# The Swampland Conjectures: A Bridge from Quantum Gravity to Particle Physics

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Swampland Conjectures

#### 2.1. Absence of Global Symmetries and Cobordisms

**No-Global Symmetries Conjecture**: There cannot be exact global symmetries in a theory of quantum gravity coupled to a finite number of degrees of freedom.

**Cobordism Conjecture**[25]: The cobordism class of any k-dimensional compact space on which a d-dimensional theory of quantum gravity can be compactified must be trivial, i.e.,

#### 2.2. The Weak Gravity Conjecture and Beyond

**Electric Weak Gravity Conjecture**[3]: The spectrum of the theory must include at least a particle with mass m and charge q that satisfies the inequality$${m}^{2}\le 2{e}^{2}{q}^{2}{M}_{P}^{2}\phantom{\rule{0.166667em}{0ex}}.$$**Magnetic Weak Gravity Conjecture**[3]: There exists an upper bound for the UV cutoff of the EFT, given by$${\Lambda}_{\mathrm{UV}}\lesssim e{M}_{P}\phantom{\rule{0.166667em}{0ex}}.$$

#### 2.2.1. The Non-Susy AdS Conjecture

**Non-susy Anti-de Sitter Conjecture**[6]: Any non-supersymmetric AdS vacuum in quantum gravity must be unstable.

#### 2.2.2. The Festina Lente Bound

**Festina Lente Bound**[66]: For a gravitational EFT on a dS background, every charged particle in the spectrum must satisfy

#### 2.3. Towers of States and the Distance Conjectures

**Swampland Distance Conjecture**(SDC) [68]: Consider a gravitational effective theory with a moduli space (i.e., a space parameterized by the massless scalar fields in the theory) and whose metric is given by the kinetic terms of the scalar fields. Starting from a point P in moduli space and moving towards a point Q an infinite geodesic distance away (i.e., $d(P,Q)\to \infty $), one encounters an infinite tower of states which become exponentially massless with the geodesic distance, i.e.,

**Anti-de Sitter Distance Conjecture**(ADC) [89]: In a d-dimensional theory of quantum gravity with cosmological constant ${\Lambda}_{d}$, there exist a tower of states that becomes light in the limit ${\Lambda}_{d}\to 0$ as

**Gravitino Distance Conjecture**(GDC) [92,93]: In a supersymmetric theory with a non-vanishing gravitino mass ${m}_{3/2}$, a tower of states becomes light in the limit ${m}_{3/2}\to 0$ according to

## 3. Implications for Particle Physics

#### 3.1. Compactifying the SM: Neutrino Masses, the Cosmological Constant and Supersymmetry

#### 3.1.1. Constraints from the Non-Susy AdS Conjecture

#### 3.1.2. Constraints from the AdS Distance Conjecture

#### 3.2. Supersymmetry Breaking and Towers of States

- ∘
- Low-energy supersymmetry breaking (${m}_{3/2}\sim 1$ TeV). In this case, if supersymmetry happens to be found at energies close to the ones that are currently being probed by the LHC, from $1\ge \delta \ge 1/3$ one would expect a tower at the scale$${10}^{3}\phantom{\rule{4pt}{0ex}}\mathrm{GeV}\phantom{\rule{0.166667em}{0ex}}\lesssim \phantom{\rule{0.166667em}{0ex}}{m}_{\mathrm{tower}}\phantom{\rule{0.166667em}{0ex}}\lesssim \phantom{\rule{0.166667em}{0ex}}{10}^{13}\phantom{\rule{4pt}{0ex}}\mathrm{GeV}\phantom{\rule{0.166667em}{0ex}}.$$Even though this is quite a wide range, it directly rules out the popular big desert scenario, which includes no new physics above the low energy supersymmetry breaking scale and until $\sim {10}^{16}$ GeV.
- ∘
- Intermediate-scale supersymmetry breaking (${m}_{3/2}\sim {10}^{10}$ GeV). This is the minimal case if one wants to prevent the Higgs potential from being unbounded from below (given the experimental value for the Higgs mass ${m}_{\mathsf{\Phi}}=125$ GeV), as restoring supersymmetry at that scale would render the potential positive and bounded. A tower is then expected at a scale$${10}^{10}\phantom{\rule{4pt}{0ex}}\mathrm{GeV}\phantom{\rule{0.166667em}{0ex}}\lesssim \phantom{\rule{0.166667em}{0ex}}{m}_{\mathrm{tower}}\phantom{\rule{0.166667em}{0ex}}\lesssim \phantom{\rule{0.166667em}{0ex}}{10}^{16}\phantom{\rule{4pt}{0ex}}\mathrm{GeV}\phantom{\rule{0.166667em}{0ex}}.$$

