cut},R,\beta,\Theta_{\rm sd}(\varepsilon),\mu\bigr{)}=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{)}\,,$ (105)

where the additional argument $\Theta_{\rm sd}(\varepsilon)$ denotes now the
shift in the soft drop _failing_ condition. The result for the ${\cal
O}(\varepsilon)$ correction to the bare global-soft function with shifted soft
drop reads

$\displaystyle S_{G,\varepsilon}^{\kappa[1],\rm bare}\bigl{(}Q_{\rm
cut},R,\beta,\mu\bigr{)}$
$\displaystyle=\frac{\alpha_{s}C_{\kappa}}{\pi}\times\left\\{\begin{array}[]{cc}\bigg{[}\frac{1}{\epsilon_{\rm
UV}}+2\log\Big{(}\frac{\mu}{Q_{\rm
cut}\tan\frac{R}{2}}\Big{)}\bigg{]}&\quad\big{(}\beta=0\big{)}\\\\[10.0pt]
\bigg{[}\frac{2}{\beta}\frac{1}{\sin^{\beta}\big{(}\frac{R}{2}\big{)}}+\ldots\bigg{]}&\quad\big{(}\beta>0\big{)}\end{array}\right.\,,$
(108)

where the additional terms not shown for $\beta>0$ are ${\cal O}(1)$ and not
relevant to the discussion here. The details of this derivation and complete
results for $\beta>0$ are provided in App. B.3. We see that the additional
single logarithmic UV divergence for $\beta=0$ case has the opposite sign to
that in Eq. (4.4.1), as required by RG consistency. Interestingly, we now find
that for $\beta=0$ the one-loop collinear-soft and global-soft functions
develop a non-zero non-cusp anomalous dimension proportional to $\varepsilon$.
With conventions for prefactors in anomalous dimensions, summarized in Eqs.
(208) and (211), we find

$\displaystyle\gamma_{0}^{S_{G}^{\kappa}}(\varepsilon,z_{\rm
cut})=-\gamma_{0}^{S_{c}^{\kappa}}(\varepsilon,z_{\rm
cut})\,\equiv\,\gamma_{0}(\varepsilon,z_{\rm
cut})\,=\,8C_{\kappa}\frac{Q\varepsilon}{Q_{\rm cut}}\,\qquad(\beta=0)\,.$
(109)

Hence, for $\beta=0$, the NLL′ resummed cross section for shifted soft drop
will involve additional running between the global-soft and collinear-soft
virtualities due to this non-cusp anomalous dimension. We now turn to
addressing how the various ${\cal O}(\varepsilon)$ corrections are combined
with the other EFT matrix elements so as to arrive at the result for resummed
cross section with shifted soft drop condition.

####  4.4.2 Assembling the ingredients

Having found that the shift in the soft drop condition induces different
${\cal O}(\varepsilon)$ modifications to the collinear-soft function on the
l.h.s. of Eq. (78), we now generalize the decomposition in the r.h.s. of that
equation to account for these additional boundary terms. Since the correction
for $\beta=0$ consists of logarithms induced by the extra anomalous dimension
in Eq. (109), we find it appropriate to combine this term with
$S_{c}^{\kappa}\big{[}\ell^{+}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta,\mu\big{]}$ in the r.h.s. of Eq. (78), which is
responsible for RG evolution. On the other hand, the non-logarithmic, $R_{g}$
dependent terms in Eq. (4.4.1) can be written as a function of the variable
$\tilde{\ell}$ and are treated in the same way as the $\Delta
S_{c}^{[C_{\kappa}]}$ piece in Eq. (81). Thus, we express the renormalized
collinear-soft function with cumulant $R_{g}$ measurement and soft drop shift
as follows:

$\displaystyle S_{c}^{\kappa}\Bigl{[}\ell^{+}Q_{\rm
cut}^{\frac{1}{1+\beta}},\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\overline{\Theta}_{\rm
sd}(\varepsilon),\beta,\mu\Bigr{]}=\int d\ell^{\prime+}\,Q_{\rm
cut}^{\frac{1}{1+\beta}}\,S_{c}^{\kappa}\Bigl{[}(\ell^{+}-\ell^{\prime+})Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta,\mu,\,\delta_{\beta,0}\gamma_{0}^{S_{c}^{\kappa}}(\varepsilon,z_{\rm
cut})\Bigr{]}$ $\displaystyle\qquad\times\int d\ell^{\prime\prime+}\Delta
S_{c}^{\kappa}\Bigl{[}(\ell^{\prime+}-\ell^{\prime\prime+})Q_{\rm
cut}^{\frac{1}{1+\beta}},\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta\Bigr{]}\>\Delta
S_{c,\varepsilon}^{\kappa}\Bigl{[}\ell^{\prime\prime+}Q_{\rm
cut}^{\frac{1}{1+\beta}},\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta\Bigr{]}\,.$ (110)

Here the collinear soft function for single differential jet mass measurement
includes the additional logarithm that is generated for $\beta=0$ in Eq.
(4.4.1), as signaled by the extra argument
$\delta_{\beta,0}\gamma_{0}^{S_{c}^{\kappa}}(\varepsilon z_{\rm cut})$, and is
modified as follows:

$\displaystyle S_{c}^{\kappa}\Bigl{[}\ell^{+}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta,\mu,\,\delta_{\beta,0}\gamma_{0}^{S_{c}^{\kappa}}(\varepsilon,z_{\rm
cut})\Bigr{]}\equiv S_{c}^{\kappa}\Bigl{[}\ell^{+}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta,\mu\Bigr{]}+\delta_{\beta,0}\frac{Q\varepsilon}{Q_{\rm
cut}}\frac{\alpha_{s}C_{\kappa}}{\pi}\frac{1}{\mu^{2}}{\cal
L}_{0}\Big{(}\frac{\ell^{+}Q_{\rm cut}}{\mu^{2}}\Big{)}\,,$ (111)

while the term $\Delta S_{c,\varepsilon}^{\kappa}$ is defined analogously to
$\Delta S_{c}^{\kappa}$ given in Eq. (79):

$\displaystyle\Delta S_{c,\varepsilon}^{\kappa}\Bigl{[}\ell^{+}Q_{\rm
cut}^{\frac{1}{1+\beta}},\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta\Bigr{]}$
$\displaystyle=\delta\Bigl{(}\ell^{+}Q_{\rm
cut}^{\frac{1}{1+\beta}}\Bigr{)}+\int\frac{d\ell^{\prime+}Q_{\rm
cut}^{\frac{1}{1+\beta}}}{\big{(}\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}}\big{)}^{2+\beta}}\>\Delta
S_{c}^{[\beta_{0}]}\Bigl{[}(\ell^{+}-\ell^{\prime+})Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta,\mu\Bigr{]}$
$\displaystyle\qquad\times\frac{Q\varepsilon}{Q_{\rm
cut}}\frac{1}{\big{(}\frac{R_{g}}{2}\big{)}^{\beta}}\Delta
S_{c,\varepsilon}^{[C_{\kappa}]}\Bigg{[}\frac{\ell^{\prime+}Q_{\rm
cut}^{\frac{1}{1+\beta}}}{\big{(}\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}}\big{)}^{2+\beta}},\beta,\mu\Bigg{]}\,.$ (112)

The $\Delta S_{c}^{[\beta_{0}]}$ piece is the same as that defined in Eq. (80)
and is again included to cancel the $\mu$ dependence due to the running
coupling at NLL′. Finally, from Eq. (4.4.1) we have

$\displaystyle\Delta
S_{c,\varepsilon}^{[C_{\kappa}]}\Bigg{[}\tilde{\ell}=\frac{\ell^{+}Q_{\rm
cut}^{\frac{1}{1+\beta}}}{\big{(}\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}}\big{)}^{2+\beta}},\,\beta,\,\mu\Bigg{]}=$ (113)
$\displaystyle\qquad\frac{2}{2+\beta}\frac{\alpha_{s}(\mu)C_{\kappa}}{\pi}\Bigg{(}\Theta(\beta>0)\>{\cal
L}_{0}^{-\frac{\beta}{2+\beta}}(\tilde{\ell})+\frac{\Theta(\tilde{\ell}-1)}{\tilde{\ell}^{\frac{2(1+\beta)}{2+\beta}}}\Bigg{)}\bigg{(}1+\frac{\alpha_{s}(\mu)}{\pi}a_{20,\varepsilon}^{S_{c}}\bigg{)}+{\cal
O}(\alpha_{s}^{2})\,.$

Here the $\Theta(\beta>0)$ signifies that the first term is only present for
the $\beta>0$ case, whereas the second term applies for all $\beta\geq 0$. As
we shall see later, this second term contributes to the bulk of the boundary
corrections, and hence to $C_{2}(m_{J}^{2})$.

Note that in defining $\Delta S_{c,\varepsilon}^{\kappa}$ in this way we
factored out the $R_{g}^{-\beta}$ power present in Eq. (4.4.1) for $\beta>0$.
Comparing this scaling with the ${\cal O}(\varepsilon)$ correction to the
global-soft function in Eq. (108), we see that the $R_{g}^{-\beta}$ power
dominates the $\sin^{-\beta}(R/2)$ power in the $R_{g}\ll R$ approximation
which is formally still valid in the large $R_{g}$ region. Thus we expect
boundary contributions to the global-soft piece (for $\beta>0$) to be
numerically insignificant, and we will leave them out from our numerical
predictions. Finally, since we lack the knowledge of the non-logarithmic two-
loop boundary corrections in $S_{c}^{\kappa}$ that cannot be predicted by the
anomalous dimensions we have included an additional parameter
$a_{20,\varepsilon}^{S_{c}}$ in spirit of Eq. (81), to facilitate estimating
the uncertainty resulting from these terms. We will come back to this again in
Sec. 6 when studying the perturbative uncertainties.

