Konatori-135 | BL4S 2026 — CERN T9 Proposal Visualizer

Pion Charge Exchange: π⁻ + p → π⁰ + n → γ + γ + n

M_{\gamma\gamma} = \sqrt{2 E_1 E_2 (1-\cos\theta)} \approx 135\,\mathrm{MeV}/c^2

Beam Momentum: 3 GeV/c | γ = 22.25 | θ_min = 90 mrad (≈5.15°)

1 10

GeV/c (adjust to see Lorentz boost effect on opening angle)

Nuclear A-Dependence Power Law for σ(A)

\sigma(A) = \sigma_0 \, A^{\alpha} \quad \Longrightarrow \quad \ln\sigma_{\rm rel} = \alpha\ln A + \ln\sigma_0

α = 2/3 (surface-dominated, opaque nucleus) | α = 1 (volume scaling, transparent nucleus) | α ≈ 0.72–0.74 (Glauber optical-limit prediction)

Glauber Multiple-Scattering Theory — Nuclear Shadowing

\lambda_\pi \approx 1\text{–}2\,\mathrm{fm} \;\lesssim\; \text{nuclear skin depth} \quad\Rightarrow\quad \alpha \approx 0.72\text{–}0.74

Inner nucleons are "shadowed" by the nuclear periphery; only surface nucleons participate in charge exchange

π⁰ → γγ Invariant Mass Reconstruction (BR = 98.82%)

M_{\gamma\gamma} = \sqrt{2E_1 E_2(1-\cos\theta)}, \quad \cos\theta = \frac{x_1 x_2+y_1 y_2+L^2}{\sqrt{(x_1^2+y_1^2+L^2)(x_2^2+y_2^2+L^2)}}

⚠ Dalitz Background (BR = 1.174%): π⁰ → γe⁺e⁻ — rejected by S₂ veto (>99.9% eff.), shower shape, and mass window. Residual: <0.02% of signal rate.

Expected: σ_m ≈ 5.6 MeV/c² (Mode A, θ=9°) | Signal purity >98% | Fit: Gaussian + 2nd-order polynomial background

Electromagnetic Shower in Lead-Glass Calorimeter (4×4 array, 10×10×37 cm³ blocks)

\frac{\sigma_E}{E} = \frac{6.3\%}{\sqrt{E\,[\mathrm{GeV}]}} \oplus 2\% \quad\Rightarrow\quad 5.5\%\text{ at }1.5\,\mathrm{GeV}\text{ (Mode A)}

Upstream Target: Lead (Pb). Target X₀ = 5.6 mm. Photon escape probability = 95.2%.

Dual-Threshold Cherenkov PID — XCET1 & XCET2 at T9

\beta = \frac{p}{\sqrt{p^2+m^2}} \)  at  \( p = 3\,\mathrm{GeV}/c

Detector Configuration & β values at p = 3 GeV/c:

XCET1 (low pressure)

β_thr ≈ 0.9999

Fires ONLY for electrons

XCET2 (higher pressure)

β_thr ≈ 0.9989

Fires for e⁻ and π⁻ (both above threshold)

Particle	Mass (MeV/c²)	β at 3 GeV/c	XCET1	XCET2	Decision
e⁻	0.511	0.99999999	ON	ON	REJECT
π⁻ (signal)	139.57	0.99892	OFF	ON	ACCEPT ✓
μ⁻ (completeness)	105.66	0.99938	OFF	ON	TRIGGER PASS*
K⁻	493.68	0.98673	OFF	OFF	REJECT
p̄	938.27	0.95441	OFF	OFF	REJECT

* μ⁻ (β=0.99938) exceeds the XCET2 threshold (β_thr=0.9989) and is NOT rejected at hardware trigger level. Muon suppression relies entirely on the S₂ charged-particle veto and offline E_total > 1.0 GeV + shower-shape cuts (see Setup §5).

π⁻ uniquely identified by: XCET2 ON ∧ XCET1 OFF ∧ S₂ silent ∧ E_total > 1 GeV

Delay Wire Chambers — Beam Tracking & Vertex Reconstruction

\sigma_{\rm vertex} \lesssim 300\,\mu\mathrm{m} \quad\text{(DWC1+DWC2, } {\sim}30\,\mathrm{cm apart)}\) \(\quad\sigma_\theta^{\rm ModeB} = \frac{\sqrt{2}\,\sigma_{\rm pos}}{L} = \frac{\sqrt{2}\times 200\text{–}250\,\mu\mathrm{m}}{30\,\mathrm{cm}} \approx 0.94\text{–}1.18\,\mathrm{mrad}

DWC3 (Mode B only): Tracks e⁺e⁻ pairs from photon conversion in 0.1X₀ Pb foil (P_conv = 7.48% per photon). Provides 3–4× better angular resolution than Mode A calorimetry (3.5 mrad). Click canvas to simulate a beam hit.

