Precise determination of ${\alpha }_s$ from the $C$-parameter distribution

We present a global fit for ${\alpha }_s(m_Z)$, analyzing the available $C$-parameter data measured at center-of-mass energies between $Q=35$ and 207 GeV. The experimental data is compared to a ${\mathrm{N}}^3{\mathrm{LL}}^{\prime }+\mathcal{O}({\alpha }_s^3)+{\mathrm{\Omega }}_1$ theoretical prediction (up to the missing four-loop cusp anomalous dimension), which includes power corrections coming from a field-theoretical nonperturbative soft function. The dominant hadronic parameter is its first moment ${\mathrm{\Omega }}_1$, which is defined in a scheme which eliminates the $\mathcal{O}({\mathrm{\Lambda }}_{\mathrm{QCD}})$ renormalon ambiguity. The resummation region plays a dominant role in the $C$-parameter spectrum, and in this region a fit for ${\alpha }_s(m_Z)$ and ${\mathrm{\Omega }}_1$ is sufficient. We find ${\alpha }_s(m_Z)=0.1123\pm 0.0015$ and ${\mathrm{\Omega }}_1=0.421\pm 0.063\mathrm{GeV}$ with ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.=0.988$ for 404 bins of data. These results agree with the prediction of universality for ${\mathrm{\Omega }}_1$ between thrust and $C$-parameter within 1-$\sigma$.

Figure 1

Bands for the profile functions for the renormalization scales ${\mu }_H$, ${\mu }_J(C)$, ${\mu }_S(C)$ when varying the profile parameters.

Figure 2

Plots of the $r$ dependence of $g_C(r){\mathrm{\Omega }}_1(R_{\mathrm{\Delta }},{\mu }_{\mathrm{\Delta }},r)$ for different values of $\theta (R_{\mathrm{\Delta }},{\mu }_{\mathrm{\Delta }})$. We normalize to ${\mathrm{\Omega }}_1^C(R_{\mathrm{\Delta }},{\mu }_{\mathrm{\Delta }})$, since it is simply an overall multiplicative factor.

Figure 3

Difference between the default cross section and the cross section varying only one parameter. We vary ${\alpha }_s(m_Z)$ by $\pm 0.001$ (solid red), $2{\mathrm{\Omega }}_1$ by $\pm 0.1$ (dashed blue) and ${\mathrm{\Omega }}_2^C$ by $\pm 0.5$ (dotted green). The three plots correspond to three different center-of-mass energies: (a) $Q=35\mathrm{GeV}$, (b) $Q=91.2\mathrm{GeV}$, (c) $Q=206\mathrm{GeV}$.

Figure 4

The evolution of the value of ${\alpha }_s(m_Z)$ adding components of the calculation. An additional $\sim 8\%$ uncertainty from not including power corrections is not included in the two left points.

Figure 5

The first two panels show the distribution of best-fit points in the ${\alpha }_s(m_Z)\text{−}2{\mathrm{\Omega }}_1$ and ${\alpha }_s(m_Z)\text{−}2{\overline{\mathrm{\Omega }}}_1$ planes. Panel (a) shows results including perturbation theory, resummation of large logs, the soft nonperturbative function and ${\mathrm{\Omega }}_1$ defined in the Rgap scheme with renormalon subtractions. Panel (b) shows the results as in panel (a), but with ${\overline{\mathrm{\Omega }}}_1$ defined in the $\overline{\mathrm{MS}}$ scheme without renormalon subtractions. In both panels the dashed lines corresponds to an ellipse fit to the contour of the best-fit points to determine the theoretical uncertainty. The respective total ($\text{experimental}+\text{theoretical}$) 39% C.L. standard uncertainty ellipses are displayed (solid lines), which correspond to $1\text{−}\sigma$ (68% C.L.) for either one-dimensional projection. The big points represent the central values in the random scan for ${\alpha }_s(m_Z)$ and $2{\mathrm{\Omega }}_1$. Likewise, the two panels at the bottom show the distribution of best-fit points in the ${\alpha }_s(m_Z)\text{−}{\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ plane. Panel (c) shows the ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ values of the points given in panel (a), whereas panel (b) shows the ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ values of the points given in panel (b).

