Two-loop $N$-jettiness soft function for $pp\to 2j$ production

We calculate the two-loop soft function for $pp\to jj$, $e^{+}{e}^{-}\to 4j$, or $ep\to 3j$ when the $N$-jettiness observable is measured. The result presented here makes up the necessary piece for realizing the full next-to-next-to-leading order predictions for these processes using the $N$-jettiness subtraction scheme.

Figure 1

Coefficients of the ${\tilde{L}}^n$ terms for the $\mathcal{O}({\alpha }_s)$ 2-jettiness soft function in $pp\to jj$ production, normalized to $\frac{{\alpha }_s}{4\pi }{T}_i·T_j$.

Figure 2

Coefficients of the ${\tilde{L}}^n$ with $n=3$, 2, 1, 0 in $S_{12}^{(2),N_F}$ (left panel) and $S_{13}^{(2),N_F}$ (right panel).

Figure 3

Coefficients of the ${\tilde{L}}^n$ with $n=3$, 2, 1, 0 in $S_{12}^{(2),C_A}$ (left panel) and $S_{13}^{(2),C_A}$ (right panel).

Figure 4

Coefficients of the ${\tilde{L}}^n$ with $n=3$, 2, 1, 0 in $S_{123}^{(2)}$ (left panel) and the $\epsilon$-pole terms (right panel).