$C$-parameter distribution at ${\mathrm{N}}^3{\mathrm{LL}}^{\prime }$ including power corrections

We compute the $e^{+}{e}^{-}$ $C$-parameter distribution using the soft-collinear effective theory with a resummation to next-to-next-to-next-to-leading-log prime accuracy of the most singular partonic terms. This includes the known fixed-order QCD results up to $\mathcal{O}({\alpha }_s^3)$, a numerical determination of the two-loop nonlogarithmic term of the soft function, and all logarithmic terms in the jet and soft functions up to three loops. Our result holds for $C$ in the peak, tail, and far tail regions. Additionally, we treat hadronization effects using a field theoretic nonperturbative soft function, with moments ${\mathrm{\Omega }}_n$. To eliminate an $\mathcal{O}({\mathrm{\Lambda }}_{\mathrm{QCD}})$ renormalon ambiguity in the soft function, we switch from the $\overline{\mathrm{MS}}$ to a short distance “Rgap” scheme to define the leading power correction parameter ${\mathrm{\Omega }}_1$. We show how to simultaneously account for running effects in ${\mathrm{\Omega }}_1$ due to renormalon subtractions and hadron-mass effects, enabling power correction universality between $C$-parameter and thrust to be tested in our setup. We discuss in detail the impact of resummation and renormalon subtractions on the convergence. In the relevant fit region for ${\alpha }_s(m_Z)$ and ${\mathrm{\Omega }}_1$, the perturbative uncertainty in our cross section is $\simeq \hspace{0.17em}2.5\%$ at $Q=m_Z$.

Figure 1

Plot of $\mathrm{d}/\mathrm{d}C\ln [(1/\sigma )\mathrm{d}\sigma /\mathrm{d}C]\simeq h^{\prime }(C)/h(C)$, computed from experimental data at $Q=m_Z$. The derivative is calculated using the central difference with neighboring points.

Figure 2

$\mathcal{O}({\alpha }_s)$ nonsingular $C$-parameter distribution, corresponding to Eq. (33).

Figure 3

$\mathcal{O}({\alpha }_s^2)$ nonsingular $C$-parameter distribution. The solid line shows our reconstruction, whereas dots with error bars correspond to the EVENT2 output with the singular terms subtracted. Our reconstruction consists of fit functions to the left of the red dashed line at $C=0.15$ and between the two red dashed lines at $C=0.75$ and $C=0.8$ and interpolation functions elsewhere.

Figure 4

$\mathcal{O}({\alpha }_s^3)$ nonsingular $C$-parameter distribution. The solid line shows our reconstruction, whereas dots with error bars correspond to the EERAD3 output with the singular terms subtracted. Our reconstruction consists of two fit functions, one for $C<0.75$ and another one for $0.75<C<0.835$, and an interpolation for $C>0.835$ (to the right of the red dashed vertical line).

Figure 5

Singular and nonsingular components of the fixed-order $C$-parameter cross section, including up to $\mathcal{O}({\alpha }_s^3)$ terms, with ${\mathrm{\Omega }}_1=0.25\mathrm{GeV}$ and ${\alpha }_s(m_Z)=0.1141$.

Figure 6

Singular and nonsingular components of the fixed-order thrust cross section, including up to $\mathcal{O}({\alpha }_s^3)$ terms, with ${\mathrm{\Omega }}_1=0.25\mathrm{GeV}$ and ${\alpha }_s(m_Z)=0.1141$.

Figure 7

Solid lines are the central results for the profile functions for the renormalization scales ${\mu }_H$, ${\mu }_J(C)$, ${\mu }_S(C)$ at $Q=m_Z$. The bands and up-down arrow indicate the results of varying the profile parameters. The result for $R(C)$ is identical to ${\mu }_S(C)$ at the resolution of this figure, differing only at small $C$. Above $C=t_s\simeq 0.8$ all the scales merge, ${\mu }_H={\mu }_J={\mu }_S=R$.

Figure 8

Theory scan for $C$-parameter [panels (a) to (f)] and thrust [panels (g) to (l)] cross section uncertainties at various orders, for few center-of-mass energies $Q$ and slopes $r_s$. The theoretical predictions include log resummation and are purely perturbative. The respective upper and lower rows use $r_s=1$ and $r_s=2$, respectively. The left, center, and right columns correspond to $Q=40\mathrm{GeV}$, 91.2 GeV, and 200 GeV, respectively. Here we use ${\alpha }_s(m_Z)=0.1141$.

Figure 9

$C$-parameter (left panel) and thrust (right panel) cross section predictions at ${\mathrm{N}}^3{\mathrm{LL}}^{\prime }+\mathcal{O}({\alpha }_s^3)+{\mathrm{\Omega }}_1(R,\mu )$ for $r_s=1$ (blue) and $r_s=2$ (red). For these plots we use our most complete setup, with power corrections in the renormalon-free Rgap scheme. Here we use ${\alpha }_s(m_Z)=0.1141$ and ${\mathrm{\Omega }}_1(R_{\mathrm{\Delta }},{\mu }_{\mathrm{\Delta }})=0.33\mathrm{GeV}$.

Figure 10

Total hadronic cross section obtained from integrating the resummed cross section. The top two panels show the prediction for $r_s=1$ and $r_s=2$ for $C$-parameter, respectively. Likewise, the bottom two panels show the thrust results. Green squares correspond to the prediction with log resummation and the power correction in the $\overline{\mathrm{MS}}$ scheme, whereas red triangles have log resummation and the power correction in the Rgap scheme. The blue points correspond to the fixed-order prediction, and the blue line shows the highest-order fixed order (FO) prediction.

