Effects of renormalon scheme and perturbative scale choices on determinations of the strong coupling from $e^{+}{e}^{-}$ event shapes

We study the role of renormalon cancellation schemes and perturbative scale choices in extractions of the strong coupling constant ${\alpha }_s(m_Z)$ and the leading nonperturbative shift parameter ${\mathrm{\Omega }}_1$ from resummed predictions of the $e^{+}{e}^{-}$ event shape thrust. We calculate the thrust distribution to $\mathrm{N}{}^3\mathrm{L}{\mathrm{L}}^{\prime }$ resummed accuracy in soft-collinear effective theory (SCET) matched to the fixed-order $\mathcal{O}({\alpha }_s^2)$ prediction, and perform a new high-statistics computation of the $\mathcal{O}({\alpha }_s^3)$ matching in EERAD3, although we do not include the latter in our final ${\alpha }_s$ fits due to some observed systematics that require further investigation. We are primarily interested in testing the phenomenological impact sourced from varying amongst three renormalon cancellation schemes and two sets of perturbative scale profile choices. We then perform a global fit to available data spanning center-of-mass energies between 35–207 GeV in each scenario. Relevant subsets of our results are consistent with prior SCET-based extractions of ${\alpha }_s(m_Z)$, but we are also led to a number of novel observations. Notably, we find that the combined effect of altering the renormalon cancellation scheme and profile parameters can lead to few-percent-level impacts on the extracted values in the ${\alpha }_s-{\mathrm{\Omega }}_1$ plane, indicating a potentially important systematic theory uncertainty that should be accounted for. We also observe that fits performed over windows dominated by dijet events are typically of a higher quality than those that extend into the far tails of the distributions, possibly motivating future fits focused more heavily in this region. Finally, we discuss how different estimates of the three-loop soft matching coefficient $c_{\tilde{S}}^3$ can also lead to measurable changes in the fitted $\{{\alpha }_s,{\mathrm{\Omega }}_1\}$ values.

Figure 1

Differential remainder function $r^3(\tau )/(2\pi {)}^3$ as extracted from EERAD3. The data points are the EERAD3 data, and the red line is a combined interpolation for large $\tau$ and a fit to a basis of logarithmic functions for small $\tau$ as explained in the text.

Figure 2

The effective shift (24) of the cross section due to different renormalon cancellation and perturbative scale profile schemes, evaluated using “central” profiles at $Q=m_Z$. The flat, dashed line corresponds to the constant shift of (1), while the vertical gray lines correspond to the two fitting windows discussed below. All functions are calculated at $\mathrm{N}{}^3\mathrm{L}{\mathrm{L}}^{\prime }+\mathcal{O}({\alpha }_s^2)$ accuracy.

Figure 3

The 2018 and 2010 profiles we implement. The top left figure compares the central values for ${\mu }_{H,J,S,R}$, while the top right figure shows the nonsingular scale ${\mu }_{\mathrm{ns}}$. Bottom plots show the variations of the 2018 and 2010 profiles from a random scan of parameters in the ranges shown in Table 1 (as ${\mu }_H$ controls the overall scale of all profiles, its variation is indicated by the black arrows). In the bottom plots ${\mu }_S<{\mu }_J$ is realized for any given set of scales, despite the fact that the overall bands overlap. The ${\mathrm{R}}^{\star }$ scales from (29) are shown in all figures, and all plots correspond to $Q=m_Z$.

Figure 4

Top Row: Normalized thrust distribution across the $\tau$ domain relevant for the ${\alpha }_s$ extractions in the R scheme at different resummed and matched accuracies as indicated by the colors. The plots are generated with 64 variations of the embedded profile parameters, with 2018 (2010) profiles shown in the left (right) column. Bottom Row: The same for the ${\mathrm{R}}^{\star }$ scheme.

Figure 5

Comparison of the differential distributions shown in Fig. 4 for four renormalon-cancellation and profile-variation schemes, normalized to the central profile of the ${\mathrm{R}}_{2010}$ scheme.

Figure 6

Extractions of $\{{\alpha }_s,{\mathrm{\Omega }}_1\}$ in the ${\mathrm{R}}_{2018}$ scheme (top panel) and the ${\mathrm{R}}_{2018}^{\star }$ scheme (bottom panel) for different resummed and matched accuracies as indicated by the colors. Here $Q=m_Z$, and we note that we have restricted ${\mathrm{\Omega }}_1\ge \mathrm{\Delta }(R_{\mathrm{\Delta }},R_{\mathrm{\Delta }})=0.1\mathrm{GeV}$ in these scans.

Figure 7

Extractions of $\{{\alpha }_s,{\mathrm{\Omega }}_1\}$ using $\mathrm{N}{}^3\mathrm{L}{\mathrm{L}}^{\prime }+\mathcal{O}({\alpha }_s^2)$ theory predictions fitted to global thrust data in our default fitting window, $6/Q\le \tau \le 0.33$. Upper panels compare the systematic impact of varying 2010 vs 2018 profile scales within a given renormalon cancellation scheme, while bottom panels vary renormalon schemes within a given set of profile functions. Dashed (solid) ellipses correspond to 68% (95%) CL. Note that ${\mathrm{\Omega }}_1$ is formally defined differently in the R and ${\mathrm{R}}^{*}$ schemes.

Figure 8

Result of a global fit to thrust data using $\mathrm{N}{}^3\mathrm{L}{\mathrm{L}}^{\prime }+\mathcal{O}({\alpha }_s^2)$ theory predictions and a fitting window $6/Q\le \tau \le 0.33$. The plots show the ${\alpha }_s-{\mathrm{\Omega }}_1$ plane (left) and ${\alpha }_s-{\chi }_{\mathrm{dof}}^2$ plane (right) for four different renormalon-cancellation and profile-variation schemes. The ellipses in the left panel correspond to 95% CL.

Figure 9

The same as in Fig. 8 for the default fit window with $6/Q\le \tau \le 0.33$ (solid contours) and a reduced dijet fit window with $6/Q\le \tau \le 0.225$ (dashed contours).

Figure 10

Comparison of two $\{{\alpha }_s,{\mathrm{\Omega }}_1\}$ extractions that either use the EERAD3 value $c_{\tilde{S}}^3=-19988\pm 5440$ (red) or the Padé approximant $c_{\tilde{S}}^3=691\pm 1000$ (brown).

Figure 11

The same as in Fig. 4 for the ${\mathrm{R}}^0$ scheme. The upper plots use the EERAD3 value for $c_{\tilde{S}}^3$ from (9), and the lower plots the Padé approximant given in (10). Left (right) plots refer to 2018 (2010) profile functions.

Figure 12

Extraction results for the ${\mathrm{R}}^0$ schemes, in analogue to those from Figs. 6, 7, 8, 9. The top-left panel represents the convergence of $\{{\alpha }_s,{\mathrm{\Omega }}_1\}$ fits at various logarithmic accuracies, and with $Q=m_Z$ data. The top-right panel represents our global $\{{\alpha }_s,{\mathrm{\Omega }}_1\}$ extractions in these schemes. The bottom panels present the ${\mathrm{R}}^0$ fits relative to ${\mathrm{R}}^{(\star )}$ results. See the text for further discussion.

Figure 13

$\{{\alpha }_s,{\mathrm{\Omega }}_1\}$ (Top) and $\{{\alpha }_s,{\chi }_{\mathrm{d}.\mathrm{o}.\mathrm{f}}^2\}$ (Bottom) fits for the $R^0$ schemes, comparing results from global $Q$ data and default $\tau$ windows (solid lines) to those with $Q=m_Z$ data restricted to a central $\tau$ region (dashed lines).