C_{x\mathfrak{m}\mathfrak{n}}\omega^{tc,i}\left[w^{2}\tilde{S}^{q\bar{q}}_{ij}(\mathfrak{m},\mathfrak{n})\right]\right\rangle_{\mathfrak{m}\mathfrak{n}}=\frac{2N_{E}}{\epsilon}\Bigg{[}-\frac{1}{3\epsilon^{2}}+\frac{19}{18\epsilon}\\\
&+\frac{\pi^{2}}{9}-\frac{83}{54}+4\ln^{2}2+\ln
2+\epsilon\Big{(}\frac{25\zeta_{3}}{3}-\frac{35\pi^{2}}{108}+\frac{1057}{162}\\\
&-\frac{1}{3}40\ln^{3}2-\frac{53\ln^{2}2}{3}+\frac{2}{9}\pi^{2}\ln 2+3\ln
2\Big{)}+{\cal O}(\epsilon^{2})\Bigg{]}.\end{split}$ (174)

The corresponding result for the double-collinear subtracted contribution
without the $N$-jettiness constraint reads

$\begin{split}&\frac{2N_{E}}{\epsilon}\int\limits_{0}^{1}\frac{{\rm
d}\omega}{\omega^{1+2\epsilon}}\left\langle\theta^{b+d}C_{\mathfrak{m}\mathfrak{n}}\left[{\rm
d}\Omega_{\mathfrak{m}\mathfrak{n}}\right]C_{x\mathfrak{m}\mathfrak{n}}\omega^{tc,i}\left[w^{2}\tilde{S}^{q\bar{q}}_{ij}(\mathfrak{m},\mathfrak{n})\right]\right\rangle_{\mathfrak{m}\mathfrak{n}}=\\\
&\frac{2N_{E}}{\epsilon}\Bigg{[}-\frac{1}{3\epsilon^{2}}+\frac{1}{\epsilon}\left(\frac{1}{2}-\frac{2\ln
2}{3}\right)-\frac{\pi^{2}}{3}+2\ln^{2}2+\frac{5\ln 2}{3}\\\
&+\epsilon\left(-\frac{10\zeta_{3}}{3}+\frac{11\pi^{2}}{18}-\frac{4}{3}-\frac{20}{9}\ln^{3}2-\frac{13}{3}\ln^{2}2+\left(\frac{8}{3}+\frac{2\pi^{2}}{9}\right)\ln
2\right)+{\cal O}(\epsilon^{2})\Bigg{]}.\end{split}$ (175)

##  13 The renormalized $N$-jettiness soft function

We are now in a position to present the renormalized $N$-jettiness soft
function through NNLO in the perturbative expansion in QCD. We write

${\tilde{S}}=1+{\tilde{S}}_{1}+{\tilde{S}}_{2}+{\cal O}(\alpha_{s}^{3}).$
(176)

It is convenient to introduce the following short-hand notations

$L_{ij}=\ln(\bar{u}\sqrt{\eta_{ij}}\mu),\;\;\;L^{\psi}_{ij,\mathfrak{m}}=\ln\left(\frac{\psi_{\mathfrak{m}}\rho_{ij}}{\rho_{i\mathfrak{m}}\rho_{j\mathfrak{m}}}\right).$
(177)

The NLO contribution reads

${\tilde{S}}_{1}=a_{s}\sum\limits_{(ij)}{\bf T}_{i}\cdot{\bf
T}_{j}\left[2L_{ij}^{2}+{\rm
Li}_{2}(1-\eta_{ij})+\frac{\pi^{2}}{12}+\left\langle\;L^{\psi}_{ij,\mathfrak{m}}\frac{\rho_{ij}}{\rho_{i\mathfrak{m}}\rho_{j\mathfrak{m}}}\right\rangle_{\mathfrak{m}}\;\right].$
(178)

The NNLO contribution is constructed from different pieces. We write

${\tilde{S}}_{2}=\frac{1}{2}\;{\tilde{S}}_{1}^{2}+a_{s}^{2}C_{A}\sum\limits_{(ij)}{\bf
T}_{i}\cdot{\bf T}_{j}\;G_{ij}+a_{s}^{2}\;n_{f}\;T_{R}\sum\limits_{(ij)}{\bf
T}_{i}\cdot{\bf
T}_{j}\;Q_{ij}+a_{s}^{2}\pi\sum\limits_{(kij)}F^{kij}\;\kappa_{kj}G_{kij}^{\text{triple}}.$
(179)

