Delta F=2 Observables and Fine-Tuning in a Warped Extra Dimension with Custodial Protection

TL;DR

This work assesses Delta F=2 transitions in a Randall–Sundrum model with custodial protection, combining KK-gluon and electroweak exchanges to derive the full NLO renormalisation-group evolved Wilson coefficients. It shows that anarchic 5D Yukawas generally require M_KK ≳ 20 TeV to satisfy ε_K, but allows regions with modest tuning where M_KK ≈ 3 TeV remain accessible to the LHC, aided by custodial protection of Z couplings. Importantly, Z_H and Z' exchanges can rival KK-gluon effects in B_d,s systems, while Z contributions remain suppressed; the model can accommodate large CP-violating effects such as S_ψφ and A_SL^s, and predicts a strong RS–Froggatt–Nielsen correspondence with explicit analytic parameterisations. Overall, the paper provides a comprehensive framework for RS flavour phenomenology, clarifying where new physics can alleviate tensions in ΔF=2 observables and identifying naturalness bounds on the KK scale relevant for current and near-future experiments.

Abstract

We present a complete study of Delta S = 2 and Delta B = 2 processes in a warped extra dimensional model with a custodial protection of Z b_L bar b_L, including epsilon_K, Delta M_K, Delta M_s, Delta M_d, A^q_SL, Delta Gamma_q, A_CP(B_d -> psi K_S) and A_CP(B_s -> psi phi). These processes are affected by tree level contributions from Kaluza-Klein gluons, the heavy KK photon, new heavy electroweak gauge bosons Z_H and Z', and in principle by tree level Z contributions. We confirm recent findings that the fully anarchic approach where all the hierarchies in quark masses and weak mixing angles are geometrically explained seems implausible and we confirm that the KK mass scale M_KK generically has to be at least ~20TeV to satisfy the epsilon_K constraint. We point out, however, that there exist regions in parameter space with only modest fine-tuning in the 5D Yukawa couplings which satisfy all existing Delta F = 2 and electroweak precision constraints for scales M_KK ~3TeV in reach of the LHC. Simultaneously we find that A_CP(B_s -> psi phi) and A^s_SL can be much larger than in the SM as indicated by recent results from CDF and D0 data. We point out that for B_{d,s} physics Delta F = 2 observables the complex (Z_H,Z') can compete with KK gluons, while the tree level Z and KK photon contributions are very small. In particular we point out that the Z d^i_L bar d^j_L couplings are protected by the custodial symmetry. As a by-product we show the relation of the RS flavour model to the Froggatt-Nielsen mechanism and we provide analytic formulae for the effective flavour mixing matrices in terms of the fundamental 5D parameters.

Figures (9)

• Figure 1: Tree level contribution of KK gluons to $K^0-\bar{K}^0$ mixing.
• Figure 2: left: $\mathop{\mathrm{Re}}\nolimits(M^K_{12})_\text{KK}/\mathop{\mathrm{Re}}\nolimits(M^K_{12})_\text{SM}$ and $\mathop{\mathrm{Im}}\nolimits(M^K_{12})_\text{KK}/\mathop{\mathrm{Im}}\nolimits(M^K_{12})_\text{SM}$, plotted on logarithmic axes. right: $\mathop{\mathrm{Re}}\nolimits(M^s_{12})_\text{KK}$ and $\mathop{\mathrm{Im}}\nolimits(M^s_{12})_\text{KK}$, normalised to $|(M^s_{12})_\text{SM}|$ and plotted on logarithmic axes.
• Figure 3: The ratio of the contribution of only $\mathcal{Q}_{LR}$ and only $\mathcal{Q}_{LL}$ to $(M^K_{12})_\text{KK}$ (left) and $(M^s_{12})_\text{KK}$ (right), as a function of $(M^i_{12})_\text{KK}/(M^i_{12})_\text{SM}$ ($i=K,s$).
• Figure 4: left: The fine-tuning $\Delta_\text{BG}(\varepsilon_K)$ plotted against $\varepsilon_K$, normalised to its experimental value. The blue line displays the average fine-tuning as a function of $\varepsilon_K$. right: The same, but displaying only the phenomenologically interesting region $0.1 < |\varepsilon_K/(\varepsilon_K)_\text{exp}|<10$.
• Figure 5: The fine-tuning $\Delta_\text{BG}(\Delta M_K)$ plotted against $\Delta M_K$, normalised to its experimental value. The blue line displays the average fine-tuning as a function of $\Delta M_K$.