Abstract
We conjecture that, in a renormalizable effective quantum field theory where the heaviest stable particle has mass mm, there are no bound states with radius below 1/m1/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/m1/m. But this minimal bound-state radius remains non-trivial as M_\mathrm{P}\to \inftyMP→∞, 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.
Highlights
It is often useful to understand which phenomena or constructions can not arise within a given theoretical framework
We are motivated by the Weak Gravity Conjecture, which can be read as a statement forbidding certain bound states
This minimal bound-state radius remains non-trivial as MP → ∞, leading us to suspect a feature of QFT rather than a quantum gravity constraint
Summary
It is often useful to understand which phenomena or constructions can not arise within a given theoretical framework. Our attempts to generalize this conjecture to all forces will lead us in an unexpected direction, making claims about the non-existence of certain bound states in quantum field theory (with little or no relation to gravity). One may formulate the WGC by saying that, for any abelian gauge force, equal-charge particles are more strongly repelled by the gauge force than they are attracted by gravity.2 By definition, this excludes a gravitational bound state of two or more such particles. Consider a collection of N free complex scalars, with the number N protected by a global U(1) Such stars become smaller as N grows and collapse to black holes at a critical (minimal) radius R ∼ 1/m, with m the mass of the elementary bosons.
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