Question

The reaction \({{\rm{\pi }}^{\rm{ + }}}{\rm{ + p}} \to {{\rm{\Delta }}^{{\rm{ + + }}}}\) (described in the preceding problem) takes place via the strong force.

(a) What is the baryon number of the \({{\rm{\Delta }}^{{\rm{ + + }}}}\)particle?

(b) Draw a Feynman diagram of the reaction showing the individual quarks involved.

Short answer

a. The baryon number of \({{\rm{\Delta }}^{{\rm{ + + }}}}\)particle is\({\rm{ + 1}}\).

b. The Feynman diagram for the reaction \({\pi ^ + } + p \to {\Delta ^{ + + }}\) is drawn.

Step-by-step solution

Definition of Concept

A graphical representation of the reactions that represent the interactions between elementary particles and the product formed in the reaction is called Feynman diagram. They are named after the famous scientist Richard Feynman.

Find the baryon number of the \({{\rm{\Delta }}^{{\rm{ +  + }}}}\) particle

(a)

The baryon number of\({{\rm{\pi }}^{\rm{ + }}}\)is 0

the baryon number of a proton is\({\rm{ + 1}}\).

Let the baryon number of\({{\rm{\Delta }}^{{\rm{ + + }}}}\)is\(b\).

for the given reaction-

\({\pi ^ + } + p \to {\Delta ^{ + + }}\)

\({\rm{Baryon number}}:0 + 1 \to b\)

In order to conserve baryon number for the given reaction, the baryon number on both sides of the reaction must be same. So, baryon number of the\({{\rm{\Delta }}^{{\rm{ + + }}}}\)particle should be equal to\({\rm{ + 1}}\).

Also, based on the quark structure,\({\Delta ^{ + + }} = uuu\)

The baryon number of a u quark is\({\rm{ + }}\frac{{\rm{1}}}{{\rm{3}}}\).

As a result, the baryon number is given as-

\(\begin{array}{c}B = \frac{1}{3} + \frac{1}{3} + \frac{1}{3}\\ = 1\end{array}\).

Therefore, the requiredbaryon number of\({{\rm{\Delta }}^{{\rm{ + + }}}}\)particle is\({\rm{ + 1}}\).

Draw a Feynman diagram of the reaction

(b)

Below is a Feynman diagram for the reaction \({\pi ^ + } + p \to {\Delta ^{ + + }}\).

Therefore, the Feynman diagram for the reaction \({\pi ^ + } + p \to {\Delta ^{ + + }}\) is drawn.