Weak Gravity Conjecture: Naturalness and UV Constraints

• Weak Gravity Conjecture is a principle requiring a charged state with gauge repulsion surpassing gravitational attraction, ensuring UV consistency.
• It generalizes to multiple U(1)s by demanding that the convex hull of charge-to-mass vectors contains the unit ball, linking black hole decay to spectrum consistency.
• The conjecture interrelates scalar naturalness with model building by enforcing a low EFT cutoff or Higgsing to resolve the hierarchy problem.

The Weak Gravity Conjecture (WGC) postulates that in any quantum theory incorporating both gravity and an Abelian gauge symmetry, there must exist a state whose gauge repulsion exceeds gravitational attraction, formally requiring a particle of charge $q$ and mass $m$ such that $q > m/m_{\rm Pl}$. The conjecture is interpreted as an ultraviolet (UV) consistency condition, directly linking effective field theory model-building with deep quantum gravitational requirements. The WGC has been generalized to multi-gauge group theories, and its tension with the principle of naturalness for scalar fields is central to understanding the hierarchy problem and the structure of potential ultraviolet completions of the Standard Model.

1. Generalization of the Weak Gravity Conjecture

For a single Abelian $U(1)$, the original WGC demands $q > m/m_{\rm Pl}$ for some state. The generalization to product gauge groups $\prod_{a=1}^N U(1)_a$ introduces for each particle a charge vector $\vec{q}_i = \{q_{i1}, q_{i2}, \ldots, q_{iN}\}$ and a charge-to-mass ratio vector $\vec{z}_i = \vec{q}_i / m_i$. The generalized WGC is formulated in terms of convex geometry: the convex hull of all $\pm \vec{z}_i$ in charge space must contain the unit ball. This requirement guarantees that every possible extremal black hole can decay to the existing spectrum.

If each species is charged under a single $U(1)$ factor (i.e., $z_a = z\; \delta_{a(i)}$), the convex hull forms an $N$-dimensional cross-polytope. The radius of its largest inscribed ball is $z/\sqrt{N}$, so enforcing the WGC gives $z > \sqrt{N}$. For the case $N=2$ and species with charge-to-mass ratios $\vec{z}_1$ and $\vec{z}_2$, the following analytic bound is derived: $$(z_1^2 - 1)(z_2^2 - 1) > (1 + |\vec{z}_1 \cdot \vec{z}_2|)^2.$$ This geometric formulation broadens the scope of the WGC to realistic models featuring multiple $U(1)$s or mixed spectra with differently charged states.

2. Tension Between the WGC and Naturalness

The paradigm of naturalness in field theory holds that scalar masses should be stable against large radiative corrections, with physical mass squared set by cutoff-scale quantum corrections: $$\delta m^2 = \frac{\Lambda^2}{16\pi^2} (a q^2 + b \lambda),$$ where $\Lambda$ is the cutoff, $q$ the gauge coupling, $\lambda$ the quartic scalar coupling, and $a,b = O(1)$. For $q^2 \ll \lambda$, naturalness implies $\delta m^2 \approx (\Lambda^2/16\pi^2) b \lambda$, and the physical mass is not much smaller than $\delta m$.

The charge-to-mass ratio for the scalar becomes

$$z = \frac{4\pi}{\Lambda} \left(1/\sqrt{a + b\lambda/q^2}\right).$$

The WGC bound $z > 1$ then sets an upper bound on the cutoff scale: $\Lambda < 4\pi \sqrt{q^2/(b\lambda)}.$ For very small $q$ and generic quartic coupling $\lambda$, $\Lambda$ must be many orders of magnitude below $m_{\rm Pl}$. This scenario demonstrates a direct conflict: demanding both naturalness (no fine-tuning of $m^2$ against $\delta m^2$) and the WGC is not possible unless either the cutoff is extremely low or new dynamics arise.

