Two- and three-loop data for the groomed jet mass

We discuss the status of resummation of large logarithmic contributions to groomed event shapes of hadronic final states in electron-positron annihilation. We identify the missing ingredients needed for next-to-next-to-next-to-leading logarithmic (NNNLL) resummation of the modified mass mass drop tagger (mMDT) groomed jet mass in $e^{+}{e}^{-}$ collisions: the low-scale collinear-soft constants at two-loop accuracy, $c_{S_c}^{(2)}$, and the three-loop noncusp anomalous dimension of the global soft function, ${\gamma }_S^{(2)}$. We present a method for extracting those constants using fixed-order codes: the event2 program to obtain the color coefficients of $c_{S_c}^{(2)}$, and mccsm for extracting ${\gamma }_S^{(2)}$. We present all necessary formulae for resummation of the mMDT groomed heavy jet mass distribution at NNNLL accuracy.

Figure 1

Plot of the numerical integral of Eq. (28) as a function of $z_{\mathrm{cut}}$ as evaluated in event2. The exact value for $z_{\mathrm{cut}}\to 0$ is plotted as the solid line.

Figure 2

Plots of the values extracted from event2 of the two-loop constants separated into distinct color channels of the collinear-soft function, as a function of $z_{\mathrm{cut}}$. Our claimed extracted value and its uncertainty is illustrated by the dotted line and shaded band, respectively.

Figure 3

Plot of the ratio of $z_{\mathrm{cut}}$ to the location of the minimum of the factorization theorem cross section at $\mathcal{O}({\alpha }_{\mathrm{s}}^3)$ logarithmic in $\rho$, ${\rho }_{\min }$. Lines corresponding to ratios of 1 and 10 have been added to guide the eye.

Figure 4

Left: Plot comparing the distribution of mMDT groomed heavy hemisphere mass at $\mathcal{O}({\alpha }_{\mathrm{s}}^3)$ from mccsm to the analytic prediction with ${\gamma }_S^{(2)}=0$ and ${\gamma }_S^{(2)}=-8400\pm 1000$. Overall coupling dependence has been stripped off. Right: Plot showing the extracted value of ${\gamma }_S^{(2)}$ from mccsm at each $\rho$ point. For both plots, $z_{\mathrm{cut}}=0.04$.

Figure 5

Plot of ${\gamma }_S^{(2)}$, with uncertainties, determined at each value of $z_{\mathrm{cut}}$. The best-fit line is dotted and its intercept at $z_{\mathrm{cut}}=0$ is the quoted extracted value of ${\gamma }_S^{(2)}$.