$Z$-boson pole mass at two-loop order in the pure $\overline{\mathrm{MS}}$ scheme

I obtain the complex pole squared mass of the $Z$ boson at full two-loop order in the Standard Model in the pure $\overline{\mathrm{MS}}$ renormalization scheme. The input parameters are the running gauge couplings, the top-quark Yukawa coupling, the Higgs self-coupling, and the vacuum expectation value that minimizes the Landau gauge effective potential. The effects of nonzero Goldstone boson mass are resummed. Within a reasonable range of renormalization scale choices, the scale dependence of the computed pole mass is found to be comparable to the current experimental uncertainty, but the true theoretical error is likely somewhat larger.

Figure 1

The computed pole mass $M_Z$ of the $Z$ boson, defined by $s_{\text{pole}}^Z=M_Z^2-i{\mathrm{\Gamma }}_Z{M}_Z$, as a function of the renormalization scale $Q$ at which it is computed, in various approximations. The dotted (green) line is the tree-level result $Z$, the short-dashed (red) line is the one-loop result, the long-dashed (blue) line is the result from the one-loop and two-loop QCD contribution, while the solid (black) line is the full two-loop order result. The input parameters $g,g^{\prime },y_t,g_3,\lambda$, and $v$ at the renormalization scale $Q$ are obtained by three-loop renormalization group running, starting from Eqs. (3.1)–(3.6). Note that the usual Breit-Wigner mass $M_Z^{\exp }$ is 0.0341 GeV larger than the $M_Z$ shown here.

Figure 2

The computed width ${\mathrm{\Gamma }}_Z$ of the $Z$ boson, defined in terms of the complex pole squared mass $s_{\text{pole}}^Z=M_Z^2-i{\mathrm{\Gamma }}_Z{M}_Z$, as in Figure 1. The short-dashed (red) line is the one-loop result, the long-dashed (blue) line is the result from the one-loop and two-loop QCD contribution, and the solid (black) line is the full two-loop order result.

Figure 3

A close-up of the renormalization scale dependence of the computed pole mass $M_Z$. The solid line is the two-loop result, just as in Fig. 1. The input parameters were chosen so that at $Q=M_Z$ the computed pole mass agrees with the experimental central value $M_Z=91.1535\mathrm{GeV}$ from Eq. (1.3). The dashed line is the same, but after expanding the one-loop part in the $\overline{\mathrm{MS}}$ squared mass $t$ to linear order about the value $T=(173.34\mathrm{GeV}{)}^2$, using Eqs. (3.7) and (3.8). This provides an alternate consistent two-loop order result. The two approximations agree near the scale $Q=77\mathrm{GeV}$ where $t=T$ (the top-quark running and pole masses coincide). Note that the Breit-Wigner mass $M_Z^{\exp }$ is 0.0341 GeV larger than the pole mass $M_Z$ shown here.