The measurement: Λ⁰ → p π⁻

Overview

Questions

• What is the p π⁻ invariant-mass observable, and its peak and background?
• Which EDM4eic collections and units does it need?

Objectives

• Compute m(p, π) by hand from the four momentum branches and the PDG masses.
• Sketch the expected spectrum — peak position, width scale, background shape — before touching data.

You install an assistant and the tool servers on the Setup page. This episode is the physics you will reconstruct in Episode 3.

The decay

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The Λ⁰ is the lightest strange baryon (uds, spin-parity ½⁺). It decays only weakly (a strangeness-changing ΔS = 1 transition), so it is long-lived: cτ ≈ 7.9 cm. Its dominant hadronic mode is

Λ⁰ → p + π⁻      (branching fraction ≈ 63.9%)

The centimeter-scale flight distance makes the decay a V0: two oppositely charged tracks from a vertex displaced from the primary interaction point.

%%{init: {'theme':'base', 'themeVariables': {'fontSize':'15px','lineColor':'#94a3b8','edgeLabelBackground':'#e2e8f0','clusterBkg':'#1f293720','clusterBorder':'#94a3b8','titleColor':'#94a3b8'}}}%%
flowchart LR
    accTitle: {Lambda to proton pion V0 decay}
    accDescr: {Lambda to proton pion V0 decay}
    PV["primary vertex<br/>e + A collision"]:::vtx -. "Λ⁰: neutral, cτ ≈ 7.9 cm" .-> DV["displaced<br/>decay vertex"]:::vtx
    DV --> P["proton<br/>PDG 2212"]:::pos
    DV --> PI["pion<br/>PDG -211"]:::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;

The observable

The Λ⁰ is neutral and not detected directly; we reconstruct it from its charged daughters. For a candidate proton \(p_1 = (E_1, \vec{p}_1)\) and candidate pion \(p_2 = (E_2, \vec{p}_2)\), the pair’s invariant mass is Lorentz invariant:

\[E_i = \sqrt{|\vec{p}_i|^2 + m_i^2}\]

with \(m_i\) the assigned proton or pion mass, and

\[m(p, \pi) = \sqrt{(E_1 + E_2)^2 - |\vec{p}_1 + \vec{p}_2|^2}\]

Assign the proton mass to one track and the pion mass to the other (using reconstructed particle ID). For true Λ⁰ decays this equals the parent mass; candidates accumulate in a peak at 1.115683 GeV.

Callout

Width: resolution, not lifetime

The Λ⁰ natural width (\(\Gamma = \hbar/\tau \approx 2.5 \times 10^{-6}\) eV) is far below any detector effect. The observed peak width — a few MeV — measures detector momentum and angular resolution, not the particle.

Background

Most proton–pion pairs do not come from a Λ⁰ at all. These random (“combinatorial”) pairs do not peak; they form a smooth distribution under the signal. The analysis extracts a yield by fitting a Gaussian peak on a low-order polynomial background (Episode 5). The charge-conjugate mode Λ̄ → p̄ π⁺ is reconstructed identically with the antiparticles.

Callout

Reference values (PDG)

Quantity	Value
m(Λ⁰)	1.115683 GeV
m(p)	0.9382720813 GeV
m(π±)	0.13957061 GeV
cτ(Λ⁰)	7.89 cm
BR(Λ⁰ → p π⁻)	63.9 %

Energies and momenta are in GeV (natural units, c = 1).

The data model

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ePIC reconstruction output uses EDM4eic, an EIC extension of EDM4hep generated with PODIO. A file contains an events tree; each entry is one event, each branch a collection. We need one collection, the reconstructed charged tracks, and four members:

events  (tree; one entry per event)
    ReconstructedChargedParticles.PDG          reconstructed particle-ID hypothesis
    ReconstructedChargedParticles.momentum.x   p_x  [GeV]
    ReconstructedChargedParticles.momentum.y   p_y  [GeV]
    ReconstructedChargedParticles.momentum.z   p_z  [GeV]

PDG is the Particle Data Group code the reconstruction assigns each track. Select protons (2212) and π⁻ (-211) for Λ⁰, antiprotons (-2212) and π⁺ (211) for Λ̄.

Callout

Caveat: PID is a hypothesis too

The PDG field is the reconstruction’s best guess, not truth. Misidentification feeds the combinatorial background — one reason a fit, not a count, is required.

You do not download a file. In Episode 3 the assistant uses the rucio tools to find a DIS dataset and xrootd to verify its files, then reads one of the dataset’s root:// URLs (e.g. root://epicxrd1.sdcc.bnl.gov:1095//...) in place with the uproot tools — pulling exactly these branches without writing any I/O code.

Key Points

• The observable is the p π⁻ invariant mass; the Λ⁰ appears as a narrow peak over a combinatorial background.
• The peak width is set by detector resolution, not the negligible Λ⁰ natural width.
• The data are EDM4eic collections in an events tree; momenta are in GeV.