Taking the Laplace transform of Eq. (4.4.2), performing an expansion up to
O($\varepsilon$), and dropping terms that are ${\cal O}(\alpha_{s}^{3})$ we
explicitly get

$\displaystyle\tilde{S}_{c}^{\kappa}\Bigl{(}s,\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\overline{\Theta}_{\rm
sd}(\varepsilon),\beta,\mu\Bigr{)}=\tilde{S}_{c}^{\kappa}\Bigl{(}s,\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta,\mu\Bigr{)}$ (114)
$\displaystyle\qquad+\frac{Q\varepsilon}{Q_{\rm
cut}}\bigg{\\{}-\delta_{\beta,0}\frac{\alpha_{s}C_{\kappa}}{\pi}\log(se^{\gamma_{E}}\mu^{2})\Delta\tilde{S}_{c}^{[C_{\kappa}]}\Bigl{(}s\,Q_{\rm
cut}^{\frac{2+\beta}{1+\beta}}\Big{(}\frac{R_{g}}{2}\Big{)}^{2+\beta},\beta,\mu\Bigr{)}$
$\displaystyle\qquad+\frac{1}{\big{(}\frac{R_{g}}{2}\big{)}^{\beta}}\>\tilde{S}_{c}^{\kappa}\bigl{(}s,\beta,\mu\bigr{)}\Delta\tilde{S}_{c}^{[\beta_{0}]}\bigl{(}s,\beta,\mu\bigr{)}\Delta\tilde{S}_{c,\varepsilon}^{[C_{\kappa}]}\Bigl{(}s\,Q_{\rm
cut}^{\frac{2+\beta}{1+\beta}}\Big{(}\frac{R_{g}}{2}\Big{)}^{2+\beta},\beta,\mu\Bigr{)}$
$\displaystyle\qquad+\frac{1}{\big{(}\frac{R_{g}}{2}\big{)}^{\beta}}\>\Delta\tilde{S}_{c}^{[C_{\kappa}]}\Bigl{(}s\,Q_{\rm
cut}^{\frac{2+\beta}{1+\beta}}\Big{(}\frac{R_{g}}{2}\Big{)}^{2+\beta},\beta,\mu\Bigr{)}\Delta\tilde{S}^{[C_{\kappa}]}_{c,\varepsilon}\Bigl{(}s\,Q_{\rm
cut}^{\frac{2+\beta}{1+\beta}}\Big{(}\frac{R_{g}}{2}\Big{)}^{2+\beta},\beta,\mu\Bigr{)}\bigg{\\}}\,,$

where we used Eqs. (83) and (84) to expand the Laplace transform of the
$\Delta S_{c}^{\kappa}$ term, while the Laplace transform of the $\Delta
S_{c,\varepsilon}^{[C_{\kappa}]}$ term is given in Eq. (262). The first line
reproduces the ${\cal O}(\varepsilon^{0})$ result in Eq. (78) and vanishes
when taking the $\varepsilon$ derivative in Eq. (5). The second line describes
boundary corrections induced by the additional logarithm present for
$\beta=0$, while the remaining lines encode boundary corrections that are
$R_{g}$ dependent. The product of the two $\Delta S_{c}^{[C_{\kappa}]}$
functions in the last line is important to ensure a smooth transition to the
regime $R_{g}\ll\theta_{g}^{\star}$, which justifies including the boundary
corrections $\Delta S_{c,\varepsilon}^{\kappa}$ multiplicatively in Eq.
(4.4.2).

Likewise, we express the global-soft function with ${\cal O}(\varepsilon)$
corrections as

$\displaystyle S_{G}^{\kappa}\bigl{(}Q_{\rm cut},R,$
$\displaystyle\overline{\Theta}_{\rm
sd}(\varepsilon),\beta,\mu\bigr{)}=S_{G}^{\kappa}\Big{(}Q_{\rm
cut}\tan^{1+\beta}\frac{R}{2},\beta,\mu_{gs},R\Big{)}+\delta_{\beta,0}\frac{Q\varepsilon}{Q_{\rm
cut}}\frac{2\alpha_{s}C_{\kappa}}{\pi}\log\Big{(}\frac{\mu}{Q_{\rm
cut}\tan\frac{R}{2}}\Big{)}$ $\displaystyle\qquad+\frac{Q\varepsilon}{Q_{\rm
cut}}\frac{\Theta(\beta>0)}{\sin^{\beta}\big{(}\frac{R}{2}\big{)}}\>S_{G}^{\kappa}\Big{(}Q_{\rm
cut}\tan^{1+\beta}\frac{R}{2},\beta,\mu_{gs},R\Big{)}\Delta
S_{G,\varepsilon}^{\kappa}\bigl{(}Q_{\rm cut},R,\beta\bigr{)}\,,$ (115)

with the result for $\mu$-independent $\Delta S_{G,\varepsilon}^{\kappa}$
piece given in Eq. (261).

####  4.4.3 Cumulative boundary cross section

With all the ingredients for the shifted soft drop condition at hand, we are
finally in the position to write resummed results for the cumulative cross
section. Eq. (4.3) is still formally valid, but the global soft and collinear-
soft functions that appear there now include boundary corrections according to
Eqs. (4.4.2) and (4.4.2). As a consequence, the equivalent of Eq. (86) has two
kind of terms:

$\displaystyle\frac{d\Sigma(R_{g},\overline{\Theta}_{\rm
sd}(\varepsilon))}{dm_{J}^{2}}$
$\displaystyle=\frac{d\Sigma\big{(}R_{g},\delta_{\beta,0}\gamma_{0}(\varepsilon,z_{\rm
cut})\big{)}}{dm_{J}^{2}}+\frac{Q\varepsilon}{Q_{\rm
cut}}\frac{d\Delta\Sigma_{\varepsilon}(R_{g})}{dm_{J}^{2}}+{\cal
O}(\varepsilon^{2})\,.$ (116)

They describe boundary corrections to the NLL RG evolution and the $R_{g}$
dependence respectively, and we now examine them in turn. The first term
reduces to the ${\cal O}(\varepsilon^{0})$ cross section for positive $\beta$
but, for $\beta=0$, includes the additional one-loop fixed order logarithms
and running due to the ${\cal O}(\varepsilon)$ one-loop non-cusp anomalous
dimension displayed in Eq. (109). Since these corrections do not depend on the
jet mass, they can be absorbed in a generalization of the operator
$d\sigma^{q}(\partial_{\Omega})$ in Eq. (4.3) yielding

$\displaystyle\frac{d\Sigma\big{(}R_{g},\delta_{\beta,0}\gamma_{0}(\varepsilon,z_{\rm
cut})\big{)}}{dm_{J}^{2}}\bigg{|}_{R_{g}\lesssim\theta_{g}^{\star}}$
$\displaystyle\equiv N^{q}_{\rm evol}\big{(}Q,Q_{\rm
cut},\beta,R,\mu;\mu_{N},\mu_{gs}\big{)}$ (117)
$\displaystyle\times\frac{d\sigma^{q}\big{[}\partial_{\Omega},\delta_{\beta,0}\gamma_{0}(\varepsilon,z_{\rm
cut})\big{]}}{dm_{J}^{2}}\>\Bigg{(}\frac{e^{\gamma_{E}\Omega}}{\Gamma(-\Omega)}+{\cal
Q}^{q}\Big{[}\Omega,\frac{R_{g}}{\theta_{g}^{\star}},\beta\Big{]}\Bigg{)}\bigg{|}_{\Omega=\tilde{\omega}(\mu_{cs},\mu_{J})}\,,$

while the evolution kernel in brackets is unchanged. The operator is obtained
from the following ${\cal O}(\varepsilon)$ modifications to its analogue in
Eq. (4.3):

  1. 1.

The collinear-soft function in Eq. (86) includes the additional
$\cal{O}(\varepsilon)$ logarithm in Eq. (114).

  2. 2.

The global-soft function includes the additional $\cal{O}(\varepsilon)$
logarithm in Eq. (4.4.2).

  3. 3.

The collinear-soft function has additional running:

$\displaystyle
K_{cs}\big{(}\mu,\mu_{cs},\delta_{\beta,0}\gamma^{S_{c}^{q}}_{0}(\varepsilon,z_{\rm
cut})\big{)}$
$\displaystyle=K_{cs}(\mu,\mu_{cs})-\delta_{\beta,0}\,\eta\Big{(}\gamma_{0}(\varepsilon,z_{\rm
cut}),\mu,\mu_{0}\Big{)}\,,$ (118)

where the LL single logarithmic evolution kernel $\eta(\Gamma,\mu,\mu_{0})$ is
given in Eq. (206) from the one-loop non-cusp anomalous dimension.

  4. 4.