Figure 11

Components of the $C$-parameter cross section with resummation at ${\mathrm{N}}^3{\mathrm{LL}}^{\prime }+\mathcal{O}({\alpha }_s^3)+{\mathrm{\Omega }}_1(R,\mu )$ with ${\mathrm{\Omega }}_1(R_{\mathrm{\Delta }},{\mu }_{\mathrm{\Delta }})=0.33\mathrm{GeV}$ and ${\alpha }_s(m_Z)=0.1141$.

Figure 12

Theory scan for cross section uncertainties in $C$-parameter. The panels are (a) fixed order, (b) resummation with no nonperturbative function, (c) resummation with a nonperturbative function using the $\overline{\mathrm{MS}}$ scheme for ${\overline{\mathrm{\Omega }}}_1^C$ without renormalon subtraction, and (d) resummation with a nonperturbative function using the Rgap scheme for ${\mathrm{\Omega }}_1^C$ with renormalon subtraction.

Figure 13

Effects of hadron masses on the differential cross section for $C$-parameter (left) and thrust (right). Curves correspond to the percent difference between the cross section with and without hadron-mass effects, for three center-of-mass energies: $Q=91.2\mathrm{GeV}$ (solid red), 200 GeV (dashed blue), and 40 GeV (dotted green). The cross section with hadron-mass effects uses $\theta (R_{\mathrm{\Delta }},{\mu }_{\mathrm{\Delta }})=0$ and ${\mathrm{\Omega }}_1(R_{\mathrm{\Delta }},{\mu }_{\mathrm{\Delta }})=0.32(0.30)\mathrm{GeV}$ for $C$-parameter (thrust), while the cross section without hadron-mass effects has ${\mathrm{\Omega }}_1(R_{\mathrm{\Delta }},{\mu }_{\mathrm{\Delta }})=0.33\mathrm{GeV}$. We set ${\alpha }_s(m_Z)=0.1141$.

Figure 14

Effects of $\theta (R_{\mathrm{\Delta }},{\mu }_{\mathrm{\Delta }})$ on the cross section for $C$-parameter (left) and thrust (right). The lines correspond to the largest variation achievable by varying $\theta (R_{\mathrm{\Delta }},{\mu }_{\mathrm{\Delta }})$ in both directions (which happen for $\theta =0.23\pi$ and $\theta =-0.27\pi$), with respect to the cross section with $\theta (R_{\mathrm{\Delta }},{\mu }_{\mathrm{\Delta }})=0$. The solid (red), dashed (blue), and dotted (green) lines correspond to $Q=91.2$, 200, and 40 GeV, respectively. We use ${\mathrm{\Omega }}_1(R_{\mathrm{\Delta }},{\mu }_{\mathrm{\Delta }})=0.32(0.30)\mathrm{GeV}$ for $C$-parameter (thrust), and ${\alpha }_s(m_Z)=0.1141$.

Figure 15

Comparison of the fixed-order analytic SCET prediction for the $\mathcal{O}({\alpha }_s^2)$ piece with the parton level Monte Carlo EVENT2. The decomposition in the three color structures $C_F^2$, $C_F{C}_A$, and $C_F{n}_f{T}_f$ is shown in panels (a), (b), and (c), respectively. The factor ${\alpha }_s^2/(2\pi {)}^2$ has been divided out. We use a log binning in the horizontal axis to emphasize the dijet region.

Figure 16

Comparison of the fixed-order analytic SCET prediction for the $\mathcal{O}({\alpha }_s^3)$ piece with the parton level Monte Carlo EERAD3. The decomposition in the three color structures $N_C^2$, 1, and $N_C^{-2}$ are shown in panels (a), (b), and (c), of the first row, respectively, and $N_C{n}_f$, $n_f/N_C$, and $n_f^2$ are shown in panels (d), (e), and (f), respectively, on the second row. The factor ${\alpha }_s^3/(2\pi {)}^3$ has been divided out. We use a log binning in the horizontal axis to emphasize the dijet region.

Figure 17

Comparison of the determination of the soft function nonlogarithmic constants at $\mathcal{O}({\alpha }_s^2)$ as explained in Sec. 6 (shown as horizontal lines), with a determination employing the method used in Ref. [47] (plateau at ${10}^{-4}\lesssim C\lesssim {10}^{-3}$). The function $s_2^{\tilde{C}}(C)$ is defined in Eq. (b1). The $C_F{C}_A$ and $C_F{n}_f{T}_f$ color structures are shown in left and right panels, respectively. The logarithmic horizontal axis emphasizes the $C\to 0$ extrapolation. Error bars are included, and the blue and red points correspond to data from a different binning.

Figure 18

Diagrams contributing to the soft function at one loop.

Figure 19

Running of the short-distance power correction ${\mathrm{\Omega }}_1^h(R,R,h)$ with respect to the reference value ${\mathrm{\Omega }}_1^h(R_{\mathrm{\Delta }},R_{\mathrm{\Delta }},1)$. The scale $R$ is set to the default profile, and the scheme parameter $h$ is set to the function displayed in Eq. (f10). Red, blue, and green correspond to center-of-mass energies of 91.2, 40, and 200 GeV, respectively. The solid lines do not include hadron-mass effects, whereas the dashed ones do.

Figure 20

Convolution $S=\widehat{S}\otimes F$ combining the perturbative and nonperturbative $C$-parameter soft functions in the $\overline{\mathrm{MS}}$ scheme (left), Rgap with $C$-parameter subtractions (center), and Rgap with thrust subtractions (right). The results at one, two, and three loops are shown in green, blue, and red, respectively, whereas the tree level result is depicted as a gray dotted line. We use $\mu =1\mathrm{GeV}$, $R=0.8\mathrm{GeV}$, ${\alpha }_s(m_Z)=0.1141$, and a fixed generic shape function of which the first moment is ${\mathrm{\Omega }}_1=0.33\mathrm{GeV}$.