Real and Complex Singlet-Scalar Benchmarks with a Vector-Like Down Quark for $B\to X_sγ$ and $B_s-\bar B_s$ Mixing

Abstract

We study a simple extension of the Standard Model with a vector-like down-type quark $D$ and a neutral singlet scalar ${\cal S}=S_R,Φ$. The scalar is considered in two forms, a real field $S_R=S_R^\dagger$ and a complex field $Φ\neqΦ^\dagger$. The interaction $-λ_i{\cal S}\bar D_L d_{Ri}+{\rm h.c.}$ generates the radiative transitions $b\to sγ$ and $b\to sg$ at one loop. Since ${\cal S}$ has no electric charge or color, the gauge boson is emitted from the internal $D$ line, giving $C_{7γ}^{\rm NP}/C_{8G}^{\rm NP}=Q_D=-1/3$ at the matching scale for $ΔB=1$ dipole transitions. For $M_D=m_{\cal S}=1~{\rm TeV}$ and $|λ_s^\astλ_b|=1$, the low scale contribution is $|C_{7γ}^{\rm NP,eff}(μ_b)|\simeq 1.1\times10^{-3}$, This is about $0.4\%$ of the Standard Model value $|C_{7γ}^{\rm SM,eff}(μ_b)|\simeq 0.30$. We also discuss \(B_s-\bar B_s\) mixing. In the real-scalar case, the direct and crossed box diagrams cancel in the exact minimal limit. In the complex-scalar case, the direct box contribution remains and gives a bound on the flavor $|λ_s^\astλ_b|$ at the level of a few tenths for TeV-scale masses. Thus, in these minimal benchmarks, $B\to X_sγ$ is radiatively safe, while $B_s-\bar B_s$ mixing gives the stronger constraint in the complex-scalar $Φ$ benchmark.