111SI-HEP-2020-11, QFET-2020-01, Nikhef 2020-007, DESY 19-157.aainstitutetext:
Theoretische Physik 1, Naturwissenschaftlich-Technische Fakultät, Universität
Siegen,
Walter-Flex-Straße 3, 57068 Siegen, Germanybbinstitutetext: Institute for
Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics,
University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, The
Netherlandsccinstitutetext: Nikhef, Theory Group, Science Park 105, 1098 XG,
Amsterdam, The Netherlandsddinstitutetext: Albert Einstein Center for
Fundamental Physics, Institut für Theoretische Physik,
Universität Bern, Sidlerstrasse 5, 3012 Bern, Switzerlandeeinstitutetext:
Niels Bohr International Academy, Niels Bohr Institute, University of
Copenhagen,
Blegdamsvej 17, DK-2100 Copenhagen, Denmarkffinstitutetext: Theory Group,
Deutsches Elektronen-Synchrotron (DESY), Notkestraße 85,
22607 Hamburg, Germany

# Generic dijet soft functions at two-loop order: uncorrelated emissions

Guido Bell b,c,d Rudi Rahn e,f and Jim Talbert<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We extend our algorithm for automating the calculation of two-loop dijet soft
functions to observables that do not obey the non-Abelian exponentiation
theorem, i.e. to those that require an independent calculation of the
uncorrelated-emission contribution. As the singularity structure of
uncorrelated double emissions differs substantially from the one for
correlated emissions, we introduce a novel phase-space parametrisation that
isolates the corresponding divergences. The resulting integrals are
implemented in SoftSERVE 1.0, which we release alongside of this work, and
which we supplement by a regulator that is consistent with the rapidity
renormalisation group framework. Using our automated setup, we confirm
existing results for various jet-veto observables and provide a novel
prediction for the soft-drop jet-grooming algorithm.

###### Keywords:

QCD, Soft-Collinear Effective Theory (SCET), NNLO Computations

##  1 Introduction

The perturbative calculation of soft functions provides insights into the
infrared structure of gauge theory amplitudes and enables the resummation of
logarithmically enhanced corrections to all orders in perturbation theory.
Starting at next-to-next-to-leading order (NNLO) and beyond, the perturbative
computations often become intricate since the divergences in the phase-space
integrations overlap. This motivated us to develop a systematic algorithm for
the calculation of two-loop soft functions in Bell:2018vaa ; Bell:2018oqa ,
which exploits the fact that the defining matrix element of the soft functions
is independent of the observable for a given hard-scattering process.

In this work we are concerned with soft functions that arise in processes with
two massless, coloured, hard partons that are in a back-to-back configuration.
These _dijet soft functions_ can be defined in terms of two light-like Wilson
lines $S_{n}$ and $S_{\bar{n}}$, which embed the eikonal form of the soft
interactions and which trace the directions $n^{\mu}$ and $\bar{n}^{\mu}$ of
the (initial or final-state) hard partons with $n^{2}=\bar{n}^{2}=0$ and
$n\cdot{\bar{n}}=2$. A generic soft function of this type can be written in
the form

$\displaystyle S(\tau,\mu)=\frac{1}{N_{c}}\;\sum_{i\in
X}\;\mathcal{M}(\tau;\\{k_{i}\\})\;\mathrm{Tr}\;|\langle
X|\,T[S^{\dagger}_{n}(0)S_{\bar{n}}(0)]\,|0\rangle|^{2}\,,$ (1)

where $\mathcal{M}\left(\tau;\\{k_{i}\\}\right)$ represents an observable-
specific measurement on the soft radiation $X$ with partonic momenta
$\\{k_{i}\\}$, which – after isolating the singularities present in the soft
matrix element – acts as a weight factor for the phase-space integrations.

In Bell:2018oqa  we specified a number of constraints that we impose on the
functional form of the measurement function
$\mathcal{M}\left(\tau;\\{k_{i}\\}\right)$, whose resulting generality was
illustrated in applications to about a dozen $e^{+}e^{-}$ and hadron-collider
soft functions. What all of these observables have in common is that they are
consistent with non-Abelian exponentiation (NAE) Gatheral:1983cz ;
Frenkel:1984pz . In a non-Abelian gauge theory this implies that for any
observable the all-order soft matrix element takes the form of an exponential,
which involves only Feynman diagrams with specific colour structures. At NNLO
this fixes one of the three colour structures to the square of the NLO
amplitude, and as long as the measurement function itself factorises into two
single-emission pieces, this contribution to the soft function does not
require a dedicated calculation since it is proportional to the square of the
NLO soft function. This allowed us in Bell:2018oqa  to present complete
results for NAE observables, although the algorithm devised in that paper
applies only to two out of three NNLO colour structures, which constitute the
so-called correlated-emission contribution.

There exist, however, interesting soft functions that do not comply with NAE,
and which require an independent calculation of the uncorrelated-emission
contribution. This applies, for instance, to soft functions that are defined
in terms of a jet algorithm, which partitions the phase space of the soft
emissions into different regions in which the partons are clustered together.
As these clustering constraints do not have an analogue at lower orders, the
respective measurement function does not factorise into single-emission pieces
and the uncorrelated-emission contribution becomes non-trivial.

The singularity structure of uncorrelated double emissions differs, on the
other hand, from the one for correlated emissions, and the phase-space
parametrisation we used in Bell:2018oqa  fails to factorise the corresponding
divergences. At first sight one may think that the calculation of the
uncorrelated-emission contribution should be simpler than the one for
correlated emissions since the underlying matrix element is trivial. As we
will see in this paper, however, the singularity structure imposes more
stringent constraints on the required phase-space parametrisation in a
generic, observable-independent approach. It therefore turns out that one
cannot apply a universal parametrisation for all observables in this case, but
one instead has to resort to specific parametrisations for different classes
of soft functions. We actually already presented the phase-space
parametrisation we use for uncorrelated emissions in Bell:2018vaa ;
Bell:2018jvf , in which we focused on the divergences of the soft functions,
whereas we present complete NNLO results in this paper.

Apart from devising a systematic algorithm for the calculation of dijet soft
functions, we developed a stand-alone program called SoftSERVE for their
numerical evaluation Bell:2018oqa . Whereas the previous version SoftSERVE 0.9
could only be used for the calculation of the correlated-emission
contribution, the new version SoftSERVE 1.0 – which we publish alongside this
paper – contains a number of new features. Most importantly, we implemented
the master formula derived in this work for the calculation of the
uncorrelated-emission contribution, such that SoftSERVE 1.0 can now handle
generic dijet soft functions that comply with our ansatz, both for NAE and
NAE-violating observables. Moreover, the new version contains a script for the
renormalisation of cumulant soft functions, which differs from the one for
Laplace-space soft functions considered in Bell:2018oqa , and we implemented
the formulae from Bell:2018vaa , which allow for a direct calculation of the
soft anomalous dimension (and also the collinear anomaly exponent
Becher:2010tm ; Becher:2011pf ), without having to calculate the complete bare
soft function. Finally, we argued in Bell:2018oqa  that the rapidity regulator
that is used in SoftSERVE 0.9 is not suited for the rapidity renormalisation
group (RRG) approach Chiu:2012ir , since it is not implemented on the level of
connected webs. In the new version we remedied this point by adding an option
which allows the user to run SoftSERVE with different rapidity regulators.
Whereas we briefly comment on all of these changes in this work, we refer to
the SoftSERVE user manual for more detailed explanations. The SoftSERVE
distribution is publicly available at https://softserve.hepforge.org/.

The remainder of the paper develops as follows: in Section 2 we introduce the
phase-space parametrisation we use for uncorrelated emissions as well as the
corresponding form of the measurement function. In Section 3 we employ this
parametrisation to obtain a master formula for the calculation of the
uncorrelated-emission contribution to a generic bare two-loop soft function,
which we then renormalise in Section 4. In Section 5 we briefly review the
technical aspects of the SoftSERVE extension, and we present sample results
for NAE and NAE-violating observables in Section 6, including a novel
calculation of an NNLO soft function for the soft-drop jet-grooming algorithm.
We finally conclude in Section 7, and we present some technical aspects of our
analysis in an appendix.

##  2 Measurement function

Following the procedure outlined in Bell:2018oqa , we restrict ourselves to
soft functions whose defining measurements are of the form

$\displaystyle\mathcal{M}(\tau;\\{k_{i}\\})=\exp\big{(}-\tau\,\omega(\\{k_{i}\\})\,\big{)}\,,$
(2)

where it is clear from the exponential that we typically evaluate the soft
functions in some space conjugate to momentum space, e.g. Laplace or Fourier
space. The variable $\tau$ then denotes the associated conjugate variable, and
the function $\omega(\\{k_{i}\\})$ characterises the specific constraint on
the final-state momenta that is provided by the observable in question. More
specifically, we assume that

  1. (A1)

the soft function is embedded in a dijet factorisation theorem and it has a
double-logarithmic evolution in the renormalisation scale $\mu$ and, possibly,
also the rapidity scale $\nu$;

  2. (A2)

$\Re\big{(}\omega(\\{k_{i}\\})\big{)}\geq 0$ and $\omega(\\{k_{i}\\})$ is
allowed to vanish only for configurations with zero weight in the phase-space
integrations, and it is furthermore supposed to be independent of the
dimensional and the rapidity regulators;

  3. (A3)

the variable $\tau$ has dimension 1/mass;

  4. (A4)

the function $\omega(\\{k_{i}\\})$ is symmetric under
$n^{\mu}\leftrightarrow\bar{n}^{\mu}$ exchange;

  5. (A5)

the soft function depends only on one variable $\tau$ in conjugate space,
although we already showed in Bell:2018oqa  how to relax this condition, which
is needed e.g. for multi-differential soft functions;

  6. (A6)

the function $\omega(\\{k_{i}\\})$ depends only on one angle $\theta_{i}$ per
emission in the $(d-2)$-dimensional transverse plane to $n^{\mu}$ and
$\bar{n}^{\mu}$ as well as on relative angles $\theta_{ij}$ between two
emissions.

For further explanations regarding these assumptions, we refer the reader to
the corresponding section in Bell:2018oqa .

In order to illustrate the implications of these assumptions, we considered
three template observables in Bell:2018oqa  relevant for $e^{+}e^{-}$ event
shapes, threshold resummation and transverse-momentum resummation that are all
consistent with NAE. We find it convenient to proceed similarly in this work,
and to highlight the salient features of NAE-violating observables with a
specific example. To this end, we consider the C-parameter-like jet veto
observable $\mathcal{T}_{C\rm cm}$ from Gangal:2014qda , whose measurement
function in Laplace space can be written in the form (2), except for a global
factor of $1/\tau$ which arises because the constraint on the soft radiation
is given in momentum space in the form of a $\theta$-function rather than a
$\delta$-function. This factor is typical for cumulant soft functions, and we
will investigate its consequences more closely when we discuss renormalisation
in Section 4. For the calculation of the bare soft function, however, this
factor is just a constant and can be ignored.

For zero and one emissions, the observable is just the usual C-parameter event
shape, which we discussed at length in Bell:2018oqa . The clustering
constraint, on the other hand, only becomes relevant for two and more
emissions. Specifically for two emissions with momenta $k$ and $l$, we have

$\displaystyle\omega^{CPV}(R;k,l)$
$\displaystyle=\theta(\Delta-R)\;\max\bigg{(}\frac{k_{+}k_{-}}{k_{+}+k_{-}},\frac{l_{+}l_{-}}{l_{+}+l_{-}}\bigg{)}+\theta(R-\Delta)\;\frac{(k_{+}+l_{+})(k_{-}+l_{-})}{k_{+}+l_{+}+k_{-}+l_{-}}\,,$
(3)

where we introduced light-cone coordinates via $k_{+}=n\cdot k$ and
$k_{-}=\bar{n}\cdot k$, and

$\displaystyle\Delta=\sqrt{\frac{1}{4}\ln^{2}\frac{k_{-}l_{+}}{k_{+}l_{-}}+\theta_{kl}^{2}}$
(4)

represents the distance measure of the jet algorithm. From (3) we see that
emissions that are closer than the jet radius $R$ are clustered together,
whereas those that are further apart are treated as individual emissions, such
that the jet veto constrains the one with a larger value of the C-parameter.
One easily verifies that the assumptions (A1)-(A6) are satisfied for this
observable, and from (3) it is obvious that $\omega^{CPV}(R;k,l)$ cannot be
written as a sum of single-emission functions, which would be required for a
factorisation of the measurement function (2). The observable therefore
violates NAE.

