⚛️ high-energy theory

Instanton Interactions and the Axion Weak Gravity Conjecture

This paper proposes a stronger formulation of the Axionic Weak Gravity Conjecture based on repulsive long-range instanton interactions mediated by massless scalars, demonstrating that while this bound implies the standard axion decay constant limit in typical UV-complete scenarios, it suggests potential modifications to that limit in theories lacking massless scalars.

Original authors: Jarod Hattab, Eran Palti

Published 2026-09-16

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CC BY 4.0

Original authors: Jarod Hattab, Eran Palti

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast, invisible architecture of the universe, physicists often look for rules that keep everything from falling apart. One such rule, known as the Weak Gravity Conjecture, suggests that gravity, which is usually the weakest force, must have a rival. In a world with electric charge, this idea implies that there must be particles heavy enough to be pulled by gravity but charged enough to push each other away with a force stronger than gravity can pull them together. This ensures that black holes, which are essentially cosmic vacuum cleaners for matter, can eventually shed their charge and evaporate rather than becoming eternal, unchanging remnants. This principle has been tested extensively for ordinary particles, but it becomes much trickier when applied to axions. Axions are hypothetical, ghostly particles that are thought to solve deep problems in physics, and they interact with instantons—fleeting, quantum events that pop in and out of existence. For years, scientists have struggled to define a clear rule for how these instantons should behave to satisfy the same cosmic balance, leading to a foggy understanding of whether the axion version of the Weak Gravity Conjecture holds true.

A new study by Jarod Hattab and Eran Palti offers a fresh way to cut through this fog. Instead of focusing on the forces between particles, the researchers turned their attention to the interactions between these fleeting instanton events. They proposed a simple, physical test: if you place two identical instantons far apart in space, do they push each other away or pull each other together? In the language of physics, they calculated the energy of this interaction. If the energy is positive, the instantons repel; if it is negative, they attract and would want to bind together. The authors suggest that for the universe to remain stable and consistent with the laws of quantum gravity, these instantons must repel each other. This repulsion acts as a cosmic speed limit, preventing the axion from becoming too heavy or too weakly coupled, which would otherwise break the delicate balance required for a consistent theory of everything.

What makes this approach so powerful is its focus on the specific messengers of these forces. In the case of ordinary particles, gravity, electric fields, and other massless fields all compete to determine whether the force is attractive or repulsive. However, Hattab and Palti discovered that for instantons, the story is different. When two instantons are separated by a large distance, the only forces that matter are those carried by massless scalar fields—fields that permeate space without a specific direction, like a temperature that varies from place to place. Gravity and electric fields, which usually play a major role, become negligible at these scales for instantons. The researchers found that the repulsive force comes entirely from the axion field itself, while the attractive force comes from other scalar fields. For the Weak Gravity Conjecture to hold, the repulsive push from the axion must be stronger than the attractive pull from the other scalars. This condition translates into a precise mathematical bound on how the instanton's properties change as the scalar fields in the universe shift.

The beauty of this new formulation is that it is sharper and more general than previous attempts. In many standard models of string theory, where the universe is described by vibrating strings in extra dimensions, this new rule automatically leads to the older, well-known limits on axions. However, it also reveals something surprising: if the universe were to exist without any of these extra massless scalar fields, the usual rules might not apply at all. In such a scenario, the constraint that limits the axion's behavior could vanish, potentially allowing for a wider range of possibilities for the axion's properties. This suggests that the existence of these invisible scalar fields is not just a detail of string theory but a fundamental requirement for the stability of the axion sector. The authors are careful to note that this is a proposal based on theoretical consistency rather than a direct experimental observation, but it provides a concrete, calculable way to test these ideas in future models of the universe.

By reframing the problem as a question of long-range repulsion between instantons, the study strips away the complexity of the usual equations and focuses on the physical intuition of forces pushing and pulling. It confirms that in the known examples of a complete theory of physics, the repulsive axion force always wins, keeping the universe in balance. But it also leaves the door open for a different kind of universe—one where the rules change if the background fields are different. This work does not claim to have solved the mystery of the axion, but it has provided a new, clearer lens through which to view it, turning a vague conjecture into a specific condition about how the universe's invisible fields must interact to keep everything from collapsing.

Technical Summary: Instanton Interactions and the Axion Weak Gravity Conjecture

Problem Statement
The Weak Gravity Conjecture (WGC) posits that in any effective field theory (EFT) with a $U(1)$ gauge symmetry admitting a UV completion in quantum gravity, there must exist a charged state with a sufficiently large charge-to-mass ratio to ensure black hole discharge. While the particle version of the WGC has been refined to include repulsive long-range forces mediated by scalars, the Axionic WGC (concerning $p=0$ forms) lacks a similarly precise, interaction-based formulation. The standard axion WGC bound, $f^2 S^2 \lesssim (M_P)^{d-2}$ , involves an ambiguous order-one coefficient and trivializes in the absence of massless scalars when viewed through standard gravitational decoupling limits. The authors seek a formulation of the Axionic WGC based on the long-range interactions between identical instantons, analogous to the repulsive force condition for particles.

