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One in a Billion: MSSM-like D-Brane Statistics

Florian Gmeiner, Ralph Blumenhagen, Gabriele Honecker, Dieter Lust, Timo Weigand

TL;DR

The paper conducts a large-scale statistical analysis of four-dimensional N=1 supersymmetric intersecting D-brane vacua on T^6/(Z_2×Z_2), solving tadpole and K-theory constraints to quantify the abundance of Standard Model features. Using a systematic computer search over factorizable branes, it analyzes distributions, correlations, and gauge-coupling relations at the string scale, highlighting both universal topological trends and moduli-dependent variations. A central result is the extreme suppression of three-generation MSSM-like models, estimated at about $1.0\times 10^{-9}$ of the viable pool, translating to roughly a few tens of such vacua in their dataset; this underscores the rarity of fully realistic vacua in this corner of the landscape. The work also finds partial universality in hidden-sector statistics and discusses how gauge couplings depend on geometric data, providing a benchmark for future cross-comparisons across string vacua classes.

Abstract

Continuing our recent work hep-th/0411173, we study the statistics of four-dimensional, supersymmetric intersecting D-brane models in a toroidal orientifold background. We have performed a vast computer survey of solutions to the stringy consistency conditions and present their statistical implications with special emphasis on the frequency of Standard Model features. Among the topics we discuss are the implications of the K-theory constraints, statistical correlations among physical quantities and an investigation of the various statistical suppression factors arising once certain Standard Model features are required. We estimate the frequency of an MSSM like gauge group with three generations to be one in a billion.

One in a Billion: MSSM-like D-Brane Statistics

TL;DR

The paper conducts a large-scale statistical analysis of four-dimensional N=1 supersymmetric intersecting D-brane vacua on T^6/(Z_2×Z_2), solving tadpole and K-theory constraints to quantify the abundance of Standard Model features. Using a systematic computer search over factorizable branes, it analyzes distributions, correlations, and gauge-coupling relations at the string scale, highlighting both universal topological trends and moduli-dependent variations. A central result is the extreme suppression of three-generation MSSM-like models, estimated at about of the viable pool, translating to roughly a few tens of such vacua in their dataset; this underscores the rarity of fully realistic vacua in this corner of the landscape. The work also finds partial universality in hidden-sector statistics and discusses how gauge couplings depend on geometric data, providing a benchmark for future cross-comparisons across string vacua classes.

Abstract

Continuing our recent work hep-th/0411173, we study the statistics of four-dimensional, supersymmetric intersecting D-brane models in a toroidal orientifold background. We have performed a vast computer survey of solutions to the stringy consistency conditions and present their statistical implications with special emphasis on the frequency of Standard Model features. Among the topics we discuss are the implications of the K-theory constraints, statistical correlations among physical quantities and an investigation of the various statistical suppression factors arising once certain Standard Model features are required. We estimate the frequency of an MSSM like gauge group with three generations to be one in a billion.

Paper Structure

This paper contains 21 sections, 33 equations, 12 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Model building ingredients
  3. Parameterisation of Type II orientifolds
  4. Gauge anomalies and K-theory constraints
  5. The $T^6/({\mathbb Z}_2 \times {\mathbb Z}_2)$ model
  6. Standard Model realisations
  7. Computer algorithms
  8. Number of solutions
  9. Complexity
  10. Statistical distributions
  11. Effect of the K-theory constraints
  12. Correlations between the total rank of the gauge group and chirality
  13. Statistics of Standard-like models
  14. Massless hypercharge
  15. Pati-Salam models
  16. ...and 6 more sections

Figures (12)

  • Figure 1: The number of different configurations of $\hat{X}^I$ for $L=2$, satisfying the constraints. The x-axis shows combinations of complex structures in an arbitrary scale.
  • Figure 2: The number of models for $L=8$, plotted against the absolute value of $\vec{U}$. The highest analysed value of $\vec{U}$ is 12.
  • Figure 3: Probability distributions for $L=8, U_I=1$. The stars represent the distribution without K-theory constraints, the boxes give the result including the constraints.
  • Figure 4: The frequency distribution of models of specific rank and chirality. $L=8, U_I=1$.
  • Figure 5: Logarithmic plot of the number of models versus the number of generations.
  • ...and 7 more figures