In analogy to the correlated-emission calculation from Bell:2018oqa , we need
to find a parametrisation of the double-emission measurement function that has
a well-defined behaviour in the singular limits of the corresponding matrix
element. The parametrisation we use for uncorrelated emissions was already
given in Bell:2018jvf ,

$\displaystyle y_{k}$ $\displaystyle=\frac{k_{+}}{k_{-}}\,,\hskip
34.14322ptq_{T}=\sqrt{k_{+}k_{-}}\left(\frac{l_{-}+l_{+}}{\sqrt{l_{+}l_{-}}}\right)^{n}+\sqrt{l_{+}l_{-}}\left(\frac{k_{-}+k_{+}}{\sqrt{k_{+}k_{-}}}\right)^{n},$
$\displaystyle y_{l}$ $\displaystyle=\frac{l_{+}}{l_{-}}\,,\hskip
42.67912ptb=\left(\frac{k_{+}k_{-}}{l_{+}l_{-}}\right)^{\frac{n+1}{2}}\left(\frac{l_{-}+l_{+}}{k_{-}+k_{+}}\right)^{n},$
(5)

where $n$ is a parameter that is related to the power counting of the modes in
the effective theory – see the discussion in Bell:2018oqa . Unlike the
correlated-emission case, we thus use specific parametrisations for classes of
observables that correspond to the same value of $n$. The parametrisation
becomes, for instance, particularly simple for SCET-2 soft functions where
$n=0$.

In physical terms, the variables $y_{k}$ and $y_{l}$ are measures of the
rapidities of the individual partons, whereas $b$ and $q_{T}$ only have a
simple interpretation for $n=0$, where they correspond to the ratio and the
scalar sum of their transverse momenta, respectively (the $n$-dependent terms
introduce rapidity-dependent weight factors). Similar to Bell:2018oqa , the
parametrisation is supplemented by the angular variables

$\displaystyle t_{k}=\frac{1-\cos\theta_{k}}{2}\,,\qquad\qquad
t_{l}=\frac{1-\cos\theta_{l}}{2}\,,\qquad\qquad
t_{kl}=\frac{1-\cos\theta_{kl}}{2}\,,$ (6)

with $\theta_{k}=\sphericalangle(\vec{v}_{\perp},\vec{k}_{\perp})$,
$\theta_{l}=\sphericalangle(\vec{v}_{\perp},\vec{l}_{\perp})$ and
$\theta_{kl}=\sphericalangle(\vec{k}_{\perp},\vec{l}_{\perp})$. The vector
$v^{\mu}$ encodes a potential azimuthal dependence of the observable around
the collinear axis – see Bell:2018oqa  for specific examples. The inverse
transformation to (5) can be found in Bell:2018vaa .

The integration ranges for the variables $y_{k}$, $y_{l}$ and $b$ span the
entire positive real axis and, similar to the correlated-emission case, they
can be mapped onto the unit hypercube using symmetry arguments. The implicit
phase-space divergences then arise in the following four limits:

  * •

$q_{T}\rightarrow 0$, which corresponds to the situation in which both emitted
partons become soft;

  * •

$b\rightarrow 0$, which implies that the parton with momentum $k^{\mu}$
becomes soft (compared to $l^{\mu}$);

  * •

$y_{k}\rightarrow 0$, which reflects the fact that the parton with momentum
$k^{\mu}$ becomes collinear to the direction $n^{\mu}$ (at fixed transverse
momentum);

  * •

$y_{l}\rightarrow 0$, which is the corresponding limit for the parton with
momentum $l^{\mu}$.

As $q_{T}$ is the only dimensionful variable in our parametrisation and the
mass dimension of the variable $\tau$ is fixed by (A3), the function
$\omega(\\{k,l\\})=\omega(q_{T},y_{k},y_{l},b,t_{k},t_{l},t_{kl})$ must be
linear in $q_{T}$. The limit $b\to 0$ is furthermore protected by infrared
safety, which means that the measurement function cannot vanish in this limit
since it must fall back to the one-emission function Bell:2018oqa . Yet, we
still have to control the measurement function in the remaining two limits to
make sure that we can properly extract the associated divergences.

The very fact that one has to control the measurement function in two
unprotected singular limits – as opposed to one for correlated emissions – is
the main complication in the present calculation. To better illustrate this
point, let us for the moment consider a generic observable that obeys NAE,
i.e. its two-emission measurement function can be written in the form

$\displaystyle\mathcal{M}_{2}(\tau;k,l)$
$\displaystyle=\mathcal{M}_{1}(\tau;k)\;\mathcal{M}_{1}(\tau;l)\,,$
$\displaystyle=\exp\Big{\\{}-\tau\,\Big{(}k_{T}\,y_{k}^{n/2}\,f(y_{k},t_{k})+l_{T}\,y_{l}^{n/2}\,f(y_{l},t_{l})\Big{)}\,\Big{\\}},$
(7)

where we have used the explicit form of the single-emission measurement
function from Bell:2018oqa , and the function $f(y,t)$ is by construction
finite and non-zero as $y\to 0$. In order to extract the collinear divergences
that arise in the limits $y_{k}\rightarrow 0$ and $y_{l}\rightarrow 0$, one
has to make sure that the term in the round parenthesis is finite and non-zero
in either of the limits and in the combined limit $y_{k},y_{l}\to 0$ as well.
Except for $n=0$ this is obviously not the case. Factoring out $y_{k}^{n/2}$,
on the other hand, would guarantee that the first term stays finite as
$y_{k}\to 0$, but at the same time the second term would blow up for $n>0$.
Similarly, factoring out powers of $y_{l}$ does not help to make the
expression in the parenthesis finite as $y_{l}\to 0$.

The problem is solved by the specific form of the parametrisation (5). In
terms of these variables, the transverse-momentum variables $k_{T}$ and
$l_{T}$ take the form

$\displaystyle
k_{T}=\sqrt{k_{+}k_{-}}=\left(\frac{\sqrt{y_{l}}}{1+y_{l}}\right)^{n}\frac{b}{1+b}\;\,q_{T}\,,\qquad
l_{T}=\sqrt{l_{+}l_{-}}=\left(\frac{\sqrt{y_{k}}}{1+y_{k}}\right)^{n}\frac{1}{1+b}\;\,q_{T}\,,$
(8)

which – when inserted into (7) – shows that both terms in the parenthesis are
proportional to $y_{k}^{n/2}y_{l}^{n/2}$. Once this term is factored out, the
remaining expression is thus finite and non-zero in the collinear limits as
desired. This explains why the phase-space parametrisation for uncorrelated
emissions must be $n$-dependent, and it motivates the following ansatz for the
double-emission measurement function:

$\displaystyle\mathcal{M}_{2}^{unc}(\tau;k,l)=\exp\big{(}-\tau\,q_{T}\,y_{k}^{n/2}\,y_{l}^{n/2}\,G(y_{k},y_{l},b,t_{k},t_{l},t_{kl})\,\big{)}\,,$
(9)

where the dependence on $q_{T}$ is fixed on dimensional grounds and the
function $G$ is supposed to be finite and non-zero as $y_{k}\to 0$ and
$y_{l}\to 0$. Although our discussion started from the specific form (7) of a
NAE observable, we expect that generic NAE-violating observables can be
written in the form (9) as well. The reason is that the soft function is by
assumption embedded in a dijet factorisation theorem – see (A1) – and the pole
cancellation between the various regions requires that a potential NAE-
violating term in the two-emission measurement function cannot upset the
scaling in the limits $y_{k}\to 0$ and $y_{l}\to 0$. The discussion is
actually similar to the one in Appendix A of Bell:2018oqa .

As an example we consider the jet-veto template from above, which corresponds
to $n=1$, $f(y_{k},t_{k})=1/(1+y_{k})$ and

$\displaystyle G(y_{k},y_{l},b,t_{k},t_{l},t_{kl})$
$\displaystyle=\theta(\Delta_{G}-R)\;\frac{\max(1,b)}{(1+b)\,(1+y_{k})\,(1+y_{l})}$
(10)
$\displaystyle\quad+\theta(R-\Delta_{G})\;\frac{\big{(}1+y_{l}+(1+y_{k})b\big{)}\,\big{(}y_{k}(1+y_{l})+by_{l}(1+y_{k})\big{)}}{\big{(}1+b\big{)}\,\big{(}1+y_{k}\big{)}\,\big{(}1+y_{l}\big{)}\,\big{(}y_{k}(1+y_{l})^{2}+by_{l}(1+y_{k})^{2}\big{)}}\,,$

where the distance measure is now given by

$\displaystyle\Delta_{G}=\sqrt{\frac{1}{4}\ln^{2}\frac{y_{k}}{y_{l}}+\arccos^{2}(1-2t_{kl})}\,.$
(11)

Due to the factorisation of $\sqrt{y_{k}y_{l}}$ in (9), we see that the
expression in (10) is indeed finite in the limits $y_{k}\to 0$ and $y_{l}\to
0$ as required. The distance measure (11) reveals, moreover, that the precise
form in which the collinear limits are evaluated matters, and we will come
back to this point at the end of this section.

Before doing so, we analyse the general constraints on the double-emission
measurement function that arise from infrared safety. Following Bell:2018oqa ,
we express the variables $b$ and $q_{T}$ in terms of those that parametrise
the one-particle phase space for each of the emitted partons,

$\displaystyle
b=\frac{k_{T}}{l_{T}}\left(\frac{\sqrt{y_{k}}}{1+y_{k}}\,\frac{1+y_{l}}{\sqrt{y_{l}}}\right)^{n},\qquad
q_{T}=k_{T}\left(\frac{1+y_{l}}{\sqrt{y_{l}}}\right)^{n}+l_{T}\left(\frac{1+y_{k}}{\sqrt{y_{k}}}\right)^{n}.$
(12)

The limit in which the parton with momentum $k^{\mu}$ becomes soft then
corresponds to $k_{T}\to 0$, which translates into $b\to 0$ and $q_{T}\to
l_{T}\big{(}(1+y_{k})/\sqrt{y_{k}}\big{)}^{n}$. Infrared safety implies that
the double-emission measurement function is related to the single-emission
function in this limit, which yields

$\displaystyle
G(y_{k},y_{l},0,t_{k},t_{l},t_{kl})=\frac{f(y_{l},t_{l})}{(1+y_{k})^{n}}\,.$
(13)

As stated above, this relation guarantees that the function $G$ does not
vanish in one of the singular limits of the uncorrelated-emission
contribution. One can derive a similar constraint in the limit in which the
two emitted partons become collinear to each other, and in this case one finds

$\displaystyle
G(y_{l},y_{l},b,t_{l},t_{l},0)=\frac{f(y_{l},t_{l})}{(1+y_{l})^{n}}\,.$ (14)

Relations (13) and (14) reflect the fact that the observable is infrared safe,
and they can easily be checked explicitly for the jet-veto template from
above.

As already mentioned, we find it convenient to map the integration region onto
the unit hypercube using symmetry arguments under $n\leftrightarrow\bar{n}$
and $k\leftrightarrow l$ exchange. Similar to Bell:2018oqa , this comes at the
price of introducing two different versions of the measurement function, which
we label by the letters “A” and “B”. As we will explain in more detail in
Section 3, they are given by

$\displaystyle G_{A}(y_{k},y_{l},b,t_{k},t_{l},t_{kl})$
$\displaystyle=G(y_{k},y_{l},b,t_{k},t_{l},t_{kl})\,,$ $\displaystyle
G_{B}(y_{k},y_{l},b,t_{k},t_{l},t_{kl})$
$\displaystyle=\begin{dcases}\,y_{k}^{-n}\;G(1/y_{k},y_{l},b,t_{k},t_{l},t_{kl})&\,\text{or}\\\
\,y_{l}^{-n}\;G(y_{k},1/y_{l},b,t_{k},t_{l},t_{kl})\,.\end{dcases}$ (15)

Physically, region A corresponds to the case in which both partons are emitted
into the same hemisphere with respect to the collinear axis, whereas region B
describes the opposite-hemisphere case.

Finally, we saw in (11) that the distance measure of the jet algorithm is
ambiguous in the double limit $y_{k}\to 0$ and $y_{l}\to 0$, since it matters
if the limit is evaluated at a fixed ratio $y_{k}/y_{l}$ or if it is evaluated
sequentially. Physically, this corresponds to a distinction between the joint
collinear limit of the emitted partons at a fixed rapidity distance and the
individual collinear limits of each of the partons. The ambiguity only arises
in the same-hemisphere case, and it can be disentangled via a sector
decomposition strategy. As we will show in the next section, this introduces
two subregions in region A with

$\displaystyle G_{A_{1}}(y,r,b,t_{k},t_{l},t_{kl})$
$\displaystyle=G_{A}(y,ry,b,t_{k},t_{l},t_{kl})\,,$ $\displaystyle
G_{A_{2}}(y,r,b,t_{k},t_{l},t_{kl})$
$\displaystyle=G_{A}(ry,y,b,t_{k},t_{l},t_{kl})\,.$ (16)

##  3 Calculation of the bare soft function

Having specified the measurement function for two uncorrelated emissions, the
calculation of the bare soft function defined in (1) proceeds along the lines
outlined for the correlated-emission contribution in Bell:2018oqa . In the
following we adopt the notation from that paper and we assume that the Wilson
lines are given in the fundamental colour representation.