The normalization factor $N^{q}_{\rm evol}$ has additional running due to the
global soft function:

$\displaystyle
K_{gs}\big{(}\mu,\mu_{gs},\delta_{\beta,0}\gamma^{S_{G}^{q}}_{0}(\varepsilon,z_{\rm
cut})\big{)}$
$\displaystyle=K_{gs}(\mu,\mu_{gs})+\delta_{\beta,0}\,\eta\big{(}\gamma_{0}(\varepsilon,z_{\rm
cut}),\mu,\mu_{0}\big{)}\,.$ (119)

The second term in Eq. (116) consists of the residual ${\cal O}(\varepsilon)$
non-logarithmic, $R_{g}$-dependent corrections to the collinear-soft and
global-soft functions in Eqs. (4.4.2) and (4.4.2), which apply for every
$\beta\geq 0$. Since these terms do not alter evolution, but have a nontrivial
dependence on the jet mass and $R_{g}$, they require a different kernel than
Eq. (4.3):

$\displaystyle{\cal
Q}^{\kappa}_{\varepsilon}\bigg{[}\Omega,\upsilon=\Big{(}\frac{R_{g}}{\theta_{g}^{\star}}\Big{)}^{2+\beta},\beta\bigg{]}$
$\displaystyle\equiv
e^{\gamma_{E}\Omega}\Delta\tilde{S}_{c}^{[\beta_{0}]}\bigg{[}\partial_{\Omega}+\log\bigg{(}\frac{Q\mu_{cs}}{m_{J}^{2}}\Big{(}\frac{\mu_{cs}}{Q_{\rm
cut}}\Big{)}^{\frac{1}{1+\beta}}\bigg{)},\beta,\alpha_{s}(\mu_{cs})\bigg{]}$
(120) $\displaystyle\times\int\frac{d\varrho}{2\pi
i}\>e^{\varrho}\varrho^{\Omega}\Delta\tilde{S}_{c,\varepsilon}^{[C_{\kappa}]}\big{(}\varrho\upsilon,\beta,\mu_{cs}\big{)}\Big{(}1+\Delta\tilde{S}_{c}^{[C_{\kappa}]}\big{(}\varrho\upsilon,\beta,\mu_{cs}\big{)}\Big{)}\,,$

but use the same function of the derivative operator,
$d\sigma^{\kappa}(\partial_{\Omega})$ defined in Eq. (4.3), valid for
$\varepsilon=0$:

$\displaystyle\frac{d\Delta\Sigma_{\varepsilon}(R_{g})}{dm_{J}^{2}}\bigg{|}_{R_{g}\lesssim\theta_{g}^{\star}}$
$\displaystyle=N^{q}_{\rm evol}\big{(}Q,Q_{\rm
cut},\beta,R,\mu;\mu_{N},\mu_{gs}\big{)}\frac{d\sigma^{q}\big{[}\partial_{\Omega}\big{]}}{dm_{J}^{2}}\Bigg{(}\frac{1}{\big{(}\frac{R_{g}}{2}\big{)}^{\beta}}\>{\cal
Q}^{q}_{\varepsilon}\Big{[}\Omega,\frac{R_{g}}{\theta_{g}^{\star}},\beta\Big{]}$
(121)
$\displaystyle\qquad+\frac{\Theta(\beta>0)}{\sin^{\beta}\big{(}\frac{R}{2}\big{)}}\>\Delta
S_{G,\varepsilon}^{q}\bigl{(}Q_{\rm
cut},R,\beta\bigr{)}\frac{e^{\gamma_{E}\Omega}}{\Gamma(-\Omega)}\Bigg{)}\bigg{|}_{\Omega=\tilde{\omega}(\mu_{cs},\mu_{J})}\,.$

The explicit expression of the evolution kernel ${\cal
Q}^{\kappa}_{\varepsilon}$ between jet and soft scales for shifted soft drop
is given in Eq. (B.3.2). Note that the term $\Delta
S_{G,\varepsilon}^{\kappa}$ for $\beta>0$ at NLL′ accuracy contributes only to
the boundary correction in the single differential jet mass cross section, and
is thus independent of $R_{g}$. In the following, we will simply ignore this
term as this is also power suppressed compared to the term in the first line
in Eq. (121).

We now briefly describe the generalization of the results above to $pp$
collisions. First off, the proportionality constant of ${\cal O}(\varepsilon)$
terms is substituted as follows

$\displaystyle\frac{Q}{Q_{\rm
cut}\big{(}\frac{R_{g}}{2}\big{)}^{\beta}}\rightarrow\frac{1}{R_{g}^{\beta}z_{\rm
cut}^{\prime}}\,.$ (122)

The same applies for the ${\cal O}(\varepsilon)$ one-loop non-cusp anomalous
dimension in Eq. (109). Thus generalization of the cross section in Eq. (117)
to the $pp$ case is simply given by Eq. (95) after including the additional
${\cal O}(\varepsilon)$ logarithms and single log kernels as outlined above.
The result for the second term in Eq. (116) for $pp$ collisions can also be
analogously derived from Eq. (121):

$\displaystyle\frac{\varepsilon}{z_{\rm
cut}^{\prime}}\frac{d\Delta\Sigma_{\varepsilon}(R_{g})}{dm_{J}^{2}d\Phi_{J}}\bigg{|}_{R_{g}\lesssim\theta_{g}^{\star}}$
$\displaystyle=\sum_{\kappa=q,g}N^{\kappa}_{\rm evol}\big{(}\Phi_{J},Q_{\rm
cut},\beta,R,\mu;\mu_{N},\mu_{gs}\big{)}$ (123)
$\displaystyle\qquad\times\frac{d\sigma^{\kappa}\big{[}\partial_{\Omega}\big{]}}{dm_{J}^{2}d\Phi_{J}}\Bigg{(}\frac{\varepsilon}{z_{\rm
cut}^{\prime}R_{g}^{\beta}}\>{\cal
Q}^{\kappa}_{\varepsilon}\Big{[}\Omega,\frac{R_{g}}{\theta_{g}^{(pp)\star}(m_{J}^{2})},\beta\Big{]}\Bigg{)}\bigg{|}_{\Omega=\tilde{\omega}(\mu_{cs},\mu_{J})}\,.$

At this point, we have the soft drop boundary cross section which is cumulant
in the groomed jet radius. In order to evaluate $C_{2}$, following Eq. (5), we
need to switch to the cross section differential in the groomed jet radius and
further take the $\varepsilon$ derivative. We perform this explicitly in App.
B.4, where we compile the results for the boundary cross section differential
in $R_{g}$. The large $R_{g}$ regime discussed so far is the only relevant
geometry in the definition of $C^{q}_{2}(m_{J}^{2})$. For completeness, we
discuss in App. B.3 the boundary corrections in the intermediate and small
$R_{g}$ regimes and show consistency with the results in the large $R_{g}$
regime as $R_{g}/\theta_{g}^{\star}(m_{J}^{2})\rightarrow 0$.

###  4.5 Recovering the Leading Logarithmic coherent branching result

In this section we make use of the results derived above for the cumulative
cross section and the boundary corrections to demonstrate that the EFT that we
have developed reproduce the computation of $C_{1}$ and $C_{2}$ at LL accuracy
within the coherent branching formulation from Hoang:2019ceu . This also makes
explicit how the EFT encodes the strong angular ordering limit used there. For
simplicity we limit ourselves to the case of $e^{+}e^{-}$ collisions here. The
LL results for $C_{1}$ and $C_{2}$ were derived by considering a chain of
emissions that are ordered in their respective contribution to the jet
mass.666Formally, the emissions are to be angular ordered, but in the
derivation the angular ordering is replaced by their ordering in the jet mass,
i.e. in the variable $\rho_{i}=z_{i}\theta_{i}^{2}$ for the $i^{\rm th}$
emission. This approximation is essential to set the resolution of the jet
mass measurement. The emission at the largest angle is required to satisfy
soft drop and set the jet mass. This leads to the following expressions for
the two moments:

$\displaystyle C_{1}^{q}(m_{J}^{2})$
$\displaystyle=\bigg{(}\frac{C_{0}^{q}(m_{J}^{2})}{m_{J}^{2}}\bigg{)}^{-1}\int_{0}^{1}dz\int_{0}^{R}\frac{d\theta^{2}}{\theta^{2}}\>\frac{\theta}{2}\frac{C_{F}}{\pi}\alpha_{s}\Big{(}\frac{Qz\theta}{2}\Big{)}p_{gq}(z)e^{-{\cal
R}(z\theta^{2})}$ (124) $\displaystyle\qquad\times\Theta\bigg{(}z-\frac{Q_{\rm
cut}}{Q}\Big{(}\frac{\theta}{2}\Big{)}^{\beta}\bigg{)}\delta\Big{(}m_{J}^{2}-\frac{1}{4}z\theta^{2}Q^{2}\Big{)}\,,$
$\displaystyle C_{2}^{q}(m_{J}^{2})$
$\displaystyle=\frac{m_{J}^{2}}{Q^{2}}\bigg{(}\frac{C_{0}^{q}(m_{J}^{2})}{m_{J}^{2}}\bigg{)}^{-1}\int_{0}^{1}dz\int_{0}^{R}\frac{d\theta^{2}}{\theta^{2}}\>\frac{2}{\theta}\frac{C_{F}}{\pi}\alpha_{s}\Big{(}\frac{Qz\theta}{2}\Big{)}p_{gq}(z)e^{-{\cal
R}(z\theta^{2})}$ $\displaystyle\qquad\times\delta\bigg{(}z-\frac{Q_{\rm
cut}}{Q}\Big{(}\frac{\theta}{2}\Big{)}^{\beta}\bigg{)}\delta\Big{(}m_{J}^{2}-\frac{1}{4}z\theta^{2}Q^{2}\Big{)}\,,$

where $C_{0}^{q}(m_{J}^{2})/m_{J}^{2}$ is the single differential jet mass
distribution and is given at this order by

