Question

(a) Is the decay \({{\rm{\Lambda }}^{\rm{0}}} \to {\rm{n + }}{{\rm{\pi }}^{\rm{0}}}\) 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 \({\Lambda ^0} \to n + {\pi ^0}\) is possible as per the conservation laws.

b. The equation \({\Lambda ^0} \to n + {\pi ^0}\)in terms of quarks is given by \(uds \to udd + (u\bar u + d\bar d)\).

Step-by-step solution

Definition of Concept

If the strangeness is not conserved for a decay process, it means that the decay has taken place through a weak reaction. As, strong interactions do not affect the strangeness of a particle.

Explain is the decay \({{\rm{\mu }}^{\rm{ - }}} \to {{\rm{e}}^{\rm{ - }}}{\rm{ + }}{{\rm{\nu }}_{\rm{e}}}{\rm{ + }}{{\rm{\nu }}_{\rm{\mu }}}\) possible considering the appropriate conservation laws

(a)

Considering the given information:

Given reaction is\({\Lambda ^0} \to n + {\pi ^0}\)

We can check the following conservation laws to see if the decay\({\Lambda ^0} \to n + {\pi ^0}\)is possible.

Decay:\({\Lambda ^0} \to n + {\pi ^0}\)

Charge:\(0 \to 0 + 0\)

\(\therefore \)the charge is conserved.

\({\rm{Baryon}}\;{\rm{number}}\left( B \right): + 1 \to + 1 + 0\)

\(\therefore \)Baryon number is conserved

\({\rm{Lepton}}\;{\rm{number}}\;\left( L \right):0 \to 0 + 0\)

\(\therefore \)lepton number is conserved

\({\rm{Strangeness}}\;\left( S \right): - 1 \to 0 + 0\)

\(\therefore \)strangeness is not conserved

While charge, lepton number\(\left( {\rm{L}} \right)\), and baryon number are all conserved, strangeness is not, for the given decay. However, the change in strangeness is\({\rm{ + 1}}\). As a result, a weak reaction is possible. As a result, according to conservation laws, decay is possible.

Therefore, the required decay\({\Lambda ^0} \to n + {\pi ^0}\)is possible as per the conservation laws.

Write the decay in terms of the quark constituents of the particles

(b)

Considering the given information:

Given reaction is\({\Lambda ^0} \to n + {\pi ^0}\)

Quark structure of\({{\rm{\Lambda }}^{\rm{0}}}{\rm{ = uds}}\)

Quark structure of\({\rm{n = udd}}\)

Quark structure of\({{\rm{\pi }}^{\rm{0}}}{\rm{ = (u\bar u + d\bar d)}}\)

As a result, the equation in terms of quarks is as follows:

\(\begin{aligned}{}{\Lambda ^0} \to n + {\pi ^0}\\uds \to udd + (u\bar u + d\bar d)\end{aligned}\)

Therefore, the required equation \({\Lambda ^0} \to n + {\pi ^0}\)in terms of quarks is given by \(uds \to udd + (u\bar u + d\bar d)\).