$\displaystyle=1+\frac{\alpha_{s}C_{\kappa}}{\pi}\bigg{[}-\Big{(}\frac{1+\beta}{2+\beta}\Big{)}\log^{2}\big{(}se^{\gamma_{E}}\mu^{\frac{2+\beta}{1+\beta}}\big{)}-\frac{\pi^{2}}{24}\frac{\beta(3\beta+4)}{(1+\beta)(2+\beta)}\bigg{]}\,,$
(240)

Likewise, Laplace transforms of other pieces appearing in Eq. (83) for NLL′
resummation in the large $R_{g}$ region are also defined with respect to the
momentum variable $\ell^{+}$ following Eq. (82), such that at one-loop we find

$\displaystyle\Delta\tilde{S}_{c}^{[\beta_{0}]}\bigl{(}s,\beta,\mu\bigr{)}$
$\displaystyle=1+\frac{\alpha_{s}(\mu)\beta_{0}}{2\pi}\Big{(}\frac{1+\beta}{2+\beta}\Big{)}\log\big{(}se^{\gamma_{E}}\mu^{\frac{2+\beta}{1+\beta}}\big{)}\,,$
(241) $\displaystyle\Delta\tilde{S}_{c}^{[C_{\kappa}]}\Bigl{(}z=s\,Q_{\rm
cut}^{\frac{2+\beta}{1+\beta}}\Big{(}\frac{R_{g}}{2}\Big{)}^{2+\beta},\beta,\mu\Bigr{)}$
$\displaystyle=-\frac{\alpha_{s}C_{\kappa}}{\pi}\frac{2}{2+\beta}\frac{\partial}{\partial
b}z^{-b}\,\Gamma[b,z]\Bigg{|}_{b=0}\,,$

where the incomplete Gamma function is defined by

$\displaystyle\Gamma[b,z]\equiv\int_{z}^{\infty}dt\>t^{b-1}e^{-t}=z^{b}\int_{1}^{\infty}dt\>t^{b-1}e^{-zt}\,,\qquad\frac{\partial}{\partial
b}z^{-b}\,\Gamma[b,z]\Bigg{|}_{b=0}=\int_{1}^{\infty}dt\>\frac{\log(t)}{t}e^{-zt}\,.$
(242)

In our alternative Laplace space notation, we have for Eqs. (240) and (241)
respectively

$\displaystyle\tilde{S}_{c}^{\kappa}\big{[}L_{cs},\beta,\alpha(\mu_{cs})\big{]}$
$\displaystyle=1+\frac{\alpha_{s}(\mu_{cs})C_{\kappa}}{\pi}\bigg{[}-\Big{(}\frac{1+\beta}{2+\beta}\Big{)}L_{cs}^{2}-\frac{\pi^{2}}{24}\frac{\beta(3\beta+4)}{(1+\beta)(2+\beta)}\bigg{]}\,,$
(243)
$\displaystyle\Delta\tilde{S}_{c}^{[\beta_{0}]}\big{[}L_{cs},\beta,\alpha_{s}(\mu_{cs})\big{]}$
$\displaystyle=1+\frac{\alpha_{s}(\mu)\beta_{0}}{2\pi}\Big{(}\frac{1+\beta}{2+\beta}\Big{)}L_{cs}\,.$
(244)

###  B.2 Derivation of the large $R_{g}$ resummed cross section

In order to derive Eq. (86) and obtain an explicit expression for the kernel
${\cal Q^{\kappa}}$ defined in Eq. (4.3) we first write down the cross section
in Laplace space in Eq. (4.3) with resummation made explicit using Eqs. (201)
and (204):

$\displaystyle\int_{0}^{\infty}\\!dm_{J}^{2}\,\,e^{-x\,m_{J}^{2}}\frac{d\Sigma(R_{g})}{dm_{J}^{2}}\bigg{|}_{R_{g}\lesssim\theta_{g}^{\star}}\\!\\!\\!=N^{q}_{\rm
evol}\big{(}Q,Q_{\rm
cut},\beta,R,\mu;\mu_{N},\mu_{gs}\big{)}\>e^{K_{cs}(\mu,\mu_{cs})+K_{J}(\mu,\mu_{J})}$
$\displaystyle\quad\times\Big{(}\frac{\mu_{J}^{2}}{m_{J}^{2}}\Big{)}^{\tilde{\omega}_{J}(\mu,\mu_{J})}\bigg{(}\frac{Q\mu_{cs}}{m_{J}^{2}}\Big{(}\frac{\mu_{cs}}{Q_{\rm
cut}}\Big{)}^{\frac{1}{1+\beta}}\bigg{)}^{\omega_{cs}(\mu,\mu_{cs})}\big{(}m_{J}^{2}xe^{\gamma_{E}}\big{)}^{\tilde{\omega}(\mu_{cs},\mu_{J})}\tilde{J}_{q}(x,\mu_{J})\,\tilde{S}_{c}^{q}\Bigl{(}xQQ_{\rm
cut}^{-\frac{1}{1+\beta}},\beta,\mu_{cs}\Bigr{)}\,$
$\displaystyle\quad\times\bigg{(}1+\Delta\tilde{S}_{c}^{[\beta_{0}]}\bigl{(}xQQ_{\rm
cut}^{-\frac{1}{1+\beta}},\beta,\mu_{cs}\bigr{)}\Delta\tilde{S}_{c}^{[C_{F}]}\Bigl{(}xQQ_{\rm
cut}\Big{(}\frac{R_{g}}{2}\Big{)}^{2+\beta},\beta,\mu_{cs}\Bigr{)}\bigg{)}\,.$
(245)

Here we have decomposed the collinear-soft function as in Eqs. (83) and (84).
We recognize in the first two lines of Eq. (B.2) the resummed, single
differential jet mass cross section, described in distribution space by the
differential operator in Eq. (4.3). We now write $xm_{J}^{2}=\varrho$ such
that the functions $\tilde{J}_{q}$ and $\tilde{S}_{c}^{q}$ depend on their
arguments only through the logarithms

$\displaystyle
L_{J}=\log\Big{(}\frac{\mu_{J}^{2}}{m_{J}^{2}}\Big{)}+\log(\varrho)+\gamma_{E}\,,\qquad
L_{S_{c}}=\log\bigg{(}\frac{Q\mu_{cs}}{m_{J}^{2}}\Big{(}\frac{\mu_{cs}}{Q_{\rm
cut}}\Big{)}^{\frac{1}{1+\beta}}\bigg{)}+\log(\varrho)+\gamma_{E}\,,$ (246)

and note that the argument of the $R_{g}$-dependent piece
$\Delta\tilde{S}_{c}^{[C_{F}]}$ in Eq. (B.2) reduces to the combination

$\displaystyle xQQ_{\rm
cut}\Big{(}\frac{R_{g}}{2}\Big{)}^{2+\beta}=\varrho\Big{(}\frac{R_{g}}{\theta_{g}^{\star}}\Big{)}^{2+\beta}\,.$
(247)

Abbreviating $\tilde{\omega}(\mu_{cs},\mu_{J})=\Omega$ and
$(R_{g}/\theta_{g}^{\star})^{2+\beta}=\upsilon$, the inverse Laplace transform
of Eq. (B.2) in terms of dummy variable $r$ conjugate to $\varrho$ for $r=1$,
yields

$\displaystyle\frac{d\Sigma(R_{g})}{dm_{J}^{2}}\bigg{|}_{R_{g}\lesssim\theta_{g}^{\star}}$
$\displaystyle=N^{q}_{\rm evol}\big{(}Q,Q_{\rm
cut},\beta,R,\mu;\mu_{N},\mu_{gs}\big{)}$ (248)
$\displaystyle\times\frac{d\sigma^{q}\big{[}\partial_{\Omega}\big{]}}{dm_{J}^{2}}e^{\gamma_{E}\Omega}\int\frac{d\varrho}{2\pi
i}\>e^{\varrho}\varrho^{\Omega}\bigg{(}1+\Delta\tilde{S}_{c}^{[\beta_{0}]}(L_{S_{c}},\beta,\alpha_{s}(\mu_{cs}))\Delta\tilde{S}_{c}^{[C_{F}]}(\rho\upsilon,\beta,\mu_{cs})\bigg{)}\,.$

Noting from Eq. (241) that $\Delta\tilde{S}_{c}^{[\beta_{0}]}$ depends on its
argument only through the logarithm defined in Eq. (246), written in the
notation of Eq. (244), allows us to rewrite it as a function of
$\partial_{\Omega}$ and recognize in the second term of Eq. (248) the
definition of ${\cal Q}^{q}$ from Eq. (4.3). This completes the derivation of
Eq. (86). The same steps carry over to the $pp$ case, leading to the
expression in Eq. (97).

