= Solution
\b[The labels X and Y cannot be uniquely assigned from the surviving text: their <decay vertices> and daughter tracks were contained in the missing figure.] The available <conservation laws> nevertheless specify how the assignment would be obtained.
The initial $\pi^-p$ state has $(Q,B,S)=(0,1,0)$, with $S$ denoting <strangeness>. At a primary <strong interaction> vertex all three numbers must be reproduced. A final <Lambda baryon> or <Sigma baryon> has $B=1$, $S=-1$ and therefore needs accompanying $S=+1$, for example a neutral <kaon> $K^0=d\bar s$ or a positive <kaon> $K^+=u\bar s$, with the remaining <electric charge> balanced. At later <weak interaction> <decay vertices>, <strangeness> need not be conserved; it would be wrong to enforce it there.
A displaced $p\pi^-$ pair with the appropriate <invariant mass> identifies $\Lambda^0\to p\pi^-$. A displaced $\pi^+\pi^-$ pair can identify $K_S^0\to\pi^+\pi^-$ after an <invariant mass> and kinematic check. For a candidate <radiative Sigma-to-Lambda transition>, the neutral parent satisfies
$$
p_\Sigma=p_\Lambda+p_\gamma,\qquad
p_\Lambda=p_p+p_{\pi^-},\qquad
p_\Sigma^2=m_\Sigma^2,\quad p_\gamma^2=0
$$
in <natural units>. Its <photon> <energy> in the parent rest frame is $(m_\Sigma^2-m_\Lambda^2)/(2m_\Sigma)$. With known charged <four-momenta>, each unobserved neutral <momentum> is the difference between the incoming and visible outgoing <momenta> at its vertex; its inferred direction must agree with the dashed segment.
This information cannot establish which dashed segment was called X. For a concrete ambiguity, both
$$
\pi^-p\to K^0\Lambda^0,\qquad
\pi^-p\to K^0\Sigma^0\to K^0\Lambda^0\gamma
$$
obey the primary <electric charge>, <baryon number>, and <strangeness> constraints and can yield a neutral <kaon> decay plus a displaced <Lambda baryon> decay. Their distinct neutral segments and the extra <photon> would be resolved by the missing topology and kinematics. The supplied possible <Sigma baryon> decay alone is insufficient to label X and Y. A labelled answer is therefore a source gap, rather than a completed identification.
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