Target	A	Thickness	X₀ fraction	Photon Escape	n (cm⁻²)	σ (mb)	Rate (Hz)
¹²C (Carbon)	12	2.65 mm	1.4%	98.9%	3.00×10²²	40.0	400
²⁷Al (Aluminium)	27	1.85 mm	2.1%	98.4%	1.115×10²²	71.7	260
²⁰⁷Pb (Lead)	207	0.353 mm	6.3%	95.2%	1.163×10²¹	310.8	120

π⁻ beam →Bending Magnet →Collimator →XCET1 →XCET2 →S1 →DWC1 →DWC2 →Target Wheel →S₂ (Veto) →[Pb Conv. + DWC3 (Mode B)] →Lead-Glass CALO 4×4

T_MASTER = S₁ ∧ XCET2 ∧ ¬XCET1 ∧ ¬S₂ᵛᵉᵗᵒ ∧ N_block≥2 ➜ WAITING

Centre-of-Gravity Cluster Position Reconstruction Algorithm

\bar{x} = \frac{\sum_i E_i\, x_i}{\sum_i E_i}, \quad \bar{y} = \frac{\sum_i E_i\, y_i}{\sum_i E_i} \quad\Rightarrow\quad \sigma_{\rm pos} \approx 2\text{–}5\,\mathrm{mm} \ll 10\,\mathrm{cm\ block}

Seed block threshold: E_seed > 50 MeV | Adjacent blocks: E > 10 MeV | Click the calorimeter face to simulate a photon hit.

Theoretical Calculations — BL4S 2026 / Konatori-135

All parameters sourced from final corrected setup document (March 10, 2026)

1. Lorentz Boost & Minimum Opening Angle

\gamma = \frac{E_{\pi^0}}{m_{\pi^0}} = \frac{\sqrt{p^2+m_{\pi^0}^2}}{m_{\pi^0}}, \quad \theta_{\min} = \frac{2}{\gamma}

At p = 3 GeV/c: γ = 22.25, θ_min = 0.090 rad = 5.15° (Worked example: M_γγ = √(2×1.5×1.5×0.00405) = 135.0 MeV/c²)

π⁰ Beam Momentum: 3.0 GeV/c

Lorentz Factor γ:22.25

θ_min = 2/γ:90.0 mrad

2. Invariant Mass Resolution (Corrected Formula)

\left(\frac{\sigma_m}{m}\right)^2 = \left(\frac{\sigma_{E_1}}{E_1}\right)^2+\left(\frac{\sigma_{E_2}}{E_2}\right)^2+\left(\frac{\sigma_\theta}{\tan(\theta/2)}\right)^2

\( \frac{\sigma_E}{E} = \frac{6.3\%}{\sqrt{E\,[\mathrm{GeV}]}} \oplus 2\% \;\Rightarrow\; \) at E_γ ≈ 1.5 GeV: 5.51%

Note: Block-to-block intercalibration via NA48 method (symmetric π⁰ decays, E₁=E₂). E₁,E₂ computed kinematically from E₁+E₂=E_π⁰ and E₁E₂=m²_π⁰/[2(1−cosθ)].

DWC Angular Resolution σ_θ: 3.5 mrad Mode A: 3.5 mrad (calo. position) | Mode B: 0.94–1.18 mrad (DWC3 track angle)

Total σ_m at θ_min 6.5 MeV/c²

3. Expected π⁰ Detection Rates

R = \Phi_{\pi^-} \cdot n(A) \cdot \sigma(A) \cdot \varepsilon \quad;\quad \Phi_{\pi^-} = 3.75\times10^5\,\mathrm{s}^{-1},\; \varepsilon = 0.88

σ(A) = σ₀ A^0.72, σ(¹²C)=40 mb | Φ_total×f_π = 5×10⁵×0.75 = 3.75×10⁵ s⁻¹

n = 3.00×10²² cm⁻²

σ_rel = R/n = 1.333×10⁻²⁰ cm² (arb. units for α fit)

Detected Rate R 1 h → 1.44×10⁶ π⁰ events

400 Hz

4. Nuclear A-Dependence Scaling Exponent α̂

\hat{\alpha} = \mathrm{slope}\!\left[\ln\sigma_{\rm rel}\,\mathrm{vs}\,\ln A\right] \;\Rightarrow\; \hat{\alpha}=0.721\;(\text{3-pt log-log LSQ})

Log-log linearisation gives proper χ² weighting for multiplicative uncertainties. Expected result: α̂ = 0.72 ± 0.045(syst) ± 0.003(stat). Barton et al. (1983): α = 0.74 ± 0.03.

R(¹²C) [Hz]: R(²⁷Al) [Hz]: R(²⁰⁷Pb) [Hz]:

Fitted α̂ α=2/3 surface | α=1 volume

0.72

Systematic Uncertainty Budget for α (from Analysis Plan §7)

Source	Estimation Method	δα
Energy scale	Vary calibration ±1% in MC	±0.015
Background model	Polynomial / exponential / Chebyshev / sideband comparison	±0.025
Selection cuts	Vary E_threshold (400–600 MeV), sep. (8–12 cm)	±0.020
Acceptance / MC	Vary π⁰ p_T spectrum within literature limits	±0.020
Trigger efficiency	MC threshold scan ±50 MeV	±0.010
Target thickness	Manufacturer tolerance ±5% → n(A)	±0.010
PID inefficiency	Vary pion purity ±2%, sideband control sample	±0.005
Total (quadrature)	—	±0.045

Final result: α̂ = 0.72 ± 0.045 (syst) ± 0.003 (stat) | Stat. precision: <0.003 with ~10⁶ reconstructed π⁰ per target

Team Konatori-135

π⁰ Invariant Mass & Nuclear A-Dependence

Mode B — Photon Conversion & DWC3 Tracking (Angular Calibration)

1. Lorentz Boost & Minimum Opening Angle

2. Invariant Mass Resolution (Corrected Formula)

3. Expected π⁰ Detection Rates

4. Nuclear A-Dependence Scaling Exponent α̂

Systematic Uncertainty Budget for α (from Analysis Plan §7)