The function $G_{ij}$ reads

$\displaystyle G_{ij}$
$\displaystyle=\frac{22}{9}L_{ij}^{3}+\left(\frac{67}{9}-\frac{\pi^{2}}{3}\right)L_{ij}^{2}+L_{ij}\Big{(}\frac{11}{3}\left\langle
L^{\psi}_{ij,\mathfrak{m}}\frac{\rho_{ij}}{\rho_{i\mathfrak{m}}\rho_{j\mathfrak{m}}}\right\rangle_{\mathfrak{m}}+\frac{11}{3}{\rm
Li}_{2}(1-\eta_{ij})$
$\displaystyle+\frac{202}{27}-7\zeta_{3}\Big{)}+\Bigg{\langle}\frac{\rho_{ij}}{\rho_{i\mathfrak{m}}\rho_{j\mathfrak{m}}}\Bigg{(}\left(L^{\psi}_{ij,\mathfrak{m}}\right)^{2}\left(\frac{11}{6}-\ln\left(\frac{\eta_{ij}}{\eta_{i\mathfrak{m}}\eta_{j\mathfrak{m}}}\right)\right)$
$\displaystyle+L^{\psi}_{ij,\mathfrak{m}}\Big{(}2\ln^{2}\left(\frac{\eta_{ij}}{\eta_{i\mathfrak{m}}\eta_{j\mathfrak{m}}}\right)+\ln\left(\frac{\eta_{ij}}{\eta_{i\mathfrak{m}}\eta_{j\mathfrak{m}}}\right)\left(-\frac{11}{3}+\ln(\eta_{i\mathfrak{m}}\eta_{j\mathfrak{m}})\right)+\frac{137}{18}-\frac{\pi^{2}}{2}$
$\displaystyle-\frac{1}{2}\ln^{2}\left(\frac{\eta_{i\mathfrak{m}}}{\eta_{j\mathfrak{m}}}\right)+{\rm
Li}_{2}(1-\eta_{ij})+\frac{11}{3}\ln
2-\frac{11}{6}\ln(\eta_{i\mathfrak{m}}\eta_{j\mathfrak{m}})-\frac{(\rho_{i\mathfrak{m}}+\rho_{j\mathfrak{m}})}{3\rho_{ij}}\Big{)}\Bigg{)}\Bigg{\rangle}_{\mathfrak{m}}$
$\displaystyle+\left\langle A^{\rm
fin}_{ij,\mathfrak{m}}\left(\frac{11}{6}L_{ij,\mathfrak{m}}^{\psi}-\frac{11}{6}\ln\frac{\eta_{ij}}{\eta_{i\mathfrak{m}}\eta_{j\mathfrak{m}}}+\frac{131}{72}\right)\right\rangle_{\mathfrak{m}}$
$\displaystyle+\left\langle B^{\rm
fin}_{ij,\mathfrak{m}}\left(-\frac{1}{6}L_{ij,\mathfrak{m}}^{\psi}+\frac{1}{6}\ln\frac{\eta_{ij}}{\eta_{i\mathfrak{m}}\eta_{j\mathfrak{m}}}-\frac{13}{72}\right)\right\rangle_{\mathfrak{m}}-2G_{-1,0,0,1}(\eta_{ij})+\frac{7}{2}G_{0,1,0,1}(\eta_{ij})$
$\displaystyle-\frac{11}{3}{\rm Ci}_{3}(2\delta_{ij})-\frac{1}{6{\rm
tan}(\delta_{ij})}{\rm
Si}_{2}(2\delta_{ij})+17S_{2,2}(\eta_{ij})-\frac{1}{2}{\rm
Li}_{4}\left(1-\eta_{ij}^{2}\right)$ $\displaystyle-2{\rm
Li}_{4}\left(\frac{1}{\eta_{ij}+1}\right)+{\rm
Li}_{4}\left(\frac{1-\eta_{ij}}{1+\eta_{ij}}\right)-{\rm
Li}_{4}\left(\frac{\eta_{ij}-1}{1+\eta_{ij}}\right)-12{\rm Li}_{4}(\eta_{ij})$
$\displaystyle+{\rm
Li}_{3}(\eta_{ij})\left(-\frac{11}{6}+10\ln(1-\eta_{ij})+10\ln(\eta_{ij})-2\ln(1+\eta_{ij})\right)$
$\displaystyle+{\rm
Li}_{3}(1-\eta_{ij})\Big{(}10\ln(\eta_{ij})+2\ln(1+\eta_{ij})\Big{)}-2{\rm
Li}_{3}(-\eta_{ij})\ln(1-\eta_{ij})$ (180) $\displaystyle+\frac{5{\rm
Li}_{2}(\eta_{ij})^{2}}{4}+{\rm Li}_{2}(\eta_{ij})\Big{(}-2{\rm
Li}_{2}(-\eta_{ij})-4\ln^{2}(\eta_{ij})$
$\displaystyle+6\ln(1-\eta_{ij})\ln(\eta_{ij})+\frac{11\ln(\eta_{ij})}{6}+\frac{\pi^{2}}{3}-11\ln
2-\frac{131}{12}\Big{)}$
$\displaystyle-\frac{1}{12}\ln^{4}(1+\eta_{ij})-\ln^{3}(\eta_{ij})\ln(1-\eta_{ij})+\frac{\pi^{2}}{12}\ln^{2}(1+\eta_{ij})$
$\displaystyle+\ln^{2}(\eta_{ij})\left(\frac{11}{2}\ln^{2}(1-\eta_{ij})+\frac{11}{12}\ln(1-\eta_{ij})-\frac{\pi^{2}}{6}+\frac{11\ln
2}{3}+\frac{32}{9}\right)$ $\displaystyle+\ln(1-\eta_{ij})\left(-11\ln
2\ln(\eta_{ij})-\frac{4\pi^{2}\ln(\eta_{ij})}{3}-\frac{131\ln(\eta_{ij})}{12}-\frac{23\zeta_{3}}{2}\right)$
$\displaystyle+\ln(\eta_{ij})\left(-\frac{11\ln^{2}2}{3}-\frac{\pi^{2}\ln
2}{3}+\frac{64\ln
2}{9}-\frac{27\zeta_{3}}{4}-\frac{11\pi^{2}}{18}+\frac{1631}{108}\right)$
$\displaystyle-\frac{7}{4}\zeta_{3}\ln(1+\eta_{ij})+\frac{1}{3}\ln^{4}2+\ln^{2}2\left(\frac{1}{3}-\frac{\pi^{2}}{3}\right)+\ln
2\left(5\zeta_{3}+\frac{11\pi^{2}}{18}-\frac{2}{9}\right)$
$\displaystyle+8{\rm
Li}_{4}\left(\frac{1}{2}\right)-\frac{11}{9}\zeta_{3}-\frac{11}{80}\pi^{4}+\frac{937}{432}\pi^{2}+\frac{403}{162}$
$\displaystyle-\left\langle{\bar{C}}_{\mathfrak{m}\mathfrak{n}}\ln\left(\frac{\psi_{m}}{\rho_{im}}\right)\ln\left(\frac{\psi_{n}}{\rho_{jn}}\right)\frac{\rho_{ij}}{\rho_{\mathfrak{m}\mathfrak{n}}\rho_{i\mathfrak{m}}\rho_{j\mathfrak{n}}}\right\rangle_{\mathfrak{m}\mathfrak{n}}+\frac{1}{2}\left\langle
L_{ij,\mathfrak{m}}^{\psi}\frac{\rho_{ij}}{\rho_{i\mathfrak{m}}\rho_{j\mathfrak{m}}}\right\rangle_{\mathfrak{m}}^{2}$
$\displaystyle+\frac{1}{2}\sum\limits_{x\in\\{i,j\\}}\int\limits_{0}^{1}\frac{{\rm
d}\omega}{\omega}\left\langle(1-\theta^{b+d}C_{\mathfrak{m}\mathfrak{n}})\left[{\rm
d}\Omega_{\mathfrak{m}\mathfrak{n}}\right]{\bar{C}}_{x\mathfrak{m}\mathfrak{n}}\;w^{x\mathfrak{m},x\mathfrak{n}}\;\ln\psi_{\mathfrak{m}\mathfrak{n}}\;{\bar{S}}_{\omega}\;[\omega^{2}{\tilde{S}}_{ij}(\mathfrak{m},\mathfrak{n})]\right\rangle_{\mathfrak{m}\mathfrak{n}}$
$\displaystyle+\frac{1}{2}\int\limits_{0}^{1}\frac{{\rm
d}\omega}{\omega}\left\langle(w^{i\mathfrak{m},j\mathfrak{n}}+w^{j\mathfrak{m},i\mathfrak{n}})\;\ln\psi_{\mathfrak{m}\mathfrak{n}}\;{\bar{S}}_{\omega}\;[\omega^{2}{\tilde{S}}_{ij}(\mathfrak{m},\mathfrak{n})]\right\rangle_{\mathfrak{m}\mathfrak{n}}.$

