$\displaystyle\eta^{\rm min}_{\rm hemi}(R)=\ln(1-R),\quad\eta_{\pi}(R)$
$\displaystyle=\ln(R-1),$ (122)

and $R_{\pi}\approx 1.28$ is the solution of the transcendental equation

$\displaystyle\eta_{\pi}(R_{\pi})=-R_{\pi}\,.$ (123)

With these explicit limits the integrals can be evaluated efficiently.

###  B.2 Power suppression of boundary integrals

We have seen in fig. -1006 that for a small jet radius the jet region from the
hemisphere decomposition with $\rho^{R}_{J}$ and the actual conical
partitioning largely overlap giving small results for the non-hemisphere
corrections. However, for $R\sim 1$ the areas in the $\eta$-$\phi$ plane begin
to differ very significantly, which might suggest that the associated
corrections become very large in this regime and the results for the small
$R$-expansion do not provide a good approximation. As we have seen in sec. 4.4
this turns out not to be the case since the deviations of the jet areas in the
beam coordinates are not representative for the size of the associated
corrections. Instead it is more meaningful to compare the jet areas in the
boosted frame where the jet and beam direction are back-to-back and soft
radiation from the beam-jet dipole $aJ$ is uniform in the respective rapidity-
azimuth coordinates $\tilde{\eta}$, $\tilde{\phi}$. The associated
transformation rules between the sets of coordinates are explicitly given in
ref. Kasemets:2015uus . In fig. -1005 we display the jet regions in these
coordinates for the conical measure (red) and for the hemisphere decomposition
with $\rho_{J}=\rho_{J}^{R}$ for different values of $R$. The areas which do
not overlap correspond directly to the integrals $I_{0,aJ}^{B}$ and
$I_{0,aJ}^{J}$, respectively, while $I_{1,aJ}^{B}$ and $I_{1,aJ}^{J}$ are
(logarithmic) moments in these regions. These are individually of
$\sim\mathcal{O}(R)$, which can be also confirmed by an analytic expansion
indicated by the black, dotted line. In total the contributions from
$F_{aJ}^{B}$ and $F_{aJ}^{J}$ cancel each other at this order leading to a net
contribution to the soft function of $\mathcal{O}(R^{2})$.111111For the
corrections $s_{aJ,B}$ and $s_{aJ,J}$ in eq. (4.3) this is obvious since only
the difference between the two mismatch areas in fig. -1005 enters. For the
correction $s_{aJ,\delta}$ this holds for measurements which are continuous
functions in $\eta$, $\phi$ due to the fact that at leading order in $R$ the
integrands are constant in these areas.

Figure -1005: Illustration of the phase space misalignment between the
hemisphere jet region with $\rho_{J}=\rho_{J}^{R}$ (blue, dashed) and a
conical partitioning (red, solid) for $\eta_{J}=0$ and $R=0.5,1.0,1.5$ in the
boosted frame where the jet and beam $a$ are back-to-back. The areas which do
not overlap scale as the integrals $I_{aJ}^{B}$ and $I_{aJ}^{J}$,
respectively. The black dotted lines indicates the analytic result for the
conical measure at $\mathcal{O}(R)$. The associated corrections to the
coefficients $s_{aJ,J}$, $s_{aJ,B}$ correspond to the areas between the dashed
and solid curves.

##  Appendix C Analytic corrections for $pp\to$ dijets

Beyond single jet production, $pp\to$ dijets is another process of
phenomenological relevance for measurements like jet mass. The full
computation of the associated soft function corrections for arbitrary jet and
beam measurements and partitionings can be carried out following the
hemisphere decompositions discussed in secs. 3 and 4. Here we compute the
analytic corrections for $pp\to$ dijets ($j_{1},j_{2}$) in a small $R$
expansion up to terms at $\mathcal{O}(R^{2})$, whereas the full $R$ dependence
can be determined numerically but now including a jet-jet dipole. For
definiteness and simplicity we consider conical jets with a jet mass
measurement (i.e. angularity in defined in eq. (59) with $\beta=2$) and a
$p_{T}$ jet veto. For generic $R<\pi/2$ we can write the renormalized one-loop
soft function as121212For $R<\pi/2$ (i.e. as long as the jet regions do not
share a common boundary) the measurements and partitioning are invariant under
boosts along the beam axis, such that this correction mainly depends on the
relative rapidity of the jets $\Delta\eta_{12}$ and the jet radius. Since the
rapidity regularization breaks boost invariance, there is, however, also a
residual dependence on the individual jet rapidities appearing in $s_{a1,B}$.

