Bubble chamber experiment on a K− beam

[Aleolomorfo]

Homework Statement

In a bubble chamber experiment on a $K^-$ beam a sample of events of the reaction $K^- + p \rightarrow \Lambda^0 + \pi^+ + \pi^-$ is selected. A resonance is detected both in the $\Lambda^0\pi^+$ and $\Lambda^0\pi^-$ mass distributions. It is called $\Sigma(1385)$. If the study of the angular distributions establishes that the orbital angular momentum of the $\Lambda^0\pi$system is $L=1$, what are the possible spin parity values $J^p$?

Relevant Equations

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Hello everybody!

Let's begin with the spin. Spin of the $\Lambda$ is $1/2$ and of the pion is $0$:
$$ \frac{1}{2} \otimes 0 = \frac{1}{2}$$

Since I know from the homework statement that $L=1$:
$$ \textbf{J} = \textbf{spin} \otimes \textbf{L} = \frac{1}{2} \otimes 1 = \frac{1}{2} \oplus \frac{3}{2} $$

Then the parity of the system:
$$P(\Lambda^0\pi) = P_{\Lambda}P_{\pi}(-1)^L= (+)(-)(-) = +1 $$

So there are two possibilities: $\frac{1}{2}^+$ and $\frac{3}{2}^+$.

But from PDG $\Sigma(1385)$ is a $\frac{3}{2}^+$ particle. I do not how to reject the other possibility of $\frac{1}{2}^+$.
Thanks in advance!

 

[mfb]

This will probably need other measurements. $\Sigma(1385) \to \Lambda \gamma$ is another possible decay, for example.
The problem statement asks about possible states (plural) based on this individual measurements.