#### 3.3. Phenomenological Implications of the Festina Lente Bound

#### 3.4. Massless Photons

#### 3.5. Constraints on the Gauge Groups

## 4. Summary and Final Comments

- ∘
- From consistency of compactifications of the SM with the Non-susy AdS conjecture, it has been argued that pure Majorana neutrinos with large Majorana masses (as in simple See-Saw models) are inconsistent with quantum gravity, leaving (pseudo-)Dirac neutrinos as the only option, with an upper bound on their mass given by the cosmological constant ${m}_{\nu}\lesssim {\Lambda}_{4}^{1/4}\sim {10}^{-3}eV\phantom{\rule{0.166667em}{0ex}},$ as argued in [6,95,97]. This applies independently of whether normal or inverse hierarchy are realized. Additionally, some new insights into the electro-weak hierarchy problem can be obtained by translating the upper bound for neutrino masses into an upper bound the electro-weak scale in terms of the cosmological constant [96], as displayed in Equation (24).
- ∘
- By considering different compactifications of the SM, supersymmetry (with no preferred supersymmetry breaking scale) is favored from requiring the destabilization of lower-dimensional AdS vacua [99]. Additionally, the Higgs vev can be related to the QCD scale as $v\gtrsim {\Lambda}_{QCD}\sim 100\phantom{\rule{0.166667em}{0ex}}\mathrm{MeV}$ [98].
- ∘
- The same upper bound for Dirac neutrino masses in terms of the cosmological constant is obtained by requiring consistency of compactifications of the SM with the AdS Distance Conjecture [101,102,106]. A possible alternative would be that the neutrinos were the light states of a tower already in 4d with ${m}_{\nu}\sim {\left|{\Lambda}_{4}\right|}^{1/4}$ [100,101,102,106]
- ∘
- Preliminary results from the Gravitino Distance Conjecture suggest that low energy supersymmetry is incompatible with the big desert scenario, as a tower with scale ${m}_{\mathrm{tower}}\lesssim {10}^{13}$ GeV is predicted. Additionally, intermediate scale supersymmetry would require ${m}_{\mathrm{tower}}\lesssim {10}^{16}$ GeV [93].
- ∘
- The Festina Lente bound applied to the SM electromagnetic $U\left(1\right)$ is satisfied by all particles in the SM [66] and it gives some insight into the cosmological constant problem by reducing the well-known 120 orders of magnitude between the cosmological constant an the Planck scale to ${\Lambda}_{4}\lesssim {10}^{-89}{M}_{P}^{4}$ [67]. It also gives a lower bound for the electro-weak hierarchy in terms of the Hubble constant ${v}^{2}\phantom{\rule{0.166667em}{0ex}}\gtrsim \phantom{\rule{0.166667em}{0ex}}{M}_{P}\phantom{\rule{0.166667em}{0ex}}H/g$ and forbids a local symmetry-preserving minimum at the origin of the Higgs potential unless extreme fine-tuning is implemented [67].
- ∘
- Additionally, when applied to non-abelian groups, the Festina Lente reasoning gives lower bounds for the masses of massive vector bosons and confinement scales in terms of the Hubble constant ${m}_{W},\phantom{\rule{4pt}{0ex}}{\Lambda}_{\mathrm{conf}}\phantom{\rule{0.166667em}{0ex}}\gtrsim H$ [67].
- ∘
- The Weak Gravity Conjecture for strings which are magnetically charged under axions giving rise to Stückelberg masses for photons allows to argue in favour of the SM photon being exactly massless. Otherwise a UV cutoff scale ${\Lambda}_{\mathrm{UV}}\lesssim 10\phantom{\rule{0.166667em}{0ex}}\mathrm{MeV}$ would be predicted, which is incompatible with observations [112].
- ∘
- ∘
- Finally, even though we have not discussed the (refined) dS Conjecture [4,5,6] here, there are particularly remarkable implications from applying it to the SM QCD vacuum [120]. In particular, for fixed Yukawa couplings, the extrapolation of large N results to $N=3$ suggest that $v\lesssim 50$ TeV is needed to avoid the formation of metastable dS vacua, even though full lattice computations have not been able to address the formation of these metastable states yet.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | Even though along this work we always have string theory in mind when referring to quantum gravity, these conjectures are believed to hold in any possible consistent theory of quantum gravity with weakly-coupled Einstein gravity as its low-energy limit, mainly because this kind of black hole arguments can equally be applied to those setups. |

