[Paper Review] Emergence of small numbers in complex systems and the origin of the electroweak scale

Radovan Dermíšek|arXiv (Cornell University)|Nov 10, 2016

Computational Physics and Python Applications3 citations

TL;DR

This paper argues that in complex models like the minimal supersymmetric standard model (MSSM), the electroweak scale can naturally emerge at up to 1,000 times smaller values than superpartner masses without requiring fine-tuning or symmetries. By specifying all parameters with only one significant figure, the model's inherent complexity—through numerous competing contributions—produces small mass splittings with probabilities comparable to larger outcomes, making small electroweak scales naturally plausible within the MSSM framework.

ABSTRACT

In sufficiently complex models with many parameters that are unknown or undetermined from first principles, a small coupling or mass can naturally arise even if it is not protected by a symmetry or a result of some dynamics. For the naturalness criterion, we advocate specifying all the model parameters with one significant figure. This automatically avoids outcomes of a given observable that would require special highly-tuned choices for parameters. The obtained range of outcomes is an attribute of a given model resulting from its specific characteristics and complexity. Using these criteria in the minimal supersymmetric model, we demonstrate that the electroweak scale up to 3 orders of magnitude below superpartner masses naturally occurs.

Motivation & Objective

• To challenge the conventional view that a small electroweak scale relative to superpartner masses implies fine-tuning.
• To propose a new naturalness criterion based on specifying model parameters with only one significant figure.
• To demonstrate that in complex models like the MSSM, small mass scales can arise naturally from the interplay of many undetermined parameters.
• To show that the observed hierarchy between the electroweak scale and superpartner masses is not unnatural if model complexity is properly accounted for.
• To argue that traditional measures like the Barbieri-Giudice sensitivity index fail to capture the role of model complexity in naturalness.

Proposed method

• Introduce a toy model where observable X ≈ A - B, with A and B drawn from overlapping intervals around 1, to simulate cancellation mechanisms in electroweak symmetry breaking.
• Use a statistical approach where parameters are varied within 50% ranges (0.5 to 1.5), with outcomes evaluated using one significant figure, avoiding high-precision tuning.
• Construct a hierarchical parameter list for the MSSM, computing contributions to Δm²Hu from Yukawa couplings and scalar masses, with decreasing magnitudes across six orders of magnitude.
• Apply a binning scheme to the distribution of outcomes, showing that small values near zero are as likely as central values when parameters are specified with one significant figure.
• Compare the probability of small outcomes (e.g., M_EW ~ 10⁻³ M_SUSY) to the most likely outcome, showing comparable likelihoods in the complex parameter space.
• Reject the use of sensitivity-based measures like |∂ln M_EW² / ∂ln p_i| as indicators of unnaturalness, arguing they ignore model complexity.

Experimental results

Research questions

• RQ1 Can a small electroweak scale arise naturally in the MSSM without requiring fine-tuning of parameters?
• RQ2 Does the conventional Barbieri-Giudice measure accurately reflect the naturalness of small mass scales in complex models?
• RQ3 What is the role of model complexity—specifically, the number and distribution of undetermined parameters—in enabling small outcomes?
• RQ4 Is a small value of M_EW² ~ 10⁻⁶ M_SUSY², corresponding to M_EW ~ 10⁻³ M_SUSY, a likely outcome when parameters are specified with only one significant figure?
• RQ5 Can the observed hierarchy between the electroweak scale and superpartner masses be considered natural if many parameters contribute with comparable likelihoods across multiple orders of magnitude?

Key findings

• In the toy model, outcomes near zero (X ≈ 0) have a probability comparable to the most probable outcome, even though they are small, when parameters are specified with one significant figure.
• In the MSSM, contributions to Δm²Hu span six orders of magnitude below M_SUSY², with 13 parameters contributing densely before a gap to 10⁻¹⁰ levels.
• The electroweak scale as small as M_EW ≈ 10⁻³ M_SUSY can occur with a probability comparable to the most likely outcome, given one-significant-figure parameter specification.
• The range 10⁻³ M_SUSY ≤ M_EW ≤ M_SUSY is a characteristic feature of the MSSM under the proposed naturalness criterion, not requiring fine-tuning.
• The Barbieri-Giudice measure overestimates the need for tuning because it does not account for the presence of many competing parameters in complex models.
• Small outcomes are not unnatural if they arise from a dense distribution of contributions across multiple scales, even if individual parameter sensitivity is high.