Reproducing $\beta_L^{32}$-vs-$m_\mathrm{LQ}$ exclusion + 5$\sigma$ discovery contours for $U_1$ at 3 TeV (1 ab$^{-1}$) and 14 TeV (20 ab$^{-1}$) muon colliders
Domain: physics (high-energy-physics, bsm-phenomenology, leptoquark, muon-collider, statistical-recast)
Topic: Reproducing $U_1$ exclusion + 5$\sigma$ discovery contours at 3 and 14 TeV muon colliders from arXiv:2104.05720
Venue: ARC-Bench Physics 2026
Source paper: arXiv 2104.05720
Headline metric: primary_metric (maximize)

Research briefing:
Reproducing the muon-collider sensitivity to a $U_1$ vector
leptoquark in the coupling-vs-mass plane via a binned-likelihood
recast of $\mu^+\mu^- \to b\bar b$, as a benchmark of the agent's
ability to combine MC parameter scans with statistical recasting
spanning two orders of magnitude in mass and four orders in coupling.
The same $U_1$ Lagrangian as P09 (only $\beta_L^{32}$ non-zero),
but here the per-bin cross section is parameterized as
$\sigma_i(m, \beta) = b_i + \beta^2 I_i(m) + \beta^4 J_i(m)$, with
$I_i(m), J_i(m)$ extracted by solving a $2\times 2$ linear system from
two reference $\beta$ values. The diagnostic figure shows 95% CL
exclusion (dashed) and 5$\sigma$ discovery (solid) contours in
($m_\mathrm{LQ}$, $\beta_L^{32}$) at $\sqrt{s}=3$ TeV (1 ab$^{-1}$,
red) and $\sqrt{s}=14$ TeV (20 ab$^{-1}$, purple).

A credible study (a) implements the $U_1$ Lagrangian (FeynRules + UFO),
(b) runs SM baselines (100k events at 3 and 14 TeV) plus 4 LQ signal
scans ($\sqrt{s}\in\{3,14\}$ TeV x $\beta_L^{32}\in\{1.0, 2.0\}$) over
17 mass points $m_\mathrm{LQ} \in \{1.0, 1.5, 2.0, 3.0, 4.0, 5.0,
6.0, 7.0, 8.0, 10, 15, 20, 30, 40, 50, 60, 70\}$ TeV with 50k events
each, (c) bins each run in 10 equal-width $|\eta|$ bins, (d) extracts
$I_i, J_i$ per (mass, $\sqrt{s}$), (e) computes the binned
log-likelihood ratio for both the exclusion null hypothesis ($n_i
= b_i^\mathrm{events}$) and the discovery null ($n_i = \mu_i$),
(f) finds the $\beta$ where $-2\log\lambda$ crosses $\chi^2(10,
0.95) = 18.307$ (exclusion) or 48.2 (5$\sigma$ discovery), and
(g) plots the four contours on log-log axes ($m_\mathrm{LQ}\in
[1, 75]$ TeV, $\beta\in[10^{-3}, 2]$). Research question: *does the
agent reproduce the muon-collider $U_1$ exclusion + discovery
contours from Figure 12 of arXiv:2104.05720?*

Hypotheses:
statement: At $\sqrt{s}=14$ TeV with 20 ab$^{-1}$, the 95% CL exclusion contour reaches $\beta_L^{32} \leq 0.01$ at $m_\mathrm{LQ} = 10$ TeV (matching the published value within a factor of 2)., statement: The 14 TeV exclusion contour extends to higher $m_\mathrm{LQ}$ than the 3 TeV exclusion contour at every fixed $\beta_L^{32}$ in $[10^{-2}, 1]$ (higher energy + luminosity gives more reach)., statement: 5$\sigma$ discovery contours always lie above the corresponding exclusion contours in $\beta_L^{32}$ at every mass (discovery requires stronger signal than exclusion).

