Heavy Quark Vacuum Polarization Function at O(alpha_s^2) and O(alpha_s^3)

TL;DR

This work develops a Padé-based reconstruction framework to obtain the full mass and $q^2$ dependence of the heavy-quark vacuum polarization function $\Pi(q^2)$ at ${\cal O}(\alpha_s^2)$ and ${\cal O}(\alpha_s^3)$ by leveraging known expansions in the threshold, high-energy, and small-$q^2$ regimes. Through conformal mapping to $\omega$ and a careful separation into logarithmic and regular parts, the authors determine previously unknown non-logarithmic high-energy terms and small-$q^2 coefficients, including the threshold constant $K^{(2)}$ and higher-order coefficients $C^{(30)}_k$, $H^{(3)}_0$, $H^{(3)}_1$, and $K^{(3)}$. They validate the method on the well-known ${\cal O}(\alpha_s^2)$ case, extracting $K^{(2)}=3.81\pm 0.02$ and confirming cross sections with earlier results, and provide new ${\cal O}(\alpha_s^3)$ information that enables fixed-order moments $M_n$ for $n\ge3$ and improved determinations of charm and bottom masses. Uncertainties are quantified via continuous variations in the logarithmic constructions and Padé choices, with prospects for substantial improvement as more multi-loop data become available. The approach offers a systematic, improvable path to precise heavy-quark vacuum polarization inputs for phenomenology and precision QCD tests.

Abstract

We determine the full mass and $q^2$ dependence of the heavy quark vacuum polarization function $Π(q^2)$ and its contribution to the total $e^+e^-$ cross section at ${\cal O}(α_s^2)$ and ${\cal O}(α_s^3)$ in perturbative QCD. We use known results for the expansions of $Π(q^2)$ at high energies, in the threshold region and around $q^2=0$, conformal mapping and the Padé approximation method. From our results for $Π(q^2)$ we determine numerically at ${\cal O}(α_s^3)$ the previously unknown non-logarithmic contributions in the high-energy expansion at order $(m^2/q^2)^i$ for $i=0,1$ and the coefficients in the expansion around $q^2=0$ at order $q^{2n}$ with $n\ge 2$. We also determine at ${\cal O}(α_s^2)$ the previously unknown ${\cal O}(v^0)$ constant term in the expansion of $Π(q^2)$ in the threshold region, where $v$ is the quark velocity. Our method allows for a quantitative estimate of uncertainties and can be systematically improved once more information in the three kinematic regions becomes available by future multi-loop computations. For the contributions to the total $e^+e^-$ cross section at ${\cal O}(α_s^2)$ we confirm results obtained earlier by Chetyrkin, Kühn and Steinhauser.

Figures (5)

• Figure 1: Results from the reconstructed $\Pi^{(2)}$ function in approximation C for the coefficients $C^{(20)}_{3,4,5,6,7}$ that arise in the expansion around $z=0$, for the coefficients $H^{(2)}_{0,1}$ that occur in the non-logarithmic terms in the high-energy expansion $|z|\to\infty$ and for the constant $K^{(2)}$ that appears in the expansion at threshold around $z=1$. The red solid lines represent the respective exact results known from computation of Feynman diagrams. The dashed blue lines represent the envelope of all results obtained from the reconstructed $\Pi^{(2)}$ functions. The individual error bars represent the range of vales obtained from the reconstructed $\Pi^{(2)}$ functions using one particular Padé approximant $P_{m,n}$. The various types of Padé approximants that have been used are indicated in the upper left panel; the same order is used for all other panels. All results are for $n_f=n_\ell+1=4$ running flavors relevant for charm production.
• Figure 2: Difference of the values for $\bar{C}^{(20)}_k$, determined from the reconstructed $\Pi^{(2)}$ functions based on Taylor-like Padé approximants, and the known exact values $\bar{C}^{(20)}_{k,\rm exact}$. The results are shown for approximation C (green triangles and light greed shaded area), D (red squared symbols and medium red shaded area) and E (blue diamonds and dark blue shaded area). The shaded areas represent the variation of the results due to the changes in the modification factors.
• Figure 3: Results for $12 \pi v \hbox{Im}[\Pi^{(2)}(q^2+i 0)]$ as a function of $v$ for $n_f=4$. The red bands represent the uncertainties. In the left panel the results are based on the reconstructed $\Pi^{(2)}$ function incorporating the coefficients in the small-$z$ expansion up to order $z^2$ (approximation C) and in the right panel the coefficients up to order $z^6$ are accounted for. The dotted and dashed black lines show the expansions in the threshold and the high-energy region up to NNLO. See the text for more details.
• Figure 4: Results from the reconstructed $\Pi^{(3)}$ function for the coefficients $C^{(30)}_{3,4,5,6,7}$ that arise in the expansion around $z=0$, for the coefficients $H^{(3)}_{0,1}$ that occur in the non-logarithmic terms in the high-energy expansion $|z|\to\infty$ and for the constant $K^{(3)}$ that appears in the expansion at threshold around $z=1$. The dashed blue lines represent the envelope of all results obtained from the reconstructed $\Pi^{(3)}$ functions. The individual error bars represent the range of vales obtained from the reconstructed $\Pi^{(3)}$ functions using one particular Padé approximant $P_{m,n}$. The various types of Padé approximants that have been used are indicated in the upper left panel; the same order is used for all other panels. All results are for $n_f=n_\ell+1=4$ running flavors relevant for charm production.
• Figure 5: Result for $12 \pi v \hbox{Im}[\Pi^{(3)}(q^2+i 0)]$ as a function of $v$ for $n_f=4$ using the currently available information for the reconstruction of $\Pi^{(3)}$.The red band represent the uncertainty. The dotted and dashed black lines show the expansions in the threshold and the high-energy region up to next-to-next-to-leading order. See the text for details.