Figure 6

Experimental $\mathrm{\Delta }{\chi }^2=1$ standard uncertainty ellipse (dotted green) at ${\mathrm{N}}^3{\mathrm{LL}}^{\prime }$ accuracy with renormalon subtractions, in the ${\alpha }_s\text{−}2{\mathrm{\Omega }}_1$ plane. The dashed blue ellipse represents the theory uncertainty which is obtained by fitting an ellipse to the contour of the distribution of the best-fit points. This ellipse should be interpreted as the $1\text{−}\sigma$ theory uncertainty for one parameter (39% confidence for two parameters). The solid red ellipse represents the total (combined experimental and perturbative) uncertainty ellipse.

Figure 7

Variations of the best-fit values for ${\alpha }_s(m_Z)$ and ${\mathrm{\Omega }}_1$ from up (dark blue) and down (light green) variations for the theory parameters according to Table 1. We do not display those parameters which do not affect the fit region (${\epsilon }_2^{\text{high}}$, ${\epsilon }_3^{\text{high}}$, ${\mu }_0$, $R_0$, $n_0$).

Figure 8

Comparison of fits to the $C$-parameter tail distribution with theory prediction which include/ignore hadron-mass effects (in red/blue). Although a direct comparison of ${\alpha }_s$ values is possible, one has to keep in mind that ${\mathrm{\Omega }}_1({\mu }_{\mathrm{\Delta }},R_{\mathrm{\Delta }})$ has a different meaning once hadron-mass running effects are included.

Figure 9

Global fit results for different choices of data set, using our best theory setup at ${\mathrm{N}}^3{\mathrm{LL}}^{\prime }$ with power corrections in the Rgap scheme. Considering the central values from left to right, the data sets read $[C_{\min },C_{\max }{]}_{\#\text{of}\text{bins}}$: $[29/Q,0.7{]}_{371}$, $[22/Q,0.75{]}_{453}$, $[23/Q,0.7{]}_{417}$, $[0.24,0.75{]}_{403}$, $[24/Q,0.7{]}_{409}$, $[25/Q,0.7{]}_{404}$ (default), $[25/Q,0.6{]}_{322}$, $[25/Q,0.75{]}_{430}$, $[27/Q,0.7{]}_{386}$, $[25/Q,0.65{]}_{349}$, $[22/Q,0.7{]}_{427}$. We accept bins which are at least 50% inside these fit regions. The ellipses correspond to total $1\text{−}\sigma$ uncertainties ($\text{experimental}+\text{theory}$) for two variables (${\alpha }_s$ and ${\mathrm{\Omega }}_1$), which are suitable for a direct comparison of the outcome of two-parameter fits. The center of the ellipses are also shown.

Figure 10

$C$-parameter distribution at ${\mathrm{N}}^3{\mathrm{LL}}^{\prime }$ order for $Q=m_Z$ showing the fit result for the values for ${\alpha }_s(m_Z)$ and ${\mathrm{\Omega }}_1$. The blue band corresponds to the theory uncertainty as described in Sec. 5b. Experimental data is also shown.

Figure 11

$C$-parameter distribution below the fit region, shown at ${\mathrm{N}}^3{\mathrm{LL}}^{\prime }$ order for $Q=m_Z$ using the best-fit values for ${\alpha }_s(m_Z)$ and ${\mathrm{\Omega }}_1$. Again the blue band corresponds to the theory uncertainty and error bars are used for experimental data.

Figure 12

$C$-parameter distribution above the fit range, shown at ${\mathrm{N}}^3{\mathrm{LL}}^{\prime }$ order for $Q=m_Z$ using the best-fit values for ${\alpha }_s(m_Z)$ and ${\mathrm{\Omega }}_1$. Again the blue band corresponds to the theory uncertainty and the error bars are used for experimental data.