The bare soft function has a double expansion in the dimensional regulator
$\epsilon=(4-d)/2$ and the rapidity regulator $\alpha$, which we implement on
the level of the phase-space integrals via the prescription Becher:2011dz

$\int d^{d}p\;\left(\frac{\nu}{n\cdot p+\bar{n}\cdot
p}\right)^{\alpha}\;\delta(p^{2})\theta(p^{0})\,.$ (17)

The rapidity regulator is required only for SCET-2 soft functions, and we will
introduce an alternative version that is compatible with the RRG framework
later in Section 4.2. Up to NNLO the bare soft function can then be written in
the form

$\displaystyle S_{0}(\tau,\nu)=1$
$\displaystyle+\left(\frac{Z_{\alpha}\alpha_{s}}{4\pi}\right)\,(\mu^{2}\bar{\tau}^{2})^{\epsilon}\;(\nu\bar{\tau})^{\alpha}\,S_{R}(\epsilon,\alpha)$
$\displaystyle+\left(\frac{Z_{\alpha}\alpha_{s}}{4\pi}\right)^{2}(\mu^{2}\bar{\tau}^{2})^{2\epsilon}\;\bigg{\\{}(\nu\bar{\tau})^{\alpha}\,S_{RV}(\epsilon,\alpha)+(\nu\bar{\tau})^{2\alpha}\,S_{RR}(\epsilon,\alpha)\bigg{\\}}+\mathcal{O}(\alpha_{s}^{3})\,,$
(18)

where $\bar{\tau}=\tau e^{\gamma_{E}}$ and $\alpha_{s}$ is the renormalised
strong coupling constant in the $\overline{\text{MS}}$ scheme. In Bell:2018oqa
we presented the calculation of the single real-emission correction
$S_{R}(\epsilon,\alpha)$, the mixed real-virtual interference
$S_{RV}(\epsilon,\alpha)$ and two out of three colour structures
($C_{F}C_{A}$, $C_{F}T_{F}n_{f}$) of the double real-emission contribution
$S_{RR}(\epsilon,\alpha)$, and the goal of the present paper consists in
computing the last missing NNLO ingredient, i.e. the $C_{F}^{2}$ contribution
to $S_{RR}(\epsilon,\alpha)$.

The starting point of our calculation is the representation

$\displaystyle S_{RR}^{(C_{F})}(\epsilon,\alpha)$ $\displaystyle=\frac{(4\pi
e^{\gamma_{E}}\tau^{2})^{-2\epsilon}\;\bar{\tau}^{-2\alpha}}{(2\pi)^{2d-2}}\,\int
d^{d}k\;\,\delta(k^{2})\,\theta(k^{0})\,\int
d^{d}l\;\,\delta(l^{2})\,\theta(l^{0})$
$\displaystyle\quad\times\frac{|\mathcal{A}_{RR}^{(C_{F})}(k,l)|^{2}}{(n\cdot
k+\bar{n}\cdot k)^{\alpha}\,(n\cdot l+\bar{n}\cdot
l)^{\alpha}}\;\,\mathcal{M}_{2}^{unc}(\tau;k,l)\,,$ (19)

where – due to NAE – the squared matrix element is given by

$\displaystyle|\mathcal{A}_{RR}^{(C_{F})}(k,l)|^{2}$
$\displaystyle=\frac{|\mathcal{A}_{R}(k)|^{2}\,|\mathcal{A}_{R}(l)|^{2}}{2}=\frac{2048\pi^{4}\,C_{F}^{2}}{k_{+}k_{-}l_{+}l_{-}}\,.$
(20)

From (19) it is evident that the calculation reduces to the square of the NLO
soft function if the observable obeys NAE, i.e. if its double-emission
measurement function is of the form (7). We do not assume here, however, that
this is the case and instead use the more general parametrisation (9) of the
measurement function.

Starting from (19), we thus switch to the variables introduced in (5) and (6)
and perform the observable-independent integrations, following Bell:2018oqa
for a convenient parametrisation of the angular integrals in the
$(d-2)$-dimensional transverse plane. In order to map the integration ranges
in the variables $y_{k}$, $y_{l}$ and $b$ onto the unit hypercube, we exploit
the fact that the variables transform under $n\leftrightarrow\bar{n}$ exchange
as

$y_{k}\rightarrow\frac{1}{y_{k}}\,,\qquad
y_{l}\rightarrow\frac{1}{y_{l}}\,,\qquad b\rightarrow b\,,\qquad
t_{k}\rightarrow t_{k}\,,\qquad t_{l}\rightarrow t_{l}\,,\qquad
t_{kl}\rightarrow t_{kl}\,,$ (21)

whereas the corresponding relations under $k\leftrightarrow l$ exchange are
given by

$y_{k}\rightarrow y_{l}\,,\qquad y_{l}\rightarrow y_{k}\,,\qquad
b\rightarrow\frac{1}{b}\,,\qquad t_{k}\rightarrow t_{l}\,,\qquad
t_{l}\rightarrow t_{k}\,,\qquad t_{kl}\rightarrow t_{kl}\,.$ (22)

Proceeding in analogy to the correlated-emission calculation in Bell:2018oqa ,
we can use these symmetry considerations to map the integration domain onto
two independent regions that are illustrated in Figure 1. In region A, which
we take to be the highlighted dashed blue cube in Figure 1(c), the integrand
is simply the original integrand in which no substitutions are made. The
second region B, on the other hand, refers to any of the white adjacent cubes
in this figure, and it can be most easily recovered from the original
integrand by inverting either of the variables $y_{k}$ or $y_{l}$.

(a) $n\leftrightarrow\bar{n}$ exchange

(b) $k\leftrightarrow l$ exchange

(c) Reduced integration region

Figure 1: Reduction of the integration domain in the variables $y_{k}$,
$y_{l}$ and $b$ for the uncorrelated-emission contribution. Cubes of the same
colour correspond to integration regions which yield the same result according
to the stated symmetries. The second integration region B, complementing the
highlighted region A in (c), can be any of the white adjacent cubes. In
practice it is most easily recovered from A by inverting either $y_{k}$ or
$y_{l}$.

After performing all of these manipulations, we arrive at the following master
formula for the calculation of the uncorrelated-emission contribution

$\displaystyle
S_{RR}^{(C_{F})}(\epsilon,\alpha)=\frac{128C_{F}^{2}\,e^{-2\gamma_{E}(\epsilon+\alpha)}\,\Gamma(-4\epsilon-2\alpha)}{\pi^{3/2}\,\Gamma(-\epsilon)\,\Gamma(1/2-\epsilon)}\;\int_{0}^{1}\\!dy_{k}\int_{0}^{1}\\!dy_{l}\int_{0}^{1}\\!db\int_{0}^{1}\\!dt_{kl}\int_{0}^{1}\\!dt_{l}\int_{0}^{1}\\!dt_{5}^{\prime}$
$\displaystyle\qquad\times
b^{-1-2\epsilon-\alpha}\,(y_{k}y_{l})^{-1+n\epsilon+(n+1)\alpha/2}\,(1+b)^{4\epsilon+2\alpha}\;\big{[}(1+y_{k})(1+y_{l})\big{]}^{2n\epsilon+(n-1)\alpha}$
$\displaystyle\qquad\times\big{(}4t_{kl}\bar{t}_{kl}\big{)}^{-1/2-\epsilon}\;\big{(}4t_{l}\bar{t}_{l}\big{)}^{-1/2-\epsilon}\;\big{(}t_{5}^{\prime}(2-t_{5}^{\prime})\big{)}^{-1-\epsilon}\;$
$\displaystyle\qquad\times\Big{\\{}G_{A}(y_{k},y_{l},b,t_{k}^{+},t_{l},t_{kl})^{4\epsilon+2\alpha}+G_{B}(y_{k},y_{l},b,t_{k}^{+},t_{l},t_{kl})^{4\epsilon+2\alpha}+(t_{k}^{+}\to
t_{k}^{-})\Big{\\}}$ (23)

with

$t_{k}^{\pm}=t_{l}+t_{kl}-2t_{l}t_{kl}\pm
2\sqrt{t_{l}\bar{t}_{l}t_{kl}\bar{t}_{kl}}\,(1-t_{5}^{\prime})$ (24)

and

$\displaystyle G_{A}(y_{k},y_{l},b,t_{k},t_{l},t_{kl})$
$\displaystyle=G(y_{k},y_{l},b,t_{k},t_{l},t_{kl})\,,$ $\displaystyle
G_{B}(y_{k},y_{l},b,t_{k},t_{l},t_{kl})$
$\displaystyle=\begin{dcases}\,y_{k}^{-n}\;G(1/y_{k},y_{l},b,t_{k},t_{l},t_{kl})&\,\text{or}\\\
\,y_{l}^{-n}\;G(y_{k},1/y_{l},b,t_{k},t_{l},t_{kl})\,.\end{dcases}$ (25)

In physical terms region A describes the emission of two soft partons into the
same hemisphere with respect to the collinear axis, whereas region B covers
the opposite-hemisphere case. Similar to Bell:2018oqa , the expression in
region B is not unique, since the symmetry arguments only guarantee that the
integrals in (23) are equal, but not necessarily the integrands. One is
therefore free to derive the functional form of $G_{B}$ using either of the
expressions on the right-hand side of (25).

From (23) we can analyse the divergence structure of the uncorrelated-emission
contribution. For SCET-1 observables with $n\neq 0$, one can set the analytic
regulator $\alpha$ to zero, and one finds an explicit divergence encoded in
$\Gamma(-4\epsilon)$ that originates from the analytic integration over the
dimensionful variable $q_{T}$. The integrand is, moreover, divergent in the
limits $b\to 0$, $y_{k}\to 0$ and $y_{l}\to 0$ as anticipated in Section 2. In
addition, there exists a spurious divergence in the limit $t_{5}^{\prime}\to
0$, which is cancelled by the prefactor $1/\Gamma(-\epsilon)$ as in
Bell:2018oqa . The overall contribution to the bare soft function therefore
starts with a $1/\epsilon^{4}$ divergence for SCET-1 observables.

For SCET-2 soft functions with $n=0$, the analytic regulator cannot be set to
zero, since the $y_{k}$ and $y_{l}$-integrations generate poles in $\alpha$ in
this case. As the $\alpha$-expansion has to be performed first, the terms
$b^{-1-2\epsilon-\alpha}$ and $\Gamma(-4\epsilon-2\alpha)$ introduce
additional $\epsilon$-divergences, and they trade $\alpha$-poles for
$\epsilon$-poles in the double expansion. The leading divergences in the
SCET-2 case are therefore of the form $1/(\alpha^{2}\epsilon^{2})$,
$1/(\alpha\epsilon^{3})$ and $1/\epsilon^{4}$.

Finally, we noted towards the end of Section 2 that the collinear limits
$y_{k}\to 0$ and $y_{l}\to 0$ can be ambiguous on the observable level. In
order to disentangle the joint collinear limit of the emitted partons from the
individual ones, we apply a sector decomposition strategy in the same-
hemisphere contribution and write

$\displaystyle\int_{0}^{1}\\!dy_{k}\,\int_{0}^{1}\\!dy_{l}\;\;\mathcal{I}(y_{k},y_{l})$
$\displaystyle=\int_{0}^{1}\\!dy\,\int_{0}^{1}\\!dr\;y\;\bigg{\\{}\mathcal{I}(y,ry)+\mathcal{I}(ry,y)\bigg{\\}}\,,$
(26)

where $\mathcal{I}(y_{k},y_{l})$ symbolically represents the integrand in
(23), which implicitly depends on the other integration variables. This
generates two subregions in region A with

$\displaystyle G_{A_{1}}(y,r,b,t_{k},t_{l},t_{kl})$
$\displaystyle=G_{A}(y,ry,b,t_{k},t_{l},t_{kl})\,,$ $\displaystyle
G_{A_{2}}(y,r,b,t_{k},t_{l},t_{kl})$
$\displaystyle=G_{A}(ry,y,b,t_{k},t_{l},t_{kl})\,.$ (27)

In the numerical implementation of our algorithm we perform a number of
additional substitutions that are designed to improve the numerical
convergence. For more details on this technical point we refer to Section 6 of
Bell:2018oqa  and the SoftSERVE user manual.