2 | |

3 | A simple example of a topological charge is the skyrmion number of solitonic solutions in theories with a global $SU\left(2\right)$ symmetry (see e.g., [27]). |

4 | We focus here in the 4d version. Note that $\mathcal{O}\left(1\right)$ factors and powers of the Planck mass would change in different dimensions. |

5 | Note that in our conventions the cosmological constant has units of mass to the power d, as opposed to the usual conventions taken in General Relativity where it is set to have units of mass squared by extracting explicitly a factor of ${M}_{P}^{d-2}$. |

6 | As a side comment, this can be seen as one of the reasons why it is harder to argue for the usual Weak Gravity Conjecture in dS space. In dS, the mere existence of a charged particle is enough to allow for the black hole decay, since such particle can be pair produced at sufficiently long distance from the black hole for the dS expansion to overcome the gravitational attraction, independently of its charge (note still that this process can be extremely slow). By contrast, in Minkowski this attraction after pair production can only be compensated if the electric repulsion is large enough, giving rise to the usual Weak Gravity Conjecture constraint. |

7 | Of course, this distance reduces to the usual distance in moduli space when only massless scalars are varied, and the Generalized Distance Conjecture reduces to the usual Swampland Distance Conjecture. |

8 | |

9 | This r is just introduced to keep the lower-dimensional metric adimensional so that the relevant component of the metric is ${g}_{33}={(R/r)}^{2}$. It gives the periodicity of the coordinate in the circle, namely $y\sim y+2\pi r$, and the physical radius of the circular dimension is then controlled by the dimensionfull R, that is $2\pi R\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\int}_{0}^{2\pi r}\phantom{\rule{0.166667em}{0ex}}dy\phantom{\rule{0.166667em}{0ex}}\sqrt{{g}_{33}}\phantom{\rule{0.166667em}{0ex}}$. |

10 | |

11 | They are expected not to be exactly massless due to electro-weak corrections, but these are numerically taken into account in [99] and do not qualitatively change the picture. Light here means below ${\Lambda}_{QCD}$. |

12 | Incidentally, this would not be the case if the number of generations were two or one, as the approximate global symmetry for n generations is $U{\left(2n\right)}_{L}\times U{\left(2n\right)}_{R}$ and it gets broken down to $U{\left(2n\right)}_{L+R}$. This yields $4{n}^{2}$ light bosonic degrees of freedom and $8n$ fermionic ones, which only lead to AdS vacua for $n>2$ (see [99] for details). |

13 | For completeness, let us mention that the analogous of Equations (20) and (21) for a massless particle in the torus is ${V}_{p}\left(a\right)=\pm {\displaystyle \frac{{n}_{p}}{{\left(2\pi a\phantom{\rule{3.33333pt}{0ex}}\right)}^{2}}}{\displaystyle \frac{\mathcal{G}}{3}}$, with $\mathcal{G}\simeq 0.915966$ the Catalan’s constant, and ${V}_{\Lambda}\left(a\right)={\left(2\pi a\phantom{\rule{3.33333pt}{0ex}}\right)}^{2}{\Lambda}_{4}$. |

14 | In the original formulation of the AdS Distance Conjecture the arguments applied to argue for a tower in the limit $|\Lambda |\to 0$ are independent of the sign of the cosmological constant, so it is equally valid for dS and AdS vacua. |

15 | To be precise, this is only valid in the IR, where all the possible heavy degrees of freedom have been integrated out. More general arguments including the analysis around the UV cutoff scale are presented in [67]. |

16 | This interplay between the Swampland Distance Conjecture and the (tower versions of the) Weak Gravity Conjecture is a recurring pattern in string compactifications and it has been suggested as a generic property of infinite distance points [73], as well as used to fix order one parameters in the conjectures [83,86]. |