$\displaystyle\frac{1}{m_{J}^{2}}C_{0}^{q}(m_{J}^{2})\equiv\frac{d\hat{\sigma}_{\rm
LL}^{q}}{dm_{J}^{2}}$
$\displaystyle=\frac{1}{m_{J}^{2}}\int_{0}^{1}dz\int_{0}^{R}\frac{d\theta^{2}}{\theta^{2}}\frac{C_{F}}{\pi}\alpha_{s}\Big{(}\frac{Qz\theta}{2}\Big{)}p_{gq}(z)e^{-{\cal
R}(z\theta^{2})}$ (125) $\displaystyle\qquad\times\Theta\bigg{(}z-\frac{Q_{\rm
cut}}{Q}\Big{(}\frac{\theta}{2}\Big{)}^{\beta}\bigg{)}\delta\Big{(}m_{J}^{2}-\frac{1}{4}z\theta^{2}Q^{2}\Big{)}\,,$

where the result for $C_{0}(m_{J}^{2})$ is valid up to LL. In Eqs. (124) and
(125) we have rewritten the soft drop condition in Eq. (8) terms of $Q_{\rm
cut}$. In the expression for $C_{1}^{q}$ and $C_{0}^{q}$, the condition is
imposed on the last emission with energy fraction $z$ and angle $\theta$. In
case of $C_{2}^{q}$ we see, however, that the soft drop $\Theta$ function is
replaced by $\delta$ function in order to capture the corrections at the soft
drop boundary. The radiator, $\mathcal{R}_{q}(\theta_{c}^{2})$, results from
all the emissions at larger angles that are either vetoed by the jet mass
measurement or that fail soft drop (and the virtual emissions). The radiator
resums double logarithms of $\theta_{g}^{\star}/\theta_{c}$, which however
cancel out in the ratio in Eqs. (127) and (125). The splitting kernel $p_{gq}$
is given by

$\displaystyle p_{gq}(z)=\frac{1+(1-z)^{2}}{2z}\underset{z\to
0}{\sim}\frac{1}{z}\,.$ (126)

Terms that are power corrections in the $z\to 0$ limit are formally beyond LL
accuracy, and we will drop them in the following; in the EFT formulation,
these are captured by the NLO corrections to the jet function. Simplifying Eq.
(124) and using the LL limit of $p_{gq}(z)$ in Eq. (126) we get

$\displaystyle C_{0}^{q}(m_{J}^{2})\big{|}_{\rm LL}$
$\displaystyle=e^{-\mathcal{R}_{q}(\theta_{c}^{2})}\int_{\theta_{c}}^{R}\frac{d\theta^{2}}{\theta^{2}}\frac{C_{F}}{\pi}\alpha_{s}\Big{(}\frac{m_{J}\theta_{c}}{\theta}\Big{)}\Theta(\theta_{g}^{\star}-\theta)\,,$
(127) $\displaystyle C_{1}^{q}(m_{J}^{2})\big{|}_{\rm LL}$
$\displaystyle=\frac{e^{-\mathcal{R}(\theta_{c}^{2})}}{C_{0}^{q}(m_{J}^{2})}\int_{\theta_{c}}^{R}\frac{d\theta^{2}}{\theta^{2}}\frac{\theta}{2}\frac{C_{F}}{\pi}\alpha_{s}\Big{(}\frac{m_{J}\theta_{c}}{\theta}\Big{)}\Theta(\theta_{g}^{\star}-\theta)\,,$
$\displaystyle C_{2}^{q}(m_{J}^{2})\big{|}_{\rm LL}$
$\displaystyle=\frac{e^{-\mathcal{R}(\theta_{c}^{2})}}{C_{0}^{q}(m_{J}^{2})}\int_{\theta_{c}}^{R}\frac{d\theta^{2}}{\theta^{2}}\>\frac{2}{\theta}\frac{C_{F}}{\pi}\alpha_{s}\bigg{(}Q_{\rm
cut}\Big{(}\frac{\theta}{2}\Big{)}^{1+\beta}\bigg{)}\frac{Q}{Q_{\rm
cut}}\Big{(}\frac{\theta}{2}\Big{)}^{-\beta}\delta\left(1-\Big{(}\frac{\theta}{\theta_{g}^{\star}}\Big{)}^{2+\beta}\right)\,,$

where we have expressed the results in terms of the angular bounds
$\theta_{c}$ and $\theta_{g}^{\star}$ defined in Eq. (12).

To establish a connection with our results, we must compare with the large
$R_{g}$ regime of Sec. 3.1, since due to the associated geometry for power
corrections in the SDOE region of the single differential jet mass
distribution, this is the primary regime that is relevant for defining the
coefficients $C_{1}$ and $C_{2}$.777We note that the coherent branching
description is based on ordering in a single variable, the jet mass
contribution of each emission, $\rho_{i}=z_{i}\theta_{i}^{2}$. This is in
correspondence with the mode picture in the large $R_{g}$ regime where the
measurement of jet mass provides a distinction between the various EFT modes
(see Fig. 2, left). This is, however, not the case with small and intermediate
$R_{g}$ regimes: in the case of small $R_{g}$ regime, the primary measurement
is the groomed jet radius which distinguishes the modes based on their angular
separation. In the intermediate $R_{g}$ regime an analogous treatment in
coherent branching would presumably involve ordering in two different
variables. Thus, the EFT prediction for the LL double differential cross
section in this regime is obtained from the cumulant in Eq. (86),

$\displaystyle\frac{d^{2}\hat{\sigma}_{\rm
LL}^{q}}{dm_{J}^{2}d\theta_{g}}\bigg{|}_{R_{g}\lesssim\theta_{g}^{\star}}$
$\displaystyle=\frac{d}{dR_{g}}\bigg{[}\frac{d\sigma^{q}_{\rm
LL}\big{[}\partial_{\Omega}\big{]}}{dm_{J}^{2}}\bigg{(}\frac{e^{\gamma_{E}\Omega}}{\Gamma(-\Omega)}+{\cal
Q}^{q}\Big{[}\Omega,\frac{R_{g}}{\theta_{g}^{\star}},\beta\Big{]}\bigg{)}\bigg{]}_{R_{g}=\theta_{g}}$
$\displaystyle=e^{-\mathcal{R}_{q}}\frac{d}{dR_{g}}{\cal
Q}^{q}\Big{[}\Omega,\frac{R_{g}}{\theta_{g}^{\star}},\beta\Big{]}\bigg{|}_{R_{g}=\theta_{g}}\,.$
(128)

In the second line we have used that the evolution in this regime is
independent of the groomed jet radius and identified the LO differential
operator with the exponential of the radiator $\mathcal{R}^{q}$. This follows
by setting the Laplace space fixed order ingredients in Eq. (4.3) to unity.

We now notice that the fixed-order corrections in Eqs. (127) and (125) are
described in the coherent branching formalism by the single splitting function
$p_{gq}$, in contrast with the convolution of the jet and collinear-soft
functions in Eq. (32). This implies that the coherent branching result ignores
some subleading terms in the kernel $\mathcal{Q}^{q}$, which are induced by
performing the RG evolution between $\mu_{J}$ and $\mu_{cs_{m}}$ in Laplace
space rather than distribution space. These terms are instead included in our
EFT approach; we can remove them by setting $\Omega\to 0$ in Eq. (251),
obtaining

$\displaystyle{\cal Q}^{q}\Big{[}\Omega\to
0,\frac{R_{g}}{\theta_{g}^{\star}},\beta\Big{]}=\Theta(\theta_{g}^{\star}-\theta_{g})\,\frac{2\alpha_{s}(\mu_{cs})C_{F}}{\pi}\,\textrm{ln}\Big{(}\frac{\theta_{g}^{\star}}{R_{g}}\Big{)}\,,$
(129)

where we have used that under the integral

$\displaystyle
x^{-1-\Omega}=-\frac{1}{\Omega}\delta(x)+\mathcal{O}(\Omega^{0})\,.$ (130)

Next, we adopt the natural scale choice for $\mu_{cs}$ in Eq. (90), noting
that

$\displaystyle m_{J}^{2}\frac{\theta_{c}}{\theta_{g}^{\star}}=\mu_{cs}^{\rm
can.}\,.$ (131)

This corresponds to setting $\theta=\theta_{g}^{\star}$ in Eq. (127), with the
difference between the two expressions being beyond LL accuracy. Substituting
Eq. (129) in Eq. (4.5) gives the integrand of Eq. (127). The LL coherent
branching result for $C_{1}$ then trivially follows by including the
additional factor $\theta/2$ in the angular integration.

