gamma_total (or 1/Gamma_total in natural units). The branching ratio for channel i is BR_i = Gamma_i/Gamma_total. This means that adding new decay channels (for example, by increasing the particle's mass so that heavier final states become accessible) increases Gamma_total and decreases the lifetime.
Question 4 Short Answer
Derive the formula for the two-body decay rate of a particle of mass M decaying into two particles of masses m1 and m2, and explain the role of each factor.
Think about your answer, then reveal below.
Model answer: In the rest frame of the decaying particle, Gamma = |p_f|/(8 pi M^2) |M|^2, where |p_f| = (1/2M)sqrt{[M^2-(m1+m2)^2][M^2-(m1-m2)^2]} is the magnitude of the final-state momentum. The factor |M|^2 encodes the dynamics (the interaction strength and structure). The factor |p_f| comes from the phase space — it vanishes at threshold (M = m1 + m2), where the decay products have zero kinetic energy, and grows as M increases. The factor 1/(8 pi M^2) combines the normalization conventions and the phase space measure. For a spin-averaged decay, an additional factor of 1/(2J+1) averages over the initial spin states.
This formula is the workhorse for computing particle lifetimes. It separates the dynamics (|M|^2, computed from Feynman diagrams) from the kinematics (phase space factors). The threshold behavior |p_f| -> 0 as M -> m1 + m2 is universal and explains why particles just barely above threshold decay slowly, while highly off-shell decays proceed rapidly.