##  4 Renormalisation

With the master formula of the uncorrelated-emission contribution at hand, we
have assembled all ingredients required for the calculation of bare NNLO dijet
soft functions. In Bell:2018oqa  we went one step ahead and extracted the
anomalous dimensions and matching corrections that are needed for resummations
within SCET. To do so, we assumed that the renormalised soft function
$S=Z_{S}S_{0}$ obeys the renormalisation group equation (RGE)

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\ln\mu}\;S(\tau,\mu)$
$\displaystyle=-\frac{1}{n}\,\bigg{[}4\,\Gamma_{\mathrm{cusp}}(\alpha_{s})\,\ln(\mu\bar{\tau})-2\gamma^{S}(\alpha_{s})\bigg{]}\;S(\tau,\mu)$
(28)

for SCET-1 observables, whereas we focused on the collinear anomaly exponent
$\mathcal{F}(\tau,\mu)$ defined via

$\displaystyle S(\tau,\mu,\nu)$
$\displaystyle=(\nu^{2}\bar{\tau}^{2})^{-\mathcal{F}(\tau,\mu)}\;W^{S}(\tau,\mu)$
(29)

in the SCET-2 case. The calculations provided in the current paper are fully
compatible with this setup, and they provide the $C_{F}^{2}$ coefficients of
the anomalous dimensions and matching corrections that were derived in
Bell:2018oqa  on the basis of NAE.

In this paper we generalise the renormalisation programme in two respects.
First, we consider soft functions that renormalise directly in momentum (or
cumulant) space rather than Laplace space, which is relevant e.g. for certain
jet-veto observables. Second, we discuss the renormalisation of SCET-2 soft
functions in the RRG approach Chiu:2012ir , which is equivalent to the
collinear anomaly framework from Becher:2010tm ; Becher:2011pf , but which
requires a specific implementation of the rapidity regulator. We will address
both of these questions in turn.

###  4.1 Cumulant soft functions

Soft functions for jet-veto observables typically involve measurement
functions that are formulated in terms of a $\theta$-function, which reflects
the fact that the jet veto provides a cutoff for the phase-space integrations
of the soft radiation. Instead of the exponential form (2), their measurement
function can be expressed as

$\displaystyle\widehat{\mathcal{M}}(\omega;\\{k_{i}\\})=\theta\big{(}\omega-\omega(\\{k_{i}\\})\,\big{)}\,,$
(30)

where $\omega$ is the cutoff variable and the function $\omega(\\{k_{i}\\})$
is assumed to obey the same constraints that were listed in detail in Section
2.

The measurement function of such _cumulant soft functions_ can easily be
brought into the form (2) via a Laplace transformation,

$\displaystyle\int_{0}^{\infty}\text{d}\omega\;e^{-\tau\omega}\;\theta\big{(}\omega-\omega(\\{k_{i}\\})\,\big{)}$
$\displaystyle=\frac{1}{\tau}\;\exp\big{(}-\tau\,\omega(\\{k_{i}\\})\,\big{)}\,.$
(31)

The factor $1/\tau$ is just a constant for the bare soft function calculation,
but it is relevant for inverting the Laplace transformation. From (18) we see
that the individual contributions to the soft function come with different
powers of the Laplace variable $\tau$, which can be transformed back to
momentum space using the relation

$\displaystyle\int_{0}^{\infty}\text{d}\omega\;e^{-\tau\omega}\;\omega^{m}$
$\displaystyle=\Gamma(1+m)\;\tau^{-1-m}\,,\qquad m>-1\,.$ (32)

Up to NNLO a generic bare cumulant soft function therefore takes the form

$\displaystyle\widehat{S}_{0}(\omega,\nu)=1$
$\displaystyle+\left(\frac{Z_{\alpha}\alpha_{s}}{4\pi}\right)\left(\frac{\mu^{2}}{\omega^{2}}\right)^{\epsilon}\left(\frac{\nu}{\omega}\right)^{\alpha}\frac{e^{\gamma_{E}(2\epsilon+\alpha)}}{\Gamma(1-2\epsilon-\alpha)}\;S_{R}(\epsilon,\alpha)$
(33)
$\displaystyle+\left(\frac{Z_{\alpha}\alpha_{s}}{4\pi}\right)^{2}\left(\frac{\mu^{2}}{\omega^{2}}\right)^{2\epsilon}\,\bigg{\\{}\left(\frac{\nu}{\omega}\right)^{\alpha}\frac{e^{\gamma_{E}(4\epsilon+\alpha)}}{\Gamma(1-4\epsilon-\alpha)}\;S_{RV}(\epsilon,\alpha)$
$\displaystyle\hskip
113.81102pt+\left(\frac{\nu}{\omega}\right)^{2\alpha}\frac{e^{\gamma_{E}(4\epsilon+2\alpha)}}{\Gamma(1-4\epsilon-2\alpha)}\;S_{RR}(\epsilon,\alpha)\bigg{\\}}+\mathcal{O}(\alpha_{s}^{3})\,,$

where the terms $S_{i}(\epsilon,\alpha)$ for $i\in\\{R,RV,RR\\}$ can be
calculated with the formulae provided in Bell:2018oqa  and the present paper,
and their prefactors in terms of Euler’s constant and Gamma functions slightly
reshuffle the coefficients in the $\epsilon$ and $\alpha$ expansions. They do
not modify, however, the divergence structure of the soft function since they
all expand to $1+\mathcal{O}(\alpha,\epsilon)$.

We now assume that the RGEs for cumulant soft functions take the same form as
(28) and the corresponding equation in the SCET-2 case, with the replacement
$\bar{\tau}\to 1/\omega$. The renormalisation procedure that we developed for
Laplace-space soft functions in Bell:2018oqa  can then be carried over to
cumulant soft functions if the prefactors in (33) are included. As we will
explain later in Section 5, SoftSERVE 1.0 contains a script for the
renormalisation of cumulant soft functions which applies these modifications
and which takes the correct error propagation into account.

###  4.2 Rapidity renormalisation group

The collinear anomaly Becher:2010tm ; Becher:2011pf  and the RRG Chiu:2012ir
provide two equivalent frameworks for the renormalisation of SCET-2 soft
functions. In the latter the soft function is renormalised via multiplication
with a Z-factor, $S=Z_{S}S_{0}$, that absorbs the divergences both in the
dimensional regulator $\epsilon$ and the rapidity regulator $\alpha$. The
renormalised soft function is furthermore assumed to satisfy the RRG equation

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\ln\nu}\;S(\tau,\mu,\nu)$
$\displaystyle=\bigg{[}4\,A_{\Gamma}(\mu_{s},\mu)-2\gamma_{\nu}^{S}(\tau,\mu_{s})\bigg{]}\;S(\tau,\mu,\nu)\,,$
(34)

where $A_{\Gamma}(\mu_{1},\mu_{2})$ is an RG kernel that was given explicitly
in Bell:2018oqa , and the $\nu$-anomalous dimension can be identified with the
collinear anomaly exponent defined in (29) via

$\displaystyle\gamma_{\nu}^{S}(\tau,\mu_{s})$
$\displaystyle=\mathcal{F}(\tau,\mu_{s})\,.$ (35)

In the RRG approach the renormalised soft function is in addition supposed to
obey a RGE in the scale $\mu$,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\ln\mu}\;S(\tau,\mu,\nu)$
$\displaystyle=\bigg{[}4\,\Gamma_{\mathrm{cusp}}(\alpha_{s})\,\ln(\mu\bar{\tau})-4\,\Gamma_{\mathrm{cusp}}(\alpha_{s})\,\ln(\nu\bar{\tau})-2\gamma^{S}_{\mu}(\alpha_{s})\bigg{]}\;S(\tau,\mu,\nu)\,,$
(36)

whereas the corresponding quantity in the collinear anomaly framework – the
soft remainder function $W^{S}(\tau,\mu)$ in (29) – does not obey a simple RGE
without its collinear counterpart. The RRG therefore makes stronger
assumptions than the collinear anomaly framework, and we argued in
Bell:2018oqa  that the RGE (36) only holds if the rapidity regulator is
implemented on the level of connected webs – a necessary requirement for the
consistency of the RRG approach that was not formulated so clearly in the
original literature.222Connected webs were discussed in Chiu:2012ir  only in
the context of gauge invariance and NAE, but it has not been stated explicitly
in that paper that the RGE (36) looses its validity if the rapidity regulator
is not implemented on the level of connected webs.

As we implement the rapidity regulator via the prescription (17) for
individual emissions, our default setup is not suited for the RRG approach. In
other words the $\alpha^{0}$-pieces calculated with SoftSERVE 0.9 cannot be
renormalised in a way that is consistent with (36) (as the problem does not
affect the $1/\alpha$ poles, all results presented in Bell:2018vaa ;
Bell:2018oqa ; Bell:2018jvf  are nevertheless correct). In SoftSERVE 1.0 we
remedy this point and implement an alternative prescription that fulfils the
requirements of the RRG approach. To do so, we add a factor $w^{2}$ to (17),
where $w$ is a bookkeeping parameter that fulfils the RRG equation
$\mathrm{d}w/\mathrm{d}\ln\nu=-\alpha w/2$ Chiu:2012ir , and we implement the
rapidity regulator for double correlated emissions via

$w^{2}\,\int d^{d}k\int
d^{d}l\;\left(\frac{\nu}{k_{+}+k_{-}+l_{+}+l_{-}}\right)^{\alpha}\;\delta(k^{2})\theta(k^{0})\;\delta(l^{2})\theta(l^{0})$
(37)

rather than

$w^{4}\int
d^{d}k\;\left(\frac{\nu}{k_{+}+k_{-}}\right)^{\alpha}\;\delta(k^{2})\theta(k^{0})\;\int
d^{d}l\;\left(\frac{\nu}{l_{+}+l_{-}}\right)^{\alpha}\;\delta(l^{2})\theta(l^{0})\,,$
(38)

whereas the remaining contributions to the bare soft function are not changed,
except for trivial factors of $w$.

We will address the technical aspects of the SoftSERVE implementation in the
following section, and show here how to extract the two-loop anomalous
dimensions and matching corrections from the bare soft function in the RRG
setup. To do so, we start from

$\displaystyle S_{0}(\tau,\nu)$
$\displaystyle=1+\left(\frac{Z_{\alpha}\alpha_{s}}{4\pi}\right)\,w^{2}\,(\mu^{2}\bar{\tau}^{2})^{\epsilon}\;(\nu\bar{\tau})^{\alpha}\bigg{\\{}\frac{1}{\alpha}\bigg{(}\frac{x^{1}_{1}}{\epsilon}+x^{1}_{0}+x^{1}_{-1}\,\epsilon+x^{1}_{-2}\,\epsilon^{2}+x^{1}_{-3}\,\epsilon^{3}\bigg{)}$
(39)
$\displaystyle+\frac{x^{0}_{2}}{\epsilon^{2}}+\frac{x^{0}_{1}}{\epsilon}+x^{0}_{0}+x^{0}_{-1}\,\epsilon+x^{0}_{-2}\,\epsilon^{2}+\alpha\bigg{(}\frac{x^{-1}_{3}}{\epsilon^{3}}+\frac{x^{-1}_{2}}{\epsilon^{2}}+\frac{x^{-1}_{1}}{\epsilon}+x^{-1}_{0}+x^{-1}_{-1}\,\epsilon\bigg{)}$
$\displaystyle+\mathcal{O}\Big{(}\frac{\epsilon^{4}}{\alpha},\epsilon^{3},\alpha\epsilon^{2},\alpha^{2}\Big{)}\bigg{\\}}+\left(\frac{Z_{\alpha}\alpha_{s}}{4\pi}\right)^{2}(\mu^{2}\bar{\tau}^{2})^{2\epsilon}\;\bigg{\\{}w^{4}\,(\nu\bar{\tau})^{2\alpha}\,\bigg{[}\frac{1}{\alpha^{2}}\bigg{(}\frac{y^{2}_{2}}{\epsilon^{2}}+\frac{y^{2}_{1}}{\epsilon}+y^{2}_{0}\bigg{)}$
$\displaystyle+\frac{1}{\alpha}\bigg{(}\frac{y^{1}_{3}}{\epsilon^{3}}+\frac{y^{1}_{2}}{\epsilon^{2}}+\frac{y^{1}_{1}}{\epsilon}+y^{1}_{0}\bigg{)}+\frac{y^{0}_{4}}{\epsilon^{4}}+\frac{y^{0}_{3}}{\epsilon^{3}}+\frac{y^{0}_{2}}{\epsilon^{2}}+\frac{y^{0}_{1}}{\epsilon}+y^{0}_{0}+\mathcal{O}\Big{(}\frac{\epsilon}{\alpha^{2}},\frac{\epsilon}{\alpha},\epsilon,\alpha\Big{)}\bigg{]}$
$\displaystyle+w^{2}\,(\nu\bar{\tau})^{\alpha}\bigg{[}\frac{1}{\alpha}\bigg{(}\frac{z^{1}_{3}}{\epsilon^{3}}+\frac{z^{1}_{2}}{\epsilon^{2}}+\frac{z^{1}_{1}}{\epsilon}+z^{1}_{0}\bigg{)}+\frac{z^{0}_{4}}{\epsilon^{4}}+\frac{z^{0}_{3}}{\epsilon^{3}}+\frac{z^{0}_{2}}{\epsilon^{2}}+\frac{z^{0}_{1}}{\epsilon}+z^{0}_{0}+\mathcal{O}\Big{(}\frac{\epsilon}{\alpha},\epsilon,\alpha\Big{)}\bigg{]}\bigg{\\}},$

where the only difference with respect to Bell:2018oqa  consists in the
presence of the bookkeeping parameter $w$. Due to (37) the correlated-emission
contribution is, moreover, now contained in the $z^{i}_{j}$ coefficients along
with the real-virtual interference term. The single real-emission and
uncorrelated double-emission contributions constitute the $x^{i}_{j}$ and
$y^{i}_{j}$ coefficients, respectively, as before. The coefficients
$x^{i}_{j}$ are thus proportional to the colour factor $C_{F}$, the
$y^{i}_{j}$ to $C_{F}^{2}$, and the $z^{i}_{j}$ consist of two contributions
with colour factors $C_{F}T_{f}n_{f}$ and $C_{F}C_{A}$.