In the above formula $\delta_{ij}=\theta_{ij}/2$ and $\theta_{ij}$ is the
relative angle between partons $i$ and $j$. The Clausen functions are defined
through their relation to complex-valued polylogarithms

$\begin{split}\text{Ci}_{n}(z)&=\frac{1}{2}\left(\text{Li}_{n}(e^{iz})+\text{Li}_{n}(e^{-iz})\right),\\\
\text{Si}_{n}(z)&=\frac{1}{2i}\left(\text{Li}_{n}(e^{iz})-\text{Li}_{n}(e^{-iz})\right).\end{split}$
(181)

The functions $S_{a_{1},a_{2}}$ and $G_{a_{1},a_{2},\dots,a_{m}}(x)$ are the
Nielsen and Goncharov Goncharov:1998kja  polylogarithms, respectively. The
auxiliary functions that we employed while writing the above formula read

$\begin{split}&A_{ij,\mathfrak{m}}^{\rm
fin}=\frac{\rho_{ij}}{\rho_{i\mathfrak{m}}\rho_{j\mathfrak{m}}}\left[\omega^{\mathfrak{m}i,\mathfrak{n}i}_{\mathfrak{m}||\mathfrak{n}}\ln\left(\frac{\eta_{j\mathfrak{m}}(1-\eta_{i\mathfrak{m}})}{\eta_{ij}}\right)+\omega^{\mathfrak{m}j,\mathfrak{n}j}_{\mathfrak{m}||\mathfrak{n}}\ln\left(\frac{\eta_{i\mathfrak{m}}(1-\eta_{j\mathfrak{m}})}{\eta_{ij}}\right)\right],\\\
&B_{ij,\mathfrak{m}}^{\rm
fin}=\omega_{\mathfrak{m}||\mathfrak{n}}^{\mathfrak{m}i,\mathfrak{n}i}\;W_{ij}^{\mathfrak{m}}+\omega_{\mathfrak{m}||\mathfrak{n}}^{\mathfrak{m}j,\mathfrak{n}j}\;W_{ji}^{\mathfrak{m}}.\end{split}$
(182)

