MS Versus Pole Masses of Gauge Bosons II: Two-Loop Electroweak Fermion Corrections

TL;DR

This work completes the Standard Model two-loop evaluation of the gauge-boson pole masses by incorporating fermion-loop and mixed corrections, establishing a gauge-invariant and infrared-finite framework. It combines the pole-mass definition with MS-bar renormalization, detailed renormalization (including tadpoles), and a comprehensive master-integral program for massless fermion contributions, aided by Tarasov reductions and hypergeometric ε-expansions. The authors verify RG consistency with the broken-to-unbroken phase relations and provide both analytic and numerical results, showing two-loop fermionic effects are small but non-negligible for high-precision electroweak tests. Their analysis yields practical insight into the MS-bar to pole-mass relations and demonstrates the robustness of the MS-bar scheme for unstable particles, with implications for precision fits and Δr-type observables.

Abstract

We have calculated the fermion contributions to the shift of the position of the poles of the massive gauge boson propagators at two-loop order in the Standard Model. Together with the bosonic contributions calculated previously the full two-loop corrections are available. This allows us to investigate the full correction in the relationship between MS and pole masses of the vector bosons Z and W. Two-loop renormalization and the corresponding renormalization group equations are discussed. Analytical results for the master-integrals appearing in the massless fermion contributions are given. A new approach of summing multiple binomial sums has been developed.

Figures (11)

• Figure 1: The prototype diagrams and their subgraphs contributing to the large mass expansion for two-loop diagrams with heavy propagators. Thick and thin lines correspond to heavy- and light-mass (massless) particle propagators, respectively. Dotted lines indicate the lines omitted in the subgraph.
• Figure 2: Diagrams corresponding to the master--integrals with massless fermion-loops contributing to the two-loop gauge-boson propagators. Bold, thin and dashed lines correspond to off-shell massive, on-shell massive and to massless propagators, respectively.
• Figure 3: Diagrams contributing to the two-loop on-shell top-quark propagator in the limit of massless gauge-bosons. Bold, thin and dashed lines correspond to off-shell massive, on-shell massive and to massless propagators, respectively.
• Figure 4: The dependence on the number of coefficients of the expansion with respect to $\sin^2 \theta_W$ of the two--loop corrections $\delta \equiv \Biggl \{ \Pi_Z^{(2)} + \Pi_Z^{(1)} \Pi_Z^{(1)}{}' \Biggr\}_{\overline{\rm MS}}$ (see \ref{['MS2:subtracted']}) as a function of the Higgs mass. The dotted, dashed, dot-dashed and full lines show results obtained with the first one, two, three and all calculated (six) coefficients, respectively. Upper plot: for intermediate Higgs masses. Lower plot: for heavy Higgs masses.
• Figure 5: The dependence on the number of coefficients of the expansion with respect to $m^2_Z/m^2_i$ ($i=H,t$) used for the evaluation of the two-loop corrections. We show $\delta \equiv \Biggl \{ \Pi_Z^{(2)} + \Pi_Z^{(1)} \Pi_Z^{(1)}{}' \Biggr\}_{\overline{\rm MS}}$ (see \ref{['MS2:subtracted']}) as a function of the Higgs mass. The dotted, dashed, dot-dashed and full lines show results obtained with the first one, two, three and all calculated (six) coefficients, respectively. Upper plot: for intermediate Higgs masses. Lower plot: for heavy Higgs masses.