To obtain the LL coherent branching result for $C_{2}$ from the EFT formalism
we can follow the same steps. We first notice that the anomalous evolution
induced by boundary corrections at $\beta=0$ affects the cross section beyond
LL, and we can thus ignore this correction. The relevant cumulant cross
section for large $R_{g}$ is then given by Eq. (121). Differentiating in
$R_{g}$, we get up to $\mathcal{O}(\varepsilon^{2})$ corrections

$\displaystyle\frac{d\hat{\sigma}_{\rm
LL}^{q}(\varepsilon)}{dm_{J}^{2}d\theta_{g}}\bigg{|}_{R_{g}\lesssim\theta_{g}^{\star}}$
$\displaystyle=e^{-\mathcal{R}_{q}}\frac{d}{dR_{g}}\bigg{[}\Big{(}\frac{R_{g}}{2}\Big{)}^{-\beta}{\cal
Q}_{\varepsilon}^{q}\Big{(}\Omega,\frac{R_{g}}{\theta_{g}^{\star}},\beta\Big{)}\bigg{]}_{R_{g}=\theta_{g}}\,$
(132) $\displaystyle=e^{-\mathcal{R}_{q}}\frac{Q\varepsilon}{Q_{\rm
cut}}\frac{\alpha_{s}(\mu_{cs})C_{F}}{\pi}\Big{(}\frac{\theta_{g}}{2}\Big{)}^{-(1+\beta)}\frac{e^{\gamma_{E}\Omega}}{\Gamma(-\Omega)}{\cal
L}_{0}^{-\Omega}\bigg{(}1-\Big{(}\frac{\theta_{g}}{\theta_{g}^{\star}(m_{J}^{2})}\Big{)}^{2+\beta}\bigg{)}\,.$

The first line is obtained from the same manipulations as in Eq. (4.5), while
the explicit expression in the second line follows from differentiating the
kernel $\mathcal{Q}^{q}_{\varepsilon}$ given in Eq. (120), which reduces to
Eq. (B.3.2) when discarding NLO corrections. In order to compare with the
coherent branching result, we again expand out this result in the limit
$\Omega\rightarrow 0$. Using the expansion in Eq. (130),

$\displaystyle\frac{1}{\Gamma(-\Omega)}{\cal
L}_{0}^{-\Omega}\bigg{(}1-\Big{(}\frac{\theta_{g}}{\theta_{g}^{\star}}\Big{)}^{2+\beta}\bigg{)}\,\rightarrow\,\delta\left(1-\Big{(}\frac{\theta_{g}}{\theta_{g}^{\star}}\Big{)}^{2+\beta}\right)$
(133)

so that

$\displaystyle\frac{d}{d\varepsilon}\bigg{|}_{\varepsilon=0}\frac{d\hat{\sigma}_{\rm
LL}^{q}(\varepsilon)}{dm_{J}^{2}d\theta_{g}}\bigg{|}_{R_{g}\lesssim\theta_{g}^{\star}}\,\rightarrow\,\frac{Q}{Q_{\rm
cut}}\frac{\alpha_{s}(\mu_{cs})C_{F}}{\pi}\Big{(}\frac{\theta_{g}}{2}\Big{)}^{-(1+\beta)}\delta\left(1-\Big{(}\frac{\theta_{g}}{\theta_{g}^{\star}(m_{J}^{2})}\Big{)}^{2+\beta}\right)\,.$
(134)

Including the factor of $m_{J}^{2}/Q^{2}$, according to our definition of
$C_{2}^{\kappa}$ from Eq. (5), this yields the LL coherent branching result of
Eq. (127). Finally, setting $\theta=\theta_{g}^{\star}$ using the $\delta$
function and noting that

$\displaystyle Q_{\rm
cut}\Big{(}\frac{\theta_{g}^{\star}}{2}\Big{)}^{1+\beta}=\mu_{cs}^{\rm
can.}\,,$ (135)

we find that the arguments of the running coupling in the two expressions also
match.

##  5 Matching the three regimes

Here we describe our strategy for consistently matching the results in various
regimes so as to obtain a smooth interpolation across the entire phase space
of $m_{J}^{2}$-$R_{g}$ measurement. Which of the three EFTs come into play
will crucially depend on the range over which the groomed jet radius varies
for given values of $z_{\rm cut}$, $\beta$ and the jet mass. In the following
we first describe the basic strategy for implementation of the matched cross
section in Sec. 5.1 using profile functions for scale choices Ligeti:2008ac ;
Abbate:2010xh  and weight functions which control the contribution from
different EFTs Lustermans:2019plv . In Sec. 5.2 we generalize the weight
function construction to depend on two variables that track where we are in
our two dimensional phase space so that the appropriate factorized formula can
be used. Next in Sec. 5.3 we describe the profile functions for each of the
large, intermediate, and small $R_{g}$ regimes, and then in Sec. 5.4 how we
vary them to obtain an estimate for the perturbative uncertainty.

###  5.1 Implementation of matched cross section

Here we describe our implementation of the complete cumulative-$R_{g}$ and
differential jet mass soft drop cross section. This will then set up the
groundwork for evaluation of the moments $C_{1,2}^{\kappa}(m_{J}^{2})$. An
important ingredient are the profile functions for our different EFT regions,
which are the functions used for the renormalization $\mu_{i}$ scales in the
factorization formulae, and which depend on both $m_{J}$ and $R_{g}$. First we
recall the set of scales that are relevant for the various kinematic regions:

$\displaystyle\text{Large $R_{g}$ region}:$ $\displaystyle
R_{g}\lesssim\theta_{g}^{\star}(m_{J}^{2},Q_{\rm cut},\beta)<R$
$\displaystyle\mu_{\rm large}\equiv\\{\mu_{J},\mu_{gs},\mu_{cs}\\}\,,$
$\displaystyle\text{Intermediate $R_{g}$ region}:$
$\displaystyle\theta_{c}(m_{J}^{2})\ll
R_{g}\ll\theta_{g}^{\star}(m_{J}^{2},Q_{\rm cut},\beta)$
$\displaystyle\mu_{\rm
int}\equiv\\{\mu_{J},\mu_{gs},\mu_{cs_{m}},\mu_{cs_{g}}\\}\,,$
$\displaystyle\text{Small $R_{g}$ region}:$
$\displaystyle\theta_{c}(m_{J}^{2})\lesssim R_{g}$ $\displaystyle\mu_{\rm
small}\equiv\\{\mu_{c},\mu_{gs},\mu_{cs_{g}}\\}\,,$ $\displaystyle\text{Plain
jet mass region}:$ $\displaystyle R_{g}\lesssim R$ $\displaystyle\mu_{\rm
plain}\equiv\\{\mu_{J},\mu_{s}\\}\,.$ (136)

In this analysis we will focus exclusively on the groomed region described by
our EFT, although we will also extrapolate our results into the ungroomed
region. In the groomed region, we construct the interpolation between
different EFT regimes via the following formula:

$\displaystyle d\Sigma_{\rm matched}\equiv d\Sigma_{\rm int}|_{\mu_{\rm
hyb}}+\big{(}d\Sigma_{\rm large}-d\Sigma_{\rm int}\big{)}|_{\mu_{\rm
large}}+\big{(}d\Sigma_{\rm small}-d\Sigma_{\rm int}\big{)}|_{\mu_{\rm
small}}\,,$ (137)

where $d\Sigma_{i}(m_{J}^{2})$ for $i=${large, int, small} stand for the
cumulative-$R_{g}$ factorized cross section formulae in the large,
intermediate and small $R_{g}$ regimes respectively. We have also indicated
the choice of profile scales used for jet, global-soft and collinear-soft
scales used in these factorization formulae utilizing the notation introduced
in Eq. (5.1). In Eq. (137) the cross sections in the large and small $R_{g}$
regimes are evaluated with their respective profile scales, whereas the
intermediate $R_{g}$ uses either the profile scales in Eq. (5.1) (in the two
subtraction terms), or a set of hybrid profiles denoted by $\mu_{\rm hyb}$.
The hybrid $\mu_{\rm hyb}$ profiles are designed such that they interpolate
between the $\mu_{\rm small}$, $\mu_{\rm int}$ and $\mu_{\rm large}$ sets as
follows:

$\displaystyle\mu_{\rm hyb}\equiv\big{(}\mu_{\rm small}\big{)}^{w_{\rm
small}}\,\big{(}\mu_{\rm int}\big{)}^{w_{\rm int}}\big{(}\mu_{\rm
large}\big{)}^{w_{\rm large}}\,,\qquad w_{\rm small}+w_{\rm int}+w_{\rm
large}=1\,,$ (138)

where the exponents $0\leq w_{i}\equiv w_{i}(m_{J},R_{g})\leq 1$ should turn
on/off in the appropriate regions and add up to one Lustermans:2019plv . We
give an explicit construction of these weight functions below for our case.

For now, we note that in the large $R_{g}$ region we will have $w_{\rm
large}=1$ (and hence $w_{\rm small}=w_{\rm int}=0$) such that the $\mu_{\rm
hyb}$ profiles are the same as the set $\mu_{\rm large}$. The term
$d\Sigma_{\rm int}|_{\mu_{\rm large}}$ in Eq. (137) corresponds to evaluating
the intermediate $R_{g}$ cross section with large $R_{g}$ profile scale
settings, i.e. by setting $\mu_{cs_{g}},\mu_{cs_{m}}\rightarrow\mu_{cs}$. This
leads to exact cancellation of the terms $d\Sigma_{\rm int}|_{\mu_{\rm hyb}}$
and $d\Sigma_{\rm int}|_{\mu_{\rm large}}$ in the
$R_{g}\lesssim\theta_{g}^{\star}$ region where $w_{\rm large}=1$, and hence
the matched cross section is given by

$\displaystyle d\Sigma_{\rm matched}=d\Sigma_{\rm large}|_{\mu_{\rm
large}}+\big{(}d\Sigma_{\rm small}-d\Sigma_{\rm int}\big{)}|_{\mu_{\rm
small}}\,,\qquad R_{g}\lesssim\theta_{g}^{\star}\,,\quad(w_{\rm large}=1)\,.$
(139)

The term $d\Sigma_{\rm int}|_{\mu_{\rm small}}$ here denotes the intermediate
$R_{g}$ cross section evaluated with the $\mu_{\rm small}$ profiles, i.e. by
replacing $\mu_{cs_{m}},\mu_{J}\rightarrow\mu_{c}$. Thus the difference
$\big{(}d\Sigma_{\rm small}-d\Sigma_{\rm int}\big{)}|_{\mu_{\rm small}}$
supplements Eq. (139) with the power corrections that are missing in both the
intermediate and large $R_{g}$ factorizations in the $\theta_{c}\lesssim
R_{g}$ region.

