Scalar fields, hierarchical UV/IR mixing and the Weak Gravity Conjecture

Journal of High Energy Physics, Feb 2018

Dieter Lüst, Eran Palti

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Scalar fields, hierarchical UV/IR mixing and the Weak Gravity Conjecture

JHE Scalar fields, hierarchical UV/IR mixing and the Dieter Lu¨st 0 1 2 Eran Palti 0 1 0 Fohringer Ring 6 , 80805 Mu ̈nchen , Germany 1 Theresienstrasse 37 , 80333 Mu ̈nchen , Germany 2 Arnold-Sommerfeld-Center for Theoretical Physics , Ludwig-Maximilians-Universita ̈t The Weak Gravity Conjecture (WGC) bounds the mass of a particle by its charge. It is expected that this bound can not be below the ultraviolet cut-off scale of the effective theory. Recently, an extension of the WGC was proposed in the presence of scalar fields. We show that this more general version can bound the mass of a particle to be arbitrarily far below the ultraviolet cut-off of the effective theory. It therefore manifests a form of hierarchical UV/IR mixing. This has possible implications for naturalness. We also present new evidence for the proposed contribution of scalar fields to the WGC by showing that it matches the results of dimensional reduction. In such a setup the UV/IR mixing is tied to the interaction between the WGC and non-local gauge operators. Black Holes; Models of Quantum Gravity; Effective Field Theories; Gauge - HJEP02( 18 )4 Weak Symmetry 1 Introduction 2 3 4 5 The Weak Gravity Conjecture and UV/IR mixing 2.1 The WGC with scalars and dimensional reduction The UV cut-off scale 4.1 Relation to a scalar Weak Gravity Conjecture Quantum corrections and naturalness 5.1 5.2 Quantum corrections to mφ Quantum corrections to β 5.2.1 5.2.2 A scalar WGC particle A fermion WGC particle 5.3 Quantum corrections in the 5-dimensional model 6 Discussion 1 Introduction conjectured property of quantum gravity is the Weak Gravity Conjecture (WGC) [3]. It is fields [4] it is, at least in principle, possible. Specifically, we propose that quantum gravity physics can restrict the mass of a particle to an IR scale far below the UV cut-off scale of an effective theory if the particle couples to gauge fields and massless scalar fields with almost precisely equal strength. The rest of the introduction is dedicated to expanding on this and forms an overview of some of the main points of this note. Consider a theory with gravity, a U( 1 ) with gauge coupling g, a charged scalar h with charge q = 1 and mass m, and a neutral scalar φ with mass mφ L = − |Dh|2 − 2 (∂φ)2 − m2h∗h − 2 1 m2φφ2 − 2mµφh ∗ h + . . . . (1.1) Here µ is a dimensionless parameter which parameterises the coupling of φ to h. The . . . denote arbitrary further terms in the Lagrangian. All the quantities in the Lagrangian (1.1) are the quantum corrected expressions having integrated out all the UV physics. The UV completion of the theory is taken to be as follows. There is a scale ΛUV ∼ gMp above which the theory can not be completed by a quantum field theory, so it is the scale where quantum gravity related physics is reached. Below this scale, but above m and mφ, there may be other scales where new degrees of freedom appear but for each one the theory can be completed by a quantum field theory. The new degrees of freedom are constrained such that the particle with the largest charge-to-mass ratio with respect to the U( 1 ) remains h. Then the claim is that in the limit mφ → 0, this theory must satisfy [4] pg2 − µ 2Mp ≡ βMp ≥ m . (1.2) We also impose g ≥ µ . For finite mφ 6= 0 the expression (1.2) will receive corrections suppressed by mmφ . β → β + O We will estimate in section 5 that this modifies (1.2) such that μ2mφ . We therefore observe a new mass scale in the theory βMp. We will βm argue that, assuming sufficiently small mφ, this mass scale can be separated arbitrarily far from the quantum gravity UV scale of the theory m ≤ βMp ≪ ΛUV. Since (1.2) is tied to quantum gravity physics, with an associated mass scale ΛUV, but is a constraint on an arbitrarily low IR scale, it manifests a form of UV/IR mixing. In other words, say the particle with the largest charge-to-mass ratio in the theory happens to couple almost precisely the same to gauge fields and massless scalar fields. From the perspective of quantum field theory this would not have any implications, but from the perspective of quantum gravity we claim that this would imply that the mass of the particle would have to be at an IR scale far below the UV scale of the theory. The reason is that if m violates the bound (1.2) then the charged particle h would form a tower of absolutely stable bound states of increasing charges. If m is increased adiabatically from satisfying (1.2) to violating it, then upon crossing the bound a tower of at least βg such stable states forms below the mass scale gMp. Since β and g are separate parameters this implies a setup where the number of stable states below a fixed mass scale is arbitrary. – 2 – HJEP02( 18 )4 Indeed, we can consider βg → ∞ inducing an infinite number of such states. We expect that this should not be possible in quantum gravity, in analogy with the argument against global symmetries. We discuss this in more detail in section 2. T (...truncated)


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Dieter Lüst, Eran Palti. Scalar fields, hierarchical UV/IR mixing and the Weak Gravity Conjecture, Journal of High Energy Physics, 2018, pp. 40, Volume 2018, Issue 2, DOI: 10.1007/JHEP02(2018)040