# A Conjecture on the Minimal Size of Bound States

### Submission summary

 As Contributors: Sascha Leonhardt Arxiv Link: https://arxiv.org/abs/1912.09485v3 Date accepted: 2020-03-30 Date submitted: 2020-03-23 Submitted by: Leonhardt, Sascha Submitted to: SciPost Physics Discipline: Physics Subject area: High-Energy Physics - Theory Approach: Theoretical

### Abstract

We conjecture that, in a renormalizable effective quantum field theory where the heaviest stable particle has mass $m$, there are no bound states with radius below $1/m$ (Bound State Conjecture). We are motivated by the (scalar) Weak Gravity Conjecture, which can be read as a statement forbidding certain bound states. As we discuss, versions for uncharged particles and their generalizations have shortcomings. This leads us to the suggestion that one should only constrain rather than exclude bound objects. In the gravitational case, the resulting conjecture takes the sharp form of forbidding the adiabatic construction of black holes smaller than $1/m$. But this minimal bound-state radius remains non-trivial as $M_\text{P}\to \infty$, leading us to suspect a feature of QFT rather than a quantum gravity constraint. We find support in a number of examples which we analyze at a parametric level.

###### Current status:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

### List of changes

- Added a comment on Hawking evaporation already in the paragraph underneath the conjecture in Sect. 3.1
- Shortened this comment underneath (13) in Sect. 3.2 correspondingly
- Added a short paragraph on the case of a bound state of mass M fulfilling R < 1/M in Sect. 3.3
- Rephrased a comment on bound states in Yang-Mills theories in the next paragraph to not include gravity

### Submission & Refereeing History

Resubmission 1912.09485v3 on 23 March 2020
Submission 1912.09485v2 on 20 January 2020