The function $Q_{ij}$ evaluates to

$\begin{split}Q_{ij}&=-\frac{8}{9}L_{ij}^{3}-\frac{20}{9}L_{ij}^{2}-L_{ij}\Bigg{(}\frac{4}{3}\left\langle
L^{\psi}_{ij,\mathfrak{m}}\frac{\rho_{ij}}{\rho_{i\mathfrak{m}}\rho_{j\mathfrak{m}}}\right\rangle_{\mathfrak{m}}+\frac{4}{3}\
{\rm Li}_{2}(1-\eta_{ij})+\frac{56}{27}\Bigg{)}\\\
&-\left\langle\frac{\rho_{ij}}{\rho_{i\mathfrak{m}}\rho_{j\mathfrak{m}}}\Bigg{(}\frac{2}{3}\left(L^{\psi}_{ij,\mathfrak{m}}\right)^{2}-L^{\psi}_{ij,\mathfrak{m}}\Big{(}\;\frac{2}{3}\ln\left(\frac{\eta^{2}_{ij}}{4\eta_{i\mathfrak{m}}\eta_{j\mathfrak{m}}}\right)-\frac{26}{9}+\frac{2(\rho_{i\mathfrak{m}}+\rho_{j\mathfrak{m}})}{3\rho_{ij}}\Big{)}\Bigg{)}\right\rangle_{\mathfrak{m}}\\\
&+\left\langle A^{\rm
fin}_{ij,\mathfrak{m}}\left(-\frac{2}{3}L^{\psi}_{ij,\mathfrak{m}}+\frac{2}{3}\ln\left(\frac{\eta_{ij}}{\eta_{i\mathfrak{m}}\eta_{j\mathfrak{m}}}\right)-\frac{23}{36}\right)\right\rangle_{\mathfrak{m}}\\\
&+\left\langle B^{\rm
fin}_{ij,\mathfrak{m}}\left(\frac{1}{3}L^{\psi}_{ij,\mathfrak{m}}-\frac{1}{3}\ln\left(\frac{\eta_{ij}}{\eta_{i\mathfrak{m}}\eta_{j\mathfrak{m}}}\right)+\frac{13}{36}\right)\right\rangle_{\mathfrak{m}}+\frac{2}{3}{\rm
Li}_{3}(\eta_{ij})\\\ &+{\rm
Li}_{2}(1-\eta_{ij})\left(\frac{2\ln(\eta_{ij})}{3}-\frac{7}{2}-4\ln
2\right)+\ln^{2}2\left(-\frac{2}{3}+\frac{4}{3}\ln(\eta_{ij})\right)\\\ &+\ln
2\left(-\frac{4}{3}\ln^{2}(\eta_{ij})-\frac{20}{9}\ln(\eta_{ij})+\frac{4\pi^{2}}{9}+\frac{4}{9}\right)\\\
&-\ln^{2}(\eta_{ij})\left(\frac{10}{9}-\frac{1}{3}\ln(1-\eta_{ij})\right)+\ln(\eta_{ij})\left(\frac{\pi^{2}}{9}-\frac{335}{54}\right)\\\
&+\frac{4\zeta_{3}}{9}+\frac{122}{81}-\frac{47\pi^{2}}{108}+\frac{4}{3}{\rm
Ci}_{3}(2\delta_{ij})+\frac{1}{3{\tan}(\delta_{ij})}{\rm
Si}_{2}(2\delta_{ij})\\\
&-\sum\limits_{x\in\\{i,j\\}}\int\limits_{0}^{1}\frac{{\rm
d}\omega}{\omega}\left\langle(1-\theta^{b+d}C_{\mathfrak{m}\mathfrak{n}})\left[{\rm
d}\Omega_{\mathfrak{m}\mathfrak{n}}\right]{\bar{C}}_{x\mathfrak{m}\mathfrak{n}}\;w^{x\mathfrak{m},x\mathfrak{n}}\;\ln\psi_{\mathfrak{m}\mathfrak{n}}\;[\omega^{2}{\tilde{S}}^{q\bar{q}}_{ij}(\mathfrak{m},\mathfrak{n})]\right\rangle_{\mathfrak{m}\mathfrak{n}}\\\
&-\int\limits_{0}^{1}\frac{{\rm
d}\omega}{\omega}\left\langle(w^{i\mathfrak{m},j\mathfrak{n}}+w^{j\mathfrak{m},i\mathfrak{n}})\;\ln\psi_{\mathfrak{m}\mathfrak{n}}\;[\omega^{2}{\tilde{S}}^{q\bar{q}}_{ij}(\mathfrak{m},\mathfrak{n})]\right\rangle_{\mathfrak{m}\mathfrak{n}},\end{split}$
(183)

The function $G_{kij}^{\text{triple}}$, defined in Eq. (62), corresponds to
the finite remainder of the triple-color correlation terms.

##  14 Numerical implementations and checks

We have implemented the above formulas into a Fortran code. The implementation
is, in principle, straightforward. To compute terms that require integration
over directions of gluons’ momenta, we use the phase space parametrization
described in Ref. Czakon:2010td ; Czakon:2011ve ; Caola:2017dug . A possible
choice of partition functions can be found in Ref. Asteriadis:2019dte .

As we already mentioned, the $N$-jettiness soft function was recently
discussed in Ref. Bell:2023yso  where a plethora of numerical results for
various kinematic configurations have been presented. We have checked many of
the presented results for various values of the parameter $N$ and found
excellent agreement. We present some examples of such a comparison in Appendix
D.