We now expand the anomalous dimensions to two-loop order,

$\displaystyle\Gamma_{\mathrm{cusp}}(\alpha_{s})$
$\displaystyle=\left(\frac{\alpha_{s}}{4\pi}\right)\Gamma_{0}+\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\Gamma_{1}\,,$
(40) $\displaystyle\gamma_{\mu}^{S}(\alpha_{s})$
$\displaystyle=\left(\frac{\alpha_{s}}{4\pi}\right)\gamma^{S}_{\mu,0}+\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\gamma^{S}_{\mu,1}\,,$
$\displaystyle\gamma_{\nu}^{S}(\tau,\mu)$
$\displaystyle=\left(\frac{\alpha_{s}}{4\pi}\right)\Big{\\{}2\Gamma_{0}L_{\mu}+\gamma^{S}_{\nu,0}\Big{\\}}+\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\Big{\\{}2\beta_{0}\Gamma_{0}L_{\mu}^{2}+2\left(\Gamma_{1}+\beta_{0}\gamma^{S}_{\nu,0}\right)L_{\mu}+\gamma^{S}_{\nu,1}\Big{\\}},$

where $L_{\mu}=\ln(\mu\bar{\tau})$ and the coefficients $\gamma^{S}_{\nu,i}$
correspond to the $d_{i+1}$ in the collinear anomaly language of Bell:2018oqa
. Using
$Z_{\alpha}=1-\beta_{0}\alpha_{s}/(4\pi\epsilon)+\mathcal{O}(\alpha_{s}^{2})$,
we can solve the RGEs (34) and (36) for the soft function and the
corresponding equations for the Z-factor $Z_{S}=S/S_{0}$ explicitly. In order
to avoid cross terms from higher orders, the latter is conveniently determined
via its logarithm, which in the $\overline{\text{MS}}$ scheme takes the form

$\displaystyle\ln
Z_{S}=\left(\frac{\alpha_{s}}{4\pi}\right)w^{2}\,\bigg{\\{}\frac{2\Gamma_{0}}{\alpha\epsilon}+\frac{4\Gamma_{0}L_{\mu}+2\gamma_{\nu,0}^{S}}{\alpha}-\frac{\Gamma_{0}}{\epsilon^{2}}+\frac{\gamma_{\mu,0}^{S}-2\Gamma_{0}(L_{\mu}-L_{\nu})}{\epsilon}\bigg{\\}}$
$\displaystyle\quad+\left(\frac{\alpha_{s}}{4\pi}\right)^{2}w^{2}\,\bigg{\\{}-\frac{\beta_{0}\Gamma_{0}}{\alpha\epsilon^{2}}+\frac{\Gamma_{1}}{\alpha\epsilon}+\Big{(}4\beta_{0}\Gamma_{0}L_{\mu}^{2}+4(\Gamma_{1}+\beta_{0}\gamma_{\nu,0}^{S})L_{\mu}+2(\gamma_{\nu,1}^{S})_{C}+w^{2}(\gamma_{\nu,1}^{S})_{U}\Big{)}\frac{1}{\alpha}$
$\displaystyle\hskip
79.66771pt+\frac{3\beta_{0}\Gamma_{0}}{4\epsilon^{3}}-\frac{1}{4}\Big{(}\Gamma_{1}+2\beta_{0}\gamma_{\mu,0}^{S}-4\beta_{0}\Gamma_{0}(L_{\mu}-L_{\nu})\Big{)}\frac{1}{\epsilon^{2}}$
$\displaystyle\hskip
79.66771pt-\frac{1}{2}\Big{(}2\Gamma_{1}(L_{\mu}-L_{\nu})-(\gamma_{\mu,1}^{S})_{C}-w^{2}(\gamma_{\mu,1}^{S})_{U}\Big{)}\frac{1}{\epsilon}\bigg{\\}}\,,$
(41)

where $L_{\nu}=\ln(\nu\bar{\tau})$ and we have split the correlated and
uncorrelated-emission contributions to the two-loop anomalous dimensions
$\gamma^{S}_{\mu,1}$ and $\gamma^{S}_{\nu,1}$ since – according to (39) – they
come with different powers of the bookkeeping parameter $w$. For the
renormalised soft function, we obtain up to the considered two-loop order

$\displaystyle\ln
S(\tau,\mu,\nu)=\left(\frac{\alpha_{s}}{4\pi}\right)\bigg{\\{}2\Gamma_{0}L_{\mu}^{2}-4\Gamma_{0}L_{\mu}L_{\nu}-2\gamma^{S}_{\mu,0}L_{\mu}-2\gamma^{S}_{\nu,0}L_{\nu}+c_{1}^{S}\bigg{\\}}$
$\displaystyle\quad+\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\bigg{\\{}\frac{4}{3}\beta_{0}\Gamma_{0}L_{\mu}^{3}-4\beta_{0}\Gamma_{0}L_{\mu}^{2}L_{\nu}+2\Big{(}\Gamma_{1}-\beta_{0}\gamma_{\mu,0}^{S}\Big{)}L_{\mu}^{2}-4\Big{(}\Gamma_{1}+\beta_{0}\gamma_{\nu,0}^{S}\Big{)}L_{\mu}L_{\nu}$
$\displaystyle\hskip
62.59605pt-2\Big{(}\gamma_{\mu,1}^{S}-\beta_{0}c_{1}^{S}\Big{)}L_{\mu}-2\gamma_{\nu,1}^{S}L_{\nu}+c_{2}^{S}-\frac{1}{2}(c_{1}^{S})^{2}\bigg{\\}}\,,$
(42)

where we have set $w=1$. As the cusp anomalous dimension and the beta function
are known to the required order,

$\Gamma_{0}=4C_{F}\,,\quad\Gamma_{1}=4C_{F}\left\\{\left(\frac{67}{9}-\frac{\pi^{2}}{3}\right)C_{A}-\frac{20}{9}T_{F}n_{f}\right\\},\quad\beta_{0}=\frac{11}{3}C_{A}-\frac{4}{3}T_{F}n_{f}\,,$
(43)

the higher poles in the product of the Z-factor and the bare soft function
provide checks of our calculation, whereas the coefficients of the $1/\alpha$
and $1/\epsilon$ poles determine the rapidity anomalous dimension
$\gamma^{S}_{\nu}$ and the $\mu$-anomalous dimension $\gamma^{S}_{\mu}$,
respectively. In terms of the coefficients introduced in (39), we obtain

$\displaystyle\gamma^{S}_{\nu,0}$ $\displaystyle=-\frac{x^{1}_{0}}{2}\,,$
$\displaystyle\gamma^{S}_{\nu,1}$
$\displaystyle=-y^{1}_{0}-\frac{z^{1}_{0}}{2}+x^{0}_{-1}x^{1}_{1}+x^{0}_{0}x^{1}_{0}+x^{0}_{1}x^{1}_{-1}+x^{0}_{2}x^{1}_{-2}+\frac{\beta_{0}x^{1}_{-1}}{2}\,,$
(44)

which is precisely the relation we found for the collinear anomaly exponent in
Bell:2018oqa . The non-logarithmic terms of the renormalised soft function
are, on the other hand, in the RRG framework given by

$\displaystyle c_{1}^{S}$ $\displaystyle=x^{0}_{0}\,,$ (45) $\displaystyle
c_{2}^{S}$
$\displaystyle=y^{0}_{0}+z^{0}_{0}-x^{0}_{2}x^{0}_{-2}-(x^{0}_{1}+\beta_{0})x^{0}_{-1}-x^{1}_{1}x^{-1}_{-1}-x^{1}_{0}x^{-1}_{0}-x^{1}_{-1}x^{-1}_{1}-x^{1}_{-2}x^{-1}_{2}-x^{1}_{-3}x^{-1}_{3}\,,$

whereas one can show that the $\mu$-anomalous dimension is unphysical for
SCET-2 soft functions since it drops out in the final expressions once the
soft and collinear RG kernels are combined. Following the procedure outlined
in Bell:2018vaa , we can actually prove that the $\mu$-anomalous dimension is
a universal number in our setup, i.e. it is independent of the observable
given by

$\displaystyle\gamma^{S}_{\mu,0}=0\,,\quad\gamma^{S}_{\mu,1}=\bigg{\\{}\frac{224}{27}-\frac{4\pi^{2}}{9}\bigg{\\}}\,C_{F}T_{F}n_{f}\,+\,\bigg{\\{}-\frac{808}{27}+\frac{11\pi^{2}}{9}+28\zeta_{3}\bigg{\\}}\,C_{F}C_{A}\,.$
(46)

Rather than extracting this quantity from the coefficient of the $1/\epsilon$
pole, we therefore turn the argument around and use these numbers in SoftSERVE
to check if the singularities cancel out as predicted by the RRG framework.

The discussion of cumulant soft functions from the previous section applies
identically to the RRG setup, with the sole exception that the correlated-
emission contribution in (33) comes with a prefactor
$e^{\gamma_{E}(4\epsilon+\alpha)}/\Gamma(1-4\epsilon-\alpha)$ rather than
$e^{\gamma_{E}(4\epsilon+2\alpha)}/\Gamma(1-4\epsilon-2\alpha)$ because of
(37). Once again, SoftSERVE 1.0 provides a script that takes these
modifications into account.

##  5 Extending the SoftSERVE distribution

The central new element of SoftSERVE 1.0 is the direct calculation of the
uncorrelated-emission contribution, whereas SoftSERVE 0.9 reconstructs this
term from the NLO correction, assuming that the observable is consistent with
NAE. For the SoftSERVE user, this means that calling make all – or calling
make without target – now generates executables for all colour structures, and
the target list is supplemented with the uncorrelated, CFA and CFB targets.
The latter correspond to the two contributions from regions A and B in (23),
and uncorrelated refers to them as a pair. For observables obeying NAE, the
correlated target now provides all the required input, skipping the
$C_{F}^{2}$ contributions.333In version 0.9 correlated was synonymous to all,
or no target at all.

In addition we implemented the new features discussed in Section 4 concerning
cumulant soft functions and the RRG. Apart from the existing script for the
renormalisation of Laplace-space soft functions (laprenorm), there now also
exists a script for the renormalisation of cumulant soft functions (momrenorm)
that applies the changes discussed in Section 4.1. Both scripts come in two
versions designed for observables that obey NAE (postfix NAE) and those that
violate NAE (no postfix). The latter require the full set of results files,
whereas the former do not need the CFA and CFB results – they reconstruct the
$C_{F}^{2}$ contribution directly from the NLO result. Execution and summary
scripts to run and refine the results now also exist in two versions for
observables that obey/violate NAE, similarly postfixed. To prevent
accidentally calling non-NAE scripts on results that are derived assuming NAE,
some safeguards are implemented.

Moreover, the SCET-2 executables can now be generated with a rapidity
regulator that is compatible with the RRG approach. As discussed in Section
4.2, this requires that one implements the regulator on the level of connected
webs rather than individual emissions. At NNLO the only difference arises in
the correlated-emission contribution for which the regulator is implemented
via (37) rather than (38). This feature is switched off by default, but it can
be used by setting a nonzero RRG variable during the make call. In other
words, to generate e.g. the $C_{F}T_{F}n_{f}$ colour structure binary for some
observable using the RRG regulator, one calls make NF RRG=1. In the SCET-2
branch, there are scripts to summarise (sftsrvres), renormalise (laprenorm or
momrenorm) and to account for Fourier phases (fourierconvert) that use the
results derived with the new regulator, and they are all postfixed RRG. These
scripts of course also exist for observables that obey NAE, and they then
simply carry both postfixes like laprenormNAERRG. Again, safeguards to avoid
calling RRG scripts on results that were derived with the default rapidity
regulator and vice-versa are implemented.

