The Generalized Uncertainty Principle. New Bounds and Trends

TL;DR

Problem: the existence of a minimal length implies a generalized uncertainty principle that modifies the canonical commutators. Approach: formulate a relativistic GUP within an effective field theory, derive the leading order QED Lagrangian and propagators with $\mathcal{O}(\beta)$ corrections, and analyze high-energy Compton scattering to bound the GUP scale $\Lambda$. Findings: Compton cross sections acquire a constant GUP term that interferes with the SM, yielding two-sided bounds $|\beta|^{-1/2}\Lambda > 0.68$ TeV for $\beta<0$ and $>0.25$ TeV for $\beta>0$; deviations reach about $40\%$ at $\sqrt{s}\sim 200$ GeV and can be as large as $50\%$ in favorable angular regions. Significance: the bounds lie below the TeV scale, motivating collider studies at the LHC and FCC and encouraging exploration of GUP effects in other high-energy processes and in RG running within an EFT framework.

Abstract

The Heisenberg uncertainty principle is one of the fundamental pillars of quantum mechanics and quantum field theory. It is normally introduced by postulating the commutation relations $[\hat{x}^i, \hat{p}^j] = i\hbar δ^{ij}$. However, as suggested by some quantum gravity models and string theory, this basic principle no longer holds true in the presence of a minimal length, possible the Plank length, and modifications of the commutation have been proposed i.e., of the form $[\hat{x}^μ, \hat{p}^ν] = -i\hbar(1 + β_0 \, \hat{p}^2/Λ^2 )η^{μν}$(plus possible additional terms). In this work we will consider the previous modified uncertainty principle in terms of an effective field theory, comment upon some theoretical subtleties that are often overlooked in the literature, and constrain, for the first time, the $Λ$ scale with the Compton high-energy experimental data. Our findings suggest that high-energy experiments are potentially sensitive to these corrections and could serve as an effective framework for probing possible violations of the Heisenberg uncertainty principle

Figures (6)

• Figure 1: Modified Feynman rules for the propagators expressed as a sum of the SM contributions and the $\mathcal{O}(\beta)$ corrections.
• Figure 2: Feynman rules of the new QED vertices at $\mathcal{O}(\beta)$ where the momentum flow is given by the corresponding arrows. Momentum conservation reads $p_1=p_2+p_3$ for the 3-point vertex and, $p_1+p_2=p_3+p_4$ for the 4-point vertex.
• Figure 3: Feynman diagrams that contribute to Compton scattering process at $\mathcal{O}(\beta)$.
• Figure 4: $\chi^2$ as function of $\beta$ and the corresponding $1\sigma$ and $2 \sigma$ bandwidths.
• Figure 5: $\Delta \sigma$ ($\text{GeV}^{-1}$) as function of $\sqrt{s}$ for two values of $\beta$ that correspond to the lower bounds \ref{['lowbo']}.