Griffiths Introduction to Elementary Particles · Problem 3.14

Symmetries — David J. Griffiths

Physics — QFT · threshold-energy

Problem

Particle AA (energy EE) hits particle BB (at rest), producing particles C1,C2,…C_1, C_2, \ldots: A+B→C1+C2+⋯+CnA + B \to C_1 + C_2 + \cdots + C_n. Calculate the threshold (i.e., minimum EE) for this reaction, in terms of the various particle masses. [Answer:E=M2−mA2−mB22mB c2,where M≡m1+m2+⋯+mn]\left[Answer: \quad E = \frac{M^2 - m_A^2 - m_B^2}{2m_B}\,c^2, \quad \text{where } M \equiv m_1 + m_2 + \cdots + m_n\right]
Intuitive explanation

At threshold, the products are created at rest in the center-of-mass frame—no kinetic energy is wasted. The key is to use four-momentum conservation: the invariant mass of the initial system must equal the rest mass of the final products. By working in the lab frame where BB is at rest, you can set up the energy-momentum relation and solve for the minimum incoming energy EE needed to reach this kinetic-energy-free configuration.

Intuitive explanation

At threshold, the products are created at rest in the center-of-mass frame—no kinetic energy is wasted. The key is to use four-momentum conservation: the invariant mass of the initial system must equal the rest mass of the final products. By working in the lab frame where BB is at rest, you can set up the energy-momentum relation and solve for the minimum incoming energy EE needed to reach this kinetic-energy-free configuration.

Related theory

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