Finally, we added the formulae derived in Bell:2018vaa  that allow for a
direct calculation of the soft anomalous dimensions and collinear anomaly
exponents without having to calculate the complete bare soft function. As the
SoftSERVE input differs slightly from the conventions of Bell:2018vaa , we
rederived these formulae in a form that is suitable for SoftSERVE and
summarise the corresponding expressions in Appendix A. To access these
formulae the user must call make with targets ADLap or ADMom, which generates
the respective executables for Laplace-space and cumulant soft functions.
These executables then reside in the Executables folder and must be called
manually. While they allow for a fast evaluation of the anomalous
dimension/anomaly exponent, we do not recommend using them for a precision
determination since they are numerically less robust. Observables which
exhibit features that reduce numerical accuracy, like integrable divergences,
slow them down disproportionately. In addition, the term (63), which is
conjectured to vanish for all observables, happens to sometimes be numerically
unstable due to the peculiar structure in its last line. For observables for
which this expression is non-trivial, the integration converges comparatively
slowly.

##  6 Results

We are now in a position to use SoftSERVE 1.0 to compute NNLO dijet soft
functions for various $e^{+}e^{-}$ event shapes and hadron-collider
observables. As in Bell:2018oqa , we present our results for SCET-1 soft
functions in the form

$\displaystyle\gamma_{0}^{S}$ $\displaystyle=\gamma_{0}^{C_{F}}\,C_{F}\,,$
$\displaystyle\gamma_{1}^{S}$
$\displaystyle=\gamma_{1}^{C_{A}}\,C_{F}C_{A}+\gamma_{1}^{n_{f}}\,C_{F}T_{F}n_{f}+\gamma_{1}^{C_{F}}\,C_{F}^{2}\,,$
$\displaystyle c_{1}^{S}$ $\displaystyle=c_{1}^{C_{F}}\,C_{F}\,,$
$\displaystyle c_{2}^{S}$
$\displaystyle=c_{2}^{C_{A}}\,C_{F}C_{A}+c_{2}^{n_{f}}\,C_{F}T_{F}n_{f}+c_{2}^{C_{F}}\,C_{F}^{2}\,,$
(47)

where the coefficients $\gamma^{S}_{i}$ of the soft anomalous dimension and
the finite terms $c_{i}^{S}$ of the renormalised soft function refer to the
conventions introduced in Section 4.1 of Bell:2018oqa . In contrast to that
work, we now use SoftSERVE to calculate the $\gamma_{1}^{C_{F}}$ and
$c_{2}^{C_{F}}$ numbers, which were derived in Bell:2018oqa  on the basis of
NAE.

For SCET-2 soft functions we quote our numbers in the RRG notation of Section
4.2. The relevant resummation ingredients are in this case the coefficients
$\gamma_{\nu,i}^{S}$ of the rapidity anomalous dimension and the finite terms
$c_{i}^{S}$ of the RRG renormalised soft function, which we decompose
analogously to (47) according to their colour structures. Whereas the former
are equivalent to the anomaly coefficients $d_{i+1}$ used in Bell:2018oqa ,
the latter are not well defined in the collinear anomaly framework and were
therefore not given in Bell:2018oqa . As explained in Section 4.2, the
$\mu$-anomalous dimension $\gamma_{\mu}^{S}$ is, moreover, unphysical for
SCET-2 soft functions and will therefore be disregarded in the following.

Similar to Bell:2018oqa , SoftSERVE 1.0 comes with a number of template files
that can be used to rederive the numbers quoted in this section. For most of
the observables the runtime of the uncorrelated-emission contribution turns
out to be comparable to the correlated-emission calculation, which can of
course be tailored to the specific needs of the user by adjusting the
respective Cuba settings.444As in Bell:2018oqa , the numbers presented in this
section were produced with the precision setting, while the plots were
produced with the standard setting. Although the focus of the present paper is
on NAE-violating observables, we first consider a few observables that respect
NAE, since this allows us to test the new algorithm and to gauge the accuracy
of our numerical predictions. We then switch to some exemplary NAE-violating
soft functions in a second step.

###  6.1 Observables that obey NAE

For all observables in this section NAE implies $\gamma_{1}^{C_{F}}=0$ and
$c_{2}^{C_{F}}=1/2(c_{1}^{C_{F}})^{2}$ for SCET-1 soft functions, and
similarly $\gamma_{\nu,1}^{C_{F}}=0$ and
$c_{2}^{C_{F}}=1/2(c_{1}^{C_{F}})^{2}$ in the SCET-2 case.

#### C-parameter

We first consider the C-parameter event shape, which was one of the template
observables we studied in Bell:2018oqa . The only new element required for the
uncorrelated-emission contribution is the function555As in Bell:2018oqa  we
suppress the angular variables in the arguments of the measurement function if
the observable does not depend on any of these angles.

$\displaystyle G(y_{k},y_{l},b)$ $\displaystyle=\frac{1}{(1+y_{k})(1+y_{l})}$
(48)

defined in (9), which can be translated into the relevant input functions
$G_{A_{1}}$, $G_{A_{2}}$ and $G_{B}$ using the relations (25) and (27). We
then find using SoftSERVE 1.0

$\displaystyle\gamma_{0}^{C_{F}}$ $\displaystyle=1\cdot 10^{-10}\pm 2\cdot
10^{-7}\,$ $\displaystyle\quad[0]\,,$ $\displaystyle\qquad\quad c_{1}^{C_{F}}$
$\displaystyle=-3.28987\pm 9\cdot 10^{-7}\,$ $\displaystyle\quad[-3.28987]\,,$
$\displaystyle\gamma_{1}^{C_{A}}$ $\displaystyle=15.7940(10)\,$
$\displaystyle\quad[15.7945]\,,$ $\displaystyle c_{2}^{C_{A}}$
$\displaystyle=-57.9814(35)$ $\displaystyle\quad[-57.9757]\,,$
$\displaystyle\gamma_{1}^{n_{f}}$ $\displaystyle=3.90983(14)\,$
$\displaystyle\quad[3.90981]\,,$ $\displaystyle c_{2}^{n_{f}}$
$\displaystyle=43.8181(4)\,$ $\displaystyle\quad[43.8182]\,,$
$\displaystyle\gamma_{1}^{C_{F}}$ $\displaystyle=-0.0004(24)$
$\displaystyle\quad[0]\,,$ $\displaystyle c_{2}^{C_{F}}$
$\displaystyle=5.41178(592)$ $\displaystyle\quad[5.41162]\,,$ (49)

which is in excellent agreement with the analytic results from Hoang:2014wka ;
Bell:2018oqa  shown in the square brackets.

#### W-production at large transverse momentum

We next consider the soft function for $W$-production at large transverse
momentum which we also discussed in detail in Bell:2018oqa . We now have

$\displaystyle G(y_{k},y_{l},b,t_{k},t_{l},t_{kl})$
$\displaystyle\\!=\\!\frac{b(1+y_{l})(1+y_{k}-2\sqrt{y_{k}}(1-2t_{k}))}{(1+b)}+\frac{(1+y_{k})(1+y_{l}-2\sqrt{y_{l}}(1-2t_{l}))}{(1+b)}$
(50)

and obtain

$\displaystyle\gamma_{0}^{C_{F}}$ $\displaystyle=-1\cdot 10^{-9}\pm 2\cdot
10^{-6}\,$ $\displaystyle\quad[0]\,,$ $\displaystyle\qquad\quad c_{1}^{C_{F}}$
$\displaystyle=9.86960(2)\,$ $\displaystyle\quad[9.86960]\,,$
$\displaystyle\gamma_{1}^{C_{A}}$ $\displaystyle=15.7945(24)\,$
$\displaystyle\quad[15.7945]\,,$ $\displaystyle c_{2}^{C_{A}}$
$\displaystyle=-2.64324(890)$ $\displaystyle\quad[-2.65010]\,,$
$\displaystyle\gamma_{1}^{n_{f}}$ $\displaystyle=3.90987(22)\,$
$\displaystyle\quad[3.90981]\,,$ $\displaystyle c_{2}^{n_{f}}$
$\displaystyle=-25.3069(10)\,$ $\displaystyle\quad[-25.3073]\,,$
$\displaystyle\gamma_{1}^{C_{F}}$ $\displaystyle=-1\cdot 10^{-7}\pm 0.003$
$\displaystyle\quad[0]\,,$ $\displaystyle c_{2}^{C_{F}}$
$\displaystyle=48.7050(96)$ $\displaystyle\quad[48.7045]\,,$ (51)

which is again in perfect agreement with the analytic results from
Becher:2012za .

#### Jet broadening

In order to illustrate the new RRG routine of SoftSERVE, we consider the
SCET-2 event-shape variable jet broadening. As in Bell:2018oqa  we consider a
recoil-free definition here and refer to that paper for more details on the
observable. The relevant input for the uncorrelated-emission contribution is
then given by

$\displaystyle G(y_{k},y_{l},b)$ $\displaystyle=\frac{1}{2}\,,$ (52)

which yields

$\displaystyle\gamma_{\nu,0}^{C_{F}}$ $\displaystyle=-5.54518(1)$
$\displaystyle\quad[-5.54518]\,,$ $\displaystyle\qquad\quad c_{1}^{C_{F}}$
$\displaystyle=-20.2930(1)$ $\displaystyle\quad[-20.2930]\,,$
$\displaystyle\gamma_{\nu,1}^{C_{A}}$ $\displaystyle=7.03652(110)$
$\displaystyle\quad[7.03605]\,,$ $\displaystyle c_{2}^{C_{A}}$
$\displaystyle=-56.6537(21)\,,$ $\displaystyle\gamma_{\nu,1}^{n_{f}}$
$\displaystyle=-11.5393(1)$ $\displaystyle\quad[-11.5393]\,,$ $\displaystyle
c_{2}^{n_{f}}$ $\displaystyle=24.1971(3)\,,$
$\displaystyle\gamma_{\nu,1}^{C_{F}}$ $\displaystyle=-0.00001(163)$
$\displaystyle\quad[0]\,,$ $\displaystyle c_{2}^{C_{F}}$
$\displaystyle=205.902(5)$ $\displaystyle\quad[205.902]\,.$ (53)

For the rapidity anomalous dimension, this agrees with the expressions found
in Becher:2012qc , and the one-loop matching coefficient
$c_{1}^{C_{F}}=-8\ln^{2}2-5\pi^{2}/3$ can be extracted from that paper as
well. Our results for the two-loop coefficients $c_{2}^{C_{A}}$ and
$c_{2}^{n_{f}}$ are, on the other hand, new.

#### Transverse-momentum resummation

We finally examine the soft function for transverse-momentum resummation in
Drell-Yan production, which is an example of a Fourier-space rather than a
Laplace-space soft function. As argued in Bell:2018oqa , these can be computed
with SoftSERVE by using the absolute value of the naive measurement function,
which in the specific case of transverse-momentum resummation is given by

$\displaystyle G(y_{k},y_{l},b,t_{k},t_{l},t_{kl})$
$\displaystyle=\frac{2}{1+b}\;\big{|}b(1-2t_{k})+1-2t_{l}\big{|}\,.$ (54)

Running the fourierconvertRRG script before renormalisation, we then obtain

$\displaystyle\gamma_{\nu,0}^{C_{F}}$ $\displaystyle=1\cdot 10^{-9}\pm 2\cdot
10^{-6}$ $\displaystyle\quad[0]\,,$ $\displaystyle\qquad\quad c_{1}^{C_{F}}$
$\displaystyle=-3.2899(1)$ $\displaystyle\quad[-3.2899]\,,$
$\displaystyle\gamma_{\nu,1}^{C_{A}}$ $\displaystyle=-3.7407(94)$
$\displaystyle\quad[-3.7317]\,,$ $\displaystyle c_{2}^{C_{A}}$
$\displaystyle=-16.749(169)$ $\displaystyle\quad[-16.507]\,,$
$\displaystyle\gamma_{\nu,1}^{n_{f}}$ $\displaystyle=-8.2963(20)$
$\displaystyle\quad[-8.2963]\,,$ $\displaystyle c_{2}^{n_{f}}$
$\displaystyle=10.338(27)$ $\displaystyle\quad[10.347]\,,$
$\displaystyle\gamma_{\nu,1}^{C_{F}}$ $\displaystyle=-0.0322(281)$
$\displaystyle\quad[0]\,,$ $\displaystyle c_{2}^{C_{F}}$
$\displaystyle=5.1718(3987)$ $\displaystyle\quad[5.4116]\,.$ (55)

While we already calculated the rapidity anomalous dimension for this
observable in Bell:2018oqa , we did not have access to the finite terms in the
RRG framework at the time, which are however known analytically from the
calculation in Luebbert:2016itl . Our SoftSERVE numbers compare well to these
results, although we observe a slightly reduced accuracy in comparison to the
prior examples, which is due to integrable divergences in the bulk of the
integration region as well as the required Fourier shuffle, which mixes
coefficients and adds up the corresponding errors. The agreement is, however,
still acceptable.

