Reproducing the normalized $|\eta|$ shape in $\mu^+\mu^- \to b\bar b$ from a $U_1$ vector leptoquark at $\sqrt{s}=3$ TeV
Domain: physics (high-energy-physics, bsm-phenomenology, leptoquark, muon-collider)
Topic: Reproducing $U_1$ leptoquark $|\eta|$ shape in $\mu^+\mu^- \to b\bar b$ at 3 TeV muon collider from arXiv:2104.05720
Venue: ARC-Bench Physics 2026
Source paper: arXiv 2104.05720
Headline metric: primary_metric (maximize)

Research briefing:
Reproducing the normalized pseudorapidity shape of the $b$ quark in
$\mu^+\mu^- \to b\bar b$ at a 3 TeV muon collider, comparing SM
($s$-channel $\gamma^*/Z^*$) against $U_1$ vector-leptoquark
$t$-channel exchange. The Lagrangian has the standard vector-LQ
kinetic term plus
$\mathcal{L} \supset (g_U/\sqrt{2})\,U_1^\mu (\beta_L^{ij}\,
\bar Q_L^i \gamma_\mu L_L^j) + \mathrm{h.c.}$, with only
$\beta_L^{32}$ (b-quark to muon) non-zero. The diagnostic observable
is the normalized $|\eta|$ distribution of the $b$ quark binned in 13
bins from 0 to 2.5 (narrow [0, 0.1] first bin then uniform 0.2-wide
bins), shown in two side-by-side panels at $\beta_L^{32} = 1.0$ and
$\beta_L^{32} = 0.1$, each overlaying SM (gray bar histogram) with
$m_\mathrm{LQ}=1$ TeV (blue dashed) and $m_\mathrm{LQ}=10$ TeV
(orange dashed).

A credible study (a) implements the $U_1$ vector-LQ Lagrangian with
$\kappa_U=\tilde\kappa_U=0$ (FeynRules + UFO), (b) runs 5 parton-level
MadGraph runs (1 SM baseline at 500k events with $\beta_L^{32}=0$ and
$m_\mathrm{LQ}=10^5$ GeV; 4 LQ signal runs at $\beta_L^{32}\in
\{1.0, 0.1\}$ x $m_\mathrm{LQ}\in\{1, 10\}$ TeV with 50k events
each), (c) extracts the $b$-quark $|\eta|$ from each event, (d)
histograms into the 13 bins and normalizes to total events per run,
(e) plots two panels side by side with the prescribed style. Research
question: *does the agent reproduce the central-vs-forward angular
redistribution induced by the $t$-channel $U_1$ exchange in
$\mu^+\mu^- \to b\bar b$ from Figure 11 of arXiv:2104.05720?*

Hypotheses:
statement: At $\beta_L^{32} = 1.0$, $m_\mathrm{LQ} = 1$ TeV, the central-bin ($|\eta| < 0.1$) normalized fraction exceeds the SM by $\geq 50\%$ (LQ $t$-channel pulls events forward+central)., statement: At $\beta_L^{32}=0.1$, the normalized $|\eta|$ shape differs from the SM baseline by $\leq 10\%$ in every bin (small-coupling regime collapses to SM)., statement: At $\beta_L^{32}=1.0$, $m_\mathrm{LQ}=10$ TeV the normalized shape lies between the SM and the $m_\mathrm{LQ}=1$ TeV curves (decoupling limit).

Experiment design:
research_question: Does the agent reproduce the normalized $|\eta|$ distributions of $\mu^+\mu^- \to b\bar b$ for SM and $U_1$ leptoquark signal at $\sqrt{s}=3$ TeV muon collider, in the two coupling regimes $\beta_L^{32} \in \{1.0, 0.1\}$ (Figure 11 of arXiv:2104.05720)?
conditions: name: sm_baseline_3tev
description: All $\beta_L^{ij} = 0$ (LQ decoupled), $m_\mathrm{LQ} = 10^5$ GeV. 500k events at $\sqrt{s}=3$ TeV, parton level., name: lq_beta1_3tev
description: $\beta_L^{32} = 1.0$, mass scan $m_\mathrm{LQ} \in \{1, 10\}$ TeV, 50k events each at $\sqrt{s}=3$ TeV., name: lq_beta01_3tev
description: $\beta_L^{32} = 0.1$, mass scan $m_\mathrm{LQ} \in \{1, 10\}$ TeV, 50k events each at $\sqrt{s}=3$ TeV.
baselines: SM $s$-channel $\mu^+\mu^- \to \gamma^*/Z^* \to b\bar b$ (the $\beta_L^{32}=0$ run)
metrics: name: central_bin_excess_fraction
direction: match_reference
description: Ratio of central-bin ($|\eta| < 0.1$) normalized event fraction in LQ signal vs SM., name: shape_deviation_l1norm
direction: match_reference
description: $L^1$ norm $\sum_i |f_i^\mathrm{LQ} - f_i^\mathrm{SM}|$ of normalized shape difference per (mass, coupling)., name: decoupling_distance
direction: match_reference
description: Shape distance between $m_\mathrm{LQ}=10$ TeV signal and SM should be smaller than for $m_\mathrm{LQ}=1$ TeV signal.
datasets: process_id: mumu_to_bbbar_U1
sqrt_s_TeV: 3
description: $\mu^+\mu^- \to b\bar b$ via SM ($s$-channel) + $U_1$ ($t$-channel) at a 3 TeV muon collider, parton level.
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 normalized central-|η|<0.1 fraction for β_L^32=1.0, m_LQ=1 TeV, AND the same fraction for SM, AND the LQ fraction MUST exceed SM by ≥50%., 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 $U_1$ vector-LQ angular shape in $\mu^+\mu^- \to b\bar b$ at 3 TeV muon collider. 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): $U_1$ vector LQ in $(\mathbf{3},\mathbf{1},2/3)$ rep with $\kappa_U=\tilde\kappa_U=0$; only $\beta_L^{32}$ (b-quark to muon) non-zero; $g_U=1$.
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 $\sqrt{s}=3$ TeV via SM $s$-channel and $U_1$ $t$-channel exchange, 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 (no PDF); 1 SM run (500k events with $\beta_L^{32}=0$, $m_\mathrm{LQ}=10^5$ GeV) + 4 LQ signal runs ($\beta_L^{32}\in\{1.0, 0.1\}$ x $m_\mathrm{LQ}\in\{1, 10\}$ TeV, 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 $\beta_L^{32}=1.0$, $m_\mathrm{LQ}=1$ TeV, central-bin $|\eta|<0.1$ normalized fraction exceeds SM by $\geq 50\%$.
task_category: Result Analysis
finegrained_task_category: Logging, Analysis & Presentation, requirements: Quantitative test of H2 — At $\beta_L^{32}=0.1$, normalized $|\eta|$ shape differs from SM by $\leq 10\%$ in every bin. Use the same graded relative-error scale as H1.
task_category: Result Analysis
finegrained_task_category: Logging, Analysis & Presentation, requirements: Quantitative test of H3 — At $\beta_L^{32}=1.0$, $m_\mathrm{LQ}=10$ TeV shape lies between SM and $m_\mathrm{LQ}=1$ TeV (decoupling). 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: Two side-by-side panels (left: $\beta_L^{32}=1.0$, right: $\beta_L^{32}=0.1$); each panel has SM gray bar histogram + $m_\mathrm{LQ}=1$ TeV blue dashed + $m_\mathrm{LQ}=10$ TeV orange dashed; 13 bins from 0 to 2.5 (narrow [0,0.1] first); $14\times 6$ figure size; y in [0, 0.20].
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