Similarly, in the small $R_{g}$ region we will have $w_{\rm small}=1$ (and
thus $w_{\rm large}=w_{\rm int}=0$). As a result, the $\mu_{\rm hyb}$ profiles
will coincide with $\mu_{\rm small}$ and the terms $d\Sigma_{\rm
int}|_{\mu_{\rm hyb}}$ and $d\Sigma_{\rm int}|_{\mu_{\rm small}}$ in Eq. (137)
will cancel exactly. Hence, the $d\Sigma_{\rm matched}$ in this region will be
given by

$\displaystyle d\Sigma_{\rm matched}=d\Sigma_{\rm small}|_{\mu_{\rm
small}}+\big{(}d\Sigma_{\rm large}-d\Sigma_{\rm int}\big{)}|_{\mu_{\rm
large}}\,,\qquad\theta_{c}\lesssim R_{g}\,,\qquad(w_{\rm small}=1)\,.$ (140)

By using $\mu_{\rm large}$ profiles in the intermediate $R_{g}$ cross section
we turn off the additional resummation relative to the large $R_{g}$ regime,
so that the second term $\big{(}d\Sigma_{\rm large}-d\Sigma_{\rm
int}\big{)}|_{\mu_{\rm large}}$ captures the power corrections that are
missing in the intermediate and small $R_{g}$ factorizations in the
$R_{g}\lesssim\theta_{g}^{\star}$ region.

Finally, if there is a contribution purely from the intermediate region where
$w_{\rm int}=1$, the $\mu_{\rm hyb}$ profiles will be the same as the bulk
intermediate $R_{g}$ profiles in Eq. (5.1). Here the term $d\Sigma_{\rm
int}|_{\mu_{\rm hyb}}$ in Eq. (137) provides for the essential intermediate
$R_{g}$ resummation, whereas the other two terms contribute the power
corrections from the large and small $R_{g}$ regions. We will find that the
intermediate regime only has a dominant weight for larger values of $\beta$,
and even then in a rather small range of phase space. When the intermediate
regime is not relevant, the formalism we develop simply implements a smooth
transition between the small and large $R_{g}$ EFTs with $w_{\rm int}=0$
throughout the entire range of $R_{g}$ (for a fixed $m_{J}$). The smooth
variation of $w_{i}(m_{J}^{2},R_{g})$ will ensure consistent interpolation
between all these regions.

###  5.2 Weight functions

Here we describe the implementation of weight functions
$w_{i}(m_{J}^{2},R_{g})$ that appear in Eq. (138). We first recall that the
transition from the large $R_{g}$ EFT,
$R_{g}\lesssim\theta^{\star}_{g}(m_{J}^{2},Q_{\rm cut},\beta)$, to the
intermediate EFT corresponds to the canonical CSg and CSm scales becoming well
separated. On the other hand, transitioning from the small $R_{g}$ EFT,
$\theta_{c}(m_{J}^{2})\lesssim R_{g}$, to the intermediate EFT happens when
$R_{g}\gg\theta_{c}(m_{J}^{2})$, resulting from separation of the canonical
CSm and the collinear scales. We can therefore characterize the regions with
two power counting parameters defined as

$\displaystyle\textrm{${\cal C^{\kappa}}$ factorizes into $J_{\kappa}$ and
$S_{c_{m}}^{\kappa}$}:$ $\displaystyle\qquad\lambda_{\rm
min}(R_{g})\equiv\frac{\mu_{cs_{m}}}{\mu_{J}}\ll 1\,,$ (141)
$\displaystyle\textrm{$S_{c}^{\kappa}$ factorizes into $S_{c_{g}}^{\kappa}$
and $S_{c_{m}}^{\kappa}$}:$ $\displaystyle\qquad\lambda_{\rm
max}(R_{g})\equiv\frac{\mu_{cs_{g}}}{\mu_{cs_{m}}}\ll 1\,.$

Here the $\mu$ scales stand for the virtuality of the modes determined by
$\\{m_{J}^{2},R_{g},Q,Q_{\rm cut},\beta\\}$ such that

$\displaystyle\lambda_{\rm
min}(R_{g})=\frac{m_{J}}{QR_{g}/2}=\frac{\theta_{c}(m_{J}^{2})}{R_{g}}\,,\quad\lambda_{\rm
max}(R_{g})=\frac{QQ_{\rm
cut}}{m_{J}^{2}}\Big{(}\frac{R_{g}}{2}\Big{)}^{2+\beta}=\Big{(}\frac{R_{g}}{\theta_{g}^{\star}(m_{J}^{2},Q_{\rm
cut},\beta)}\Big{)}^{2+\beta},$ (142)

which is consistent with Eqs. (54) and (50).888Although the power corrections
between the small and intermediate $R_{g}$ regimes are of the form
$\theta_{c}^{2}/R_{g}^{2}$, it is the power counting parameter $\lambda_{\rm
min}=\theta_{c}/R_{g}$ that is the important one, since it controls the size
of the logarithms in the transition between leading power contributions in
different regimes. The $pp$ case involves precisely the same expressions but
with the $pp$ versions of the angles $\theta_{c}$ and $\theta_{g}^{\star}$
used in Eq. (14). Furthermore, since the power counting parameters are defined
via ratios of angles, the factors of $\cosh\eta_{J}$ cancel, and thus all the
formulae derived in this section will apply for the $pp$ case with the
corresponding definition of $Q_{\rm cut}$ in Eq. (10).

Note that $\lambda_{\rm min}$ grows with $R_{g}$ while $\lambda_{\rm max}$
decreases with $R_{g}$, and in particular they become equal when
$R_{g}=\theta_{t}$, where

$\displaystyle\theta_{t}(m_{J}^{2},Q_{\rm
cut},\beta)\equiv\big{[}\theta_{c}\,(\theta_{g}^{\star})^{2+\beta}\big{]}^{\frac{1}{3+\beta}}=2\bigg{[}\Big{(}\frac{m_{J}}{Q}\Big{)}^{3}\frac{Q}{Q_{\rm
cut}}\bigg{]}^{\frac{1}{3+\beta}}\,.$ (143)

Whether or not the additional resummation in the intermediate $R_{g}$ region
is essential is given by the condition $\lambda_{\rm
min}(\theta_{t})=\lambda_{\rm max}(\theta_{t})\ll 1$. In our numerical
analysis below we replace the strong inequality by a factor of 3, such that
for intermediate EFT resummation to be significant we require

$\displaystyle\lambda_{\rm min}(\theta_{t})=\lambda_{\rm
max}(\theta_{t})=\bigg{[}\Big{(}\frac{m_{J}}{Q}\Big{)}^{\beta}\frac{Q_{\rm
cut}}{Q}\bigg{]}^{\frac{1}{3+\beta}}\leq\frac{1}{3}\,.$ (144)

When this condition is not satisfied, we directly match the cross section in
region $R_{g}\lesssim\theta_{g}^{\star}$ to $\theta_{c}\lesssim R_{g}$. We
refer to this as the “two-EFT scheme” since it does not include the
intermediate $R_{g}$ regime. In the opposite situation, when the condition in
Eq. (144) is valid, then the additional resummation in the intermediate
$R_{g}$ region is needed, thus defining a “three-EFT” scheme. For a fixed
value of $\\{Q,Q_{\rm cut},\beta\\}$, this happens at smaller jet masses. In
the three-EFT scheme we can further identify the two angles where respectively
the small-$R_{g}$ and large-$R_{g}$ power corrections are equal to $1/3$,

$\displaystyle\theta_{t,\rm min}(m_{J}^{2})\equiv
3\theta_{c}(m_{J}^{2})\,,\qquad\theta_{t,\rm max}(m_{J}^{2},Q_{\rm
cut},\beta)\equiv 3^{\frac{-1}{2+\beta}}\theta_{g}^{\star}(m_{J}^{2},Q_{\rm
cut},\beta)\,.$ (145)

This suggests that intermediate $R_{g}$ resummation is important for
$\theta_{t,\rm min}\leq R_{g}\leq\theta_{t,\rm max}$.

