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Weak Gravity Conjecture, Multiple Point Principle and the Standard Model Landscape

Yuta Hamada, Gary Shiu

TL;DR

This work investigates how the Standard Model vacua behave under compactification to $S^1$ and $T^2$, motivated by the Weak Gravity Conjecture and the Multiple Point Principle. By deriving 3D and 2D effective actions and computing the one-loop Casimir energies (with general boundary conditions and fluxes), it maps the landscape of lower-dimensional vacua arising from both electroweak and high-scale Higgs vacua. A key result is that, under the MPP, the lightest neutrino is likely Dirac with mass in the $ ext{O}(1{-}10) ext{ meV}$ range, a prediction that can be tested by future cosmological and 21 cm observations. The analysis also reveals rich flux- and axion-induced vacua, particularly AdS minima, and establishes stability criteria (BF bounds) for the resulting lower-dimensional vacua, offering insights into UV consistency constraints on the SM. Overall, the paper links vacuum structure in compactified SM settings to phenomenology and swampland criteria, highlighting testable neutrino physics and potential signatures of lower-dimensional landscapes.

Abstract

The requirement for an ultraviolet completable theory to be well-behaved upon compactification has been suggested as a guiding principle for distinguishing the landscape from the swampland. Motivated by the weak gravity conjecture and the multiple point principle, we investigate the vacuum structure of the standard model compactified on $S^1$ and $T^2$. The measured value of the Higgs mass implies, in addition to the electroweak vacuum, the existence of a new vacuum where the Higgs field value is around the Planck scale. We explore two- and three-dimensional critical points of the moduli potential arising from compactifications of the electroweak vacuum as well as this high scale vacuum, in the presence of Majorana/Dirac neutrinos and/or axions. We point out potential sources of instability for these lower dimensional critical points in the standard model landscape. We also point out that a high scale $AdS_4$ vacuum of the Standard Model, if exists, would be at odd with the conjecture that all non-supersymmetric $AdS$ vacua are unstable. We argue that, if we require a degeneracy between three- and four-dimensional vacua as suggested by the multiple point principle, the neutrinos are predicted to be Dirac, with the mass of the lightest neutrino O(1-10) meV, which may be tested by future CMB, large scale structure and $21$cm line observations.

Weak Gravity Conjecture, Multiple Point Principle and the Standard Model Landscape

TL;DR

This work investigates how the Standard Model vacua behave under compactification to and , motivated by the Weak Gravity Conjecture and the Multiple Point Principle. By deriving 3D and 2D effective actions and computing the one-loop Casimir energies (with general boundary conditions and fluxes), it maps the landscape of lower-dimensional vacua arising from both electroweak and high-scale Higgs vacua. A key result is that, under the MPP, the lightest neutrino is likely Dirac with mass in the range, a prediction that can be tested by future cosmological and 21 cm observations. The analysis also reveals rich flux- and axion-induced vacua, particularly AdS minima, and establishes stability criteria (BF bounds) for the resulting lower-dimensional vacua, offering insights into UV consistency constraints on the SM. Overall, the paper links vacuum structure in compactified SM settings to phenomenology and swampland criteria, highlighting testable neutrino physics and potential signatures of lower-dimensional landscapes.

Abstract

The requirement for an ultraviolet completable theory to be well-behaved upon compactification has been suggested as a guiding principle for distinguishing the landscape from the swampland. Motivated by the weak gravity conjecture and the multiple point principle, we investigate the vacuum structure of the standard model compactified on and . The measured value of the Higgs mass implies, in addition to the electroweak vacuum, the existence of a new vacuum where the Higgs field value is around the Planck scale. We explore two- and three-dimensional critical points of the moduli potential arising from compactifications of the electroweak vacuum as well as this high scale vacuum, in the presence of Majorana/Dirac neutrinos and/or axions. We point out potential sources of instability for these lower dimensional critical points in the standard model landscape. We also point out that a high scale vacuum of the Standard Model, if exists, would be at odd with the conjecture that all non-supersymmetric vacua are unstable. We argue that, if we require a degeneracy between three- and four-dimensional vacua as suggested by the multiple point principle, the neutrinos are predicted to be Dirac, with the mass of the lightest neutrino O(1-10) meV, which may be tested by future CMB, large scale structure and cm line observations.

Paper Structure

This paper contains 30 sections, 117 equations, 22 figures, 9 tables.

Table of Contents

  1. Introduction
  2. The SM vacua in four dimension
  3. The SM vacua from $S^1$ compactification
  4. Effective action
  5. boundary condition
  6. $U(1)$ gauge theory on $S^1$
  7. with charged matter
  8. with neutral matter
  9. SM on $S^1$
  10. Multiple point principle and prediction on the neutrino mass
  11. Flux vacua
  12. The SM vacua from $T^2$ compactification
  13. Effective action
  14. $U(1)$ gauge theory on $T^2$
  15. with charged matter
  16. ...and 15 more sections

Figures (22)

  • Figure 1: The $4$-dimensional Higgs potential as a function of the physical Higgs field $h$, Eq. \ref{['Eq:Higgs potential']}. In the left panel, we put $c_6=0$. The potential has AdS, flat or dS vacua depending on the value of the top mass. In the right panel, the $c_6$ term is included while $M_t$ is fixed. We again have AdS, flat or dS minima corresponding to the value of $c_6$.
  • Figure 2: Left: The potential of the $U(1)$ gauge theory with a charged Dirac fermion, Eq. \ref{['Eq:charged S1 potential']}, is plotted as a function of the Wilson line. The potential takes minimum at $q_e+(1-z_e)/2=1/2$. Here we take $\Lambda_4=0$. For the illustration, the vertical axis is not the potential itself, but the potential multiplied by $L_0^{-2}L^6$. Right: The potential as a function of $L$, the radius of $S^1$. The value of the Wilson line is set to be at the minimum of the potential.
  • Figure 3: The potential of compactified $U(1)$ gauge theory with neutral matter. In the left figure, $\Lambda_4$ is set to be zero. In the right figure, periodic boundary condition, $z_e=1$, is taken.
  • Figure 4: $S^1$ compactification of the SM. The effective potential as a function of the radion $L$. Here the Wilson lines are fixed at the potential minimum. "$\nu_{M(D)}$" represents Majorana (Dirac) neutrino, and $z$ is the boundary condition of fermion $\psi\to -e^{-i\pi z}\psi$. The shaded region is close to the QCD scale, $0.3\text{--}1$ GeV, around which perturbation theory is not good. Right figures are enlarged view of the left figures. The solid and dashed line correspond to the normal and inverted hierarchy, respectively. We can see that there is vacuum at around the neutrino mass scale if the boundary condition is close to the periodic one.
  • Figure 5: The potential is the same as the upper right panel of Fig. \ref{['Fig:S1_result']}, but the vertical axes are $V$(left), and $\log_{10} |V|$(right), respectively. Here the scale $L_0$ is taken to be $1\,\text{GeV} ^{-1}$.
  • ...and 17 more figures