GeckOpt: Detector Geometry Optimization

• GeckOpt is a branch-and-bound geometry optimization algorithm that selects optimal detector panels for LLP experiments.
• It minimizes instrumented area while achieving high track and vertex reconstruction efficiency, as demonstrated on the CODEX-b design.
• The algorithm utilizes pruning heuristics and surrogate estimators to efficiently explore combinatorial designs, enabling scalable and modular detector deployment.

GeckOpt refers to a branch-and-bound based geometry optimization algorithm developed specifically for the design and instrumentation of long-lived particle (LLP) detectors, as well as the broader computational and experimental landscape where efficient selection or arrangement of discrete elements is required. Its foundational context is in high energy physics (HEP), exemplified by its application to the CODEX-b detector for the LHC, but the methodology is generalizable to comparable modular design or subset-selection problems (Gorordo et al., 2022).

1. Problem Formulation: Optimization of Instrumented Detector Geometry

GeckOpt models instrumentation as a discrete selection problem over a set of candidate detector panels within a fixed fiducial volume $V$. Each potential panel $i$ (such as an RPC triplet) can be installed ($x_i=1$) or omitted ($x_i=0$), forming a binary decision vector $x\in\{0,1\}^N$, where $N$ is the total number of panels. The objective is to optimize detector performance—specifically, LLP track and vertex reconstruction efficiency—while minimizing total instrumented area or meeting resource constraints.

Key elements:

• Instrumented area: $S(x) = \sum_{i=1}^N a_i x_i$ with $a_i$ as the area of panel $i$
• Reconstruction efficiency: For LLP physics benchmark $b$, the efficiency is $i$0, measuring the fraction of events passing track and vertex criteria given $i$1.
• Total efficiency: $i$2, where $i$3 is the probability of decay inside $i$4 (geometry-dependent).

GeckOpt allows for two dual formulations:

• Area minimization at efficiency floor:

$i$5

• Efficiency maximization at panel-count budget:

$i$6

This structure provides a Pareto frontier of tradeoffs between resource expenditure and signal efficiency (Gorordo et al., 2022).

2. Branch-and-Bound Optimization Algorithm

The critical computational innovation behind GeckOpt is a branch-and-bound search on the partially ordered power set of panel subsets. The optimization exploits the non-additive nature of the efficiency function, arising from requirements like "at least two panels per track".

Algorithmic details:

• Search Tree: Each node is a subset $i$7 of active panels; descendants are generated by adding a new panel.
• Bounding Function: An additive surrogate, the "any-single-hit" estimator $i$8, computes the fraction of events where at least one track hits any panel in $i$9. The bound $x_i=1$0 on further gain is used to prune branches.
• Pruning Heuristic: If $x_i=1$1 plus the current set's efficiency is below the best-so-far, the descendants of $x_i=1$2 are skipped.

Pseudo-code sketch: $N$3 Empirical scaling is $x_i=1$3 for $x_i=1$4 (often $x_i=1$5–$x_i=1$6), sharply less than brute-force $x_i=1$7 (Gorordo et al., 2022).

3. Generalized Simulation and Efficiency Calculation Framework

Panel set efficiency is evaluated by a fast hybrid Python/C++ toolkit ("hepgk" + "hepymc"). This supports arbitrary detector shapes (via constructive solid geometry), various LLP models and final states, and complex reconstruction criteria (e.g., $x_i=1$8 hits per track, $x_i=1$9, hit separations $x_i=0$0 cm).

Simulation workflow:

• Pythia8 for LLP event generation and decay
• Propagation of charged tracks through all candidate panels
• Event filtering by geometric and kinematic cuts
• Calculation of $x_i=0$1 for every $x_i=0$2 subset visited during optimization

Efficiency integration is performed across LLP models and lifetimes to construct robust instrumentation strategies (Gorordo et al., 2022).

4. Application to CODEX-b and Performance Benchmarks

Applied to the CODEX-b layout (400 RPC triplet panels, $x_i=0$3 m$x_i=0$4), GeckOpt produced the following results:

Panel Count $x_i=0$5 (Fraction $x_i=0$6)	$x_i=0$7 \; ρ$x_i=0$8	$x_i=0$9 ρ$x\in\{0,1\}^N$0b\to s S,\;S\to 4π$x\in\{0,1\}^N$1
150 ($x\in\{0,1\}^N$20.29)	0.8–0.9	0.6–0.8
250 ($x\in\{0,1\}^N$30.71)	$x\in\{0,1\}^N$40.9	$x\in\{0,1\}^N$50.9

Heuristic (half-surface) layouts at the same area yield substantially lower signal efficiency, especially for softer LLP decay modes. Efficiency loss as a function of reduced instrumentation is quantifiably offset by GeckOpt-guided panel selection. For long LLP lifetimes ($x\in\{0,1\}^N$7), efficiency plateaus and dependence on $x\in\{0,1\}^N$8 nearly factorizes out (Gorordo et al., 2022).

5. Scalability, Modular Deployment, and Practical Usage

GeckOpt supports seamless scaling up to $x\in\{0,1\}^N$9 panels; for $N$0, a multiscale approach (grouped "superpanels") is practical. The algorithm naturally produces a nested ordering of panel groups, facilitating modular installation: highest-yield groups are installed first.

For new detector design:

• Key inputs: track-hit requirements, per-panel area costs, objective weights for LLP models.
• Modular installation plans are directly enabled by optimization output, providing phase-wise guidance for commissioning and cost control.
• Early-installed panels are concentrated on detector faces/edges with highest expected LLP track density (Gorordo et al., 2022).

6. Significance and Impact for Detector Design

GeckOpt demonstrated $N$1 reductions in instrumented area with only $N$2 signal loss, assisting both cost minimization and staged deployment for next-generation LLP experiments. The method is general for any subset selection problem with non-additive, geometry-dependent objective functions and can inform installation order and resource allocation in high-dimensional, discrete design landscapes.

By providing a flexible, efficient workflow for complex detector assembly under stringent physics-driven constraints, GeckOpt represents a critical advance in instrumentation design for HEP and potentially for other modular large-scale experiments with combinatorial subset selection challenges (Gorordo et al., 2022).