Standard Model and Beyond · Problem 3.29

Gauge Theories

Physics — Particle Physics · explicit-o2-breaking-charge-nonconservation

Problem

The O(2)O(2) invariance of (3.146) is explicitly broken by the mass term in (3.151). (a) Identify any unbroken symmetries and describe their consequences. (b) Show that charge is not conserved and draw a Feynman diagram that demonstrates this, e.g., by allowing the reaction ϕ+ϕ−→ϕ+ϕ+ϕ+ϕ−\phi^+\phi^- \to \phi^+\phi^+\phi^+\phi^-, where ϕ+=(ϕ−)†\phi^+ = (\phi^-)^\dagger is the particle created by ϕ†\phi^\dagger. Treat the symmetry breaking term as a mass insertion on an internal line. (There can also be mixing associated with the external legs, leading to processes such as ϕ+ϕ−→ϕ+ϕ+\phi^+\phi^- \to \phi^+\phi^+. However, the degeneracy of the states for ϵ→0\epsilon \to 0 makes this case more complicated to treat. It is very much like K0K^0-Kˉ0\bar{K}^0 mixing treated in Chapter 8.) Note that this is a spin-0 analog of a Majorana neutrino mass term.
Intuitive explanation

The mass term breaks the global U(1)U(1) charge symmetry that would normally protect particle number, but residual discrete symmetries may survive. By treating the symmetry-breaking mass insertion as a vertex in Feynman diagrams, you can construct processes that violate charge conservation—such as converting ϕ+ϕ−\phi^+\phi^- pairs into states with unequal numbers of particles and antiparticles. This is the scalar analog of how a Majorana mass term mixes particles with their antiparticles, allowing processes forbidden in the symmetric theory.

Intuitive explanation

The mass term breaks the global U(1)U(1) charge symmetry that would normally protect particle number, but residual discrete symmetries may survive. By treating the symmetry-breaking mass insertion as a vertex in Feynman diagrams, you can construct processes that violate charge conservation—such as converting ϕ+ϕ−\phi^+\phi^- pairs into states with unequal numbers of particles and antiparticles. This is the scalar analog of how a Majorana mass term mixes particles with their antiparticles, allowing processes forbidden in the symmetric theory.

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