17 | As opposed to this, in the Higgs case there are typically semiclassical strings at whose core the Higgs vev goes to zero (i.e., the value for which the symmetry is not broken). This difference is the reason why the claim in [112] is only about Stückelberg masses. |

18 | This comes about because the I-fold is the fixed plane of a symmetry that involves charge conjugation. |

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**Figure 1.**(

**a**) Two cobordant d-dimensional manifolds, which at the level of a lower dimensional EFT give rise to a domain wall separating a theory compactified on both of them. (

**b**) A d-dimensional manifold in the trivial cobordism class. Compactifying a theory in such a manifold thus supports an end of the world membrane separating it from nothing.

**Figure 2.**(

**a**) Shift in a flux ${f}_{0}$ as a consequence of crossing a membrane with charge q under the dual 3-form. (

**b**) Vacuum decay by nucleation of a bubble that expands and is surrounded by a membrane with charge q, such that the flux outside the bubble, ${f}_{0}$, is reduced in the internal region of the bubble.

**Figure 3.**(

**a**) Illustration of an extra circular dimension, the x-direction represents the non-compact space. (

**b**) If the size of the extra dimension (i.e., the circle) is reduced along the non-compact direction until it eventually reaches zero size, a region of space is removed (the one that corresponds to the values of x that are between the two black dots). (

**c**) A bubble of nothing in more dimensions is obtained when the size of the extra dimension becomes zero at the boundary of such a region. Here we have represented this schematically only at two points in the boundary of the bubble, but in reality there would be one cigar-like blue figure attached to each point on the boundary of the bubble.

**Figure 4.**Figures taken from [66]. (

**a**) Phase space of Reissner–Nordström–de Sitter black holes. (

**b**) Decay of Nariai black holes by emission of particles with ${m}^{2}\ll qE$, so that they discharge almost instantaneously without effectively loosing mass.

**Figure 5.**Figures taken from [98]. The vertical axis represents the 3d effective potential for the radion, $V\left(R\right)$ (in units of with $r=1$ GeV), multiplied times ${R}^{6}$ so as to give a constant profile when $R\to 0$, and it is normalized by the contribution of one degree of freedom (

**a**) Radion effective potential for Majorana neutrinos (with heavy Majorana masses) where an AdS minimum always develops. (

**b**) Radion effective potential for (pseudo-)Dirac neutrinos with different values for the lightest neutrino mass ${m}_{{\nu}_{1}}$ (in mili-electron volts). The formation of an AdS vacuum can be avoided if the neutrino masses are light enough.

**Figure 6.**Figures taken from [98]. The red band indicates the region with energies around the QCD scale where no perturbative description is available. In analogy to the previous figures, the vertical axis shows the 3d scalar potential for the area modulus of the torus, $V\left(a\right)$, multiplied by ${a}^{2}$ to give a constant profile as $a\to 0$, and normalized by the contribution of one degree of freedom. (

**a**) Effective potential of the SM compactified in ${T}^{2}/{Z}_{4}$ embedded into a discrete subgroup of the $SU\left(3\right)$ colour Cartan subalgebra, where an AdS vacuum forms just below the QCD threshold. (

**b**) Same plot for the MSSM, where the extra bosonic degrees of freedom create a runaway behaviour destabilizing the AdS vacua and rescuing the SM from the swampland.

**Figure 7.**Figure adapted from [67]. The Mexican hat potential in the right, with an unstable maximum at the (symmetry restoring) point at the origin is not in tension with the Festina Lente bound. A local minimum as in the figure on the right is nevertheless in tension with the Festina Lente bound unless extreme fine-tuning is included.

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Graña, M.; Herráez, A.
The Swampland Conjectures: A Bridge from Quantum Gravity to Particle Physics. *Universe* **2021**, *7*, 273.
https://doi.org/10.3390/universe7080273

**AMA Style**

Graña M, Herráez A.
The Swampland Conjectures: A Bridge from Quantum Gravity to Particle Physics. *Universe*. 2021; 7(8):273.
https://doi.org/10.3390/universe7080273

**Chicago/Turabian Style**

Graña, Mariana, and Alvaro Herráez.
2021. "The Swampland Conjectures: A Bridge from Quantum Gravity to Particle Physics" *Universe* 7, no. 8: 273.
https://doi.org/10.3390/universe7080273