Finally, we introduce a transition function which smoothly interpolates
between $0$ and $1$,

$\displaystyle
X(\theta,\theta_{t})\equiv\frac{1}{2}\bigg{[}1+\tanh\Big{(}r_{t}\frac{\theta-\theta_{t}}{\theta_{\rm
max}-\theta_{\rm min}}\Big{)}\bigg{]}\,,$ (146)

where $r_{t}$ controls the rate of the transition and is normalized with
respect to the range of integration. In our numerical analysis in Sec. 6 we
use $r_{t}=20$. This transition function is used to construct the weights
$w_{\rm int}$ and $w_{\rm large}$ for the two- and three-EFT schemes:

2-EFT scheme: $\displaystyle w_{\rm large}=X(R_{g},\theta_{t})\,,$
$\displaystyle w_{\rm int}=0\,,$ (147) 3-EFT scheme: $\displaystyle w_{\rm
large}=X(R_{g},\mathrm{max}\\{\theta_{t,\rm max},\theta_{t}\\})\,,$
$\displaystyle w_{\rm int}=\big{(}1-X(R_{g},\theta_{t,\rm
max})\big{)}X(R_{g},\theta_{t,\rm min})\,,$

and $w_{\rm small}=1-w_{\rm large}-w_{\rm int}$ in either of these cases.

In the 2-EFT scheme, the large $R_{g}$ EFT is employed for $R_{g}>\theta_{t}$
and small $R_{g}$ for $R_{g}<\theta_{t}$. Here the intermediate $R_{g}$
resummation is not present and the piece $d\Sigma_{\rm int}|_{\mu_{\rm hyb}}$
in Eq. (137) serves mainly to provide an interpolation between these two
regimes. In the 3-EFT scheme, the intermediate $R_{g}$ resummation is
effective in the range $\theta_{t,\rm min}<R_{g}<\theta_{t,\rm max}$. Note
that $w_{\rm large}$ in this scheme reduces to the corresponding weight in the
2-EFT scheme when Eq. (144) is valid with equality sign. The transition
function $X(R_{g},\theta_{t,\rm min})$ in $w_{\rm int}$ turns off the
intermediate $R_{g}$ resummation for $R_{g}\ll\theta_{t,\rm min}$.
Furthermore, in the 2-EFT scheme, we have $\theta_{t,\rm min}>\theta_{t,\rm
max}$ which automatically drives $w_{\rm int}$ to $0$.

###  5.3 Profile functions

With the exception of the hard and global-soft functions, the logarithms that
appear in the various matrix elements in the soft drop cross sections depend
on the jet mass and groomed jet radius, and thus vary along the spectrum.
Hence, to minimize these logarithms, the renormalization scales must also vary
accordingly in the resummation region as discussed in Sec. 4.2. Additionally,
for larger jet masses where $R_{g}\lesssim R$ we enter the ungroomed region
where the soft drop resummation must be turned off. For smaller jet masses,
when we approach the nonperturbative region the soft scale must be frozen to
an ${\cal O}(1)$ GeV value. This requires some care since the boundary of the
nonperturbative region is a non-trivial function of both $m_{J}$ and $R_{g}$
measurements. Hence, in order to consistently describe the entire jet mass
spectrum, we make use of profile functions Abbate:2010xh ; Ligeti:2008ac
which depend on the jet mass and groomed jet radius. In the following we will
state results for both $e^{+}e^{-}$ and $pp$ cases.

####  5.3.1 General strategy and canonical relations

In this section we first review the canonical relations between the scales
that form the basis for implementation of the profile functions. To set up the
notation first consider a relatively simple scenario of plain jet mass
resummation for a jet with radius $R$ in $e^{+}e^{-}$ collisions. The relevant
canonical scales are the hard, jet, and the (ungroomed) soft scales:

$\displaystyle\mu_{N}^{\rm
can.}=Q\tan\Big{(}\frac{R}{2}\Big{)}\,,\qquad\mu_{J}^{\rm
can.}=m_{J}\,,\qquad\mu_{s}^{\rm
can.}=\frac{m_{J}^{2}}{Q\tan\big{(}\frac{R}{2}\big{)}}\,,$ (148)

such that the endpoint of the spectrum is at $m_{J}=m^{ee}_{\rm
max}=Q\tan(R/2)$, where the jet and the soft scales merge with the hard scale.
Thus, the seesaw relation

$\displaystyle\mu_{J}^{2}=\mu_{N}\>\mu_{s}$ (149)

between the three scales is at the heart of the soft-collinear factorization
in what is often called ${\rm SCET}_{\rm I}$.

When we include soft drop grooming, the canonical hard and the jet scales
remain the same, whereas only the soft sector is modified, splitting into the
global-soft and collinear-soft scales:

$\displaystyle\mu_{gs}^{\rm can.}=Q_{\rm
cut}\tan^{1+\beta}\Big{(}\frac{R}{2}\Big{)}\,,\qquad\mu_{cs}^{\rm
can.}=\Big{(}\frac{m_{J}^{2}}{Q}\Big{)}^{\frac{1+\beta}{2+\beta}}Q_{\rm
cut}^{\frac{1}{2+\beta}}\,.$ (150)

We see that $\mu_{cs}$ is independent of the jet radius $R$. These scales are
relevant for the single differential jet mass distribution as well as in the
large $R_{g}$ region. For the single differential jet mass case the canonical
scales were first discussed in Ref. Frye:2016aiz , with the generalization to
include $R$ dependence given in Refs. Kang:2018jwa ; Chien:2019osu .

We note that the collinear-soft and global-soft scales are related by the
following condition:

$\displaystyle(\mu_{cs})^{2+\beta}$
$\displaystyle=\big{(}\mu_{s}\big{)}^{1+\beta}\>\mu_{gs}$
$\displaystyle=\Big{(}\frac{\mu_{J}^{2}}{\mu_{N}}\Big{)}^{1+\beta}\mu_{gs}\,.$
(151)

This relation can be understood from RG consistency, noting that the $z_{\rm
cut}$ and $\beta$ dependence must cancel within the soft sector, as jet and
hard functions do not depend on these parameters. The relation in the second
line is obtained using Eq. (149) and we will use it below in implementing the
profile for our jet scale.

In the intermediate $R_{g}$ region, the collinear-soft sector further splits
into CSg and CSm modes, with the canonical values of the corresponding scales
being

$\displaystyle\mu_{cs_{m}}^{\rm
can.}=\frac{m_{J}^{2}}{Q(R_{g}/2)}\,,\qquad\mu_{cs_{g}}^{\rm can.}=Q_{\rm
cut}\Big{(}\frac{R_{g}}{2}\Big{)}^{1+\beta}\,.$ (152)

They satisfy the following see-saw relation:

$\displaystyle\big{(}\mu_{cs}\big{)}^{2+\beta}=\big{(}\mu_{cs_{m}}\big{)}^{1+\beta}\mu_{cs_{g}}\,,$
(153)

which holds for all values of $R_{g}$. This relation will be used below to
define the $\mu_{cs_{m}}$ profile. It is analogous to the relation in Eq.
(5.3.1) as can be seen by setting $R_{g}=R$. We will elaborate on this more
below.

Lastly, we consider the canonical scales in the small $R_{g}$ region. In
addition to the $\mu_{gs}$ and $\mu_{cs_{g}}$ scales given above, the jet and
the CSm scale in the intermediate region merge into a single collinear scale.
As we saw in Sec. 3.3, the logarithms in the collinear function ${\cal
C}^{\kappa}$ are minimized for $\mu\sim QR_{g}/2$, hence the canonical scale
is given by

$\displaystyle\mu_{c}^{\rm can.}=\frac{QR_{g}}{2}\,,$ (154)

and the canonical relation between three scales in the small $R_{g}$ region
reads

$\displaystyle\mu_{c}=\mu_{N}\>\bigg{(}\frac{\mu_{cs_{g}}}{\mu_{gs}}\bigg{)}^{\frac{1}{1+\beta}}\,.$
(155)

In deriving this relation we have replaced $R_{g}/2\rightarrow\tan(R_{g}/2)$,
capturing some of the power corrections in $R_{g}/2$ that are important in
this region. Again, we will use the see-saw relation in Eq. (155) to fix the
$\mu_{c}$ profile. Furthermore, given the consistency between the small and
intermediate regime, the following canonical relation also holds:

$\displaystyle\mu_{J}^{2}=\mu_{c}\>\mu_{cs_{m}}\,.$ (156)

This relation is relevant in the ungroomed region, as can be seen by
substituting $R_{g}\rightarrow R$ in Eq. (149).

#### Consistency of the intermediate $R_{g}$ with the small and large $R_{g}$
regions

It is worthwhile to explore the behavior of the scales in the intermediate
$R_{g}$ region as $R_{g}$ approaches its upper and lower kinematic bounds. At
the upper limit $R_{g}\rightarrow\theta_{g}^{\star}(m_{J}^{2},Q_{\rm
cut},\beta)$ (in the groomed region) the scales merge into the $\mu_{cs}$
scale,

$\displaystyle\mu_{cs_{m}}(m_{J}^{2},R_{g}=\theta_{g}^{\star})=\mu_{cs_{g}}(Q_{\rm
cut},R_{g}=\theta_{g}^{\star})$ $\displaystyle=\mu_{cs}(m_{J}^{2},Q_{\rm
cut})\,,\qquad\quad\text{for $m_{J}<m_{J}^{0}$}\,,$ (157)

as can be seen by substituting $R_{g}=\theta_{g}^{\star}(m_{J}^{2},Q_{\rm
cut},\beta)$ in Eq. (152), with $\theta_{g}^{\star}$ defined in Eq. (12). In
the ungroomed region the upper limit is set by the jet radius, and the
collinear-soft scales for $R_{g}\rightarrow
R<\theta_{g}^{\star}(m_{J}^{2},Q_{\rm cut},\beta)$ behave differently:

$\displaystyle\mu_{cs_{m}}(m_{J}^{2},R_{g}=R)$
$\displaystyle=\mu_{s}(m_{J}^{2})\,,\qquad\hskip 1.0pt\qquad\quad\text{for
$m_{J}>m_{J}^{0}$}\,,$ (158) $\displaystyle\mu_{cs_{g}}(Q_{\rm cut},R_{g}=R)$
$\displaystyle=\mu_{gs}(Q_{\rm cut},R)\,,\qquad\quad\text{for
$m_{J}>m_{J}^{0}$}.$ (159)

This suggests that, unlike for groomed to ungroomed transition for the single
differential jet mass distribution, there is a smooth transition between the
intermediate and the ungroomed region owing to the additional $R_{g}$
measurement. Here the soft drop resummation is automatically turned off as
$\mu_{cs_{g}}$ approaches $\mu_{gs}$, whereas $\mu_{cs_{m}}\rightarrow\mu_{s}$
provides for the essential plain jet mass resummation in the ungroomed region.
We stress, however, that the intermediate $R_{g}$ regime still lacks the power
corrections of ${\cal O}(m_{J}^{2}/(QQ_{\rm cut}))$ in the ungroomed region.