Methodology
The authors adopt a "binding energy" perspective, reformulating the WGC condition as a requirement that the interaction between two identical charged objects be non-attractive (i.e., the real part of the interaction action must be non-negative).

Interaction Definition: For two identical objects (particles or instantons) with collective coordinates $x_1$ and $x_2$ , the interaction action $I_{int}$ is defined as the difference between the effective action of the two-object system and the sum of the single-object actions: $I_{int} \equiv I_{eff}(x_1, x_2) - I_{eff}(x_1) - I_{eff}(x_2)$ .
Probe Calculation: The authors treat the objects as probes in a background of massless fields (gravity, gauge fields, and scalars). They expand the Euclidean action to quadratic order in field fluctuations around a constant background.
Force Mediation Analysis:
- Particles: At large separation, particles experience interactions mediated by scalars, gauge fields, and gravitons. The leading order interaction yields the known repulsive force condition: $\left(\frac{d-3}{d-2}\right)m^2 + g_{MN}(\partial_M m)(\partial_N m) \leq q^2 g^2$ .
- Instantons: The authors demonstrate that for instantons, the leading long-range interaction is mediated purely by massless scalar fields. Gauge and gravitational interactions enter only at sub-leading orders (dipole or higher multipole) because gauge invariance and diffeomorphism invariance forbid monopole couplings for these fields. The interaction is dominated by the scalar monopole coupling derived from the instanton action $I_{inst} = S + i\Theta$ .
Derivation of the Bound: By integrating out the scalar fluctuations, the interaction term is found to be proportional to $-g_{MN} \partial_M(S+i\Theta) \partial_N(S+i\Theta)$ . Requiring the real part of the interaction to be non-negative ( $\text{Re } I_{int} \geq 0$ ) leads to the inequality:
$g_{MN} (\partial_M S) (\partial_N S) \leq g_{MN} (\partial_M \Theta) (\partial_N \Theta)$
where $S$ is the real part of the instanton action (associated with attractive forces) and $\Theta$ is the imaginary part (associated with repulsive axionic forces).

Key Contributions and Results

Scalar-Only Formulation: The paper establishes that the Axionic WGC is fundamentally a constraint on the scalar field dependence of the complex instanton action. Unlike the particle WGC, which balances gauge repulsion against gravity and scalar attraction, the instanton WGC is a competition solely between attractive scalar interactions (mediated by non-axionic scalars) and repulsive axionic interactions.
Sharp Bound: The derived inequality provides a sharp bound without the order-one coefficient ambiguity present in the standard axion WGC formulation.
Recovery of Standard Bounds: In specific ultraviolet-complete examples (such as dimensional reduction and Type II string theory c-map scenarios), the authors show that their scalar-based bound reduces to the standard axion WGC form ( $f^2 S^2 \lesssim (M_P)^{d-2}$ $f^{2} S^{2} ≲ (M_{P})^{d - 2}$ ) but is generally stronger.
- In dimensional reduction, the radion (circle radius) acts as the dilaton-like scalar, and the bound matches the particle WGC under reduction.
- In $N=2$ supergravity (via the c-map), the bound is saturated by BPS instantons, where the repulsive axion interaction exactly balances the attractive interactions from the dilaton and complex structure moduli.
Implication for Massive Scalars: A critical result is that if all scalar fields (other than the axion itself) acquire a mass, the leading long-range scalar interaction vanishes. Consequently, the constraint trivializes in the absence of massless scalars. This suggests that the standard axion WGC bound on the decay constant $f$ might be modified or violated in settings where no massless scalars exist beyond the axion.

Significance and Claims
The authors claim that this interaction-based approach offers a more general and precise formulation of the Axionic WGC. By isolating the role of massless scalars, the paper highlights that the usual axion WGC bound is a consequence of the presence of additional massless scalars (like dilatons) that couple with gravitational strength.

The paper posits that in the absence of such massless scalars, the standard bound may not hold, raising the possibility of violating the usual axion WGC in specific EFT settings. The authors note that while wormhole-based approaches suggest constraints even with only a light axion, the interaction approach presented here implies that without massless scalars, the constraint on the axion decay constant could be absent. The paper concludes by identifying the behavior of the conjecture when scalars are massive as a central open question, suggesting that the constraint might need to be re-evaluated at finite separation distances or that higher-derivative gravitational/gauge interactions might become relevant.

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