To find an explicit expression for ${\cal Q^{\kappa}}$, valid for both
$\kappa=q,g$, we take the inverse Laplace transform of
$\Delta\tilde{S}_{c}^{[C_{\kappa}]}$ from Eq. (241), which gives the integral

$\displaystyle\int\frac{d\varrho}{2\pi
i}\>e^{\varrho}\varrho^{\Omega}\big{(}\varrho\upsilon\big{)}^{-b}\Gamma\big{[}b,\varrho\upsilon\big{]}$
$\displaystyle=\,\int_{1}^{\infty}dt\>t^{b-1}\>\int\frac{d\varrho}{2\pi
i}\>e^{\varrho(1-\upsilon t)}\varrho^{\Omega}$
$\displaystyle=\,\frac{\Theta(1-\upsilon)}{\Gamma[-\Omega]}\upsilon^{-b}B[1-\upsilon;-\Omega,b]\,.$
(249)

Here we used the incomplete Beta function

$\displaystyle
B[1-\upsilon,-\Omega,b]\equiv\int_{0}^{1-\upsilon}dx\>x^{-\Omega-1}(1-x)^{b-1}\,.$
(250)

Including the constant factors, we get the result

$\displaystyle{\cal
Q}^{\kappa}\Big{[}\Omega,\upsilon=\Big{(}\frac{R_{g}}{\theta_{g}^{\star}}\Big{)}^{2+\beta},\beta\Big{]}=\Delta\tilde{S}_{c}^{[\beta_{0}]}\bigg{[}\partial_{\Omega}+\log\bigg{(}\frac{Q\mu_{cs}}{m_{J}^{2}}\Big{(}\frac{\mu_{cs}}{Q_{\rm
cut}}\Big{)}^{\frac{1}{1+\beta}}\bigg{)},\beta,\alpha_{s}(\mu_{cs})\bigg{]}$
(251)
$\displaystyle\qquad\times\frac{e^{\gamma_{E}\Omega}}{\Gamma(-\Omega)}\frac{2}{2+\beta}\frac{\alpha_{s}(\mu_{cs})C_{\kappa}}{\pi}\bigg{(}-\Theta(1-\upsilon)\frac{\partial}{\partial
b}\upsilon^{-b}B\big{[}1-\upsilon;\,-\Omega,b\big{]}\bigg{|}_{b=0}\bigg{)}\,,$

where we can equivalently write

$\displaystyle\partial_{b}\>\upsilon^{-b}B[1-\upsilon;-\Omega,b]\Big{|}_{b=0}=\int_{0}^{1-\upsilon}\frac{dx}{1-x}\>x^{-\Omega-1}\log\frac{1-x}{\upsilon}\,.$
(252)

###  B.3 Results for the boundary cross section

We now provide results and derivations for boundary corrections to soft matrix
elements. We first consider the case of global-soft function. We then present
results for the boundary corrections to the $S_{c_{g}}$ function, that would
be needed to compute the boundary cross section in the small and intermediate
$R_{g}$ regions. Finally, we give an explicit result for the kernel ${\cal
Q}^{\kappa}_{\varepsilon}$ defined in Eq. (120).

####  B.3.1 Global-soft function with shifted soft drop

Here we derive the one-loop expression of the global-soft function with
shifted soft drop condition, following the analogous derivation preformed for
the collinear-soft function in Sec. 4.4. In analogy with Eq. (101) we can
write

$\displaystyle S_{G}^{\kappa[1],\rm bare}\bigl{(}Q_{\rm
cut},R,\beta,\Theta_{\rm
sd}(\varepsilon),\mu\bigr{)}=\frac{\alpha_{s}C_{\kappa}}{\pi}\frac{(\mu^{2}e^{\gamma_{E}})^{\epsilon}}{\Gamma(1-\epsilon)}\int\frac{dp^{+}dp^{-}}{(p^{+}p^{-})^{1+\epsilon}}\Theta_{\rm
sd}(\varepsilon)\overline{\Theta}_{R}$
$\displaystyle\qquad=S_{G}^{\kappa[1],\rm bare}\bigl{(}Q_{\rm
cut},R,\beta,\mu\bigr{)}+\frac{Q\varepsilon}{Q_{\rm
cut}}S_{G,\varepsilon}^{\kappa[1],\rm bare}\bigl{(}Q_{\rm
cut},R,\beta,\mu\bigr{)}+{\cal O}(\varepsilon^{2})\,,$ (253)

where the first term is the global soft function for the cross section
cumulant in $R_{g}$ that we computed in Eq. (219), the constraint
$\overline{\Theta}_{R}$ was given in Eq. (221), while $\Theta_{\rm
sd}(\varepsilon)$ refers to the soft drop failing condition with $\varepsilon$
shift (without small angle approximation):

$\displaystyle\Theta_{\rm sd}(\varepsilon)$
$\displaystyle\equiv\Theta\bigg{(}\tilde{z}_{\rm
cut}\Big{(}\sin\frac{\theta_{p}}{2}\Big{)}^{\beta}-\frac{p^{+}+p^{-}}{Q}-\varepsilon\bigg{)}$
(254) $\displaystyle=\Theta_{\rm
sd}-\varepsilon\delta\bigg{(}\frac{p^{+}+p^{-}}{Q}-\tilde{z}_{\rm
cut}\Big{(}\sin\frac{\theta_{p}}{2}\Big{)}^{\beta}\bigg{)}+{\cal
O}(\varepsilon^{2})\,.$

The ${\cal O}(\varepsilon)$ correction in Eq. (B.3.1) is then given by
substituting the soft drop condition with $-\delta_{\rm sd}$ term in Eq.
(254), which fixes $p^{-}$ to the following value:

$\displaystyle p^{-}=Q_{\rm
cut}^{\frac{2}{2+\beta}}(p^{+})^{\frac{\beta}{2+\beta}}\bigg{[}1-\Big{(}\frac{p^{+}}{Q_{\rm
cut}}\Big{)}^{\frac{2}{2+\beta}}\bigg{]}\,,\qquad p^{+}<Q_{\rm cut}\,.$ (255)

This leads to

$\displaystyle S_{G,\varepsilon}^{\kappa[1],\rm bare}$
$\displaystyle\bigl{(}Q_{\rm
cut},R,\beta,\mu\bigr{)}=-\frac{2}{2+\beta}\frac{\alpha_{s}C_{\kappa}}{\pi}\Big{(}\frac{\mu}{Q_{\rm
cut}}\Big{)}^{2\epsilon}\frac{(e^{\gamma_{E}})^{\epsilon}}{\Gamma(1-\epsilon)}$
(256) $\displaystyle\times\int\frac{dp^{+}}{Q_{\rm
cut}}\Big{(}\frac{p^{+}}{Q_{\rm
cut}}\Big{)}^{-\frac{2(1+\beta)}{2+\beta}(1+\epsilon)}\bigg{[}1-\Big{(}\frac{p^{+}}{Q_{\rm
cut}}\Big{)}^{\frac{2}{2+\beta}}\bigg{]}^{-(1+\epsilon)}\Theta\bigg{(}\sin\frac{R}{2}-\Big{(}\frac{p^{+}}{Q_{\rm
cut}}\Big{)}^{\frac{1}{2+\beta}}\bigg{)}\,.$

We first consider the case of $\beta=0$ for which the result is given by

$\displaystyle S_{G,\varepsilon}^{\kappa[1],\rm bare}$
$\displaystyle\bigl{(}Q_{\rm cut},R,\beta=0,\mu\bigr{)}+\text{virtual
term}=\frac{\alpha_{s}C_{\kappa}}{\pi}\bigg{[}\frac{1}{\epsilon_{\rm
UV}}+2\log\Big{(}\frac{\mu}{Q_{\rm cut}\tan\frac{R}{2}}\Big{)}\bigg{]}\,.$
(257)

Note that we only considered the real emission term in the measurement
function in Eq. (B.3.1). We then demand that the virtual term, which we have
not explicitly included, appropriately turns the IR pole in Eq. (257) into a
UV pole.

