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Mass Hierarchies and Dynamical Field Range

Aitor Landete, Gary Shiu

TL;DR

This work clarifies that swampland limits on field ranges must be applied to the dynamical field range traced by the true equations-of-motion trajectory in multi-field systems, not merely to a kinematic geodesic distance. It develops a quantitative framework to assess when heavy-field kinetics can be neglected and when the trajectory remains geodesic, highlighting the role of mass hierarchies. Using a concrete axion model in Type II string compactifications, it shows that without a strong mass hierarchy the dynamical range can be sub-Planckian and the single-field approximation breaks down, while a large hierarchy can extend the dynamical range and align with geodesic motion. The results imply that EFT validity and realized field ranges hinge on the light/heavy mass spectrum, informing how to test swampland conjectures in inflationary contexts and guiding constructions to achieve extended field ranges.

Abstract

Several swampland conjectures suggest that there is a critical field range beyond which the effective field theory (EFT) description breaks down in quantum gravity. In applications of these conjectures, however, the field range of interest is the field space distance traced by the physical trajectory that solves the equations of motion. We refer to this field space distance as the dynamical field range. We show that in the absence of a mass hierarchy between the light and heavy fields, the trajectory of the light field does not, in general, follow a geodesic in field space. Then, stabilizing the heavy fields at the minimum of their potential does not accurately describe the dynamics of the light field in general. A mass hierarchy can delay the breakdown of the EFT, and extend the effective field range. We illustrate these subtleties of multi-field dynamics with axions in Type II string compactifications.

Mass Hierarchies and Dynamical Field Range

TL;DR

This work clarifies that swampland limits on field ranges must be applied to the dynamical field range traced by the true equations-of-motion trajectory in multi-field systems, not merely to a kinematic geodesic distance. It develops a quantitative framework to assess when heavy-field kinetics can be neglected and when the trajectory remains geodesic, highlighting the role of mass hierarchies. Using a concrete axion model in Type II string compactifications, it shows that without a strong mass hierarchy the dynamical range can be sub-Planckian and the single-field approximation breaks down, while a large hierarchy can extend the dynamical range and align with geodesic motion. The results imply that EFT validity and realized field ranges hinge on the light/heavy mass spectrum, informing how to test swampland conjectures in inflationary contexts and guiding constructions to achieve extended field ranges.

Abstract

Several swampland conjectures suggest that there is a critical field range beyond which the effective field theory (EFT) description breaks down in quantum gravity. In applications of these conjectures, however, the field range of interest is the field space distance traced by the physical trajectory that solves the equations of motion. We refer to this field space distance as the dynamical field range. We show that in the absence of a mass hierarchy between the light and heavy fields, the trajectory of the light field does not, in general, follow a geodesic in field space. Then, stabilizing the heavy fields at the minimum of their potential does not accurately describe the dynamics of the light field in general. A mass hierarchy can delay the breakdown of the EFT, and extend the effective field range. We illustrate these subtleties of multi-field dynamics with axions in Type II string compactifications.

Paper Structure

This paper contains 12 sections, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The dynamical field range
  3. The Refined Swampland Conjecture for axions
  4. Proper field range and validity of the approximation
  5. Is the trajectory a geodesic?
  6. An Illustrative example
  7. Validity of the approach
  8. The RSDC in different regimes
  9. Large mass hierarchies.
  10. No mass hierarchies.
  11. Discussion
  12. Appendix: Refined Swampland Conjecture for Closed String Models

Figures (3)

  • Figure 1: Field ranges obtained by \ref{['rangebetter']} (blue) and \ref{['defrange2']} (orange) for $\frac{\mathfrak{f}_{0}}{\mu} \sim 10^{4}$ and $G_{\varphi \bar{\varphi}}|_{0} \sim 2\cdot10^{-1}$.
  • Figure 2: Field range obtained by \ref{['rangebetter']} (blue) and \ref{['defrange2']} (orange) for $\frac{\mathfrak{f}_{0}}{\mu} \sim 1$ and $G_{\varphi \bar{\varphi}}|_{0} \sim 2\cdot10^{-1}$. In this case $\hat{\varphi}_{\text{app}} \sim 0.1 M_{\text{Pl}}$ for $\varepsilon = 0.1$
  • Figure 3: Field ranges obtained by \ref{['naive2']} (blue) and \ref{['naiveclosed1']} (orange) for a stabilization with $h=1, \, q=1, \, \mathfrak{f}_{0} = -2, \mathfrak{f}_{0} = -4,$.