Figure 13

Comparison of determinations of ${\alpha }_s(m_Z)$ and ${\mathrm{\Omega }}_1$ with the corresponding total 1-$\sigma$ uncertainty ellipses. As an illustration we display the determination of ${\mathrm{\Omega }}_1^C$ obtained from fits to the $C$-parameter distribution (green), which is clearly different from ${\mathrm{\Omega }}_1^{\tau }$ obtained from thrust fits (blue), and the determination of ${\mathrm{\Omega }}_1^{\tau }$ as obtained from $C$-parameter distribution fits (red). All fits have been performed with ${\mathrm{N}}^3{\mathrm{LL}}^{\prime }$ theoretical predictions with power corrections and in the Rgap scheme. The dashed vertical lines indicate the PDG 2014 [23] determination of ${\alpha }_s(m_Z)$.

Figure 14

Distribution of best-fit points in the ${\alpha }_s(m_Z)$-$2{\mathrm{\Omega }}_1$ plane for both thrust (blue) and $C$-parameter (red) at ${\text{N}}^3{\mathrm{LL}}^{\prime }+\mathcal{O}({\alpha }_s^3)+{\mathrm{\Omega }}_1(R,\mu )$. The outer solid ellipses show the $\mathrm{\Delta }{\chi }^2=2.3$ variations, representing $1\text{−}\sigma$ uncertainties for two variables. The inner dashed ellipses correspond to the $1\text{−}\sigma$ theory uncertainties for each one of the fit parameters. The dotted ellipses correspond to $\mathrm{\Delta }{\chi }^2=1$ variations of the total uncertainties. All fits have been performed with ${\mathrm{N}}^3{\mathrm{LL}}^{\prime }$ theoretical predictions with power corrections and in the Rgap scheme. This plot zooms in on the bottom two ellipses of Fig. 13.

Figure 15

Distribution of best-fit points in the ${\alpha }_s(m_Z)$-$2{\overline{\mathrm{\Omega }}}_1$ plane for both thrust (blue) and $C$-parameter (red) at ${\text{N}}^3{\mathrm{LL}}^{\prime }+\mathcal{O}({\alpha }_s^3)+{\overline{\mathrm{\Omega }}}_1$. The meaning of the different ellipses is the same as in Fig. 14.

Figure 16

Comparison of our determination of ${\alpha }_s(m_Z)$ (red) with similar analyses from thrust [9, 10] (green) and other determinations from fits to event-shape distributions using $\mathcal{O}({\alpha }_s^3)$ theory predictions and different levels of resummation. Results shown below the lower dashed line include power corrections as predicted by MC generators, and results above this line treat power corrections either from a shape function (red and green) or from the dispersive model (orange [4] and purple [11]). Determinations above the upper dashed line correspond to fits to moments of the distributions, and those below to fits to the tail of the differential distribution. The translucent green band corresponds to the world average from Ref. [23].

Figure 17

Comparison of our determination of ${\alpha }_s(m_Z)$ (red) with similar previous analyses at ${\mathrm{N}}^3{\mathrm{LL}}^{\prime }$ for thrust (green) [9, 10] and other selected determinations: lattice [24] and static energy potential [29] (both use lattice input, in blue), electroweak precision observables fits [87] (black), deep inelastic scattering [28] and global PDF fits [30, 31], and hadron $\tau$ decays [27] [fixed-order perturbation theory (FOPT), and contour-improved perturbation theory (CIPT), both in gray]. The current world average [23] is shown as a translucent green band.

Figure 18

Comparison of ${\alpha }_s$ determinations from $C$-parameter tail fits in the thrust Rgap scheme (lower red ellipse) and the $C$-parameter Rgap scheme (upper blue ellipse). The leading power correction ${\mathrm{\Omega }}_1^C$ in the $C$-parameter Rgap scheme is converted to ${\mathrm{\Omega }}_1$ in the thrust Rgap scheme in order to have a meaningful comparison. Theoretical uncertainty ellipses are shown which are suitable for projection onto one dimension to obtain the $1\text{−}\sigma$ uncertainty, without experimental uncertainties.

Figure 19

Comparison of thrust ${\alpha }_s$ determinations using our new profiles (left red ellipse) and the profiles of Ref. [9] (right blue ellipse). Theoretical uncertainty ellipses are shown which are suitable for projection onto one dimension to obtain the $1\text{−}\sigma$ uncertainty, without experimental uncertainties.