Next we consider the case of $\beta>0$. Here, we note that the integral is
power law IR divergent and will be set to zero in dimensional regularization.
This results from the fact that we are trying to factorize the soft function
at subleading power but without considering the full set of operators that
appear at this order. Hence, in order to resolve this ambiguity it is helpful
to consider the combined measurement resulting from the shifted soft drop
condition in Eq. (254), the $R_{g}$ constraint in Eq. (99) (without small-
angle approximation), the jet radius constraint in Eq. (221), along with
$p^{+}$ measurement applied to the matrix element defined by a single soft
mode (i.e. not factorizing it into a global-soft and a collinear-soft mode).
In this case, the measurement function is given by

$\displaystyle\delta_{\rm full}\big{(}p^{+},\ell^{+},R_{g}\big{)}$
$\displaystyle\equiv\overline{\Theta}_{R}\bigg{[}\overline{\Theta}_{\rm
sd}(\varepsilon)\overline{\Theta}_{R_{g}}\delta(\ell^{+}-p^{+})+\Theta_{\rm
sd}(\varepsilon)\delta(\ell^{+})\bigg{]}+\Theta_{R}\delta(\ell^{+})-\delta(\ell^{+})$
$\displaystyle=\overline{\Theta}_{R}\overline{\Theta}_{\rm
sd}(\varepsilon)\big{[}\overline{\Theta}_{R_{g}}\delta(\ell^{+}-p^{+})-\delta(\ell^{+})\big{]}\,,$
(258)

where the three terms in the first line correspond to emissions that pass the
jet radius constraint, that fail and the virtual term. The ones that are
clustered in the jet radius are then tested for (shifted) soft drop condition
and the $R_{g}$ constraint as in Eq. (98). Writing $\overline{\Theta}_{\rm
sd}(\varepsilon)=\overline{\Theta}_{\rm sd}+\varepsilon\delta_{\rm sd}$ and
using the $\delta$ function to fix $p^{-}$ to Eq. (255), we find

$\displaystyle\int_{0}^{\infty}dp^{-}\>\overline{\Theta}_{R}\big{[}\overline{\Theta}_{R_{g}}\delta(\ell^{+}-p^{+})-\delta(\ell^{+})\big{]}\delta\bigg{(}p^{-}-Q_{\rm
cut}^{\frac{2}{2+\beta}}(p^{+})^{\frac{\beta}{2+\beta}}\bigg{[}1-\Big{(}\frac{p^{+}}{Q_{\rm
cut}}\Big{)}^{\frac{2}{2+\beta}}\bigg{]}\bigg{)}$ (259)
$\displaystyle=\Theta\bigg{(}\sin\frac{R_{g}}{2}-\Big{(}\frac{p^{+}}{Q_{\rm
cut}}\Big{)}^{\frac{1}{2+\beta}}\bigg{)}\big{[}\delta(\ell^{+}-p^{+})-\delta(\ell^{+})\big{]}+\Theta\bigg{(}\sin\frac{R_{g}}{2}<\Big{(}\frac{p^{+}}{Q_{\rm
cut}}\Big{)}^{\frac{1}{2+\beta}}<\sin\frac{R}{2}\bigg{)}\delta(\ell^{+})\,.$

The first term in the second line when expanded in small-angle approximation
matches with the corresponding term in the measurement function that we used
above for the $\ell^{+}$-differential collinear-soft function $S_{c}$ in Eq.
(98). The second term proportional to $\delta(\ell^{+})$ is of interest to us
and when integrated over $p^{+}$ leads precisely to the bare ${\cal
O}(\varepsilon)$ correction in Eq. (256) but now with $p^{+}$ integral bounded
below due to $R_{g}$ measurement, which then renders it IR finite. This
$R_{g}$-dependent term in the factorized cross section is captured in the
${\cal O}(\varepsilon)$ modification to the collinear-soft functions: the
$\ell^{+}$-differential $S_{c}$, as seen in Eq. (102), or equivalently in the
cumulative-$R_{g}$ $S_{c_{g}}^{\kappa}$ function if we are considering
intermediate $R_{g}$ regime as we show below. Thus, we can consider the
integration in Eq. (256) including the lower bound on $p^{+}$ due to $R_{g}$
measurement, and identify the pieces depending on $R$ and $R_{g}$ as parts of
the global-soft and the collinear-soft functions respectively. We then arrive
at the following result for $\beta>0$:

$\displaystyle S_{G,\varepsilon}^{\kappa[1],\rm bare}$
$\displaystyle\bigl{(}Q_{\rm cut},R,\beta,\mu\bigr{)}=$ (260)
$\displaystyle\Big{(}\frac{\beta}{2}\not\in\mathbb{N}\Big{)}\qquad$
$\displaystyle=-\frac{\alpha_{s}C_{\kappa}}{\pi}B\Big{[}\sin^{2}\Big{(}\frac{R}{2}\Big{)},-\frac{\beta}{2},0\Big{]}$
$\displaystyle\Big{(}\frac{\beta}{2}\in\mathbb{N}\Big{)}\qquad$
$\displaystyle=-\frac{\alpha_{s}C_{\kappa}}{\pi}\Bigg{[}\log\Big{[}\tan^{2}\Big{(}\frac{R}{2}\Big{)}\Big{]}+\sum_{k=1}^{\beta/2}\frac{1}{k}\Big{[}\sin^{2}\Big{(}\frac{R}{2}\Big{)}\Big{]}^{-k}\Bigg{]}\,.$

The result for $S_{c_{g},\varepsilon}^{\kappa[1]\rm bare}$ is simply negative
of the result for $S_{G,\varepsilon}^{\kappa[1],\rm bare}$ with $R$ replaced
by $R_{g}$ and dropping the subleading terms in $R_{g}\ll R$. This leads to
the result presented below in Eq. (265).

The terms in Eq. (260) are included in the $\Delta S_{G,\varepsilon}^{\kappa}$
piece in Eq. (4.4.2) for NLL′ resummation as follows:

$\displaystyle\Delta S_{G,\varepsilon}^{\kappa}\bigl{(}Q_{\rm
cut},R,\beta\bigr{)}=\Theta(\beta>0)\Bigg{[}1+\frac{\alpha_{s}(\mu_{gs})\beta_{0}}{2\pi}\log\Big{(}\frac{\mu}{Q_{\rm
cut}\tan^{1+\beta}\frac{R}{2}}\Big{)}\Bigg{]}$ (261)
$\displaystyle\qquad\times\frac{\alpha_{s}(\mu_{gs})C_{\kappa}}{\pi}\Bigg{[}\frac{2}{\beta}-\sin^{\beta}\frac{R}{2}\bigg{(}\Theta(0<\beta<2)\>B\Big{[}\sin^{2}\frac{R}{2},1-\frac{\beta}{2},0\Big{]}+\delta_{\beta,2}\log\tan^{2}\frac{R}{2}\bigg{)}\Bigg{]}\,,$

where as in the case of $R_{g}$-dependent collinear-soft function in Eq. (79)
we have included a $\beta_{0}$ term to cancel the $\mu$ dependence due to the
running coupling.

####  B.3.2 Boundary corrections to the collinear-soft kernel

Here we explicitly solve the Laplace transforms in Eq. (120), which define the
evolution kernel ${\cal Q}^{\kappa}_{\varepsilon}$ governing boundary
corrections to the cross section in the large $R_{g}$ region. The $\Delta
S_{c,\varepsilon}^{[C_{\kappa}]}$ piece defined in Eq. (113) transforms into

$\displaystyle\Delta\tilde{S}_{c,\varepsilon}^{[C_{\kappa}]}\Bigl{(}z=s\,Q_{\rm
cut}^{\frac{2+\beta}{1+\beta}}\Big{(}\frac{R_{g}}{2}\Big{)}^{2+\beta},\beta,\mu\Bigr{)}$
(262)
$\displaystyle\qquad=\frac{2}{2+\beta}\frac{\alpha_{s}(\mu)C_{\kappa}}{\pi}z^{\frac{\beta}{2+\beta}}\bigg{[}\Theta(\beta>0)\>\Gamma\Big{(}\\!-\frac{\beta}{2+\beta}\Big{)}-\Gamma\bigg{(}\\!-\frac{\beta}{2+\beta},z\bigg{)}\bigg{]}\,,$

where the incomplete gamma function was defined in Eq. (242). Using this
result we find

