A Conjecture on the Minimal Size of Bound States

TL;DR

This paper proposes a conjecture that in certain quantum field theories, bound states cannot have a radius smaller than the inverse of the heaviest stable particle's mass, with implications for quantum gravity and effective theories.

Contribution

It introduces the Bound State Conjecture, linking minimal bound state size to particle mass and exploring its implications for quantum gravity and effective field theories.

Findings

Support from analyzed examples at a parametric level

Connection to the Weak Gravity Conjecture and black hole bounds

Suggests a feature of QFT independent of quantum gravity constraints

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.