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The Impact of Universal Extra Dimensions on the Unitarity Triangle and Rare K and B Decays

Andrzej J. Buras, Michael Spranger, Andreas Weiler

TL;DR

This work analyzes the Appelquist–Cheng–Dobrescu model with one universal extra dimension to quantify how Kaluza–Klein states modify loop-induced flavor processes. By computing the KK contributions to the Inami–Lim functions S, X, and Y, the authors show that S and C are enhanced, yielding shifts in CKM parameters and an increased ΔM_s, while ΔM_K remains SM-like. The study finds hierarchical enhancements in branching ratios for rare K and B decays and demonstrates that a low compactification scale (1/R ≲ 400 GeV) can be consistent with current data, albeit with sizable theoretical uncertainties in nonperturbative parameters. The work emphasizes the MFV nature of the model, provides complete 5D Feynman rules, and outlines future tests including B→X_sγ and related decays to distinguish ACD from other new physics scenarios.

Abstract

We calculate the contributions of the Kaluza-Klein (KK) modes to the K_L-K_S mass difference ΔM_K, the parameter ε_K, the B^0_{d,s}-\bar B^0_{d,s} mixing mass differences ΔM_{d,s} and rare decays K^+\to π^+ν\barν, K_L\to π^0ν\barν, K_L\toμ\barμ, B\to X_{s,d}ν\barνand B_{s,d}\toμ\barμin the Appelquist, Cheng and Dobrescu (ACD) model with one universal extra dimension. For the compactification scale 1/R= 200 GeV the KK effects in these processes are governed by a 17% enhancement of the ΔF=2 box diagram function S(x_t,1/R) and by a 37% enhancement of the Z^0 penguin diagram function C(x_t/1/R) relative to their Standard Model (SM) values. This implies the suppressions of \vtd by 8%, of \barηby 11% and of the angle γin the unitarity triangle by 10^\circ. ΔM_s is increased by 17%. ΔM_K is essentially uneffected. All branching ratios considered in this paper are increased with a hierarchical structure of enhancements: K^+\to π^+ν\barν(16%), K_L\to π^0ν\barν(17%), B\to X_{d}ν\barν(22%), (K_L\toμ\barμ)_{\rm SD} (38%), B\to X_{s}ν\barν(44%), B_{d}\toμ\barμ(46%) and B_{s}\toμ\barμ(72%). For 1/R= 250 (300) GeV all these effects are decreased roughly by a factor of 1.5 (2.0). We give the relevant Feynman rules for the five dimensional ACD model. We also emphasize that a consistent calculation of branching ratios has to take into account the modifications in the values of the CKM parameters. As a byproduct we confirm the dominant \ord (g_2 G_F m_t^4 R^2) correction from the KK modes to the $Z^0 b\bar b$ vertex calculated recently in the large m_t limit.

The Impact of Universal Extra Dimensions on the Unitarity Triangle and Rare K and B Decays

TL;DR

This work analyzes the Appelquist–Cheng–Dobrescu model with one universal extra dimension to quantify how Kaluza–Klein states modify loop-induced flavor processes. By computing the KK contributions to the Inami–Lim functions S, X, and Y, the authors show that S and C are enhanced, yielding shifts in CKM parameters and an increased ΔM_s, while ΔM_K remains SM-like. The study finds hierarchical enhancements in branching ratios for rare K and B decays and demonstrates that a low compactification scale (1/R ≲ 400 GeV) can be consistent with current data, albeit with sizable theoretical uncertainties in nonperturbative parameters. The work emphasizes the MFV nature of the model, provides complete 5D Feynman rules, and outlines future tests including B→X_sγ and related decays to distinguish ACD from other new physics scenarios.

Abstract

We calculate the contributions of the Kaluza-Klein (KK) modes to the K_L-K_S mass difference ΔM_K, the parameter ε_K, the B^0_{d,s}-\bar B^0_{d,s} mixing mass differences ΔM_{d,s} and rare decays K^+\to π^+ν\barν, K_L\to π^0ν\barν, K_L\toμ\barμ, B\to X_{s,d}ν\barνand B_{s,d}\toμ\barμin the Appelquist, Cheng and Dobrescu (ACD) model with one universal extra dimension. For the compactification scale 1/R= 200 GeV the KK effects in these processes are governed by a 17% enhancement of the ΔF=2 box diagram function S(x_t,1/R) and by a 37% enhancement of the Z^0 penguin diagram function C(x_t/1/R) relative to their Standard Model (SM) values. This implies the suppressions of \vtd by 8%, of \barηby 11% and of the angle γin the unitarity triangle by 10^\circ. ΔM_s is increased by 17%. ΔM_K is essentially uneffected. All branching ratios considered in this paper are increased with a hierarchical structure of enhancements: K^+\to π^+ν\barν(16%), K_L\to π^0ν\barν(17%), B\to X_{d}ν\barν(22%), (K_L\toμ\barμ)_{\rm SD} (38%), B\to X_{s}ν\barν(44%), B_{d}\toμ\barμ(46%) and B_{s}\toμ\barμ(72%). For 1/R= 250 (300) GeV all these effects are decreased roughly by a factor of 1.5 (2.0). We give the relevant Feynman rules for the five dimensional ACD model. We also emphasize that a consistent calculation of branching ratios has to take into account the modifications in the values of the CKM parameters. As a byproduct we confirm the dominant \ord (g_2 G_F m_t^4 R^2) correction from the KK modes to the vertex calculated recently in the large m_t limit.

Paper Structure

This paper contains 30 sections, 132 equations, 12 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The Five Dimensional ACD Model
  3. Kaluza-Klein mode expansion
  4. Universal Extra Dimensions
  5. Gauge fixing and Goldstone mixing
  6. Fermion-Higgs coupling
  7. $B^0_{d,s}-\bar{B}^0_{d,s}$ Mixing and $\varepsilon_K$
  8. $B^0_{d,s}-\bar{B}^0_{d,s}$ Mixing
  9. $\varepsilon_K$
  10. Unitarity Triangle in the ACD Model
  11. Rare K and B Decays
  12. Preliminaries
  13. Effective Hamiltonians for $K\to\pi\nu\bar{\nu}$ and $B\to X_s\nu\bar{\nu}$
  14. Effective Hamiltonians for $B_{s,d}\to\mu\bar{\mu}$ and $K_L\to \mu\bar{\mu}$
  15. $Z^0$-Penguin Diagrams
  16. ...and 15 more sections

Figures (12)

  • Figure 1: Box diagrams contributing to $S_n(x_t,x_n)$. We suppress the KK mode number.
  • Figure 2: (a) Contribution $S_n$ of the $n^{\rm th}$ KK mode to $S(x_t,1/R)$. The contributions with $a^\pm$ dominate, those with only $G^\pm$ and $W^\pm$ are negligible and not shown. (b) The functions $S(x_t,1/R)$ and $S_0(x_t)$.
  • Figure 3: Unitarity Triangle.
  • Figure 4: Results for various CKM parameters in the ACD model and in the SM: (a) $\left|V_{td}\right|$ (b) $\gamma$ (c) $\bar{\eta}$ (d) $\bar{\rho}$.
  • Figure 5: Unitarity Triangle in the ACD model for $1/R=200\, {\rm GeV}$ and in the SM.
  • ...and 7 more figures