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The Fundamental Need for a SM Higgs and the Weak Gravity Conjecture

Eduardo Gonzalo, Luis E. Ibáñez

TL;DR

This paper uses quantum gravity constraints, via a sharpened Weak Gravity Conjecture, to explore the necessity of the Standard Model Higgs. By examining SM compactifications to lower dimensions and the associated radion Casimir potential, the authors derive a lower bound on the Higgs vev around the QCD scale and an upper bound tied to the lightest neutrino mass and the 4D cosmological constant. The combined bounds carve out an AdS-safe region for the Higgs vev and offer predictions for neutrino masses if the upper bound is saturated, reframing the electroweak hierarchy as a question of quantum-gravity consistency. All conclusions depend on the assumed UV stability of the AdS vacua; thus, UV completions (e.g., SUSY or string theory) are crucial for the robustness of these constraints.

Abstract

Compactifying the SM down to 3D or 2D one may obtain AdS vacua depending on the neutrino mass spectrum. It has been recently shown that, by insisting in the absence of these vacua, as suggested by {\it Weak Gravity Conjecture} (WGC) arguments, intriguing constraints on the value of the lightest neutrino mass and the 4D cosmological constant are obtained. For fixed Yukawa coupling one also obtains an upper bound on the EW scale $\left\langle H\right\rangle\lesssim {Λ_4^{1/4}} /{Y_{ν_{i}}}$,where $Λ_4$ is the 4D cosmological constant and $Y_{ν_{i}}$ the Yukawa coupling of the lightest (Dirac) neutrino. This bound may lead to a reassessment of the gauge hierarchy problem. In this letter, following the same line of arguments, we point out that the SM without a Higgs field would give rise to new AdS lower dimensional vacua. Absence of latter would require the very existence of the SM Higgs. Furthermore one can derive a lower bound on the Higgs vev $\left\langle H\right\rangle\gtrsim Λ_{\text{QCD}}$ which is required by the absence of AdS vacua in lower dimensions. The lowest number of quark/lepton generations in which this need for a Higgs applies is three, giving a justification for family replication. We also reassess the connection between the EW scale, neutrino masses and the c.c. in this approach. The EW fine-tuning is here related to the proximity between the c.c. scale $Λ_4^{1/4}$ and the lightest neutrino mass $m_{ν_i}$ by the expression $ \frac {ΔH}{H} \lesssim \frac {(aΛ_4^ {1/4} -m_{ν_i})} {m_{ν_i}}. $ We emphasize that all the above results rely on the assumption of the stability of the AdS SM vacua found.

The Fundamental Need for a SM Higgs and the Weak Gravity Conjecture

TL;DR

This paper uses quantum gravity constraints, via a sharpened Weak Gravity Conjecture, to explore the necessity of the Standard Model Higgs. By examining SM compactifications to lower dimensions and the associated radion Casimir potential, the authors derive a lower bound on the Higgs vev around the QCD scale and an upper bound tied to the lightest neutrino mass and the 4D cosmological constant. The combined bounds carve out an AdS-safe region for the Higgs vev and offer predictions for neutrino masses if the upper bound is saturated, reframing the electroweak hierarchy as a question of quantum-gravity consistency. All conclusions depend on the assumed UV stability of the AdS vacua; thus, UV completions (e.g., SUSY or string theory) are crucial for the robustness of these constraints.

Abstract

Compactifying the SM down to 3D or 2D one may obtain AdS vacua depending on the neutrino mass spectrum. It has been recently shown that, by insisting in the absence of these vacua, as suggested by {\it Weak Gravity Conjecture} (WGC) arguments, intriguing constraints on the value of the lightest neutrino mass and the 4D cosmological constant are obtained. For fixed Yukawa coupling one also obtains an upper bound on the EW scale ,where is the 4D cosmological constant and the Yukawa coupling of the lightest (Dirac) neutrino. This bound may lead to a reassessment of the gauge hierarchy problem. In this letter, following the same line of arguments, we point out that the SM without a Higgs field would give rise to new AdS lower dimensional vacua. Absence of latter would require the very existence of the SM Higgs. Furthermore one can derive a lower bound on the Higgs vev which is required by the absence of AdS vacua in lower dimensions. The lowest number of quark/lepton generations in which this need for a Higgs applies is three, giving a justification for family replication. We also reassess the connection between the EW scale, neutrino masses and the c.c. in this approach. The EW fine-tuning is here related to the proximity between the c.c. scale and the lightest neutrino mass by the expression We emphasize that all the above results rely on the assumption of the stability of the AdS SM vacua found.

Paper Structure

This paper contains 4 sections, 12 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A world with no Higgs is in the swampland
  3. A lower bound on the Higgs vev
  4. Combined constraints on the Higgs vev

Figures (3)

  • Figure 1: Effective radion potential for different numbers of quark/lepton generations $n_g$ in the absence of a Higgs. For $n_g\geq 3$ an AdS vacuum develops slightly below the corresponding QCD scale.
  • Figure 2: Effective radion potential for different values of the Higgs vev $\left\langle H\right\rangle$ in units of the SM value $v=246$ GeV, with $n_g=3$. The Yukawa couplings are fixed at their SM values. For Higgs vevs larger than $10^{-3}v$ the AdS vacua ceases to develop.
  • Figure 3: Constraints on the Higgs vev as a function of the c.c. scale $\Lambda^{1/4}$ for fixed neutrino Yukawa couplings. The vertical(horizontal) dashed line gives the experimental value of the c.c.(Higgs vev) respectively. We also show with a vertical band bounds on the cosmological constant from anthropic constraints anthropic. We have divided the plot in AdS safe (in blue) and unsafe (in red) zones. For fixed values of the c.c. an upper bound on the Higgs vev is obtained.