[Paper Review] On the geometric phase and the scattering at the LHC

Sergey Troshin, N. E. Tyurin|arXiv (Cornell University)|May 27, 2013

Particle physics theoretical and experimental studies2 references3 citations

TL;DR

This paper proposes that a geometric (Berry-type) phase emerges in high-energy hadronic scattering at the LHC when the absorption factor κ performs a cyclic, adiabatic loop in energy and impact parameter space, leading to a non-zero phase δ = π/2 at small impact parameters. This phase, linked to reflective scattering (S → −1), signals non-factorized extra dimensions in the Randall-Sundrum model and manifests experimentally as a change in the elastic-to-total cross-section ratio, with geometric dominance at small b and increased albedo at high energies.

ABSTRACT

We discuss appearance at the LHC energies of the geometric phase, which in its turn can reflect a presence of the non-factorizable background geometry proposed in the RS-scenario with extra spatial dimensions.

Motivation & Objective

• To investigate the emergence of a geometric phase in relativistic high-energy scattering, analogous to the Berry phase in quantum mechanics.
• To explore the connection between such a geometric phase and the presence of non-factorized extra dimensions in the Randall-Sundrum model.
• To identify observable signatures of this phase in LHC data, particularly in impact parameter-dependent cross-section ratios.
• To distinguish geometric effects from dynamical phases by assuming a pure imaginary scattering amplitude.
• To provide a theoretical framework linking unitary S-matrix evolution to geometric phase emergence via cyclic variation of the absorption factor κ.

Proposed method

• Uses the impact parameter representation of the S-matrix: S(s,b) = κ(s,b)exp[2iδ(s,b)], with κ ∈ [0,1] as the absorption factor.
• Models the S-matrix via the U-matrix formalism: S = (1−U)/(1+U), where U is a pure imaginary generalized reaction matrix.
• Imposes a cyclic, adiabatic variation of κ along a closed path in (s,b) space, with κ_i = κ_f, to induce a geometric phase independent of dynamics.
• Analyzes the behavior of the elastic-to-total cross-section ratio R(s,b) = σ_el(s,b)/σ_tot(s,b), showing distinct forms for b > R(s) and b < R(s).
• Identifies the energy s₀ where S(s₀,0) = 0 (black disk limit), and studies the regime s > s₀ where reflective scattering (S → −1) and δ = π/2 emerge.
• Uses the ratio R̄(s,b) = σ_el/σ_inel = f(s,b)/(1−f(s,b)) to reconstruct U(s,b) from experimental data, linking it to the geometric phase.

Experimental results

Research questions

• RQ1 Can a geometric phase analogous to the Berry phase emerge in relativistic high-energy scattering, despite the absence of a known Hamiltonian?
• RQ2 What physical conditions lead to the appearance of a non-zero geometric phase in the S-matrix at high energies?
• RQ3 How does the presence of a geometric phase distinguish non-factorized extra dimensions (e.g., RS-model) from factorized ones (e.g., ADD-model)?
• RQ4 What observable signatures in impact parameter-dependent cross-sections can signal the onset of a geometric phase at LHC energies?
• RQ5 How does the geometric phase affect the energy dependence of the albedo and the ridge/double-ridge effects in two-particle correlations?

Key findings

• A geometric phase δ = π/2 emerges when the absorption factor κ performs a cyclic loop in (s,b) space, even if κ_i = κ_f, due to the topological nature of the path.
• At high energies s > s₀, the S-matrix approaches −1 at small impact parameters, indicating reflective scattering and a dominant geometric contribution.
• The ratio R(s,b) = σ_el/σ_tot changes from [1−κ]/2 (for b > R(s)) to [1+κ]/2 (for b < R(s)), signaling geometric dominance in the central region.
• At fixed b and s → ∞, R(s,b) → 1, indicating that geometric elastic scattering saturates the total cross-section in the high-energy limit.
• The albedo increases with energy in the geometric regime, consistent with enhanced reflection at small impact parameters.
• The presence of the geometric phase is interpreted as a signature of non-factorized extra dimensions in the Randall-Sundrum model, distinguishing it from factorized models like ADD.