###  6.2 Observables that violate NAE

Having established that SoftSERVE 1.0 satisfactorily reproduces known results
for sample NAE observables, we now turn to soft functions that do not respect
the NAE theorem and which require an independent calculation of the
uncorrelated-emission contribution. Whereas we already presented our results
for the corresponding anomalous dimensions in Bell:2018vaa ; Bell:2018jvf , we
compute the matching coefficients in this work for the first time.

#### Rapidity-dependent jet vetoes

Figure 2: Two-loop anomalous dimension and finite term of the renormalised
C-parameter jet-veto soft function. Red dots indicate values calculated with
SoftSERVE and green dashed lines represent the interpolating functions from
Gangal:2016kuo .

The first family of NAE-violating observables are the rapidity-dependent jet
vetoes from Gangal:2014qda . Specifically, we consider the beam-thrust and
C-parameter-like jet-veto variables $\mathcal{T}_{B\rm cm}$ and
$\mathcal{T}_{C\rm cm}$ defined in that paper, which are both SCET-1
observables with $n=1$. For the C-parameter jet veto, one further has
$f(y_{k},t_{k})=1/(1+y_{k})$ and

$\displaystyle F(a,b,y,t_{k},t_{l},t_{kl})$
$\displaystyle=\theta(\Delta_{F}-R)\;\text{max}\bigg{(}\frac{ab}{a(a+b)+(1+ab)y}~{},~{}\frac{a}{a+b+a(1+ab)y}\bigg{)}$
$\displaystyle\quad+\theta(R-\Delta_{F})\;\frac{1}{1+y}\,,$ (56)

where $R$ is the jet radius and
$\Delta_{F}=\sqrt{\ln^{2}a+\arccos^{2}(1-2t_{kl})}$, and the corresponding
expression for the uncorrelated-emission measurement function was given in
(10). The jet-veto observables renormalise multiplicatively in cumulant space,
and therefore the formalism from Section 4.1 applies in this case.
Furthermore, as the jet algorithm has no effect on a single emission, the NLO
coefficients $\gamma_{0}^{C_{F}}=0$ and $c_{1}^{C_{F}}=\pi^{2}$ are
independent of the jet radius $R$, whereas the NNLO coefficients are displayed
in the range $0\leq R\leq 1$ in Figure 2. From the plots it is evident that
our SoftSERVE numbers agree well with the numerical results from
Gangal:2016kuo  indicated by the dashed lines.

Figure 3: The same as in Figure 2 for the finite term of the renormalised
beam-thrust jet-veto soft function.

For the beam-thrust jet veto, the input functions are slightly more
complicated and we refer to the SoftSERVE manual for their explicit
expressions. As the two jet vetoes have the same anomalous dimension, we
refrain from showing the corresponding plots in this case, since they are – in
view of the negligible numerical uncertainties – literally identical to the
upper plots in Figure 2. The one-loop matching coefficient is, moreover, now
given by $c_{1}^{C_{F}}=\pi^{2}/3$, and the two-loop coefficients are
displayed as a function of the jet radius in Figure 3. Our numbers are once
more in perfect agreement with the results from Gangal:2016kuo .

#### Standard jet veto

The standard way of implementing a jet veto uses a cutoff on the transverse
momenta of the emissions. The corresponding soft function is in this case
defined in SCET-2, and the required SoftSERVE input is given by $n=0$,
$f(y_{k},t_{k})=1$ and

$\displaystyle F(a,b,y,t_{k},t_{l},t_{kl})$
$\displaystyle=\sqrt{\frac{a}{(1+ab)(a+b)}}\bigg{\\{}\\!\theta(\Delta_{F}-R)+\theta(R-\Delta_{F})\sqrt{1+b^{2}+2b(1-2t_{kl})}\bigg{\\}}\,,$
$\displaystyle\\!\\!\\!\\!G(y_{k},y_{l},b,t_{k},t_{l},t_{kl})$
$\displaystyle=\theta(\Delta_{G}-R)\,\frac{\max(1,b)}{1+b}+\theta(R-\Delta_{G})\,\frac{\sqrt{1+b^{2}+2b(1-2t_{kl})}}{1+b}\,.$
(57)

As for the rapidity-dependent jet vetoes, the soft function renormalises
multiplicatively in cumulant space, and the respective NLO coefficients are
now given by $\gamma_{\nu,0}^{C_{F}}=0$ and $c_{1}^{C_{F}}=-\pi^{2}/3$. Our
numbers for the two-loop rapidity anomalous dimension are shown in the upper
plots of Figure 4, and they confirm the existing results from Banfi:2012yh ;
Becher:2013xia ; Stewart:2013faa  indicated by the dashed lines. In the RRG
setup the two-loop matching corrections can furthermore be compared to
Stewart:2013faa , which gives these numbers in an expansion in $R\ll 1$ up to
terms of $\mathcal{O}(R^{0})$. As is evident from the lower plots in Figure 4,
this expansion works surprisingly well for the $c_{2}^{C_{A}}$ and
$c_{2}^{n_{f}}$ coefficients even for large values $R\simeq 1$, but it misses
the leading $\mathcal{O}(R^{2})$ correction to $c_{2}^{C_{F}}$.

Figure 4: Two-loop rapidity anomalous dimension and finite term of the RRG
renormalised $p_{T}$ veto soft function. Red dots indicate values calculated
with SoftSERVE and green dashed lines show the interpolating functions from
Becher:2013xia  (upper plots) and Stewart:2013faa  (lower plots).

#### Soft-drop jet groomer

Finally, we present novel results for the soft-drop groomed jet mass discussed
in Frye:2016aiz . According to this definition, the groomer depends on a
parameter $\beta$, and for values $\beta>0$ considered here, the soft function
is defined in SCET-1 with $n=-1-\beta$. As the formulae for the measurement
functions are rather lengthy, we refer to the SoftSERVE distribution for their
explicit expressions. The renormalisation of the soft function is, moreover,
again performed in cumulant space, and the one-loop coefficients are found to
be $\gamma_{0}^{C_{F}}=0$ and
$c_{1}^{C_{F}}=-\pi^{2}(3+3\beta+\beta^{2})/3/(1+\beta)$. Our results for the
two-loop coefficients are shown in Figure 5 together with the numbers from
Frye:2016aiz  for the anomalous dimension. For $\beta=0$ these values have
been extracted from an analytic calculation, whereas the $\beta=1$ numbers
stem from a fit to the EVENT2 generator. From the plots we see that our
results confirm these numbers, but they are far more precise than the EVENT2
extraction. Our results for other values of the grooming parameter $\beta$ are
new, as are the finite terms of the renormalised soft function which are shown
in the lower plots of the figure.666As in Bell:2018oqa  we validated these
predictions with independent pySecDec runs Borowka:2017idc . Our numbers have
actually already been used to extend the resummation for the soft-drop groomed
jet mass to next-to-next-to-next-to-leading logarithmic (N3LL) accuracy
Kardos:2020gty ; Kardos:2020ppl .

Figure 5: Two-loop anomalous dimensions and finite term of the renormalised
soft function for the soft-drop jet groomer. Red dots indicate values
calculated with SoftSERVE and the green diamonds show the numbers from
Frye:2016aiz .

##  7 Conclusions

We have extended our automated approach for calculating NNLO dijet soft
functions to the uncorrelated-emission ($C_{F}^{2}$) contribution. While one
can trivially obtain this term from the NLO calculation for observables that
obey the NAE theorem, one must calculate it explicitly for NAE-violating
observables like those that depend on a jet algorithm. From the technical
point of view, the divergence structure of the $C_{F}^{2}$ matrix element
differs from the other colour structures treated in Bell:2018oqa , and we have
devised a novel phase-space parametrisation that isolates these singularities.

Our algorithm permits a systematic numerical evaluation of NNLO dijet soft
functions, and it is implemented in SoftSERVE 1.0 which we release alongside
of this paper at https://softserve.hepforge.org/. In addition to the new core
routine for calculating the uncorrelated-emission contribution to bare dijet
soft functions, SoftSERVE 1.0 includes novel renormalisation scripts that are
compatible with the RRG formalism and observables that renormalise directly in
momentum space rather than Laplace space.

SoftSERVE has therefore become a powerful program for calculating NNLO dijet
soft functions, and we have used it to cross-check existing calculations for
multiple $e^{+}e^{-}$ and hadron-collider observables, as well as to obtain
some novel predictions. In particular, our results for the angularity event
shape derived in Bell:2018oqa  enabled NNLL Procura:2018zpn  and NNLL′
Bell:2018gce  resummations, and our novel predictions for the soft-drop
groomed jet mass have recently been employed in a precision N3LL resummation
in Kardos:2020gty ; Kardos:2020ppl . While we hope that SoftSERVE will prove
useful for many further applications, an extension of our algorithm to soft
functions that depend on more than two light-like directions is currently in
progress Bell:2018mkk .

###### Acknowledgements.

We thank Andrew Larkoski and Frank Tackmann for discussions. G.B. is supported
by the Deutsche Forschungsgemeinschaft (DFG) within Research Unit FOR 1873.
J.T. acknowledges support from the Villum Fund, project number 00010102, prior
funding from DESY Hamburg, and also thanks the Albert Einstein Center in Bern
for its support and hospitality. R.R. is supported by ERC grant ERC-
STG-2015-677323, and acknowledges prior funding from the Swiss National
Science Foundation (SNF) under grants CRSII2$\\_$160814 and
200020$\\_$182038\.

##  Appendix A Anomalous dimensions

In this appendix we rederive the integral representations from Bell:2018vaa ,
which allow for a fast evaluation of the soft anomalous dimension $\gamma^{S}$
and the collinear anomaly exponent $\mathcal{F}(\tau,\mu)$ for SCET-1 and
SCET-2 observables, respectively. While our derivation follows the conventions
from Bell:2018vaa , it differs in one aspect from that work; namely the sector
decomposition step in (27) is performed only for the same-hemisphere
contribution for uncorrelated emissions (region A), while it was also applied
to the opposite-hemisphere case (region B) in Bell:2018vaa .

We start with the $C_{F}^{2}$ contribution to the soft anomalous dimension for
SCET-1 observables, for which (22) of Bell:2018vaa  is replaced by

$\displaystyle\gamma^{C_{F}}_{1}$
$\displaystyle=\frac{128}{\pi}\,\int_{0}^{1}\\!dy\int_{0}^{1}\\!dt_{l}\;\,\frac{1}{\sqrt{\rule{0.0pt}{9.95845pt}4t_{l}\bar{t}_{l}}}\;\,\frac{1}{y}\;\ln^{2}\left(\frac{(1+y)^{n}\,f(y,t_{l})}{f(0,t_{l})}\right)$
$\displaystyle\quad+\frac{256}{\pi}\,\int_{0}^{1}\\!dy\int_{0}^{1}\\!dt_{l}\;\,\frac{\ln
f(0,t_{l})}{\sqrt{\rule{0.0pt}{9.95845pt}4t_{l}\bar{t}_{l}}}\;\frac{\ln
f(y,t_{l})}{y_{+}}$
$\displaystyle\quad-\frac{512}{\pi^{2}}\,\int_{0}^{1}\\!dt_{k}\;\,\frac{\ln
f(0,t_{k})}{\sqrt{\rule{0.0pt}{9.95845pt}4t_{k}\bar{t}_{k}}}\;\int_{0}^{1}\\!dy\int_{0}^{1}\\!dt_{l}\;\,\frac{1}{\sqrt{\rule{0.0pt}{9.95845pt}4t_{l}\bar{t}_{l}}}\;\frac{\ln
f(y,t_{l})}{y_{+}}$
$\displaystyle\quad-\frac{128}{\pi^{2}}\int_{0}^{1}\\!dy\int_{0}^{1}\\!db\int_{0}^{1}\\!dt_{l}\int_{0}^{1}\\!dt_{kl}\;\frac{1}{\sqrt{\rule{0.0pt}{9.95845pt}16t_{l}\bar{t}_{l}t_{kl}\bar{t}_{kl}}}\;\,\frac{\mathcal{H}_{1}(y,b,t_{l},t_{kl})}{y_{+}b_{+}}\,$
$\displaystyle\quad-\frac{64}{\pi^{2}}\int_{0}^{1}\\!dr\int_{0}^{1}\\!db\int_{0}^{1}\\!dt_{l}\int_{0}^{1}\\!dt_{kl}\;\frac{1}{\sqrt{\rule{0.0pt}{9.95845pt}16t_{l}\bar{t}_{l}t_{kl}\bar{t}_{kl}}}\;\,\frac{\mathcal{H}_{2}(r,b,t_{l},t_{kl})}{r_{+}b_{+}}$
$\displaystyle\quad-\frac{128}{\pi^{2}}\int_{0}^{1}\\!dy_{k}\int_{0}^{1}\\!db\int_{0}^{1}\\!dt_{l}\int_{0}^{1}\\!dt_{kl}\;\frac{1}{\sqrt{\rule{0.0pt}{9.95845pt}16t_{l}\bar{t}_{l}t_{kl}\bar{t}_{kl}}}\;\,\frac{\mathcal{H}_{3}(y_{k},b,t_{l},t_{kl})}{{y_{k}}_{+}b_{+}}\,$
$\displaystyle\quad-\frac{128}{\pi^{2}}\int_{0}^{1}\\!dy_{l}\int_{0}^{1}\\!db\int_{0}^{1}\\!dt_{l}\int_{0}^{1}\\!dt_{kl}\;\frac{1}{\sqrt{\rule{0.0pt}{9.95845pt}16t_{l}\bar{t}_{l}t_{kl}\bar{t}_{kl}}}\;\,\frac{\mathcal{H}_{4}(y_{l},b,t_{l},t_{kl})}{{y_{l}}_{+}b_{+}}\,,$
(58)