Experiment design:
research_question: Does the agent reproduce the 95% CL exclusion and 5$\sigma$ discovery contours for the $U_1$ leptoquark in the $(m_\mathrm{LQ}, \beta_L^{32})$ plane at 3 TeV (1 ab$^{-1}$) and 14 TeV (20 ab$^{-1}$) muon colliders (Figure 12 of arXiv:2104.05720)?
conditions: name: sm_baselines
description: Two SM-only runs (100k events each) at $\sqrt{s}\in\{3, 14\}$ TeV, $\beta_L^{ij}=0$. Define per-bin SM cross section $b_i$ in 10 $|\eta|$ bins., name: lq_signal_3tev
description: Mass scan over 17 points $m_\mathrm{LQ} \in [1, 70]$ TeV at $\sqrt{s}=3$ TeV, with $\beta_L^{32}\in\{1.0, 2.0\}$, 50k events each., name: lq_signal_14tev
description: Same mass scan at $\sqrt{s}=14$ TeV with $\beta_L^{32}\in\{1.0, 2.0\}$, 50k events each.
baselines: SM $\mu^+\mu^- \to \gamma^*/Z^* \to b\bar b$ at $\sqrt{s} = 3$ TeV and 14 TeV
metrics: name: exclusion_beta_at_10tev_14tev
direction: match_reference
description: $\beta_L^{32}$ at the 95% CL exclusion contour at $m_\mathrm{LQ}=10$ TeV, $\sqrt{s}=14$ TeV., name: discovery_beta_at_10tev_14tev
direction: match_reference
description: $\beta_L^{32}$ at the 5$\sigma$ discovery contour at $m_\mathrm{LQ}=10$ TeV, $\sqrt{s}=14$ TeV., name: I_J_coefficients_per_mass
direction: match_reference
description: $(I_i, J_i)$ per $|\eta|$ bin and mass point, extracted from the $\beta\in\{1, 2\}$ runs; positive-definite check on $J_i$.
datasets: process_id: mumu_to_bbbar_U1_3and14tev
sqrt_s_TeV: 14
description: $\mu^+\mu^- \to b\bar b$ via SM + $U_1$ exchange at muon colliders, two energy points.
compute_requirements: gpu_required: no
estimated_wall_clock_sec: 3600

Requirements:
type: artifact
description: A canonical results.json file exists at the workspace root with at least the keys: primary_metric (number), metric_key (string), metrics (object with numeric keys), hypotheses (object with h1/h2/h3 entries each carrying a `supported` boolean), summary (non-empty string)., type: numeric
description: results.json metrics MUST contain at least 3 numeric (non-null, finite) values directly relevant to the headline physics observable named in the experiment_design.metrics list above — these are the numbers the paper will report in its Results section., type: discussion
description: results.json hypotheses.h1/h2/h3 each MUST have an explicit `supported` boolean AND a `details` string ≥ 40 characters quoting the numerical evidence (specific values + their source artifact) used to reach the verdict., type: artifact
description: At least one publication-quality figure file (PDF or PNG, ≥150 DPI for raster) exists under figures/ or output/figures/ with axes labeled in physical units (GeV / pb / fb / dimensionless) and a legend if multiple series are plotted.  The figure must directly support a hypothesis verdict., type: artifact
description: The BSM Lagrangian is implemented either as a FeynRules .fr file (models/*.fr) with a matching UFO directory (models/*_UFO/ containing at least particles.py, parameters.py, couplings.py, vertices.py), OR as analytic Python code that explicitly computes the cross sections from the Lagrangian terms.  A pure SM baseline with no BSM piece is NOT sufficient., type: numeric
description: results.json metrics MUST report the 95% CL exclusion β_L^32 reach at m_LQ=10 TeV, √s=14 TeV, 20 ab^-1; the value MUST be ≤ 0.02 (published ≤0.01 within a factor of 2)., type: discussion
description: The summary or structured_results section provides a one-paragraph mechanistic interpretation of WHY the headline observable comes out the way it does (which interference / propagator structure / cut effect drives the result).  Nice-to-have, not blocking proceed., type: discussion
description: results.json or a sibling reproducibility section names: (a) the MadGraph5_aMC@NLO version, (b) the PDF set used (if applicable), (c) at least one explicit random seed.  Required for full reproducibility but not for scientific correctness.