$\displaystyle\widehat{S}^{\kappa(1)}_{2}(\\{k_{i}\\},R,\eta_{1},\eta_{2},\mu,\nu)=\frac{\alpha_{s}(\mu)}{4\pi}\biggl{\\{}{\bf
T}_{a}\cdot{\bf
T}_{b}\biggl{[}\frac{16}{\mu}\Bigl{(}{\mathcal{L}}_{1}\Bigl{(}\frac{k_{B}}{\mu}\Bigr{)}-{\mathcal{L}}_{0}\Bigl{(}\frac{k_{B}}{\mu}\Bigr{)}\ln\Bigl{(}\frac{\nu}{\mu}\Bigr{)}\Bigr{)}\,\delta(k_{1})\,\delta(k_{2})$
$\displaystyle\quad+s_{ab,B}(R)\Bigl{(}\frac{2}{\mu}\,{\mathcal{L}}_{0}\Bigl{(}\frac{k_{B}}{\mu}\Bigr{)}\,\delta(k_{1})\,\delta(k_{2})-\frac{1}{\mu}\,{\mathcal{L}}_{0}\Bigl{(}\frac{k_{1}}{\mu}\Bigr{)}\,\delta(k_{B})\,\delta(k_{2})-\frac{1}{\mu}\,{\mathcal{L}}_{0}\Bigl{(}\frac{k_{2}}{\mu}\Bigr{)}\,\delta(k_{B})\,\delta(k_{1})\Bigr{)}$
$\displaystyle\quad+s_{ab,\delta}(R)\,\delta(k_{B})\,\delta(k_{1})\,\delta(k_{2})\biggr{]}+{\bf
T}_{1}\cdot{\bf
T}_{2}\biggl{[}\frac{8}{\mu}\,{\mathcal{L}}_{1}\Bigl{(}\frac{k_{1}}{\mu}\Bigr{)}\,\delta(k_{2})\,\delta(k_{B})+\frac{8}{\mu}\,{\mathcal{L}}_{1}\Bigl{(}\frac{k_{2}}{\mu}\Bigr{)}\,\delta(k_{1})\,\delta(k_{B})$
$\displaystyle\quad+s_{12,J}(R,\Delta\eta_{12})\Bigl{(}\frac{1}{\mu}\,{\mathcal{L}}_{0}\Bigl{(}\frac{k_{1}}{\mu}\Bigr{)}\,\delta(k_{2})+\frac{1}{\mu}\,{\mathcal{L}}_{0}\Bigl{(}\frac{k_{2}}{\mu}\Bigr{)}\,\delta(k_{1})\Bigr{)}\delta(k_{B})$
$\displaystyle\quad+s_{12,B}(R,\Delta\eta_{12})\,\frac{1}{\mu}\,{\mathcal{L}}_{0}\Bigl{(}\frac{k_{B}}{\mu}\Bigr{)}\,\delta(k_{1})\,\delta(k_{2})+s_{12,\delta}(R,\Delta\eta_{12})\,\delta(k_{1})\,\delta(k_{2})\,\delta(k_{B})\biggr{]}$
$\displaystyle\quad+{\bf T}_{a}\cdot{\bf
T}_{1}\biggl{[}\frac{8}{\mu}\,{\mathcal{L}}_{1}\Bigl{(}\frac{k_{1}}{\mu}\Bigr{)}\,\delta(k_{B})\,\delta(k_{2})+\frac{8}{\mu}\,{\mathcal{L}}_{1}\Bigl{(}\frac{k_{B}}{\mu}\Bigr{)}\,\delta(k_{1})\,\delta(k_{2})-\frac{8}{\mu}\,\mathcal{L}_{0}\Bigl{(}\frac{k_{B}}{\mu}\Bigr{)}\ln\Bigl{(}\frac{\nu}{\mu}\Bigr{)}\,\delta(k_{1})\,\delta(k_{2})$
$\displaystyle\quad+s_{a1,1}(R)\,\frac{1}{\mu}\,{\mathcal{L}}_{0}\Bigl{(}\frac{k_{1}}{\mu}\Bigr{)}\,\delta(k_{B})\,\delta(k_{2})+s_{a1,2}(R,\Delta\eta_{12})\,\frac{1}{\mu}\,{\mathcal{L}}_{0}\Bigl{(}\frac{k_{2}}{\mu}\Bigr{)}\,\delta(k_{B})\,\delta(k_{1})$
$\displaystyle\quad+s_{a1,B}(R,\eta_{1},\Delta\eta_{12})\,\frac{1}{\mu}\,{\mathcal{L}}_{0}\Bigl{(}\frac{k_{B}}{\mu}\Bigr{)}\,\delta(k_{1})\,\delta(k_{2})+s_{a1,\delta}(R,\Delta\eta_{12})\,\delta(k_{B})\,\delta(k_{1})\,\delta(k_{2})\biggr{]}$
$\displaystyle\quad+{\bf T}_{b}\cdot{\bf
T}_{1}\biggl{[}\Delta\eta_{12}\to-\Delta\eta_{12}\biggr{]}+{\bf
T}_{b}\cdot{\bf
T}_{2}\biggl{[}(k_{1},k_{2},\eta_{1})\to(k_{2},k_{1},\eta_{2})\biggr{]}$
$\displaystyle\quad+{\bf T}_{a}\cdot{\bf
T}_{2}\biggl{[}(k_{1},k_{2},\eta_{1},\Delta\eta_{12})\to(k_{2},k_{1},\eta_{2},-\Delta\eta_{12})\biggr{]}\biggr{\\}}\,,$
(124)

