Real and Complex Singlet-Scalar Benchmarks with a Vector-Like Down Quark for $B\to X_s\gamma$ and $B_s-{\bar{B}}_s$ Mixing

We study a simple extension of the Standard Model with a vector-like down-type quark $D$ and a neutral singlet scalar $\mathcal{S}=S_R,\mathrm{\Phi }$. The scalar is considered in two forms, a real field $S_R=S_R^{†}$ and a complex field $\mathrm{\Phi }\mathrm{\ne }{\mathrm{\Phi }}^{†}$. The interaction $-{\lambda }_i\mathcal{S}{\bar{D}}_L{d}_{Ri}+\mathrm{h}\mathrm{.}\mathrm{c}\mathrm{.}$ generates the radiative transitions $b\to s\gamma$ and $b\to sg$ at one loop. Since $\mathcal{S}$ has no electric charge or color, the gauge boson is emitted from the internal $D$ line, giving $C_{7\gamma }^{\mathrm{N}\mathrm{P}}\mathrm{/}{C}_{8G}^{\mathrm{N}\mathrm{P}}=Q_D=-1\mathrm{/}3$ at the matching scale for $\mathrm{\Delta }B=1$ dipole transitions. For $M_D=m_{\mathcal{S}}=1\mathrm{T}\mathrm{e}\mathrm{V}$ and $\mathrm{\mid }{\lambda }_s^{\ast }{\lambda }_b\mathrm{\mid }=1$, the low scale contribution is $\mathrm{\mid }{C}_{7\gamma }^{\mathrm{N}\mathrm{P},\mathrm{e}\mathrm{f}\mathrm{f}}({\mu }_b)\mathrm{\mid }\simeq 1.1\times {10}^{-3}$, This is about $0.4\mathrm{\%}$ of the Standard Model value $\mathrm{\mid }{C}_{7\gamma }^{\mathrm{S}\mathrm{M},\mathrm{e}\mathrm{f}\mathrm{f}}({\mu }_b)\mathrm{\mid }\simeq 0.30$. 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 $\mathrm{\mid }{\lambda }_s^{\ast }{\lambda }_b\mathrm{\mid }$ at the level of a few tenths for TeV-scale masses. Thus, in these minimal benchmarks, $B\to X_s\gamma$ is radiatively safe, while $B_s-{\bar{B}}_s$ mixing gives the stronger constraint in the complex-scalar $\mathrm{\Phi }$ benchmark.