$\displaystyle\int\frac{d\varrho}{2\pi
i}\>e^{\varrho}\varrho^{\Omega}\Delta\tilde{S}_{c,\varepsilon}^{[C_{\kappa}]}(\varrho\upsilon,\beta,\mu_{cs})=\frac{2}{2+\beta}\frac{\alpha_{s}(\mu_{cs})C_{\kappa}}{\pi}\frac{1}{\Gamma(-\Omega)}$
$\displaystyle\qquad\times\upsilon^{\frac{\beta}{2+\beta}}\Bigg{(}\Theta(\beta>0)\>B\Big{[}-\Omega,-\frac{\beta}{2+\beta}\Big{]}-\Theta(1-\upsilon)B\Big{[}1-\upsilon;\,-\Omega,-\frac{\beta}{2+\beta}\Big{]}\Bigg{)}\,.$
(263)

We now state the result for the second term in Eq. (120), which involves again
Eq. (241)

$\displaystyle\int\frac{d\varrho}{2\pi
i}\>e^{\varrho}\varrho^{\Omega}\Delta\tilde{S}_{c,\varepsilon}^{[C_{\kappa}]}(\varrho\upsilon,\beta,\mu_{cs})\Delta\tilde{S}_{c}^{[C_{\kappa}]}(\varrho\upsilon,\beta,\mu_{cs})=-\Bigg{(}\frac{2}{2+\beta}\frac{\alpha_{s}(\mu)C_{\kappa}}{\pi}\Bigg{)}^{2}\frac{1}{\Gamma(-\Omega)}$
(264)
$\displaystyle\qquad\times\upsilon^{-b}\Bigg{(}\Theta(\beta>0)B[-\Omega,b]\Theta(1-\upsilon)\int_{0}^{1-\upsilon}\frac{dx}{1-x}\>x^{b-\Omega-1}\log\frac{1-x}{\upsilon}$
$\displaystyle\qquad\qquad-\Theta\Big{(}\frac{1}{2}-\upsilon\Big{)}\int_{\upsilon}^{1-\upsilon}\frac{dx}{1-x}\>x^{b-\Omega-1}\log\frac{1-x}{\upsilon}\>B\Big{[}1-\frac{\upsilon}{x},-\Omega,b\Big{]}\Bigg{)}\Bigg{|}_{b=\frac{-\beta}{2+\beta}}\,.$

####  B.3.3 Boundary Corrections for intermediate and small $R_{g}$

Here we discuss the soft drop boundary cross section in the intermediate and
small $R_{g}$ regimes. In both of the regimes where
$R_{g}\ll\theta_{g}^{\star}$ it is the $S_{c_{g}}^{\kappa}$ function that
receives ${\cal O}(\varepsilon)$ corrections upon shift to the soft drop
condition. This allows to treat the two cases where on equal footing and
results in considerable simplifications, since none of the boundary
corrections now depend on the jet mass. In this work we only provide the
details of the boundary corrections in the soft sector in these regimes and
leave further studies of the corresponding resummation to future work.

As commented below Eq. (260), the results for boundary corrections to
$S_{c_{g}}^{\kappa}$ are closely related to the corresponding corrections to
the global-soft function. Following the reasoning described there, we find

$\displaystyle S_{c_{g}}^{\kappa[1],\rm bare}$
$\displaystyle\Bigl{[}\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\overline{\Theta}_{\rm
sd}(\varepsilon),\beta,\mu\Bigr{]}-S_{c_{g}}^{\kappa[1],\rm
bare}\Bigl{[}\frac{R_{g}}{2}Q_{\rm cut}^{\frac{1}{1+\beta}},\beta,\mu\Bigr{]}$
(265) $\displaystyle(\beta=0)\qquad=-\frac{Q\varepsilon}{Q_{\rm
cut}}\frac{\alpha_{s}(\mu)C_{\kappa}}{\pi}\bigg{[}\frac{1}{\epsilon}+2\log\Big{(}\frac{\mu}{Q_{\rm
cut}(\frac{R_{g}}{2})}\Big{)}\bigg{]}\,,$
$\displaystyle(\beta>0)\qquad=-\frac{Q\varepsilon}{Q_{\rm
cut}}\Big{(}\frac{R_{g}}{2}\Big{)}^{-\beta}\frac{2}{\beta}\frac{\alpha_{s}(\mu)C_{\kappa}}{\pi}\,.$

Again, as required by consistency with RGE and the large $R_{g}$ case, for
$\beta=0$ we find a UV pole, which cancels against the global-soft function in
Eq. (108). Hence, we have an additional running between the scales $\mu_{gs}$
and $\mu_{cs_{g}}$ resulting from the non-cusp anomalous dimension

$\displaystyle\gamma_{0}^{S_{c_{g}}^{\kappa}}(\varepsilon,z_{\rm
cut})=-\gamma_{0}(\varepsilon,z_{\rm
cut})=-8C_{\kappa}\frac{Q\varepsilon}{Q_{\rm cut}}\,.\qquad(\beta=0)$ (266)

In analogy with the treatment of the boundary corrections to the global-soft
function in Eq. (4.4.2), we write the ${\cal O}(\varepsilon)$ corrections to
the CSg function as

$\displaystyle S_{c_{g}}^{\kappa}\Bigl{[}\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\overline{\Theta}_{\rm
sd}(\varepsilon),\beta,\mu\Bigr{]}$
$\displaystyle=S_{c_{g}}^{\kappa}\Bigl{[}\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta,\mu\Bigr{]}-2\delta_{\beta,0}\frac{Q\varepsilon}{Q_{\rm
cut}}\frac{\alpha_{s}(\mu)C_{\kappa}}{\pi}L_{cs_{g}}$ (267)
$\displaystyle+\frac{Q\varepsilon}{Q_{\rm
cut}}\frac{\Theta(\beta>0)}{\big{(}\frac{R_{g}}{2}\big{)}^{\beta}}S_{c_{g}}^{\kappa}\Bigl{[}\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta,\mu\Bigr{]}\Delta
S_{c_{g},\varepsilon}^{\kappa}\Bigl{[}\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta\Bigr{]}\,,$

where

$\displaystyle\Delta
S_{c_{g},\varepsilon}^{\kappa}\Bigl{[}\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta\Bigr{]}$
$\displaystyle=\Theta(\beta>0)\frac{-2}{\beta}\frac{\alpha_{s}(\mu)C_{\kappa}}{\pi}\bigg{(}1+\frac{\alpha_{s}(\mu)\beta_{0}}{2\pi}L_{cs_{g}}\bigg{)}$
(268)

and the logarithm being

$\displaystyle L_{cs_{g}}\equiv\log\Bigg{(}\frac{\mu}{Q_{\rm
cut}\big{(}\frac{R_{g}}{2}\big{)}^{1+\beta}}\Bigg{)}\,.$ (269)

Like in Eq. (79) for the large-$R_{g}$ collinear-soft function, Eq. (267)
includes an additional $\beta_{0}$ term to compensate for the $\mu$ dependence
of the running coupling, such that the non-logarithmic correction $\Delta
S_{c_{g},\varepsilon}^{\kappa}\Bigl{[}\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta\Bigr{]}$ is explicitly $\mu$ independent at
NLL′. This result consistently matches with the boundary correction in the
large $R_{g}$ region stated above in Eq. (B.3.2). To see this we can take
$\upsilon\rightarrow 0$ limit in Eq. (B.3.2) finding

$\displaystyle\lim\limits_{\upsilon\rightarrow 0}\int\frac{d\varrho}{2\pi
i}\>e^{\varrho}\varrho^{\Omega}\Delta\tilde{S}_{c,\varepsilon}^{[C_{\kappa}]}(\varrho\upsilon,\beta,\mu_{cs})$
$\displaystyle=\frac{-2}{\beta}\frac{\alpha_{s}(\mu_{cs})C_{\kappa}}{\pi}\frac{1}{\Gamma(-\Omega)}\,,$
(270)

which results in the right coefficient so as to match onto the boundary
correction in Eq. (268).