Scoring rubric:
requirements: A credible HEP collider study reproducing the published reference figure for $U_1$ exclusion + 5$\sigma$ discovery contours at 3 TeV (1 ab$^{-1}$) and 14 TeV (20 ab$^{-1}$) muon colliders. The agent must (a) implement the BSM Lagrangian, (b) generate Monte Carlo events for the specified process with appropriate PDFs and generation cuts, (c) run the necessary detector / shower simulation if required, (d) reconstruct the diagnostic kinematic distribution, and (e) compare against the published reference figure with quantitative residuals tied to H1, H2, H3.
judging_note: Physics topics are scored on whether the agent (i) assembles a working HEP MC pipeline AND (ii) reproduces the published numbers within stated tolerance. Partial credit is allowed; the agent may use analytic shortcuts in place of MadGraph for cases where it is overkill, as long as the answer matches the published reference within the per-leaf tolerance. Quantitative result leaves are scored on a graded scale: 100% if relative error < 3%, 67% if < 10%, 33% if < 30%, 0% otherwise.
sub_tasks: requirements: Code-development bucket: BSM model + process card + PDF/cuts + full pipeline are wired correctly.
sub_tasks: requirements: BSM Lagrangian implemented (FeynRules .fr file + UFO output, or equivalent analytic Python module): Same $U_1$ Lagrangian as P09; per-bin cross section parameterized as $\sigma_i(m, \beta) = b_i + \beta^2 I_i(m) + \beta^4 J_i(m)$.
task_category: Code Development
finegrained_task_category: Method Implementation, requirements: MadGraph generate-process card or analytic process module matches the paper's process string: $\mu^+\mu^- \to b\bar b$ at $\sqrt{s}\in\{3, 14\}$ TeV, parton-level.
task_category: Code Development
finegrained_task_category: Method Implementation, requirements: PDF set and generation-level cuts match the paper exactly: Lepton-collider beams; 2 SM baselines (100k events each at 3 and 14 TeV) + 4 LQ signal scans ($\sqrt{s}\in\{3,14\}$ x $\beta_L^{32}\in\{1.0, 2.0\}$) over 17 mass points $m_\mathrm{LQ}\in\{1.0,\ldots,70\}$ TeV with 50k events each.
task_category: Code Development
finegrained_task_category: Experimental Setup, requirements: Full HEP chain wired together (model -> events -> [shower -> detector] -> reconstruction -> analysis -> figure). For analytic shortcuts the chain may be simpler but must still terminate in the diagnostic figure.
task_category: Code Development
finegrained_task_category: Method Implementation, requirements: Execution-validity bucket: MC sample passes nevents threshold, physics-validity gates clear, MC stats are sufficient.
sub_tasks: requirements: MC sample(s) generated successfully with at least the requested number of events per cell (or a documented partial count if compute-limited), with no fatal MadGraph or downstream errors.
task_category: Code Execution
finegrained_task_category: Evaluation, Metrics & Benchmarking, requirements: Physics-validity gate: no negative cross sections; gauge invariance / unitarity sanity checks pass; mass spectrum and decay widths printed by MadGraph match the input parameter card; no obviously unphysical kinematics in the generated events.
task_category: Code Execution
finegrained_task_category: Evaluation, Metrics & Benchmarking, requirements: MC statistics are sufficient: relative MC error in the signal region is < 10% OR at least 3 random seeds are run with std reported across them.
task_category: Code Execution
finegrained_task_category: Evaluation, Metrics & Benchmarking, requirements: Results bucket: quantitative comparison of three hypothesis-driven observables to the published figure, plus a reproduced figure artifact and a per-hypothesis writeup.
sub_tasks: requirements: Quantitative test of H1 — At $\sqrt{s}=14$ TeV with 20 ab$^{-1}$, 95% CL exclusion reaches $\beta_L^{32} \leq 0.01$ at $m_\mathrm{LQ}=10$ TeV (within factor 2 of published).
task_category: Result Analysis
finegrained_task_category: Logging, Analysis & Presentation, requirements: Quantitative test of H2 — 14 TeV exclusion contour extends to higher $m_\mathrm{LQ}$ than 3 TeV contour at every fixed $\beta_L^{32}\in[10^{-2}, 1]$. Use the same graded relative-error scale as H1.
task_category: Result Analysis
finegrained_task_category: Logging, Analysis & Presentation, requirements: Quantitative test of H3 — 5$\sigma$ discovery contours always lie above (i.e. require larger $\beta_L^{32}$ than) corresponding exclusion contours at every mass. Use the same graded relative-error scale as H1.
task_category: Result Analysis
finegrained_task_category: Logging, Analysis & Presentation, requirements: A reproduced figure exists in artifacts (PDF or PNG) with axes, units, ranges, and legend matching the published reference: Log-log plot, x-axis $m_\mathrm{LQ}\in[1,75]$ TeV log, y-axis $\beta_L^{32}\in[10^{-3}, 2]$ log, 1:1 aspect; 3 TeV (red) and 14 TeV (purple) contours; dashed = exclusion, solid = discovery (4 contours total).
task_category: Result Analysis
finegrained_task_category: Logging, Analysis & Presentation, requirements: Writeup discusses the outcome of each hypothesis (supported / refuted / inconclusive) tied to numeric residuals, identifies the dominant systematic uncertainty (PDF set, scale variation, k-factor, Delphes card, statistical MC error), and states explicitly any analytic shortcuts taken in place of full MC.
task_category: Result Analysis
finegrained_task_category: Logging, Analysis & Presentation, requirements: Reproducibility bucket (new for physics topics; absent in T-rubrics): the artifacts contain everything needed to rerun the analysis from scratch.
sub_tasks: requirements: UFO model directory (or analytic model Python module) is checked into the artifacts and is loadable by MadGraph (or directly importable for analytic models).
task_category: Code Development
finegrained_task_category: Reproducibility, requirements: MadGraph run_card / param_card, Pythia8 settings, Delphes card, and MadAnalysis5 (or equivalent analysis) cards are saved alongside the run with explicit parameter values per cell.
task_category: Code Development
finegrained_task_category: Reproducibility, requirements: RNG seeds are documented; rerunning the run from saved seeds should give matching central values within the reported MC error. For analytic shortcuts, the seed requirement is replaced by an explicit list of input numerical parameters.
task_category: Code Development
finegrained_task_category: Reproducibility