Heavy Quark Current Correlators to O(α_s^2)

TL;DR

The paper computes heavy-quark current correlators at next-to-next-to-leading order (O(α_s^2)) for vector, axial-vector, scalar, and pseudoscalar currents, incorporating full quark-mass dependence across all kinematics. It adapts a Padé-approximation strategy, using high-energy expansions, threshold behavior, and near-q^2=0 Taylor coefficients to reconstruct the complete q^2-dependence, including non-singlet contributions and mass effects. The results agree with known limits and exact fermionic inputs, providing reliable cross-section and decay-rate predictions and practical formulae for top-pair production and Higgs decays to heavy quarks. Singlet (double-triangle) contributions are deferred to future work, while non-abelian pieces are constrained via a ξ=4 gluonic-double-bubble approach.

Abstract

In this paper the three-loop polarization functions Π(q^2) are calculated for the cases of an external vector, axial-vector, scalar or pseudo-scalar current. Results are presented for the imaginary part which directly leads to the cross section $σ(e^+e^- \to Z \to hadrons)$ and to the Higgs decay rates, respectively.

Figures (7)

• Figure 1: $R^{(2),v}$ and $R^{(2),p}$ plotted against $v$. The dashed curves represent the threshold and the high energy approximations, respectively. Whereas for the vector case also the terms of order $(m^2/s)^6$ are available CheHarKueSte97 for the pseudo-scalar correlator terms of order $(m^2/s)^4$HarSte97 are plotted.
• Figure 2: $R^{(2),v}$ and $R^{(2),p}$ plotted against $x$. The dashed curves represent the high energy approximations including terms up to ${\cal O}(x^{12})$ for the vector correlator and ${\cal O}(x^8)$ for the pseudo-scalar case.
• Figure 3: $R^{(2),a}$ and $R^{(2),s}$ plotted against $v$. The dashed curves represent the threshold and the high energy approximations, respectively. For the axial-vector case terms of order $(m^2/s)^4$ are available CheKue94 and for the scalar correlator terms of order $(m^2/s)^4$HarSte97 are plotted.
• Figure 4: $R^{(2),a}$ and $R^{(2),s}$ plotted against $x$. The dashed curves represent the high energy approximations including terms up to ${\cal O}(x^{4})$ for the axial-vector correlator and ${\cal O}(x^8)$ for the scalar case.
• Figure 5: $R^{(2),v}$ and $R^{(2),p}$ plotted against $v$. The leading threshold terms are subtracted. The dashed lines contain only information up to $C_6$ whereas for the full curves also $C_7$ and $C_8$ is used. The obvious exceptions are represented by the dash-dotted curves. They are described in the text.