where $\Delta\eta_{12}\equiv\eta_{1}-\eta_{2}$ is the difference between the
rapidities of the two jets and $R_{1}=R_{2}\equiv R<\pi/2$. The replacements
in the last line are always with respect to the terms with the color factor
${\bf T}_{a}\cdot{\bf T}_{1}$.

The contributions from the beam-beam dipole are equivalent to the case of
single production given in eq. (4.3) and app. A.2, i.e.

$\displaystyle s_{ab,B}(R)=4R^{2}+\mathcal{O}(R^{4})\,,\quad
s_{ab,\delta}(R)=-\frac{\pi^{2}}{3}+4R^{2}(2\ln R-1)+\mathcal{O}(R^{4})\,.$
(125)

The contributions from the beam-jet dipoles are also closely related to the
ones for single production given in eq. (4.3) and app. A.2 with the difference
that starting at $\mathcal{O}(R^{2})$ there is now also a correction due to
emissions into the phase space region of the second jet, which concerns the
coefficients $s_{a1,2}$, $s_{a1,B}$ and $s_{a1,\delta}$ and can be easily
computed analytically in analogy to app. A.2. We get

$\displaystyle s_{a1,1}(R)$ $\displaystyle=-8\ln
R-R^{2}+\mathcal{O}(R^{4})\,,$ (126) $\displaystyle
s_{a1,2}(R,\Delta\eta_{12})$
$\displaystyle=-R^{2}\frac{e^{-\Delta\eta_{12}}}{\cosh^{2}\Bigl{(}\frac{\Delta\eta_{12}}{2}\Bigr{)}}+\mathcal{O}(R^{4})\,,$
$\displaystyle s_{a1,B}(R,\eta,\Delta\eta_{12})$ $\displaystyle=8\ln
R+8\eta+R^{2}\biggl{[}1+\frac{e^{-\Delta\eta_{12}}}{\cosh^{2}\Bigl{(}\frac{\Delta\eta_{12}}{2}\Bigr{)}}\biggr{]}+\mathcal{O}(R^{4})\,,$
$\displaystyle s_{a1,\delta}(R,\Delta\eta_{12})$
$\displaystyle=4\ln^{2}R(2-\theta(R-1))+R^{2}(2\ln
R-1)\biggl{[}1+\frac{e^{-\Delta\eta_{12}}}{\cosh^{2}\Bigl{(}\frac{\Delta\eta_{12}}{2}\Bigr{)}}\biggr{]}+\mathcal{O}(R^{4})\,.$