We note that to obtain the high-precision numbers shown in Appendix D, we have
used a very large number of sample points which results in runtimes that can
be as large as a few minutes per dipole per kinematic point. However, to
obtain the same numbers with a better-than-percent precision, we need a
relatively small number of sampling points which results in runtimes of an
order of a few seconds per dipole per phase space point. We note that runtimes
for individual dipoles are largely _independent_ of $N$. Since $(N+2)(N+1)/2$
dipoles need to be calculated to determine the $N$-jettiness soft function for
a given phase space point, ${\cal O}(1-2)$ minutes will be required to do that
in case of three- or four-jet production at a hadron collider, or six-jet
production at an $e^{+}e^{-}$ collider.

The situation with triple-color correlated terms is quite similar. Since in
this case integration over the direction of a single gluon is required, the
integration converges rather fast. In general, for an $N$-jettiness case, the
evaluation of $(N+2)(N+1)N$ independent functions $G_{kij}^{\rm triple}$ is
required from which a smaller number of independent color-correlated
contributions can be constructed. We need about twenty seconds to compute all
the required $G_{kij}^{\rm triple}$ functions for $N=2$ and, therefore, we
will need about a minute to compute them for $N=3$ and about two minutes for
$N=4$.

##  15 Conclusions

We have described the computation of the $N$-jettiness soft function through
NNLO in QCD. Keeping $N$ as the parameter, we demonstrated the cancellation of
all $1/\epsilon$ poles _analytically_ against the soft-function
renormalization matrix. Furthermore, we derived a simple representation for
the finite, jettiness-dependent remainder valid for an arbitrary number of
hard partons $N$. We compared our numerical results for $N=1,2$ and $N=3$ with
the ones recently presented in Ref. Bell:2023yso  and found excellent
agreement. We have also found that the representation of the finite remainder
that is derived in this paper leads to fast and rapidly convergent integration
which is important if the results for $N$-jettiness soft function are to be
used for the computation of higher-order corrections for multi-jet production
at colliders.

From a more general perspective, this study was motivated by a question of how
advances in developing NNLO subtraction schemes for generic processes at the
LHC can be used to derive representations for the building blocks of modern
slicing methods that rely on choosing observables to define suitable slicing
variables. We have shown that, at least for $N$-jettiness, the benefits of
applying subtraction-inspired methods are significant. We note that the
application of the subtraction-inspired methods for computing the
$N$-jettiness soft function becomes possible if one departs from the (by now)
standard approach Jouttenus:2011wh ; Boughezal:2015eha ; Campbell:2017hsw  to
such computations where hemisphere soft functions are used as elementary
building blocks and, instead, interprets $N$-jettiness as one of the many
infrared safe observables that can be studied using available subtraction
schemes.

## Acknowledgments

We are grateful to G. Bell for useful exchanges and R. K. Ellis for helpful
conversations. This research was supported by the German Research Foundation
(DFG, Deutsche Forschungsgemeinschaft) under grant 396021762-TRR 257.

##  Appendix A Renormalization of the soft function

A suitable expression for the renormalization of the $N$-jettiness soft
function was recently given in Ref. Bell:2023yso . It reads

$Z=1+Z_{1}+\frac{1}{2}Z_{1}Z_{1}+Z_{2,r},$ (184)

where

$Z_{1}=a_{s}\sum\limits_{(ij)}{\bf T}_{i}\cdot{\bf
T}_{j}\left(\frac{1}{2\epsilon^{2}}+\frac{2L_{ij}+i\pi\lambda_{ij}}{2\epsilon}\right),$
(185)

and

$Z_{2,r}=a_{s}^{2}\sum\limits_{(ij)}{\bf T}_{i}\cdot{\bf
T}_{j}\left(-\frac{3\beta_{0}}{8\epsilon^{3}}+\frac{\Gamma_{1}-4\beta_{0}(2L_{ij}+i\pi\lambda_{ij})}{16\epsilon^{2}}+\frac{\Gamma_{1}(2L_{ij}+i\pi\lambda_{ij})+\gamma^{S}_{1}}{8\epsilon}\right),$
(186)

with

$L_{ij}=\ln(\bar{u}\sqrt{\eta_{ij}}\mu),$ (187)

$\Gamma_{1}=\left(\frac{67}{9}-\frac{\pi^{2}}{3}\right)C_{A}-\frac{20}{9}T_{R}n_{f},$
(188)

and

$\gamma^{S}_{1}=\left(\frac{202}{27}-\frac{11\pi^{2}}{36}-7\zeta_{3}\right)C_{A}+\left(-\frac{56}{27}+\frac{\pi^{2}}{9}\right)T_{R}n_{f}.$
(189)

where $\Gamma_{1}$ is the cusp anomalous dimension and the non-cusp anomalous
dimension $\gamma^{S}_{1}$ is known to two-loop order Stewart:2010qs .
Furthermore, $\lambda_{ij}=1$ if both $i$ and $j$ are incoming or outgoing and
$\lambda_{ij}=0$ otherwise.

