On the determination of alpha_s from hadronic tau decays with contour-improved, fixed order and renormalon-chain perturbation theory

TL;DR

This work analyzes how three perturbative frameworks—FOPT, CIPT, and RCPT—affect alpha_s extraction from hadronic tau decays. By introducing a Generalized FOPT approach that shifts the expansion point on the contour, it shows that logarithmic effects dominate the uncertainty and that CIPT remains the more reliable method. It demonstrates that RCPT discrepancies can be significantly reduced through 2-loop scheme matching, leading to convergence among the approaches when updated input (K4) is used. The resulting alpha_s is evolved to the Z mass, yielding alpha_s(m_Z^2) ≈ 0.1213 with modest theoretical uncertainty, reinforcing higher tau-based determinations and resolving previous method-based spreads.

Abstract

One of the largest theoretical uncertainties assigned to the strong coupling constant alpha_s as determined from hadronic tau decays stems from the differences in the results for Fixed Order Perturbation Theory (FOPT), Contour Improved Perturbation Theory (CIPT) and Renormalon Chain Perturbation Theory (RCPT). It is often argued that the three methods differ in the treatment of higher orders only and therefore the full difference should be treated as theoretical error. Recently other arguments either in favor of FOPT, CIPT or RCPT have been given, but none of those is able to combine all three to a single value in the strong coupling constant. In this note I will show that FOPT alone has a much larger uncertainty than previously assumed and therefore agrees within error with CIPT. Furthermore a more appropriate matching of the different schemes used in RCPT reduces the difference to the CIPT result by a factor of 6. Together with recently published results for the 4th order term K_4 this reduces the theoretical error on alpha_s by a factor of 2.5 compared to the previously assumed spread of the three perturbative approaches.

Figures (5)

• Figure 1: Quality of the Taylor expansion of $\mathrm{\alpha_s}(m_\tau^2 \exp(i\varphi))$. The two plots show the absolute value $|\mathrm{\alpha_s}|$ on the complex circle $s=m_\tau^2 \exp(i\varphi)$. The left plot shows with long-dashed, dash-dotted, narrow dotted and wide dotted lines the Taylor (FOPT) expansion up to 5th, 4th, 3rd, and 2nd order, respectively. The 4-loop result for the numerically solved $\beta$-function (CIPT) is drawn as solid line for comparison. The right plot shows with dahsed, dotted, and dash-dotted lines the numerical solutions for 3, 2 and 1 loop $\beta$-functions, respectively. Again the same 4-loop result as in the left plot is shown as a solid line. The reference value $\mathrm{\alpha_s}(m_\tau^2) = 0.35$ was used for all curves.
• Figure 2: $\delta_{\mathrm{pert}}$ as function of the development point $\varphi_0$. The solid, medium-dashed and short-dashed lines show the CIPT result to 5th order for $K_5 = 400$, $0$, and $800$, respectively. The long-dashed, dash-dotted and dotted lines show the generalized FOPT result to 5th order for $K_5 = 400$, $0$, and $800$, respectively. The reference value $\mathrm{\alpha_s}(m_\tau^2) = 0.35$ was used for all curves.
• Figure 3: $\delta_{\mathrm{pert}}$ as function of the development point $\varphi_0$. The solid, medium-dashed and short-dashed lines show the CIPT result to 5th order for $K_5 = 400$, $0$, and $800$, respectively. The long-dashed, dash-dotted and dotted lines show the RCPT result with 0-loop matching and generalized FOPT correction to 5th order for $K_5 = 400$, $0$, and $800$, respectively. The reference value $\mathrm{\alpha_s}(m_\tau^2) = 0.35$ was used for all curves.
• Figure 4: $\delta_{\mathrm{pert}}$ as function of the development point $\varphi_0$. The solid, medium-dashed and short-dashed lines show the CIPT result to 5th order for $K_5 = 400$, $0$, and $800$, respectively. The long-dashed, dash-dotted and dotted lines show the RCPT result with 2-loop matching and generalized FOPT correction to 5th order for $K_5 = 400$, $0$, and $800$, respectively. The reference value $\mathrm{\alpha_s}(m_\tau^2) = 0.35$ was used for all curves.
• Figure 5: $\mathrm{\alpha_s}(m_\tau^2)$ as function of the development point $\varphi_0$. The solid, medium-dashed and short-dashed lines show the CIPT result to 5th order for $K_4 = 400$, $0$, and $800$, respectively. The long-dashed, dash-dotted and dotted lines show the RCPT result with 2-loop matching and generalized FOPT correction to 5th order for $K_5 = 400$, $0$, and $800$, respectively. All curves are obtained with $\delta_{\rm pert} = 0.21179$ as reference value.