Evidence for the Multiverse in the Standard Model and Beyond

TL;DR

This work introduces a quantitative naturalness framework based on a probability measure $P$ over ensembles of theories and applies it to fundamental questions in the Standard Model and cosmology. It argues that three distinct observer-naturalness problems—the cosmological constant, nuclear stability, and electroweak symmetry breaking—emerge as natural consequences of a multiverse with environmental selection. By modeling how distributions of Lagrangian parameters and observer boundaries shape effective distributions, the authors show that the observed near-boundary values can arise without invoking new symmetries, and that both little and large hierarchies for $v$ and $M$ can be generated. The paper also outlines concrete predictions for first-generation masses $m_u$, $m_d$, and $m_e$, and discusses how forthcoming collider data, especially from the LHC, could strengthen or challenge the environmental selection narrative, potentially providing indirect evidence for a multiverse. All mathematical notation is presented within $...$ delimiters, including $P$, $P_{ m nuc}$, $P_{ m EWSB}$, $m_h^2$, $M$, $v$, and $\Lambda_{\rm QCD}$, to ensure precise interpretation in analytic and search contexts.

Abstract

In any theory it is unnatural if the observed parameters lie very close to special values that determine the existence of complex structures necessary for observers. A naturalness probability, P, is introduced to numerically evaluate the unnaturalness. If P is small in all known theories, there is an observer naturalness problem. In addition to the well-known case of the cosmological constant, we argue that nuclear stability and electroweak symmetry breaking (EWSB) represent significant observer naturalness problems. The naturalness probability associated with nuclear stability is conservatively estimated as P_nuc < 10^{-(3-2)}, and for simple EWSB theories P_EWSB < 10^{-(2-1)}. This pattern of unnaturalness in three different arenas, cosmology, nuclear physics, and EWSB, provides evidence for the multiverse. In the nuclear case the problem is largely solved even with a flat multiverse distribution, and with nontrivial distributions it is possible to understand both the proximity to neutron stability and the values of m_e and m_d - m_u in terms of the electromagnetic contribution to the proton mass. It is reasonable that multiverse distributions are strong functions of Lagrangian parameters due to their dependence on various factors. In any EWSB theory, strongly varying distributions typically lead to a little or large hierarchy, and in certain multiverses the size of the little hierarchy is enhanced by a loop factor. Since the correct theory of EWSB is unknown, our estimate for P_EWSB is theoretical. The LHC will determine P_EWSB more robustly, which may remove or strengthen the observer naturalness problem of EWSB. For each of the three arenas, the discovery of a natural theory would eliminate the evidence for the multiverse; but in the absence of such a theory, the multiverse provides a provisional understanding of the data.

Figures (23)

• Figure 1: Illustrations of the definition of the naturalness probability $P$ in multi-dimensional parameter space.
• Figure 2: The contour of $m_h^2/M_*^2 = \pm 1, \pm 0.1, \pm 0.01, \cdots$ in the $M_1$-$M_2$ plane. The special line of $m_h^2 = 0$ is visible at $M_1 = M_2$. The observed value of $M_2 = M_1 + O(10^{-32})$ is denoted by the little dot for an arbitrary value of $M_1 = 1.2 M_*$.
• Figure 3: The contour of $m_h^2/M_*^2 = 10^{-4n}$ with $n=3,4,5,\cdots$ in the $|b|g_*^2$-$r$ plane. The observed value of $v \approx \sqrt{-m_h^2} \approx 100~{\rm GeV}$ can correspond to a completely generic point in the plane, which is indicated by the little dot for an arbitrary value of $r = 1.4$.
• Figure 4: Characteristic situations for complexity and observer boundaries in 2-dimensional parameter space. Complexity boundaries that are and are not relevant for the existence of observers are depicted by solid and dashed lines, respectively. The shaded region indicates an observer region ${\cal O}$, which (a) may or (b) may not be a small region around the observed point, which is denoted by the dot. The observer boundary, $O(x_i) = 1$, is given by the border of ${\cal O}$ and is represented by the thick solid line.
• Figure 5: The location of the observer boundary in $m_u$-$m_d$-$m_e$ space. The neutron and complex nuclei boundaries of Eqs. (\ref{['eq:boundary-H-2']}) and (\ref{['eq:boundary-cn-2']}) are depicted by solid lines (below and above, respectively). Dashed lines represent the deuteron boundary of Eq. (\ref{['eq:boundary-D-2']}) (for $a = 5.5$, $2.2$ and $1.3~{\rm MeV}$ from below). The observed point is represented by little dots.