##  Appendix B Useful phase space integrals

We define measures of angular integration as

$[{\rm d}\Omega^{(d-1)}]=\frac{{\rm d}\Omega^{(d-1)}}{\Omega^{(d-2)}}.$ (190)

Then

$\frac{{\rm
d}\Omega^{(d-1)}}{2(2\pi)^{d-1}}=\frac{\Omega^{(d-2)}}{2(2\pi)^{d-1}}\;[{\rm
d}\Omega^{(d-1)}],$ (191)

where

$\frac{\Omega^{(d-2)}}{2(2\pi)^{d-1}}=\left[\frac{(4\pi)^{\epsilon}}{8\pi^{2}\Gamma(1-\epsilon)}\right].$
(192)

With this integration measure, we find Somogyi:2011ir

$\left\langle\frac{1}{\rho_{i\mathfrak{m}}^{\alpha}}\right\rangle_{\mathfrak{m}}=2^{1-\alpha-2\epsilon}\frac{\Gamma(1-\epsilon)\Gamma(1-\alpha-\epsilon)}{\Gamma(2-\alpha-2\epsilon)},$
(193)

$\begin{split}\left\langle\frac{1}{\rho_{i\mathfrak{m}}^{\alpha}\rho_{j\mathfrak{m}}^{\beta}}\right\rangle_{\mathfrak{m}}=2^{1-\alpha-\beta-2\epsilon}&\frac{\Gamma(1-\beta-\epsilon)\Gamma(1-\alpha-\epsilon)}{\Gamma(2-\alpha-\beta-2\epsilon)}\times{}_{2}F_{1}(\alpha,\beta,1-\epsilon,1-\eta_{ij}),\end{split}$
(194)

where brackets $\langle...\rangle_{\mathfrak{m}}$ indicate integration over
$[{\rm d}\Omega^{(d-1)}_{\mathfrak{m}}]$.

##  Appendix C Eikonal functions

For completeness, we provide the double real-emission eikonal functions that
we use to compute the relevant contributions to the $N$-jettiness soft
function. Following Ref. Catani:1999ss , for two soft gluons, we define

${\tilde{S}}_{ij}^{gg}(\mathfrak{m},\mathfrak{n})=2S_{ij}^{gg}(\mathfrak{m},\mathfrak{n})-S_{ii}^{gg}(\mathfrak{m},\mathfrak{n})-S_{jj}^{gg}(\mathfrak{m},\mathfrak{n}).$
(195)

The function $S_{ij}^{gg}(\mathfrak{m},\mathfrak{n})$ reads

$\begin{split}S_{ij}^{gg}(\mathfrak{m},\mathfrak{n})&=S_{ij}^{gg,\rm
so}(\mathfrak{m},\mathfrak{n})-\frac{2p_{i}\cdot p_{j}}{p_{\mathfrak{m}}\cdot
p_{\mathfrak{n}}[p_{i}\cdot(p_{\mathfrak{m}}+p_{\mathfrak{n}})][p_{j}\cdot(p_{\mathfrak{m}}+p_{\mathfrak{n}})]}\\\
&+\frac{(p_{i}\cdot p_{\mathfrak{m}})(p_{j}\cdot p_{\mathfrak{n}})+(p_{i}\cdot
p_{\mathfrak{n}})(p_{i}\cdot
p_{\mathfrak{m}})}{[p_{i}\cdot(p_{\mathfrak{m}}+p_{\mathfrak{n}})p_{j}\cdot(p_{\mathfrak{m}}+p_{\mathfrak{n}})]}\left[\frac{1-\epsilon}{(p_{\mathfrak{m}}\cdot
p_{\mathfrak{n}})^{2}}-\frac{1}{2}S_{ij}^{gg,\rm
so}(\mathfrak{m},\mathfrak{n})\right],\end{split}$ (196)

where

$\begin{split}S_{ij}^{gg,\rm so}(\mathfrak{m},\mathfrak{n})&=\frac{p_{i}\cdot
p_{j}}{p_{\mathfrak{m}}\cdot p_{\mathfrak{n}}}\left(\frac{1}{(p_{i}\cdot
p_{\mathfrak{m}})(p_{j}\cdot p_{\mathfrak{n}})}+\frac{1}{(p_{i}\cdot
p_{\mathfrak{n}})(p_{j}\cdot p_{\mathfrak{m}})}\right)\\\ &-\frac{(p_{i}\cdot
p_{j})^{2}}{(p_{i}\cdot p_{\mathfrak{m}})(p_{j}\cdot
p_{\mathfrak{m}})(p_{i}\cdot p_{\mathfrak{n}})(p_{j}\cdot
p_{\mathfrak{n}})}.\end{split}$ (197)

In case the soft partons are a quark and an anti-quark, we write

${\tilde{S}}_{ij}^{q\bar{q}}(\mathfrak{m},\mathfrak{n})=2S_{ij}^{q\bar{q}}(\mathfrak{m},\mathfrak{n})-S_{ii}^{q\bar{q}}(\mathfrak{m},\mathfrak{n})-S_{jj}^{q\bar{q}}(\mathfrak{m},\mathfrak{n}),$
(198)

where Catani:1999ss

$\begin{split}S_{ij}^{q\bar{q}}(\mathfrak{m},\mathfrak{n})=\frac{(p_{i}\cdot
p_{\mathfrak{m}})(p_{j}\cdot p_{\mathfrak{n}})+(p_{i}\cdot
p_{\mathfrak{n}})(p_{j}\cdot p_{\mathfrak{m}})-(p_{i}\cdot
p_{j})(p_{\mathfrak{m}}\cdot p_{\mathfrak{n}})}{(p_{\mathfrak{m}}\cdot
p_{\mathfrak{n}})^{2}[p_{i}\cdot(p_{\mathfrak{m}}+p_{\mathfrak{n}})][p_{j}\cdot(p_{\mathfrak{m}}+p_{\mathfrak{n}})]}.\end{split}$
(199)

