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The Point of E_8 in F-theory GUTs

Jonathan J. Heckman, Alireza Tavanfar, Cumrun Vafa

TL;DR

This work argues that realistic flavor hierarchies in F-theory GUTs require a single $E_{8}$ enhancement point, with monodromy enforcing the needed matter-curves and Yukawa couplings. The authors classify all minimal, $E_{8}$-driven scenarios, showing that messenger fields for minimal gauge-mediated SUSY breaking typically arise in the $10igoplusar{10}$ of $SU(5)$ and often share curves with MSSM matter, tightly constraining low-energy spectra. Cosmology naturally favors a light gravitino ($m_{3/2}\sim 10-100 ext{ MeV}$) as dark matter, while saxion dynamics dilute thermal relics and complicate non-gravitino DM scenarios; all GUT-singlet DM candidates built from the $E_{8}$ point are found unstable on cosmological timescales. Consequently, recent PAMELA/ATIC/FERMI signals are more plausibly explained by astrophysical sources (e.g., pulsars) rather than new TeV-scale DM within the minimal framework. The paper thus presents a highly predictive, tightly constrained picture where flavor physics, SUSY breaking, and dark matter are interwoven by the geometry of an $E_{8}$ point, and it highlights the need to explore non-minimal constructions for alternative dark-sector possibilities.

Abstract

We show that in F-theory GUTs, a natural explanation of flavor hierarchies in the quark and lepton sector requires a single point of E_8 enhancement in the internal geometry, from which all Yukawa couplings originate. The monodromy group acting on the seven-brane configuration plays a key role in this analysis. Moreover, the E_8 structure automatically leads to the existence of the additional fields and interactions needed for minimal gauge mediated supersymmetry breaking,__and almost nothing else__. Surprisingly, we find that in all but one Dirac neutrino scenario the messenger fields in the gauge mediated supersymmetry breaking sector transform as vector-like pairs in the 10 + 10* of SU(5). We also classify dark matter candidates available from this enhancement point, and rule out both annihilating and decaying dark matter scenarios as explanations for the recent experiments PAMELA, ATIC and FERMI. In F-theory GUT models, a 10-100 MeV mass gravitino remains as the prime candidate for dark matter, thus suggesting an astrophysical origin for recent experimental signals.

The Point of E_8 in F-theory GUTs

TL;DR

This work argues that realistic flavor hierarchies in F-theory GUTs require a single enhancement point, with monodromy enforcing the needed matter-curves and Yukawa couplings. The authors classify all minimal, -driven scenarios, showing that messenger fields for minimal gauge-mediated SUSY breaking typically arise in the of and often share curves with MSSM matter, tightly constraining low-energy spectra. Cosmology naturally favors a light gravitino () as dark matter, while saxion dynamics dilute thermal relics and complicate non-gravitino DM scenarios; all GUT-singlet DM candidates built from the point are found unstable on cosmological timescales. Consequently, recent PAMELA/ATIC/FERMI signals are more plausibly explained by astrophysical sources (e.g., pulsars) rather than new TeV-scale DM within the minimal framework. The paper thus presents a highly predictive, tightly constrained picture where flavor physics, SUSY breaking, and dark matter are interwoven by the geometry of an point, and it highlights the need to explore non-minimal constructions for alternative dark-sector possibilities.

Abstract

We show that in F-theory GUTs, a natural explanation of flavor hierarchies in the quark and lepton sector requires a single point of E_8 enhancement in the internal geometry, from which all Yukawa couplings originate. The monodromy group acting on the seven-brane configuration plays a key role in this analysis. Moreover, the E_8 structure automatically leads to the existence of the additional fields and interactions needed for minimal gauge mediated supersymmetry breaking,__and almost nothing else__. Surprisingly, we find that in all but one Dirac neutrino scenario the messenger fields in the gauge mediated supersymmetry breaking sector transform as vector-like pairs in the 10 + 10* of SU(5). We also classify dark matter candidates available from this enhancement point, and rule out both annihilating and decaying dark matter scenarios as explanations for the recent experiments PAMELA, ATIC and FERMI. In F-theory GUT models, a 10-100 MeV mass gravitino remains as the prime candidate for dark matter, thus suggesting an astrophysical origin for recent experimental signals.

Paper Structure

This paper contains 48 sections, 289 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Building Blocks and Mass Scales of F-theory GUTs
  3. Gravitino Dark Matter and the Cosmology of F-theory GUTs
  4. Flavor Implies $E_{8}$
  5. Monodromy Groups: Generalities
  6. $E_{7}$ Does Not Suffice
  7. $G_{mono}\simeq\mathbb{Z} _{3}$ or $S_{3}$
  8. $G_{mono}\simeq\mathbb{Z} _{2}$
  9. Matter and Monodromy in $E_{8}$
  10. Flux and Monodromy
  11. Dirac Scenarios
  12. $\mathbb{Z} _{2}$ and $\mathbb{Z} _{2}\times\mathbb{Z} _{2}$ Orbits
  13. $\mathbb{Z} _{3}$ and $S_{3}$ Orbits
  14. Majorana Scenarios: $\mathbb{Z}_{2} \times\mathbb{Z}_{2}$ and $Dih_{4}$ Orbits
  15. $E_{8}$ and the Absence of Exotica
  16. ...and 33 more sections

Figures (3)

  • Figure 1: Starting from a GUT seven-brane with $SU(5)$ symmetry, each additional phenomenological condition leads to a further jump in the rank at a point of the geometry. Including the $5_{H}10_{M}10_{M}$ interaction requires an $E_{6}$ point, and the $\overline{5}_{H}\overline{5}_{M}10_{M}$ interaction requires an $SO(12)$ point. A hierarchical CKM matrix then suggests that these points should also unify to an $E_{7}$ point of enhancement. Incorporating leptonic mixing structure pushes this all the way up to $E_{8}$.
  • Figure 2: Depiction of the $SU(5)_{GUT}$ seven-brane, the $U(1)_{PQ}$ seven-brane, and other possible branes and bulk modes which can in principle appear as “ dark objects” in F-theory GUTs. Here, the $7_{\text{Near}}$-branes share a mutual intersection with the $SU(5)_{GUT}$ seven-brane along the PQ seven-brane. As in HVGMSB, supersymmetry breaking occurs due to dynamics localized on the PQ seven-brane. In addition, we have also included the possibility of additional three-branes and seven-branes, $3_{\text{Far}}$ and $7_{\text{Far}}$, as well as bulk modes, such as the gravitino. These latter possibilities are less directly connected to the supersymmetry breaking sector, and so typically do not contain weak to TeV scale dark matter candidates.
  • Figure 3: Depiction of an F-theory GUT in which all of the necessary interaction terms descend from a single point of $E_{8}$ enhancement. In all but one Dirac neutrino scenario, accommodating messenger fields in the gauge mediated supersymmetry breaking sector turns out to force the messengers ($Y_{10}$ and $Y_{\overline{10}}^{\prime}$) to transform in the $10\oplus \overline{10}$ of $SU(5)_{GUT}$. GUT singlets such as the right-handed neutrinos $N_{R}$ and $X$ field localize on curves normal to the GUT seven-brane. Here we have also included a dark matter candidate $D$ which is localized on a curve.