###  B.4 Calculation of $C_{2}^{q}$ from the differential boundary cross
section

Here we set up the calculation of $C_{2}^{q}$ based on the cross section
differential in groomed jet radius, i.e. by taking the $R_{g}$ derivative of
the large $R_{g}$ cumulative cross section with boundary corrections described
in Sec. 4.4. Although the extensions to $pp$ collisions is straightforward,
here we limit ourselves to the $e^{+}e^{-}$ case. We first consider the case
of $\beta>0$ and then include the additional terms that appear for $\beta=0$
at NLL. Based on the relation with the moment $M_{-1}^{q\varocircle}$ defined
in Eq. (5) adapted for the $e^{+}e^{-}$ case considered here and following our
prescription for computing the large $R_{g}$ differential boundary cross
section in Eq. (195), $C_{2}^{q}(m_{J}^{2})$ is given by

$\displaystyle C^{q}_{2}(m_{J}^{2})\frac{d\sigma}{dm_{J}^{2}}$
$\displaystyle=\frac{m_{J}^{2}}{Q^{2}}\int_{\theta_{\rm min}}^{\theta_{\rm
max}}\\!\\!\\!\\!d\theta_{g}\,\frac{2}{\theta_{g}}w_{\rm
large}(m_{J}^{2},\theta_{g})\frac{d}{d\varepsilon}\>\frac{d\sigma(\overline{\Theta}_{\rm
sd}(\varepsilon))}{dm_{J}^{2}d\theta_{g}}\bigg{|}_{{\rm
large}\,R_{g},\varepsilon\to 0}\,,$ (271)

where the double differential boundary cross section is computed in the large
$R_{g}$ theory and is matched to zero at small angle using the weight function
$w_{\rm large}$ as described in Sec. 5. We switch to the auxiliary variable
$\upsilon=(\theta_{g}/\theta_{g}^{\star})^{2+\beta}$ introduced in Eq. (120),

$\displaystyle C^{q}_{2}(m_{J}^{2})\frac{d\sigma}{dm_{J}^{2}}$
$\displaystyle=\frac{m_{J}^{2}}{Q^{2}}\int_{\upsilon_{\rm min}}^{\upsilon_{\rm
max}}\\!\\!\\!\\!d\upsilon\,\frac{2}{\theta_{g}(\upsilon)}w_{\rm
large}(m_{J}^{2},\theta_{g}(\upsilon))\frac{d}{d\varepsilon}\>\frac{d\sigma(\overline{\Theta}_{\rm
sd}(\varepsilon))}{dm_{J}^{2}d\upsilon}\bigg{|}_{{\rm
large}\,R_{g},\varepsilon\to 0}\,,$ (272)

and following the decomposition of the cumulant in Eq. (116) we write

$\displaystyle\frac{d\sigma(\overline{\Theta}_{\rm
sd}(\varepsilon))}{dm_{J}^{2}d\upsilon}\equiv\frac{d}{d\upsilon}\frac{d\Sigma(R_{g}(\upsilon),\overline{\Theta}_{\rm
sd}(\varepsilon))}{dm_{J}^{2}}$
$\displaystyle=\frac{d}{d\upsilon}\frac{d\Sigma\big{(}R_{g}(v),\delta_{\beta,0}\gamma_{0}(\varepsilon,z_{\rm
cut})\big{)}}{dm_{J}^{2}}+\frac{Q\varepsilon}{Q_{\rm
cut}}\frac{d\Delta\sigma_{\varepsilon}}{dm_{J}^{2}d\upsilon}\,,$ (273)

where each of the two terms in the r.h.s. is defined as the
$\upsilon$-derivative of the equivalent contribution in Eq. (116). The first
term is only present for $\beta=0$, while the second contributes for each
$\beta$.

We now compute the two terms in turn, starting from the second. This requires
differentiating Eq. (121), where we note that the term involving $S^{q}_{G}$
is independent of $\theta_{g}$ (or $\upsilon$),

$\displaystyle\frac{d\Delta\sigma_{\varepsilon}}{dm_{J}^{2}d\upsilon}$
$\displaystyle=\Big{(}\frac{2}{\theta_{g}^{\star}}\Big{)}^{\beta}N^{q}_{\rm
evol}\frac{d\sigma^{q}\big{[}\partial_{\Omega}\big{]}}{dm_{J}^{2}}\frac{d}{d\upsilon}\Big{(}v^{-\frac{\beta}{2+\beta}}{\cal
Q}_{\varepsilon}^{q}[\Omega,\upsilon,\beta]\Big{)}$ (274)
$\displaystyle=v^{\frac{-\beta}{2+\beta}}\Big{(}\frac{2}{\theta_{g}^{\star}}\Big{)}^{\beta}N^{q}_{\rm
evol}\frac{d\sigma^{q}\big{[}\partial_{\Omega}\big{]}}{dm_{J}^{2}}\tilde{S}_{c}^{[\beta_{0}]}\big{[}\partial_{\Omega}\big{]}\bigg{[}\frac{e^{\gamma_{E}\Omega}}{\Gamma(-\Omega)}\sum_{i=1,2}\Big{(}\frac{2}{2+\beta}\frac{\alpha_{s}(\mu_{cs})C_{F}}{\pi}\Big{)}^{i}{\cal
R}^{[i]}_{\varepsilon}\big{(}\Omega,\upsilon,\beta\big{)}\bigg{]}\,.$

In the second step we extracted the operator
$\tilde{S}_{c}^{[\beta_{0}]}\big{[}\partial_{\Omega}\big{]}$ from the
definition on ${\cal Q}_{\varepsilon}^{q}$ in Eq. (120) (omitting for brevity
the remaining arguments) and we organized the remaining $\upsilon$ dependent
terms as a perturbative expansion in $\alpha_{s}$. For simplicity we suppress
the arguments of the normalization factor $N^{q}_{\rm evol}$ defined in Eq.
(61). By taking derivatives of the inverse Laplace transforms of respectively
Eqs. (B.3.2) and (264), we determine the coefficients ${\cal
R}^{[1]}_{\varepsilon}$ and ${\cal R}^{[2]}_{\varepsilon}$:

$\displaystyle{\cal R}^{[1]}_{\varepsilon}\big{(}\Omega,\upsilon,\beta\big{)}$
$\displaystyle={\cal L}_{0}^{-\Omega}(1-\upsilon)\,,$ (275)
$\displaystyle{\cal R}^{[2]}_{\varepsilon}\big{(}\Omega,\upsilon,\beta\big{)}$
$\displaystyle=B[-\Omega,b]\Theta(1-\upsilon)\upsilon^{-b}B[1-\upsilon,b-\Omega,0]-\Theta(\beta>0)\Theta\Big{(}\frac{1}{2}-\upsilon\Big{)}$
$\displaystyle\times\int_{\upsilon}^{1-\upsilon}\\!\\!\\!\frac{dx}{1-x}\>\frac{1}{x^{1+\Omega}}\Bigg{(}\log\frac{1-x}{\upsilon}\Big{(}1-\frac{\upsilon}{x}\Big{)}^{-(1+\Omega)}+\Big{(}\frac{\upsilon}{x}\Big{)}^{-b}B\Big{[}1-\frac{\upsilon}{x},-\Omega,b\Big{]}\Bigg{)}\Bigg{|}_{b=\frac{-\beta}{2+\beta}}\,.$
(276)

Let us now move on to the first term in Eq. (273), which is the
$\upsilon$-derivative of Eq. (117),

$\displaystyle\frac{d}{d\upsilon}\frac{d\Sigma\big{(}R_{g}(v),\delta_{\beta,0}\gamma_{0}(\varepsilon,z_{\rm
cut})\big{)}}{dm_{J}^{2}}=N^{q}_{\rm
evol}\frac{d\sigma^{q}\big{[}\partial_{\Omega},\delta_{\beta,0}\gamma_{0}(\varepsilon,z_{\rm
cut})\big{]}}{dm_{J}^{2}}\>\frac{d}{d\upsilon}{\cal
Q}^{q}\big{[}\Omega,\upsilon,\beta\big{]}$ (277)
$\displaystyle\qquad=N^{q}_{\rm
evol}\frac{d\sigma^{q}\big{[}\partial_{\Omega},\delta_{\beta,0}\gamma_{0}(\varepsilon,z_{\rm
cut})\big{]}}{dm_{J}^{2}}\>\tilde{S}_{c}^{[\beta_{0}]}\big{[}\partial_{\Omega}\big{]}\frac{e^{\gamma_{E}\Omega}}{\Gamma(-\Omega)}\bigg{(}\frac{2}{2+\beta}\frac{\alpha_{s}(\mu_{cs})C_{F}}{\pi}\bigg{)}\frac{1}{\upsilon}B\big{[}1-\upsilon,-\Omega,0\big{]}\,.$

