A Weak Gravity Conjecture for Scalar Field Theories

Scalars, such as Higgs and inflaton, play significant roles in both particle physics and cosmology. Recently, Vafa et.al. conjectured an upper bound on the strength of gravity relative to gauge forces in quantum gravity [4]. Their conjecture enriches the criteria of consistent effective field theory proposed in [2, 3]. These criteria help to constrain the string landscape [1] in the vast vacua of string theory. After a careful investigation, we find that a similar bound exists in a class of scalar field theories. In this class of theories at least, “gravity is the weakest” even in the appearance of scalars. We will study scalar field theories with soliton solutions. we will focus our attention on a special class of such theories. In these theories, the coefficient of the mass term is ± 1 2 μ with μ > 0, and there are higher order terms controlled by a coupling constant λ in addition to μ. So the theories are described by only two parameters: the mass parameter μ and the coupling constant λ. When the scalar is coupled to gravity, we find an interesting constraint on μ and λ. The idea is to study a scalar system coupled to twodimensional dilaton gravity, which is assumed to be dimensional reduction of some consistent quantum gravity system in higher dimensions. In two dimensions, the system contains solitons in the weak gravity limit (plus some other massive particles in a sector not considered here). These solitons appear as asymptotic scattering states so there is a well-defined S-matrix. As one increases the gravity coupling, these solitons disappear, thus there are no more massive particles and the S-matrix ceases to exist. We assume that in this case the quantum system becomes inconsistent, because without the S-matrix, only correlation functions exist which are not observables in a gravity theory. We therefore conjecture that the quantum theory exists only in the weak gravity region. It is a matter subject to debate that whether this is also true in higher dimensions, but we feel that a consistent theory in higher dimensions should result in a consistent theory upon dimensional reduction, thus we lift this conjecture to higher dimensions, in particular, to four dimensions. When the scalar is coupled to two-dimensional dilaton gravity, the action is generally taken to be