Geometric algebra as the input language of collider foundation models

TL;DR

This paper introduces a geometric algebra framework to represent collider events as multivectors, organizing observables by grade and providing a uniform, symmetry-aware input for foundation models in collider physics.

Contribution

It develops a novel geometric algebra-based event representation, including a comprehensive dictionary of observables and a grade-resolved pre-training strategy for collider analysis models.

Findings

Explicit multivector representation encodes all current collider observables.

No new Lorentz-invariant scalars are found beyond known invariants.

The approach improves resonance-topology separation using a Lorentz-equivariant transformer.

Abstract

A hard hadron-collider event is treated here as a single geometric object - the kinematics and the discrete object-type labels of all reconstructed final-state particles encoded in one multivector $\evMV\in\Cl(1,3)\otimes\Vflav$ - rather than as the customary list of four-momenta with separate label fields attached. The natural mathematical setting for this view is geometric algebra, whose grade decomposition is shown to organise essentially every observable in current use for collider analyses: inner products and invariant masses at grade zero, four-momenta at grade one, decay-plane bivectors at grade two, oriented three-volumes at grade three, and the CP-odd pseudoscalar at grade four. The high-level invariants, the low-level recipe, and the equivariant-network inputs are recovered as projections onto specific grades. An explicit per-grade dictionary of $34$ classical observables is provided, and the spacetime, discrete and approximate symmetries acting on $\evMV$ are listed. The Cayley--Menger lemma settles the question of new Lorentz-invariant scalars: none are unlocked beyond $\{p_i\!\cdot\!p_j,\,m_i^2\}$; the genuine non-trivial channel is the CP-odd sign of the pseudoscalar. The event-as-geometric-object representation is intended as a uniform input layer for foundation models of collider physics, and a grade-resolved pre-training strategy is outlined. The methodology is illustrated on the resonance-topology separation with a Lorentz-equivariant multivector transformer type whose per-particle grade-$0\!\oplus\!1$ tokens are complemented by event-level pairing tokens that surface the grade-two and grade-three candidate-pairing content of the multi-resonance topology at the input layer.