# Feynman Toy Model Constraints on Number of Each Kind of External Lines

(From my homework) In our toy model we have three kinds of spinless particles: $A$, $B$, and $C$. The primitive vertex of decay/interaction is shown below:

My actual problem statement says: "Suppose a diagram in our toy model for the Feynman Calculus has $N_A$ external $A$ lines, $N_B$ external $B$ lines, and $N_C$ external $C$ lines. Develop a simple rule for determining whether it is an allowed reaction."

If all the initial particles are $A$ then I was able to determine that $2N_A\geq N_B + N_C$ is a necessary restriction but I am not sure if this basic toy model restricts all initial particles to be $A$. Additionally, if the model restricts all final particles to be $B$ or $C$ then the inequality in my restriction becomes an equality. Does anyone have enough experience with this toy model to know if the only initial particles are $A$? If not, can you offer some suggestions to help me figure out an appropriate restriction for the general case, where $A$ $B$ and $C$ particles can all be initial or final particles?

EDIT: After talking with my professor he told me that the masses of each particle obey $m_A>m_B+m_C$, so that $A$ particles are able to decay into $B$ and $C$ but that reactions that produced $A$ particles as real (instead of virtual) particles are not allowed.

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