The principal decay mode of the sigma zero is \[{{\rm{\Sigma }}^{\rm{0}}}{\rm{ }} \to {\rm{ }}{{\rm{\Lambda }}^{\rm{0}}}{\rm{ + \gamma }}\]. (a) What energy is released? (b) Considering the quark structure of the two baryons, does it appear that the \[{{\rm{\Sigma }}^{\rm{0}}}\]is an excited state of the \[{{\rm{\Lambda }}^{\rm{0}}}\]? (c) Verify that strangeness, charge, and baryon number are conserved in the decay. (d) Considering the preceding and the short lifetime, can the weak force be responsible? State why or why not.

The capacity to work is defined as energy. It can take the form of potential energy, kinetic energy, thermal energy, electrical energy, chemical energy, nuclear energy, or any other form. Law of conservation of energy states that the total energy of the system should be conserved during a reaction.

Principal decay mode of \[{{\rm{\Sigma }}^{\rm{0}}}\] is: \[{{\rm{\Sigma }}^{\rm{0}}}{\rm{ }} \to {\rm{ }}{{\rm{\Lambda }}^{\rm{0}}}{\rm{ + \gamma }}\;\; \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \left( 1 \right)\]

a)

The release energy\(\left( {\Delta E} \right)\)is the difference between the\[{{\rm{\Sigma }}^{\rm{0}}}\]particle's rest mass energy\(\left( {{E_\Sigma }} \right)\)and the two rest mass energies of the\[{{\rm{\Lambda }}^{\rm{0}}}\;{\rm{particle}}\left( {{E_\Lambda }} \right){\rm{ and \gamma }}\;{\rm{particle }}\left( {{E_\gamma }} \right)\].

\[\begin{array}{c}\Delta E = {E_\Sigma } - {E_\Lambda } - {E_\gamma }\\ = 1192.6\;{\rm{MeV}} - 1115.7\;{\rm{MeV}} - (0)\\ = 76.90\;{\rm{MeV}}\end{array}\]

Therefore, the required solution is 76.90 MeV.

b)

The \[{{\rm{\Sigma }}^{\rm{0}}}\] particle is an excited state of the \[{{\rm{\Lambda }}^{\rm{0}}}\]particle because both of them share the identical quark structure. When \[{{\rm{\Sigma }}^{\rm{0}}}\]decays to \[{{\rm{\Lambda }}^{\rm{0}}}\], only energy is released in the form of -rays.

c)

Both the \[{{\rm{\Sigma }}^{\rm{0}}}\] and \[{{\rm{\Lambda }}^{\rm{0}}}\] have strangeness quantum number equal to\( - 1\), whereas for -rays, its value is zero. The conservation of the strangeness:

\[\begin{array}{c}{S_L} = - 1\\{S_R} = - 1 + 0 = - 1\\{S_R} = {S_L}\end{array}\]

Here\[{S_R}\]is the total strangeness on the right side and\[{S_L}\]is the total strangeness on the left side of the reaction (1). As they are equal, Strangeness is conserved in the reaction.

Both the \[{{\rm{\Sigma }}^{\rm{0}}}\] and \[{{\rm{\Lambda }}^{\rm{0}}}\] have charge equal to \( + e\), whereas for -rays, its value is zero. The conservation of the charge is shown below:

\[\begin{array}{c}{Q_1} = + e\\{Q_2} = + e + 0 = + e\\{Q_1} = {Q_2}\end{array}\]

Here \[{{\rm{Q}}_{\rm{zzz1 is the total charge on the left side and Q2 is the total charge on the right side of the reaction (1) and e is the elementary charge. As they are equal, the charge is conserved in the reaction.

Both the \[{{\rm{\Sigma }}^{\rm{0}}}\] and \[{{\rm{\Lambda }}^{\rm{0}}}\] have Baryon number equal to \( + 1\), whereas for -rays, its value is zero. The conservation of the Baryon number is shown below:

\[\begin{array}{c}{B_1} = 1\\{B_2} = 1 + 0 = 1\\{B_1} = {B_2}\end{array}\]

Here \[{{\rm{B}}_{\rm{1}}}\] is the total Baryon number on the right side and \[{{\rm{B}}_2}\]is the total Baryon number on the left side of the reaction (1). As they are equal, Baryon number is conserved in the reaction.

d)

The weak force cannot be responsible as the strangeness is conserved. Also, the lifetime is short and this allows us to state that the responsible force is the strong nuclear force.