Here we used that
$d\sigma\big{[}\partial_{\Omega},\delta_{\beta,0}\gamma_{0}(\varepsilon,z_{\rm
cut})\big{]}$ does not depend on the groomed jet radius, extracted
$S_{c}^{[\beta_{0}]}\big{[}\partial_{\Omega}\big{]}$ from the definition of
${\cal Q}^{q}$ in Eq. (251), and took the $\upsilon$ derivative of the inverse
Laplace transform in Eq. (241). Expanding the operator
$d\sigma\big{[}\partial_{\Omega},\delta_{\beta,0}\gamma_{0}(\varepsilon,z_{\rm
cut})\big{]}$ to ${\cal O}(\varepsilon)$, we get

$\displaystyle\frac{d\sigma^{q}\big{[}\partial_{\Omega},\delta_{\beta,0}\gamma_{0}(\varepsilon,z_{\rm
cut})\big{]}}{dm_{J}^{2}}=\frac{d\sigma^{q}\big{[}\partial_{\Omega}\big{]}}{dm_{J}^{2}}\bigg{[}1+\delta_{\beta,0}\frac{Q\varepsilon}{Q_{\rm
cut}}\eta\big{(}\gamma_{0}(\varepsilon,z_{\rm
cut}),\mu_{cs},\mu_{gs}\big{)}\bigg{]}$ (278)
$\displaystyle\qquad+\delta_{\beta,0}\frac{Q\varepsilon}{Q_{\rm
cut}}\frac{d\sigma^{q}\big{[}\partial_{\Omega}\rightarrow
0\big{]}}{dm_{J}^{2}}\bigg{[}\frac{2\alpha_{s}(\mu_{gs})C_{F}}{\pi}\log\Big{(}\frac{\mu_{gs}}{Q_{\rm
cut}\tan\frac{R}{2}}\Big{)}-\frac{\alpha_{s}(\mu_{cs})C_{F}}{\pi}\partial_{\Omega}\bigg{]}\,,$

where the kernel $\eta\big{(}\gamma_{0}(\varepsilon,z_{\rm cut})\big{)}$ in
Eq. (206) arises from the ${\cal O}(\varepsilon)$ expansion of the Sudakov
exponent, and the additional terms in the second line result from the ${\cal
O}(\varepsilon)$ logarithms in the global-soft and collinear soft function in
Eqs. (4.4.2) and (111) respectively. Note that after taking inverse Laplace
transform the log in the collinear-soft function is replaced by the derivative
$\partial_{\Omega}$. Additionally, the two terms are evaluated at different
$\mu$ arguments. Furthermore, $d\sigma\big{[}\partial_{\Omega}\rightarrow
0\big{]}$ denotes setting to zero ${\cal O}(\alpha_{s})$ (or higher)
corrections in the term $d\sigma[\partial_{\Omega}]$ so as to account purely
for the evolution factor in Eq. (4.3). We can now finally combine Eqs. (274)
and (277) into the double differential boundary cross section in Eq. (273).
Substituting in Eq. (272) yields our final formula

$\displaystyle C^{q}_{2}(m_{J}^{2})\frac{d\sigma}{dm_{J}^{2}}$
$\displaystyle=\frac{m_{J}^{2}}{QQ_{\rm cut}}N^{q}_{\rm
evol}\int_{\upsilon_{\rm min}}^{\upsilon_{\rm
max}}\frac{d\upsilon}{\upsilon}\,\frac{2}{\theta_{g}(\upsilon)}w_{\rm
large}(m_{J}^{2},\theta_{g}(\upsilon))$ $\displaystyle\times$
$\displaystyle\Bigg{(}\delta_{\beta,0}\frac{d\sigma\big{[}\partial_{\Omega}\rightarrow
0\big{]}}{dm_{J}^{2}}\bigg{[}\frac{2\alpha_{s}(\mu_{gs})C_{F}}{\pi}\log\Big{(}\frac{\mu_{gs}}{Q_{\rm
cut}\tan\frac{R}{2}}\Big{)}-\frac{\alpha_{s}(\mu_{cs})C_{F}}{\pi}\partial_{\Omega}\bigg{]}B\big{[}1-\upsilon,-\Omega,0\big{]}$
$\displaystyle+\frac{2}{2+\beta}\frac{\alpha_{s}(\mu_{cs})C_{F}}{\pi}\frac{d\sigma\big{[}\partial_{\Omega}\big{]}}{dm_{J}^{2}}\tilde{S}_{c}^{[\beta_{0}]}\big{[}\partial_{\Omega}\big{]}\bigg{\\{}\delta_{\beta,0}\,\eta\big{(}\gamma_{0}(\varepsilon,z_{\rm
cut}),\mu_{cs},\mu_{gs}\big{)}B\big{[}1-\upsilon,-\Omega,0\big{]}$
$\displaystyle+v^{\frac{1+\beta}{2+\beta}}\Big{(}\frac{2}{\theta_{g}^{\star}}\Big{)}^{\beta}\Big{[}{\cal
R}_{\varepsilon}^{[1]}(\Omega,\upsilon,\beta)+\Big{(}\frac{2}{2+\beta}\frac{\alpha_{s}(\mu_{cs})C_{F}}{\pi}\Big{)}{\cal
R}_{\varepsilon}^{[2]}(\Omega,\upsilon,\beta)\Big{]}\bigg{\\}}\Bigg{)}\frac{e^{\gamma_{E}\Omega}}{\Gamma(-\Omega)}\,.$
(279)

As discussed in the introduction, the geometry relevant to the non-
perturbative corrections multiplying $C_{2}^{q}$ requires computing this
Wilson coefficient using only the large $R_{g}$ theory, which is enforced by
the weight function $w_{\rm large}(m_{J}^{2},\theta(\upsilon))$ in Eq. (B.4).
This is consistent with the idea that $C_{2}^{q}$ should describe only
perturbative physics. We note however that the integrand has an overall
behavior $\sim 1/\upsilon$, which enhances NP effects at low jet mass. To make
sure that there are no residual NP effects in $C_{2}^{q}$, we adopt a
prescription analogous to the one used for freezing the scale $\mu_{c_{g}}$ in
Eq. (174),

$\displaystyle\frac{1}{\upsilon}\simeq\Big{(}\frac{\psi^{\star}}{\psi}\Big{)}^{2+\beta}\;\to\;\frac{(\psi^{\star})^{2+\beta}}{\big{[}f_{\rm
run}(\psi^{1+\beta})\big{]}^{\frac{2+\beta}{1+\beta}}}\,,$ (280)

where $\psi,\psi^{\star}$ are the auxiliary variables defined in Eqs. (161)
and (167), and the first equality is exact in the $R_{g}\ll 1$ limit. The
profile function $f_{\rm run}$ defined in Eq. (177) smoothly freezes its
argument when approaching non-perturbative values.

As a final remark, we notice that in the region $m_{J}^{2}<m_{0}^{2}$ where
grooming is active, $\upsilon_{\rm max}=1$ and Eq. (275) requires solving the
class of integrals

$\displaystyle I_{n}(\upsilon_{\rm
min},\Omega)=\partial_{\Omega}^{n}\big{[}I_{0}(\upsilon_{\rm
min},\Omega)\big{]}\equiv\int_{\upsilon_{\rm
min}}^{1}\textrm{d}\upsilon\frac{\big{(}-\textrm{ln}(1-\upsilon)\big{)}^{n}}{(1-\upsilon)^{\Omega+1}}f_{n}(\upsilon)\,,$
(281)

where the derivatives arise from the operators
$d\sigma\big{[}\partial_{\Omega}\big{]}$ and
$\tilde{S}_{c}^{[\beta_{0}]}[\partial_{\Omega}\big{]}$ in Eq. (B.4). To avoid
numerical instabilities, we deal with these integrals using the subtraction

$\displaystyle I_{n}(\upsilon_{\rm min},\Omega)=\int_{\upsilon_{\rm
min}}^{1}\textrm{d}\upsilon\frac{\big{(}-\textrm{ln}(1-\upsilon)\big{)}^{i}}{(1-\upsilon)^{\Omega+1}}\big{[}f_{i}(\upsilon)-f_{i}(1)\big{]}+f_{i}(1)\bigg{(}\int_{\upsilon_{\rm
min}}^{1}\textrm{d}\upsilon\frac{\big{(}-\textrm{ln}(1-\upsilon)\big{)}^{i}}{(1-\upsilon)^{\Omega+1}}\bigg{)}\,.$
(282)

