Impact of Warped Extra Dimensions on the Dipole Coefficients in $\boldsymbol{b\to s\gamma}$ Transitions

Abstract

We calculate the electro- and chromomagnetic dipole coefficients $C_{7\gamma,8g}$ and $\tilde C_{7\gamma,8g}$ in the context of the minimal Randall-Sundrum (RS) model with a Higgs sector localized on the IR brane using the five-dimensional (5D) approach, where the coefficients are expressed in terms of integrals over 5D propagators. Since we keep the full dependence on the Yukawa matrices, the integral expressions are formally valid to all orders in $v^2/M_{\rm KK}^2$. In addition we relate our results to the expressions obtained in the Kaluza-Klein (KK) decomposed theory and show the consistency in both pictures analytically and numerically, which presents a non-trivial cross-check. In Feynman-'t Hooft gauge, the dominant corrections from virtual KK modes arise from the scalar parts of the $W^\pm$-boson penguin diagrams, including the contributions from the scalar component of the 5D gauge-boson field and from the charged Goldstone bosons in the Higgs sector. The size of the KK corrections depends on the parameter $y_\ast$, which sets the upper bound for the anarchic 5D Yukawa matrices. We find that for $y_\ast\gtrsim1$ the dominant KK corrections are proportional to $y_\ast^2$. We discuss the phenomenological implications of our results for the branching ratio ${\rm Br}(\bar B\to X_s\gamma)$, the time-dependent CP asymmetry $S_{K^*\gamma}$, the direct CP asymmetry $A_{\rm CP}^{b\to s\gamma}$ and the CP asymmetry difference $\Delta A_{\rm CP}^{b\to s\gamma}$. We can derive a lower bound on the first KK gluon resonance of $3.8\,$TeV for $y_\ast=3$, requiring that at least $10\%$ of the RS parameter space covers the experimental $2\sigma$ error margins. We further discuss the branching ratio ${\rm Br}(\bar B\to X_sl^+l^-)$ and compare our predictions for $C_{7\gamma,9,10}$ and $\tilde C_{7\gamma,9,10}$ with phenomenological results derived from model-independent analyses.