##  Appendix D Numerical Checks of N-Jettiness Soft Functions

In this section, we compare results for the $N$-Jettiness soft function
obtained in this paper with those in Ref. Bell:2023yso . For this comparison,
we define two functions $G_{ij}^{nl}$ and $Q_{ij}^{nl}$ that are obtained from
$G_{ij}$ and $Q_{ij}$ by setting all terms with $L_{ij}$ to zero

$\begin{split}G_{ij}&=\frac{22}{9}L_{ij}^{3}+\left(\frac{67}{9}-\frac{\pi^{2}}{3}\right)L_{ij}^{2}+L_{ij}\Big{(}\frac{11}{3}\left\langle
L^{\psi}_{ij,\mathfrak{m}}\frac{\rho_{ij}}{\rho_{i\mathfrak{m}}\rho_{j\mathfrak{m}}}\right\rangle_{\mathfrak{m}}+\frac{11}{3}{\rm
Li}_{2}(1-\eta_{ij})\\\
&+\frac{202}{27}-7\zeta_{3}\Big{)}+G_{ij}^{nl}\end{split},$ (200)
$Q_{ij}=-\frac{8}{9}L_{ij}^{3}-\frac{20}{9}L_{ij}^{2}-L_{ij}\Bigg{(}\frac{4}{3}\left\langle
L^{\psi}_{ij,\mathfrak{m}}\frac{\rho_{ij}}{\rho_{i\mathfrak{m}}\rho_{j\mathfrak{m}}}\right\rangle_{\mathfrak{m}}+\frac{4}{3}\
{\rm Li}_{2}(1-\eta_{ij})+\frac{56}{27}\Bigg{)}+Q_{ij}^{nl}.$ (201)

We also note that for the comparison of the triple-color correlated
contributions, we do not set these logarithms to zero but take $\bar{u}=\mu=1$
instead. As the result, we should set $L_{ki}\to 1/2\ln\eta_{ki}$ in Eq. (62).

###  D.1 1-Jettiness

In order to compare our numerical results, we parameterize our phase space in
a similar way as done in Ref. Bell:2023yso . In the case where there are two
back-to-back beams and one jet confined in a plane, we parameterize the
scattering by the angle $\theta_{13}$ using the following momenta,

$n_{1}=(0,0,1),\quad n_{2}=(0,0,-1),\quad
n_{3}=(\sin\theta_{13},0,\cos\theta_{13}).$ (202)

In order to compare a phase space point where the jet is separated from the
beams, we take

$\theta_{13}=\frac{12\pi}{25}.$ (203)

In Table 1 we present the results for the non-logarithmic coefficient of the
renormalized $1$-jettiness soft function, which are in agreement with the ones
in Ref. Bell:2023yso .

Dipoles | Gluons | Quarks
---|---|---
$G_{ij}^{nl}$ | Ref. Bell:2023yso  | $Q_{ij}^{nl}$ | Ref. Bell:2023yso
12 | 48.10 $\pm$ 0.01 | 48.04 $\pm$ 0.07 | -23.504 $\pm$ 0.001 | -23.503 $\pm$ 0.010
13 | 36.81 $\pm$ 0.02 | 36.82 $\pm$ 0.04 | -21.871 $\pm$ 0.003 | -21.875 $\pm$ 0.009
23 | 38.14 $\pm$ 0.01 | 38.13 $\pm$ 0.05 | -22.039 $\pm$ 0.002 | -22.031 $\pm$ 0.009
Table 1: Comparison of the selected results for the 1-jettiness soft function
for the kinematic point in Eq. (203). When quoting results for functions
$G_{ij}^{nl}$, $Q_{ij}^{nl}$, we show Vegas integration errors which, most
likely, underestimate the true uncertainties of the result.

###  D.2 2-jettiness

We again consider the configuration where there are two back-to-back beams. We
parameterize the scattering of the two jets by $\theta_{13}$, $\theta_{14}$
and $\phi_{4}$ as

$\begin{split}n_{1}=(0,0,1),&\quad n_{2}=(0,0,-1),\\\
n_{3}=(\sin\theta_{13},0,\cos\theta_{13}),&\quad
n_{4}=(\sin\theta_{14}\cos\phi_{4},\sin\theta_{14}\sin\phi_{4},\cos\theta_{14}).\end{split}$
(204)

Considering a generic phase space point where the beams and jets are separated
from each other, we take

$\theta_{13}=\frac{6\pi}{25},\quad\theta_{14}=\frac{13\pi}{25},\quad\phi_{4}=\frac{\pi}{5}.$
(205)

Results for all possible dipole coefficients are presented on Table 2. In the
case of $2$-jettiness, the tripole contribution can be reduced to a single
color structure, $f_{ABC}\ T_{1}^{A}T_{2}^{B}T_{3}^{C}$. The renormalized sum
of all tripole contributions in this configuration is $1064.77\pm 0.08$, which
is in agreement with the value ($1064.6\pm 0.1$) reported in Bell:2023yso .

