Physics reference: Λ⁰ → p π⁻

Reference for the decay reconstructed in this lesson.

The Λ⁰ baryon

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The Λ⁰ is the lightest strange baryon (quark content uds, spin-parity ½⁺). With no lighter strange state to decay into strongly or electromagnetically, it decays weakly (strangeness-changing ΔS = 1). Weak decays are slow, so the Λ⁰ is long-lived on detector scales (cτ ≈ 7.9 cm) with a dominant two-body hadronic mode.

Quantity	Value (PDG)
m(Λ⁰)	1.115683 GeV
cτ(Λ⁰)	7.89 cm
BR(Λ⁰ → p π⁻)	63.9 %
m(p)	0.9382720813 GeV
m(π±)	0.13957061 GeV
Q-value	≈ 37.8 MeV
natural width Γ = ħ/τ	≈ 2.5 × 10⁻⁶ eV

The charge conjugate Λ̄ → p̄ π⁺ is reconstructed identically with the antiproton (PDG -2212) and π⁺ (211).

The V0 signature

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Traveling ≈ 8 cm before decaying, the Λ⁰ produces a V0: two oppositely charged tracks from a vertex displaced from the primary interaction point. The neutral Λ⁰ leaves no track.

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flowchart LR
    accTitle: {Lambda V0 decay topology}
    accDescr: {Lambda V0 decay topology}
    PV["primary vertex<br/>e + A collision"]:::vtx -. "Λ⁰: neutral, cτ ≈ 7.9 cm" .-> DV["displaced<br/>decay vertex"]:::vtx
    DV --> P["proton track (+)"]:::pos
    DV --> PI["pion track (−)"]:::neg
    classDef vtx fill:#e7efff,stroke:#4c6ef5,stroke-width:1.5px,color:#10204a;
    classDef pos fill:#ffe3e3,stroke:#e03131,stroke-width:1.5px,color:#5c0a0a;
    classDef neg fill:#e7f5ff,stroke:#1971c2,stroke-width:1.5px,color:#0a3d62;

This lesson skips the dedicated V0 finder and uses a simpler combination of particle identification and invariant mass, enough to expose a clear peak.

Observable: invariant mass

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For a candidate proton p₁ = (E₁, p₁) and candidate pion p₂ = (E₂, p₂), the pair invariant mass is Lorentz invariant:

   E_i = sqrt(|p_i|^2 + m_i^2)            (m_i = assigned proton or pion mass)

   m(p, π) = sqrt( (E_1 + E_2)^2 − |p_1 + p_2|^2 )

Forming this for proton–π⁻ pairs produces:

• a peak at 1.115683 GeV from true Λ⁰ decays, and
• a smooth combinatorial background from pairs with no common parent.

Callout

Width is resolution, not lifetime

The natural width (Γ ≈ 2.5 × 10⁻⁶ eV) is negligible next to any instrumental effect. The observed peak width — a few MeV — measures detector momentum and angular resolution, not an intrinsic Λ⁰ property.

Signal extraction

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Fit a Gaussian signal on a second-order polynomial background over a window centered on the peak:

   f(m) = A · exp( −½ ((m − μ)/σ)^2 )  +  ( c0 + c1 (m − m_Λ) + c2 (m − m_Λ)^2 )

The fitted μ should lie within a few MeV of 1.115683 GeV (a residual offset reflects momentum calibration), σ measures the mass resolution, and the integrated signal is \(S = A\sqrt{2\pi}\,\sigma / (\text{bin width})\). Build the proton–π⁻ mass spectrum by asking opencode to run the uproot execute_kernel_dataset tool (tree events, with the proton and pion branches) across the dataset’s root:// files, then fit it with the form above. Reference result below: μ ≈ 1.1163 GeV, σ ≈ 2.7 MeV, χ²/ndf ≈ 1.2.

Fitted Λ⁰ and Λ̄ invariant-mass spectra with Gaussian-plus-polynomial fits