We demonstrate in fig. -1004 that including the terms up to
$\mathcal{O}(R^{2})$ gives a very good approximation of the full results, even
for $R\sim 1$.

Figure -1004: The full coefficients $s_{a1,2}$ (top) and $s_{a1,\delta}$
(bottom) together with the small $R$ for conical (anti-$k_{T}$) jets for a jet
mass measurement ($\beta=2$) and a $p_{T}$ veto, for $\Delta\eta_{12}=1$ in
terms of $R$ (left) and for $R=1$ in terms of $\Delta\eta_{12}$ (right).

The only remaining ingredient is the correction from the jet-jet dipole. The
leading small-$R$ results have been computed in ref. Hornig:2016ahz , which we
have reproduced.131313Reference Hornig:2016ahz  considers a $p_{T}$-veto with
a rapidity cutoff $\eta_{\rm cut}$. For the jet-jet dipole the effect due to
$\eta_{\rm cut}$ is power suppressed in $1/e^{\eta_{\rm cut}}$, while for the
other dipole contributions it leads to different results than those given
above. The $\mathcal{O}(R^{2})$ corrections can be computed following app.
A.2. This gives

$\displaystyle s_{12,J}(R,\Delta\eta_{12})$ $\displaystyle=-8\ln
R-R^{2}\tanh^{2}\frac{\Delta\eta_{12}}{2}+\mathcal{O}(R^{4})\,,$ (127)
$\displaystyle s_{12,B}(R,\Delta\eta_{12})$
$\displaystyle=-16\ln\Bigl{(}2\cosh\frac{\Delta\eta_{12}}{2}\Bigr{)}-2s_{12,J}(\Delta\eta_{12},R)\,,$
$\displaystyle s_{12,\delta}(R,\Delta\eta_{12})$
$\displaystyle=16\ln^{2}R-8\ln^{2}\Bigl{(}2\cosh\frac{\Delta\eta_{12}}{2}\Bigr{)}+2(\Delta\eta_{12})^{2}-\frac{\pi^{2}}{3}$
$\displaystyle\quad+R^{2}\Bigl{[}2(2\ln
R-1)\tanh^{2}\frac{\Delta\eta_{12}}{2}\biggr{]}+\mathcal{O}(R^{4})\,.$

In fig. -1003 we compare the full numeric results for these coefficients to
the analytic expressions. Again the small $R$ expansion provides an excellent
approximation of the full result for the jet-jet dipole contribution. Together
with the findings for the beam-beam and beam-jet dipole corrections this
indicates that keeping terms up to $\mathcal{O}(R^{2})$ is likely sufficient
for phenomenological purposes.

Figure -1003: The full coefficients $s_{12,B}$ (top) and $s_{12,\delta}$
(bottom) together with the small $R$ results for conical (anti-$k_{T}$) jets
for a jet mass measurement ($\beta=2$) and a $p_{T}$ veto, for
$\Delta\eta_{12}=1$ in terms of $R$ (left) and for $R=1$ in terms of
$\Delta\eta_{12}$ (right).

We remark that for jet vetoes which are not boost invariant, all of the
dipoles, in particular also the jet-jet dipole, depend on the individual jet
rapidities. For multijet processes or an additional recoiling color singlet
state the soft function depends in addition on the separation of the jets in
azimuth. The analytic computation for these cases is significantly more
involved.

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