Question

(a) Is the decay \({\sum ^{\rm{ - }}} \to {\rm{n + }}{\pi ^{\rm{ - }}}\) possible considering the appropriate conservation laws? State why or why not.

(b) Write the decay in terms of the quark constituents of the particles.

Short answer

(a) The decay\({\sum ^{\rm{ - }}} \to {\rm{n + }}{\pi ^{\rm{ - }}}\)is possible considering the appropriate conservation laws as all quantities are conserved.

(b) The decay in terms of the quark constituents of the particles is \(\left( {{\rm{dds}}} \right) \to \left( {{\rm{udd}}} \right){\rm{ + }}\left( {{\rm{\bar ud}}} \right)\).

Step-by-step solution

Concept Introduction

The sum of the baryon numbers of all entering particles equals the sum of the baryon numbers of all particles produced by the reaction, according to the rule of conservation of baryon number.

Conservation of Lepton says that when a lepton of a given generation is formed or destroyed in a reaction, a matching antilepton of the same generation must also be created or destroyed.

The concept of conservation of charge states that in an isolated system, the total electric charge never changes. The net quantity of electric charge in the cosmos, which is equal to the amount of positive charge minus the amount of negative charge, is constantly preserved.

Decay of \({\sum ^{\rm{ - }}} \to {\rm{n + }}{\pi ^{\rm{ - }}}\)

(a)

Check the conservation laws and if the reaction violates one of them it won't occur. The conservation laws are as follows –

• Conservation of Baryon number.
• Conservation of Charge.
• Conservation of Lepton numbers.

Now, the equation is written as –

\({\sum ^{\rm{ - }}} \to {\rm{n + }}{\pi ^{\rm{ - }}}\)

	Before	After
\({\rm{B}}\)	\({\rm{1}}\)	\({\rm{1 + 0 = 1}}\)
\({\rm{Q}}\)	\({\rm{ - 1}}\)	\({\rm{0 + ( - 1) = - 1}}\)
\({{\rm{L}}_{\rm{e}}}\)	\({\rm{0}}\)	\({\rm{0}}\)
\({{\rm{L}}_{\rm{\mu }}}\)	\({\rm{0}}\)	\({\rm{0}}\)
\({{\rm{L}}_{\rm{\tau }}}\)	\({\rm{0}}\)	\({\rm{0}}\)

Therefore, the reaction is allowed.

Decay in terms of quark compositions

(b)

According to table \({\rm{33}}{\rm{.4}}\), the quark composition for each particle is –

\(\begin{aligned}{c}{\sum ^{\rm{ - }}}{\rm{ = }}\left( {{\rm{dds}}} \right)\\{\rm{n = }}\left( {{\rm{udd}}} \right)\\{\pi ^{\rm{ - }}}{\rm{ = }}\left( {{\rm{\bar ud}}} \right)\end{aligned}\)

Substituting the values in the reaction –

\(\left( {{\rm{dds}}} \right) \to \left( {{\rm{udd}}} \right){\rm{ + }}\left( {{\rm{\bar ud}}} \right)\)

Therefore, the decay is written as \(\left( {{\rm{dds}}} \right) \to \left( {{\rm{udd}}} \right){\rm{ + }}\left( {{\rm{\bar ud}}} \right)\).