with

$\displaystyle\mathcal{H}_{1}(y,b,t_{l},t_{kl})$ $\displaystyle=\ln
G_{A_{1}}(y,0,b,t_{k}^{+},t_{l},t_{kl})+\ln
G_{A_{2}}(y,0,b,t_{k}^{+},t_{l},t_{kl})+(t_{k}^{+}\to t_{k}^{-})\,,$
$\displaystyle\mathcal{H}_{2}(r,b,t_{l},t_{kl})$ $\displaystyle=\ln
G_{A_{1}}(0,r,b,t_{k}^{+},t_{l},t_{kl})+\ln
G_{A_{2}}(0,r,b,t_{k}^{+},t_{l},t_{kl})+(t_{k}^{+}\to t_{k}^{-})\,,$
$\displaystyle\mathcal{H}_{3}(y_{k},b,t_{l},t_{kl})$ $\displaystyle=\ln
G_{B}(y_{k},0,b,t_{k}^{+},t_{l},t_{kl})+(t_{k}^{+}\to t_{k}^{-})\,,$
$\displaystyle\mathcal{H}_{4}(y_{l},b,t_{l},t_{kl})$ $\displaystyle=\ln
G_{B}(0,y_{l},b,t_{k}^{+},t_{l},t_{kl})+(t_{k}^{+}\to t_{k}^{-})\,,$ (59)

and

$\displaystyle t_{k}^{\pm}=t_{l}+t_{kl}-2t_{l}t_{kl}\pm
2\sqrt{t_{l}\bar{t}_{l}t_{kl}\bar{t}_{kl}}\,.$ (60)

Similar to Bell:2018vaa , we find that this result only holds if the following
constraint

$\displaystyle\frac{8}{\pi}\,\int_{0}^{1}\\!dt_{l}\;\,\frac{\ln^{2}f(0,t_{l})}{\sqrt{\rule{0.0pt}{9.95845pt}4t_{l}\bar{t}_{l}}}-\frac{16}{\pi^{2}}\,\int_{0}^{1}\\!dt_{k}\;\,\frac{\ln
f(0,t_{k})}{\sqrt{\rule{0.0pt}{9.95845pt}4t_{k}\bar{t}_{k}}}\;\int_{0}^{1}\\!dt_{l}\;\,\frac{\ln
f(0,t_{l})}{\sqrt{\rule{0.0pt}{9.95845pt}4t_{l}\bar{t}_{l}}}$
$\displaystyle\qquad-\frac{4}{\pi^{2}}\int_{0}^{1}\\!db\int_{0}^{1}\\!dt_{l}\int_{0}^{1}\\!dt_{kl}\;\frac{1}{\sqrt{\rule{0.0pt}{9.95845pt}16t_{l}\bar{t}_{l}t_{kl}\bar{t}_{kl}}}\;\,\frac{\mathcal{H}_{0}(b,t_{l},t_{kl})}{b_{+}}\,=\,0$
(61)

is satisfied, where

$\displaystyle\mathcal{H}_{0}(b,t_{l},t_{kl})$ $\displaystyle=\ln
G_{A_{1}}(0,0,b,t_{k}^{+},t_{l},t_{kl})+\ln
G_{A_{2}}(0,0,b,t_{k}^{+},t_{l},t_{kl})$ $\displaystyle\quad+2\ln
G_{B}(0,0,b,t_{k}^{+},t_{l},t_{kl})+(t_{k}^{+}\to t_{k}^{-})\,.$ (62)

Moreover, we find an additional contribution to the soft anomalous dimension,
which we conjecture to vanish for all observables, given by

$\displaystyle\Delta\gamma^{C_{F}}_{1}$
$\displaystyle=\frac{64}{n}\,\bigg{\\{}\frac{4}{\pi}\,\int_{0}^{1}\\!dt_{l}\;\,\frac{\ln^{3}f(0,t_{l})}{\sqrt{\rule{0.0pt}{9.95845pt}4t_{l}\bar{t}_{l}}}-\frac{2}{\pi}\,\int_{0}^{1}\\!dt_{l}\;\,\frac{\ln(16t_{l}\bar{t}_{l})}{\sqrt{\rule{0.0pt}{9.95845pt}4t_{l}\bar{t}_{l}}}\;\ln^{2}f(0,t_{l})$
(63)
$\displaystyle\qquad\quad-\frac{8}{\pi^{2}}\,\int_{0}^{1}\\!dt_{k}\;\,\frac{\ln
f(0,t_{k})}{\sqrt{\rule{0.0pt}{9.95845pt}4t_{k}\bar{t}_{k}}}\;\int_{0}^{1}\\!dt_{l}\;\,\frac{\ln
f(0,t_{l})}{\sqrt{\rule{0.0pt}{9.95845pt}4t_{l}\bar{t}_{l}}}\;\ln\frac{f(0,t_{l})}{16t_{l}\bar{t}_{l}}$
$\displaystyle\qquad\quad+\frac{1}{\pi^{2}}\int_{0}^{1}\\!db\int_{0}^{1}\\!dt_{l}\int_{0}^{1}\\!dt_{kl}\;\frac{1}{\sqrt{\rule{0.0pt}{9.95845pt}16t_{l}\bar{t}_{l}t_{kl}\bar{t}_{kl}}}\;\,\bigg{[}\frac{1}{b}\,\ln\frac{256\,t_{l}\bar{t}_{l}t_{kl}\bar{t}_{kl}\,b^{2}}{(1+b)^{4}}\bigg{]}_{+}\;\mathcal{H}_{0}(b,t_{l},t_{kl})$
$\displaystyle\qquad\quad-\frac{2}{\pi^{2}}\int_{0}^{1}\\!db\int_{0}^{1}\\!dt_{l}\int_{0}^{1}\\!dt_{kl}\;\frac{1}{\sqrt{\rule{0.0pt}{9.95845pt}16t_{l}\bar{t}_{l}t_{kl}\bar{t}_{kl}}}\;\,\frac{\mathcal{H}_{5}(b,t_{l},t_{kl})}{b_{+}}$
$\displaystyle\qquad\quad+\frac{2}{\pi^{2}}\int_{0}^{1}\\!db\int_{0}^{1}\\!dt_{l}\int_{0}^{1}\\!dt_{kl}\int_{0}^{1}\\!ds\;\frac{1}{\sqrt{\rule{0.0pt}{9.95845pt}16t_{l}\bar{t}_{l}t_{kl}\bar{t}_{kl}}}\;\,\frac{1}{b}\bigg{[}\frac{1}{s\sqrt{1-s^{2}}}\bigg{]}_{+}\;\mathcal{H}_{6}(b,t_{l},t_{kl},s)\bigg{\\}},$

with

$\displaystyle\mathcal{H}_{5}(b,t_{l},t_{kl})$
$\displaystyle=\ln^{2}G_{A_{1}}(0,0,b,t_{k}^{+},t_{l},t_{kl})+\ln^{2}G_{A_{2}}(0,0,b,t_{k}^{+},t_{l},t_{kl})$
$\displaystyle\quad+2\ln^{2}G_{B}(0,0,b,t_{k}^{+},t_{l},t_{kl})+(t_{k}^{+}\to
t_{k}^{-})\,,$ $\displaystyle\mathcal{H}_{6}(b,t_{l},t_{kl},s)$
$\displaystyle=\ln G_{A_{1}}(0,0,b,t_{k}^{\oplus},t_{l},t_{kl})+\ln
G_{A_{2}}(0,0,b,t_{k}^{\oplus},t_{l},t_{kl})$ $\displaystyle\quad+2\ln
G_{B}(0,0,b,t_{k}^{\oplus},t_{l},t_{kl})+(t_{k}^{\oplus}\to
t_{k}^{\ominus})\,,$ (64)

and

$\displaystyle t_{k}^{\oplus}$
$\displaystyle=t_{l}+t_{kl}-2t_{l}t_{kl}+2\sqrt{t_{l}\bar{t}_{l}t_{kl}\bar{t}_{kl}(1-s^{2})}\,,$
$\displaystyle t_{k}^{\ominus}$
$\displaystyle=t_{l}+t_{kl}-2t_{l}t_{kl}-2\sqrt{t_{l}\bar{t}_{l}t_{kl}\bar{t}_{kl}(1-s^{2})}\,.$
(65)

For SCET-2 observables the relevant formulae are
$d_{2}^{\,C_{F}}=-\gamma^{C_{F}}_{1}$,

$\displaystyle\Delta d^{\,C_{F}}_{2}$
$\displaystyle=64\,\bigg{\\{}-\frac{4}{\pi}\,\int_{0}^{1}\\!dt_{l}\;\,\frac{\ln^{3}f(0,t_{l})}{\sqrt{\rule{0.0pt}{9.95845pt}4t_{l}\bar{t}_{l}}}+\frac{8}{\pi^{2}}\,\int_{0}^{1}\\!dt_{k}\;\,\frac{\ln
f(0,t_{k})}{\sqrt{\rule{0.0pt}{9.95845pt}4t_{k}\bar{t}_{k}}}\;\int_{0}^{1}\\!dt_{l}\;\,\frac{\ln^{2}f(0,t_{l})}{\sqrt{\rule{0.0pt}{9.95845pt}4t_{l}\bar{t}_{l}}}\;$
$\displaystyle\qquad\quad-\frac{1}{\pi^{2}}\int_{0}^{1}\\!db\int_{0}^{1}\\!dt_{l}\int_{0}^{1}\\!dt_{kl}\;\frac{1}{\sqrt{\rule{0.0pt}{9.95845pt}16t_{l}\bar{t}_{l}t_{kl}\bar{t}_{kl}}}\;\,\bigg{[}\frac{1}{b}\,\ln\frac{b^{2}}{(1+b)^{4}}\bigg{]}_{+}\;\mathcal{H}_{0}(b,t_{l},t_{kl})$
$\displaystyle\qquad\quad+\frac{2}{\pi^{2}}\int_{0}^{1}\\!db\int_{0}^{1}\\!dt_{l}\int_{0}^{1}\\!dt_{kl}\;\frac{1}{\sqrt{\rule{0.0pt}{9.95845pt}16t_{l}\bar{t}_{l}t_{kl}\bar{t}_{kl}}}\;\,\frac{\mathcal{H}_{5}(b,t_{l},t_{kl})}{b_{+}}\bigg{\\}},$
(66)

and the same constraint (61) has to be fulfilled. According to Bell:2018vaa ,
these relations are slightly modified for cumulant soft functions, and we will
not repeat the required changes here.

While the above formulae hold under the assumptions specified in Section 2,
our SoftSERVE implementation is subject to one additional constraint, i.e. the
measurement function $\omega(\\{k_{i}\\})$ must be strictly real and non-
negative. The SoftSERVE routines ADLap and ADMom therefore cannot immediately
be applied to Fourier-space soft functions, but as we explained in Appendix B
of Bell:2018oqa , there exists a workaround in SoftSERVE, which consists in
replacing the complex-valued measurement functions by their absolute values,
and by multiplying the result with appropriate factors that reshuffle the
expansion in the dimensional and rapidity regulators. For the anomalous
dimensions considered here, there exists a similar workaround, and in the
SCET-1 case one finds that the anomalous dimension in (58) is not changed,
whereas (61) and (63) receive additional contributions in this case given by
$(-\pi^{2})$ and
$-128\pi/n\int_{0}^{1}\\!dt_{l}/\sqrt{\rule{0.0pt}{9.95845pt}4t_{l}\bar{t}_{l}}\,\ln
f(0,t_{l})$, respectively. For SCET-2 soft functions, we find that the
collinear anomaly exponent itself is shifted by $-2\pi^{2}\beta_{0}C_{F}$,
whereas (61) and (66) are changed by $(-\pi^{2})$ and
$128\pi\int_{0}^{1}\\!dt_{l}/\sqrt{\rule{0.0pt}{9.95845pt}4t_{l}\bar{t}_{l}}\,\ln
f(0,t_{l})$.

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