3. Reconciliation Mechanisms: Higgsing and Low Cutoff

The paper identifies two mechanisms to reconcile the naturalness principle with the WGC:

• Higgsing: If radiative corrections drive $m^2$ negative, the scalar condenses and the $U(1)$ is Higgsed. In the broken phase, the WGC cannot be stated in terms of stable extremal black holes, as the original definition in terms of asymptotic charge becomes ambiguous.
• Lowering the Cutoff: Insist that new physics enters at $\Lambda \sim q m_{\rm Pl}$. The effective theory is valid only up to this scale, naturally keeping the scalar mass light. There is no naturalness problem as the hierarchy between $m$ and $m_{\rm Pl}$ does not occur within the validity of the EFT.

With $q^2 \ll \lambda$, the WGC translates to $\Lambda < 4\pi\sqrt{q^2/(b\lambda)}$, practically requiring $\Lambda$ to be low. For $q^2 \gg \lambda$ (a more weakly coupled scalar), the bound relaxes to $\Lambda \lesssim 4\pi/\sqrt{a} \sim m_{\rm Pl}$. Thus, the only generic options are Higgsing or a low EFT cutoff.

4. Implications for the Electroweak Scale

Applying this analysis to extensions of the Standard Model, such as those with an unbroken $U(1)_{B-L}$, the tension sharpens. Assigning an ultralight right-handed neutrino a minuscule $B-L$ charge $q \sim m_\nu / m_{\rm Pl} \sim 10^{-29}$ (for $m_\nu \sim 0.1$ eV) places it just above the WGC threshold for the minimal charge-to-mass ratio. However, if the electroweak scale were natural (i.e., neutrino mass set by the cutoff $\Lambda \sim m_{\rm Pl}$), then $q$ would be much less than $m_\nu / m_{\rm Pl}$, violating the WGC. Therefore, models that allow or predict a “naturally" large weak scale run afoul of the WGC unless new states or symmetry-breaking phases intervene.

This interplay renders models that “forbid” a natural weak scale (i.e., that explain why weak interactions are weak rather than Planckian) especially relevant for both the hierarchy problem and quantum gravity consistency.

5. Observational Consequences and Experimental Probes

Observation of a submillielectron $q$-charged, stable, light state (such as a neutrino charged under a new $U(1)$ gauging $B-L$ with $q \sim 10^{-29}$) would have major implications:

• If no additional ultralight states are found below the cutoff, such an observation would directly rule out naturalness—as there cannot be a natural (radiatively stable) electroweak scale within the allowed EFT.
• If the Higgs phase is experimentally excluded, then the only viable option from WGC consistency is a new, very low cutoff scale $\Lambda \sim q m_{\rm Pl}$, potentially in the keV range. This would shift the expected energy scale for new physics well below the traditional TeV window.
• Discovery of a fifth force at such couplings would force a paradigm shift for the hierarchy problem, removing the expectation of TeV scale new physics and favoring UV completions that intrinsically accommodate an unnaturally low electroweak scale.
• Inconsistency between the experimentally measured cutoff and WGC predictions would even challenge the internal consistency of certain string compactification scenarios.

Thus, the experimental search for extremely weak gauge forces or ultralight charged states is a direct probe of both naturalness and the quantum gravity swampland.

6. Mathematical Summary Table

Scenario	Bound from WGC	Consequence
Charged scalar, small $q$	$\Lambda < 4\pi\sqrt{q^2/(b\lambda)}$	Only low cutoff or Higgsing
$N$-fold $U(1)$, orthogonal charges	$z > \sqrt{N}$	Convex hull contains unit ball
Natural Higgs sector + ultralight $q$	$q \gtrsim m_\nu / m_{\rm Pl}$	Weak scale not naturally large

7. Synthesis: WGC as a UV Consistency and its Impact on Model Building

The WGC, when rigorously applied, both generalizes to product gauge groups and fundamentally intertwines with scalar naturalness. The requirement that all extremal black holes are unstable to decay strongly constrains viable UV completions: any theory with a “natural” unsuppressed scalar mass in the presence of an ultralight gauge coupling must either feature a Higgsed $U(1)$ or a low effective cutoff, eliminating the standard window for a natural electroweak scale. These constraints directly extend to bottom-up model-building, favoring models that either break the problematic gauge symmetry or admit new light charged fermions below the weak scale. The WGC thus offers concrete phenomenological targets—such as searches for fifth forces or deviations in electroweak observables—and presents an avenue by which quantum gravitational consistency may rule out otherwise compelling extensions of the Standard Model.