Dipoles | Gluons | Quarks
---|---|---
$G_{ij}^{nl}$ | Ref. Bell:2023yso  | $Q_{ij}^{nl}$ | Ref. Bell:2023yso
12 | 71.15 $\pm$ 0.05 | 71.11 $\pm$ 0.12 | -27.837 $\pm$ 0.001 | -27.841 $\pm$ 0.011
13 | 36.27 $\pm$ 0.02 | 36.17 $\pm$ 0.07 | -21.719 $\pm$ 0.005 | -21.724 $\pm$ 0.009
23 | 75.79 $\pm$ 0.01 | 75.62 $\pm$ 0.09 | -27.804 $\pm$ 0.002 | -27.807 $\pm$ 0.011
14 | 65.48 $\pm$ 0.02 | 65.38 $\pm$ 0.09 | -25.660 $\pm$ 0.003 | -25.666 $\pm$ 0.010
24 | 46.25 $\pm$ 0.01 | 46.15 $\pm$ 0.06 | -22.933 $\pm$ 0.005 | -22.908 $\pm$ 0.009
34 | 44.86 $\pm$ 0.02 | 44.72 $\pm$ 0.09 | -22.513 $\pm$ 0.004 | -22.518 $\pm$ 0.009
Table 2: Comparison of the selected results for the 2-jettiness soft function
for the kinematic point in Eq. (205). When quoting results for functions
$G_{ij}^{nl}$, $Q_{ij}^{nl}$, we show Vegas integration errors which, most
likely, underestimate the true uncertainties of the result.

###  D.3 3-jettiness

We parameterize the scattering of the additional jet by $\theta_{15}$ and
$\phi_{5}$

$n_{5}=(\sin\theta_{15}\cos\phi_{5},\sin\theta_{15}\sin\phi_{5},\cos\theta_{15}).$
(206)

We compare in Table 3 our 3-jettiness dipole contributions to the benchmark
result in Bell:2023yso  by taking the following phase space point

$\theta_{13}=\frac{3\pi}{10},\quad\theta_{14}=\frac{6\pi}{10},\quad\theta_{15}=\frac{9\pi}{10},\quad\phi_{4}=\frac{3\pi}{5},\quad\phi_{5}=\frac{6\pi}{5}.$
(207)

Dipoles | Gluons | Quarks
---|---|---
$G_{ij}^{nl}$ | Ref. Bell:2023yso  | $Q_{ij}^{nl}$ | Ref. Bell:2023yso
12 | 116.20 $\pm$ 0.01 | 116.20 $\pm$ 0.16 | -36.249 $\pm$ 0.001 | -36.244 $\pm$ 0.009
13 | 38.13 $\pm$ 0.03 | 37.63 $\pm$ 0.03 | -21.717 $\pm$ 0.007 | -21.732 $\pm$ 0.005
14 | 63.63 $\pm$ 0.01 | 63.66 $\pm$ 0.06 | -25.189 $\pm$ 0.003 | -25.192 $\pm$ 0.006
15 | 107.17 $\pm$ 0.01 | 106.99 $\pm$ 0.12 | -35.268 $\pm$ 0.001 | -35.256 $\pm$ 0.009
23 | 97.11 $\pm$ 0.01 | 96.97 $\pm$ 0.10 | -32.875 $\pm$ 0.002 | -32.872 $\pm$ 0.008
24 | 67.36 $\pm$ 0.02 | 67.51 $\pm$ 0.08 | -26.821 $\pm$ 0.003 | -26.815 $\pm$ 0.007
25 | 30.87 $\pm$ 0.03 | 30.73 $\pm$ 0.04 | -21.561 $\pm$ 0.009 | -21.561 $\pm$ 0.005
34 | 69.43 $\pm$ 0.01 | 69.24 $\pm$ 0.07 | -25.854 $\pm$ 0.002 | -25.861 $\pm$ 0.006
35 | 106.13 $\pm$ 0.02 | 105.97 $\pm$ 0.13 | -34.799 $\pm$ 0.002 | -34.796 $\pm$ 0.008
45 | 74.45 $\pm$ 0.02 | 74.36 $\pm$ 0.09 | -28.247 $\pm$ 0.004 | -28.251 $\pm$ 0.007
Table 3: Comparison of the selected results for the 3-jettiness soft function
for the kinematic point in Eq. (207). When quoting results for functions
$G_{ij}^{nl}$, $Q_{ij}^{nl}$, we show Vegas integration errors which, most
likely, underestimate the true uncertainties of the result.

For the tripole contribution in this configuration, there are four independent
color structures. In Table 4 we present our results for each of these
configurations, as defined in Bell:2023yso .

| $\tilde{c}_{\text{tripoles}}^{(2,124)}$ | $\tilde{c}_{\text{tripoles}}^{(2,125)}$ | $\tilde{c}_{\text{tripoles}}^{(2,145)}$ | $\tilde{c}_{\text{tripoles}}^{(2,245)}$
---|---|---|---|---
$\tilde{c}_{\text{tripoles}}$ | -683.25 $\pm$ 0.01 | -2203.3 $\pm$ 0.2 | -6.324 $\pm$ 0.004 | -0.837 $\pm$ 0.008
Ref. Bell:2023yso  | -683.23 $\pm$ 0.04 | -2203.5 $\pm$ 0.1 | -6.325 $\pm$ 0.04 | -0.830 $\pm$ 0.039
Table 4: Same as in Table 3 for the four independent triple-color correlated
contributions.

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