Similarly, we handle the singular integrand in the first term in Eq. (276)
using

$\displaystyle{\cal I}(v,\Omega)$
$\displaystyle\equiv\int_{\upsilon}^{1-\upsilon}\frac{dx}{1-x}\>\frac{1}{x^{1+\Omega}}\log\frac{1-x}{\upsilon}\Big{(}1-\frac{\upsilon}{x}\Big{)}^{-(1+\Omega)}$
$\displaystyle=\int_{\upsilon}^{1-\upsilon}\frac{dx}{1-x}\>\frac{1}{(x-v)^{1+\Omega}}\log\frac{1-x}{\upsilon}$
$\displaystyle=\int_{\upsilon}^{1-\upsilon}\frac{f(x)-f(\upsilon)}{(x-\upsilon)^{1+\Omega}}-f(\upsilon)\frac{(1-2\upsilon)^{-\Omega}}{(-\Omega)}\,,$
(283)

where

$\displaystyle
f(x)\equiv\frac{1}{1-x}\log\Big{(}\frac{1-x}{\upsilon}\Big{)}\,.$ (284)

##  Appendix C Calculation of NGLs and Clustering effects

###  C.1 Nonglobal logarithms

The functions $\Xi_{S}^{q},\Xi_{C}^{q}$ introduced in Eqs. (42) and (52) can
be expanded as a perturbative series in $\alpha_{s}$. Here we compute the
first nontrivial correction, which arise at ${\cal O}(\alpha_{s}^{2})$,
starting from the function $\Xi_{S}^{q}$ that describes the NGLs in the
intermediate $R_{g}$ region. To avoid plus distributions we switch to the
function cumulative in jet mass. Following Kang:2019prh , when the clustering
effects are ignored we can write

$\displaystyle\Xi_{S}^{q}(t_{S})=1-C_{F}C_{A}\left(\frac{\alpha_{s}}{2\pi}\right)^{2}\int\frac{dx_{1}}{x_{1}}\int\frac{dx_{2}}{x_{2}}\int_{1\in
J}dc_{1}\frac{d\phi_{1}}{2\pi}\int_{2\not\in J}dc_{2}\frac{d\phi_{2}}{2\pi}$
$\displaystyle\qquad\times\Theta(x_{1}-x_{2})\,\Theta(x_{2}-\tilde{z}_{\rm
cut}R_{g}^{\beta})\,\Theta\left(\frac{\theta_{c}^{2}}{4}-\frac{x_{1}}{2}(1-c_{1})\right)\frac{\cos\phi_{2}}{(1-c_{1}c_{2}-s_{1}s_{2}\cos\phi_{2})s_{1}s_{2}}$
(285)

where $c_{i}=\cos\theta_{i}$, $s_{i}=\sin\theta_{i}$ and the dependence on the
(cumulative) jet mass variable is implicit in
$(\theta_{c}/2)^{2}=(\theta_{g}^{\star}/2)^{2+\beta}Q_{\rm cut}/Q$. Here the
integration variables $x_{i},\theta_{i}$ and $\phi_{i}$ represent respectively
the energy fractions, colatitudes, and azimuths of the strongly ordered
emissions. The geometry relevant to the NGLs requires the largest emission to
lie inside the groomed jet and the second emission to lie out of it.

The energy integrals can be done easily and yield

$\displaystyle\Xi_{S}^{q}(t_{S})$
$\displaystyle=1-C_{F}C_{A}\left(\frac{\alpha_{s}}{2\pi}\right)^{2}\int_{1\in
J}dc_{1}\frac{d\phi_{1}}{2\pi}\int_{2\not{\in}J}dc_{2}\frac{d\phi_{2}}{2\pi}\Theta\left(\frac{\theta_{c}^{2}}{2\tilde{z}_{\rm
cut}R_{g}^{\beta}}-(1-c_{1})\right)$
$\displaystyle\quad\times\frac{\cos\phi_{2}}{(1-c_{1}c_{2}-s_{1}s_{2}\cos\phi_{2})s_{1}s_{2}}\frac{1}{2}\textrm{ln}^{2}\bigg{(}\frac{\theta_{c}^{2}}{2(1-c_{1})\tilde{z}_{\rm
cut}R_{g}^{\beta}}\bigg{)}\,.$ (286)

Since the gluon 1 is restricted to be inside the jet, $\theta_{1}/2\leq
R_{g}/2\ll 1$, we can approximate the factor of $1-c_{1}$ inside the logarithm
as $\theta_{1}^{2}/2$. At the same time, taking into account clustering
effects as in Kang:2019prh , the dominant term in the $R_{g}/2\ll 1$ limit
gives the following result at two loops:

$\displaystyle\Xi_{S}^{q}(t_{S})\,=\,1-\frac{4}{9}C_{F}C_{A}\frac{\pi^{2}}{3}t_{S}^{2}\,.$
(287)

The leading NGLs in the other two regions can be obtained from the
intermediate $R_{g}$ expression in Eq. (45) by taking appropriate limits. For
the large groomed jet radius regime $R_{g}=\theta_{g}^{\star}$ and the NGL
contribution goes to 0 as was expected. For the small groomed jet radius
regime we can formally set $\theta_{c}=R_{g}$ and the dominant contribution in
this EFT is

$\displaystyle\Xi_{C}^{q}(t_{C})$
$\displaystyle=1-\frac{4}{9}C_{F}C_{A}\frac{\pi^{2}}{3}t_{C}^{2}\,,$ (288)

which is larger than the contribution for the case of intermediate groomed jet
radius. Written with $t_{C}$ see that Eq. (288) has the same form as Eq.
(287).

###  C.2 Abelian clustering logarithms

In this section we account for the leading logarithms arising from clustering
effects of the C/A algorithm while implementing soft drop for independent
emissions. To this aim, we first consider a naive implementation of the soft
drop algorithm where clustering logarithms are absent and see how clustering
effects modify this result. The naive implementation simply tests for the SD
condition for each particle independently, i.e. the allowed phase space for
each emission is

$\displaystyle\Theta(z_{i}-\tilde{z}_{\rm
cut}\theta_{i}^{\beta})\Theta(R_{g}-\theta_{i})+\Theta(\tilde{z}_{\rm
cut}\theta_{i}^{\beta}-z_{i})\,.$ (289)

However, we know that the C/A algorithm recursively clusters partons with
minimum angular separation; since the SD algorithm works backwards through the
history of the clustered jet, the grooming condition is then effectively
applied on subjets instead of individual partons. Therefore, (in the case of 2
collinear-soft emissions that gives the leading configuration) we can have a
scenario where one of the partons would fail soft drop and the other would
pass if tested independently, but the two get clustered with each other first
and with the collinear parton later. The SD condition is now tested on the
entire collinear-soft subjet, which will pass the test. In this case the
softest parton has been pulled inside the groomed jet, violating the naive
measure in Eq. (289). Instead, in case the most energetic collinear-soft
emission is clustered with the collinear parton first, then the naive
implementation of the measurement is correct.

