Three Loop Top Quark Contributions to the rho Parameter

TL;DR

This work resolves the three-loop top-quark contributions to the electroweak $\rho$ parameter in the large-$m_t$ limit, focusing on mixed $\mathcal{O}\left( (G_F m_t^2)^2 \alpha_s \right)$ and purely electroweak $\mathcal{O}\left( (G_F m_t^2)^3 \right)$ terms. The authors treat the Higgs-mass dependence with two expansions—around $M_H \approx m_t$ and in the $M_H \gg m_t$ regime—and provide consistent MSbar- and on-shell-mass formulations by combining with the known $M_H=0$ results. They present explicit coefficient functions for $\Delta \rho$ up to $x_t^3$ and $\alpha_s x_t^2$, including cross-checks that demonstrate smooth matching across Higgs-mass regions. The findings indicate that the three-loop corrections are non-negligible for precision electroweak fits, yielding shifts in the $W$-boson mass at the level of a few MeV and impacting inferred Higgs-mass constraints, with particularly sizable $\alpha_s X_t^2$ contributions and comparatively small $X_t^3$ terms. Overall, the work provides a robust, mass-range–covering prediction for $\Delta \rho$ that strengthens precision tests of the Standard Model and Higgs-sector inferences.

Abstract

We present results for the three-loop top quark contributions to the rho parameter in the limit of large top quark mass. The simultaneous dependence on the mass of the Higgs boson M_H and the mass of the top quark m_t is obtained from expansions in the range of M_H around m_t and in the limit M_H >> m_t. In combination with the previous result for M_H = 0 the dependence of the rho parameter on the leading Yukawa contributions, i.e. on m_t and M_H, is well under control for all mass values of practical importance. The effects lead to a shift in the W mass in the order of 5 MeV and are relevant for precision measurements at TESLA.

Figures (5)

• Figure 1: Contributions of order $x_t^3$ to $\Delta \rho$ in the $\overline{\hbox{MS}}$ definition of the top quark mass. The black squares indicate the points where the exact result is known.
• Figure 3: Contributions of order $x_t^2$ to the relation between the $\overline{\hbox{MS}}$ and on-shell top quark mass. The black squares indicate the points where the exact result is known.
• Figure 5: Contributions of order $X_t^3$ to $\Delta \rho$ in the on-shell definition of the top quark mass. The black squares indicate the points where the exact result is known.
• Figure 7: Effect of several contributions to $\Delta \rho$ in the on-shell definition of the top quark mass.
• Figure 8: Shifts in $M_W$ and in $\sin^2 \theta_W$ from the various corrections to $\Delta \rho$.