Reconstruction of heavy quark current correlators at O(α_s^3)

TL;DR

The paper develops a Padé-approximation framework, augmented by conformal mapping and careful subtraction of known logarithmic pieces, to reconstruct the full energy dependence of heavy-quark current correlators at O(αs^3) across vector, axial-vector, scalar, and pseudoscalar channels. It integrates low-energy, threshold, and high-energy expansions as input constraints and uses rigorous criteria to discard unphysical approximants, yielding predictions for additional expansion coefficients with quantified uncertainties. The approach achieves high accuracy for the low-energy coefficients and provides improved estimates for the other regions, with important implications for QCD sum rules and heavy-quark phenomenology. Overall, the method demonstrates that sparse high-order information can be extrapolated to full energy behavior in a controlled, uncertainty-quantified way.

Abstract

We construct approximate formulas for the O(α_s^3) QCD contributions to vector, axial-vector, scalar and pseudo-scalar quark current correlators, which are valid for arbitrary values of momenta and masses. The derivation is based on conformal mapping and the Pade approximation procedure and incorporates known expansions in the low energy, threshold and high energy regions. We use our results to estimate additional terms in these expansions.

Figures (4)

• Figure 1: Conformal mapping of the complex plane onto the unit circle. The points $z=0$ and $z=1$ correspond to $\omega=0$ and $\omega=1$, respectively. $z\pm \infty$ goes to $\omega=-1$. The branch cut starting at $z=1$ is mapped onto the perimeter of the circle.
• Figure 2: Imaginary part of the four loop contributions to the vector, pseudo--scalar, axial-vector and scalar polarisation functions above the charm threshold. The plots show $v R^{(3),v}=v 12\pi \text{Im}(\Pi^{(3),v})$, $v R^{(3),p}=v 8\pi \text{Im}(\Pi^{(3),p})$, $R^{(3),a}=12\pi \text{Im}(\Pi^{(3),a})$ and $R^{(3),s}=8\pi \text{Im}(\Pi^{(3),s})$ as functions of $v=\sqrt{1-1/z}$. The solid black line is the mean from all approximants, the area covered by three standard deviations is shown by bands. The dashed lines show the expansions in the threshold and high energy regions (see Eqs. \ref{['eq:15']} and \ref{['eq:18']}).
• Figure 3: Real part of the four loop contribution to the vector polarisation function for the case of charm quarks. On the left hand side the region below threshold is shown, on the right hand side the behaviour above threshold is plotted as a function of $v=\sqrt{1-1/z}$. The dashed lines show the known expansions in the respective regions (see Eqs. \ref{['eq:15']} and \ref{['eq:22']}), the solid lines are the mean values of all approximants. The shaded areas show the variation given by three standard deviations. In order to obtain finite values at threshold, $\text{Re}\,\Pi^{(3),v}$ is plotted with an extra factor $1-z$ below and a factor $v^2$ above threshold.
• Figure 4: Distribution of the values of $C_4^{(3),v}$ in on-shell scheme from different Padé approximants to the charm vector correlator.