Let us denote the portion of phase space where the naive measure is violated
by

$\displaystyle\Theta_{C/A}=\Theta(d_{1}-R_{g}^{2})\Theta(R_{g}^{2}-d_{2})\Theta(d_{2}-d_{12})\,,$
(290)

where $d_{12}$ is the relative distance of the collinear soft emissions and
$d_{i}$ their distance from the jet axis, so that parton 1 is at a distance
greater then $R_{g}$ from the jet axis while parton 2 is within $R_{g}$ from
it. For 1 and 2 to be clustered first, we then need $d_{2}>d_{12}$. Expressed
in the small angle approximation, this becomes

$\displaystyle\Theta_{C/A}=\Theta(\theta^{2}_{1}-R_{g}^{2})\Theta(R_{g}^{2}-\theta_{2}^{2})\Theta(-\theta_{1}^{2}+2\theta_{1}\theta_{2}\cos\phi_{2})\,,$
(291)

where we have assumed without loss of generality that $\phi_{12}=\phi_{2}$.
The allowed phase space requires that parton 1 must fail SD while parton 2 can
either pass or fail (with energy fractions $x_{2}\gg x_{1}$). However, for
parton 1 to be pulled inside the jet, we need that parton 2 passes soft drop.
If we ignore for the time being the mass measurement, and consider only a
cumulative $R_{g}$ measurement, the naive result without clustering effects
reads

$\displaystyle M_{\rm indep.}(R_{g})$ $\displaystyle\equiv\int
d\Pi_{2}\,\Theta(\tilde{z}_{\rm
cut}\theta_{1}^{\beta}-x_{1})\Theta(x_{2}-\tilde{z}_{\rm
cut}\theta_{2}^{\beta})\Theta_{C/A}\,,$ (292)

where we have abbreviated the two-parton phase space in the strongly ordered
limit as

$\displaystyle\int
d\Pi_{2}=\left(\alpha_{s}\frac{C_{\kappa}}{\pi}\right)^{2}\int\frac{dx_{1}}{x_{1}}\int\frac{dx_{2}}{x_{2}}\int\frac{d\theta_{1}}{\theta_{1}}\int\frac{d\theta_{2}}{\theta_{2}}\int\frac{d\phi_{1}}{2\pi}\int\frac{d\phi_{2}}{2\pi}\,.$
(293)

Let us now account for clustering effects. The idea is that since parton 1 is
pulled inside the jet, it never gets individually tested for SD and can in
fact also be allowed to pass SD in this region of phase space. Therefore the
measurement in Eq. (292) gets modified to give us

$\displaystyle M_{\rm cluster}(R_{g})$ $\displaystyle\equiv\int
d\Pi_{2}\,\Theta(x_{2}-\tilde{z}_{\rm cut}\theta_{2}^{\beta})\Theta_{C/A}\,.$
(294)

We now express the result as a correction on top of the naive result by
subtracting it out the from the full result to isolate the clustering effects:

$\displaystyle\Delta M_{\rm alg}(R_{g})$ $\displaystyle\equiv M_{\rm
cluster}-M_{\rm indep.}=\frac{1}{2!}\int d\Pi_{2}\,\Theta(x_{1}-\tilde{z}_{\rm
cut}\theta_{2}^{\beta})\Theta(x_{2}-\tilde{z}_{\rm
cut}\theta_{2}^{\beta})\Theta_{C/A}\,,$ (295)

which agrees with the integrand quoted in Kang:2019prh . The factor of $1/2!$
takes into account the scenario where we interchange the labels 1 and 2 in the
strongly ordered phase space measure in Eq. (293).

We can now extend this result to our case where we also have a jet mass
measurement on the groomed jet. The naive value in Eq. (292) with a cumulative
jet mass measurement becomes

$\displaystyle M_{\rm indep.}(m_{J}^{2},R_{g})$ $\displaystyle\equiv\int
d\Pi_{2}\,\Theta(\tilde{z}_{\rm
cut}\theta_{1}^{\beta}-x_{1})\Theta(x_{2}-\tilde{z}_{\rm
cut}\theta_{2}^{\beta})\Theta\Big{(}(\theta_{c}(m_{J}^{2}))^{2}-x_{2}\theta_{2}^{2}\Big{)}\Theta_{C/A}\,,$
(296)

which upon including clustering corrections gives

$\displaystyle M_{\rm cluster}(m_{J}^{2},R_{g})$ $\displaystyle\equiv\int
d\Pi_{2}\,\Theta(\theta_{c}^{2}-x_{1}\theta_{1}^{2})\Theta(x_{2}-\tilde{z}_{\rm
cut}\theta_{2}^{\beta})\Theta(\theta_{c}^{2}-x_{2}\theta_{2}^{2})\Theta_{C/A}\,,$
(297)

where we note that the parton 1 will contribute to the jet mass regardless of
whether it passes or fails since it is part of our groomed jet. Therefore the
correction from the algorithm becomes

$\displaystyle\Delta M_{\rm alg}(m_{J}^{2},R_{g})\equiv\int
d\Pi_{2}\,\Big{[}\Theta(\theta_{c}^{2}-x_{1}\theta_{1}^{2})-\Theta(\tilde{z}_{\rm
cut}\theta_{1}^{\beta}-x_{1})\Big{]}\Theta(x_{2}-\tilde{z}_{\rm
cut}\theta_{2}^{\beta})\Theta(\theta_{c}^{2}-x_{2}\theta_{2}^{2})\Theta_{C/A}\,.$
(298)

To extract out the leading logarithmic behavior, we notice that none of the
terms in Eq. (298) are sensitive to the difference between $\theta_{1}$ and
$\theta_{2}$, and hence can approximate $\theta_{1}\sim\theta_{2}\sim R_{g}$
in the integrand, such that

$\displaystyle\Theta(\theta_{c}^{2}-x_{1}\theta_{1}^{2})-\Theta(\tilde{z}_{\rm
cut}\theta_{1}^{\beta}-x_{1})$
$\displaystyle\approx\Theta(\theta_{c}^{2}-x_{1}R_{g}^{2})-\Theta(\tilde{z}_{\rm
cut}R_{g}^{\beta}-x_{1})\,,$ (299) $\displaystyle\Theta(x_{2}-\tilde{z}_{\rm
cut}\theta_{2}^{\beta})\,\Theta(\theta_{c}^{2}-x_{2}\theta_{2}^{2})$
$\displaystyle\approx\Theta(x_{2}-\tilde{z}_{\rm
cut}R_{g}^{\beta})\,\Theta(\theta_{c}^{2}-x_{2}R_{g}^{2})\,,$

which allow us to factorize the angular and energy integrals,

$\displaystyle\Delta M_{\rm alg}(m_{J}^{2},R_{g})$
$\displaystyle=\frac{1}{2!}\left(\frac{\alpha_{s}C_{\kappa}}{\pi}\right)^{2}\int\frac{d\theta_{1}}{\theta_{1}}\int\frac{d\theta_{2}}{\theta_{2}}\int\frac{d\phi_{1}}{2\pi}\frac{d\phi_{2}}{2\pi}\Theta_{C/A}$
(300)
$\displaystyle\times\int\frac{dx_{1}}{x_{1}}\Big{[}\Theta(\theta_{c}^{2}-x_{1}R_{g}^{2})-\Theta(\tilde{z}_{\rm
cut}R_{g}^{\beta}-x_{1})\Big{]}\int\frac{dx_{2}}{x_{2}}\
\Theta(x_{2}-\tilde{z}_{\rm
cut}R_{g}^{\beta})\Theta(\theta_{c}^{2}-x_{2}R_{g}^{2})$
$\displaystyle=\frac{1}{2!}\left(\frac{\alpha_{s}C_{\kappa}}{\pi}\right)^{2}\textrm{ln}^{2}\bigg{(}\frac{\theta_{c}^{2}}{\tilde{z}_{\rm
cut}R_{g}^{\beta+2}}\bigg{)}\int\frac{d\theta_{1}}{\theta_{1}}\int\frac{d\theta_{2}}{\theta_{2}}\int\frac{d\phi_{1}}{2\pi}\int\frac{d\phi_{2}}{2\pi}\Theta_{C/A}\,.$

The angular integral is now identical to the one in Kang:2019prh , which can
be solved to yield

$\displaystyle\Delta M_{\rm alg}(m_{J}^{2},R_{g})$
$\displaystyle=\frac{1}{2!}\left(\frac{\alpha_{s}C_{\kappa}}{\pi}\right)^{2}\frac{\pi^{2}}{54}\textrm{ln}^{2}\left(\frac{\big{(}\theta_{c}(m_{J}^{2})\big{)}^{2}}{\tilde{z}_{\rm
cut}R_{g}^{\beta+2}}\right)\,.$ (301)

We see that the leading abelian clustering logarithms have the same functional
form as the NGLs in Eq. (45), but are further suppressed by the factor
$\frac{1}{36}$. Furthermore, this result shows that just like the NGLs, the
leading clustering logarithms also vanish in the large $R_{g}$ regime,
consistent with the existence of a single collinear-soft mode rather than two
modes that know about $R_{g}$ separated in virtuality. On the other hand, they
are maximized in the small $R_{g}$ limit, $\theta_{c}=R_{g}$, where the effect
of the mass measurement drops out and we find back the result of Kang:2019prh
.

We also note that in this calculation we have not considered the scenario when
both partons naively fail the SD condition but pass if clustered together.
This is evaluated in Frye:2016aiz  and would lead to a sub-leading single
logarithmic contribution $\alpha_{s}^{2}\log X$ compared to the correction
explicitly considered here.

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