Leptonic Decay of the $\mathrm{\Upsilon }(1S)$ Meson at Third Order in QCD

We present the complete next-to-next-to-next-to-leading order short-distance and bound-state QCD correction to the leptonic decay rate $\mathrm{\Gamma }\mathbf{(}\mathrm{\Upsilon }(1S)\to {\ell }^{+}{\ell }^{-}\mathbf{)}$ of the lowest-lying spin-1 bottomonium state. The perturbative QCD prediction is compared to the measurement $\mathrm{\Gamma }\mathbf{(}\mathrm{\Upsilon }(1S)\to e^{+}{e}^{-}\mathbf{)}=1.340(18)\mathrm{keV}$.

Figure 1

The decay rate in the PS scheme as a function of the renormalization scale $\mu$. Dotted (red), dash-dotted (green), short-dashed (blue), and solid (black) lines correspond to LO, NLO, NNLO, and ${\mathrm{N}}^3\mathrm{LO}$ prediction.

Figure 2

The decay rate as a function of ${\alpha }_s(M_Z)$ at LO (red, bottom), NLO (green, middle), NNLO (blue, top), and ${\mathrm{N}}^3\mathrm{LO}$ (black, inner top band). The bands denote the variation of $\mu$ between 3 and 10 GeV. The horizontal bar denotes the experimental value, while the vertical bar denotes the world average of the strong coupling constant, ${\alpha }_s(M_Z)=0.1184(10)$.