Next, we note that upon decreasing $R_{g}$ the $\mu_{cs_{m}}$ scale meets the
jet scale as $R_{g}$ approaches the lower limit of the angle of the collinear
mode:

$\displaystyle\mu_{cs_{m}}(m_{J}^{2},R_{g}=\theta_{c})=\mu_{J}(m_{J}^{2})\,.$
(160)

On the other hand, from Eq. (152) we see that $\mu_{cs_{g}}$ keeps decreasing
on reducing $R_{g}$. Being the smallest of all the scales, for a given jet
mass it becomes nonperturbative first for $R_{g}\sim(R_{g})_{\rm NP}$ defined
in Eq. (18). Thus we will first implement the $\mu_{cs_{g}}$ profile so as to
consistently freeze it to an ${\cal O}(1{\rm GeV})$ value in the
nonperturbative region.

#### Rescaled variables

To simplify the expressions we will work with the following unit-interval
variables corresponding to jet mass and $R_{g}$. For the case of $e^{+}e^{-}$
collisions, we have

$\displaystyle\xi(m_{J}^{2})\equiv\frac{m_{J}^{2}}{Q^{2}\tan^{2}\frac{R}{2}}=\frac{m_{J}^{2}}{(\mu_{N}^{\rm
can.})^{2}}\,,\qquad\psi(R_{g})\equiv\frac{\tan\big{(}\frac{R_{g}}{2}\big{)}}{\tan\big{(}\frac{R}{2}\big{)}}\,\qquad(\text{$e^{+}e^{-}$
case})\,.$ (161)

Here we use $\mu_{N}^{\rm can.}=Q\tan(R/2)$ to keep it distinct from $\mu_{N}$
which we will vary to study perturbative uncertainty. Analogously, the
corresponding variables in the $pp$ case are defined as

$\displaystyle\xi(m_{J}^{2})\equiv\frac{m_{J}^{2}}{p_{T}^{2}R^{2}}\,,\qquad\psi(R_{g})\equiv\frac{R_{g}}{R}\,\qquad(\text{$pp$
case})\,.$ (162)

From Eq. (15) we see that for $m_{J}^{2}>m_{0}^{2}$ we enter the ungroomed
resummation region. Here we turn off the soft-drop dependent resummation in
the soft sector by merging the collinear-soft and global-soft scales to the
ungroomed soft scale. In terms of $\xi$, the transition point is given by

$\displaystyle\tilde{\xi}_{0}\equiv\frac{m_{0}^{2}}{\mu_{N}^{2}}=\frac{Q_{\rm
cut}}{Q}\tan^{\beta}\Big{(}\frac{R}{2}\Big{)}\cos^{(2+\beta)}\Big{(}\frac{R}{2}\Big{)}=\frac{\mu_{gs}}{\mu_{N}}\cos^{(2+\beta)}\Big{(}\frac{R}{2}\Big{)}\,.$
(163)

We note that as a result of the $\cos^{2+\beta}(R/2)$ the transition to
ungroomed region happens slightly earlier than the value of $\xi=\xi_{0}$ for
which the global-soft and collinear-soft scales merge:

$\displaystyle\xi_{0}\equiv\xi\Big{|}_{\mu_{gs}=\mu_{cs}}=\frac{\mu_{gs}}{\mu_{N}}=\frac{Q_{\rm
cut}}{Q}\tan^{\beta}\Big{(}\frac{R}{2}\Big{)}\,\qquad(\text{$e^{+}e^{-}$
case})\,.$ (164)

In the following we will, however, ignore the difference between
$\tilde{\xi}_{0}$ and $\xi_{0}$ and take $\xi_{0}$ in Eq. (2) to correspond to
the groomed and ungroomed transition point. For the $pp$ case we have

$\displaystyle\xi_{0}\equiv z_{\rm cut}^{\prime}R^{\beta}\,\qquad(\text{$pp$
case})\,.$ (165)

From Eq. (11) the phase space in the variables $\xi$ and $\psi$ is then given
by

Figure 3: The range of values of allowed $\psi$ values as a function of $\xi$
are shown by the region between the solid (magenta and blue) lines. The dot-
dashed (green) line indicates where the transition to the ungroomed regime
takes place. The maximum value of $\psi$ corresponds to
$R_{g}=\theta_{g}^{\star}(\xi)$, or $\psi=R$ in the ungroomed regime.

$\displaystyle\psi_{\rm min}(\xi)\equiv\sqrt{\xi}\>\leq\>\psi\>\leq\>\psi_{\rm
max}(\xi)\equiv\min\big{\\{}\psi^{\star}(\xi),\>1\big{\\}}\>\,,\qquad
0\leq\xi\leq 1\,,$ (166)

and is shown in Fig. 3. Here $\psi^{\star}(\xi)$ corresponds to $\psi$ for
$R_{g}=\theta_{g}^{\star}(m_{J}^{2},Q_{\rm cut},\beta)$, and is given by

$\displaystyle\psi^{\star}(\xi)\equiv\Big{(}\frac{\xi}{\xi_{0}}\Big{)}^{\frac{1}{2+\beta}}\,.$
(167)

For $\xi>\xi_{0}$, i.e. in the ungroomed region, $R_{g}$ is bounded above by
the jet radius $R$, which limits $\psi\leq 1$. We note that Eqs. (166) and
(167) also hold for the $pp$ case.

We now summarize the canonical scales in terms of $\xi$ and $\psi$ variables:

  1. 1.

Canonical scales for plain jet mass:

$\displaystyle\mu_{N}^{\rm can.}=Q\tan\frac{R}{2}\,$
$\displaystyle(\text{$e^{+}e^{-}$ case})\,,$ (168) $\displaystyle\mu_{N}^{\rm
can.}=p_{T}R\,$ $\displaystyle(\text{$pp$ case})\,,$
$\displaystyle\mu_{s}^{\rm can.}=\mu_{N}^{\rm can.}\xi\,,\qquad\qquad\mu^{\rm
can.}_{J}=\mu_{N}^{\rm can.}\sqrt{\xi}\,.$ (169)

  2. 2.

Canonical scales for large $R_{g}$ region (with $\mu_{N}$ and $\mu_{J}$ same
as above):

$\displaystyle\mu_{gs}^{\rm can.}=\mu_{N}^{\rm
can.}\xi_{0}\,,\qquad\mu_{cs}^{\rm can.}=\mu_{N}^{\rm
can.}\xi_{0}^{\frac{1}{2+\beta}}\xi^{\frac{1+\beta}{2+\beta}}=\mu_{gs}^{\rm
can.}\big{[}\psi^{\star}(\xi)\big{]}^{1+\beta}\,,\qquad\xi<\xi_{0}\,,$ (170)

where $\xi_{0}$ is defined as

$\displaystyle\xi_{0}=\frac{Q_{\rm
cut}}{Q}\tan^{\beta}\Big{(}\frac{R}{2}\Big{)}\,$
$\displaystyle(\text{$e^{+}e^{-}$ case})\,,$ $\displaystyle\xi_{0}=z_{\rm
cut}^{\prime}R^{\beta}\,$ $\displaystyle(\text{$pp$ case})\,.$ (171)

  3. 3.

Canonical scales for intermediate $R_{g}$ region (with $\mu_{N}$, $\mu_{J}$
and $\mu_{gs}$ same as above):

$\displaystyle\mu_{cs_{m}}^{\rm can.}=\mu_{N}^{\rm
can.}\frac{\xi}{\psi}\,,\qquad\mu^{\rm can.}_{cs_{g}}=\mu_{gs}^{\rm
can.}\psi^{1+\beta}\,.$ (172)

  4. 4.

Canonical scales for the small $R_{g}$ region (with $\mu_{N}$ and $\mu_{gs}$
same as above):

$\displaystyle\mu_{c}^{\rm can.}=\mu_{N}^{\rm can.}\psi\,.$ (173)

We will find the expression of $\mu_{cs}$ in terms of $\psi^{\star}$ in Eq.
(170) useful to construct the $\mu_{cs}$ profile below.

Lastly, we note that for the case of $e^{+}e^{-}$ collisions in the small
angle limit, we have $\psi\tan(R/2)=\tan(R_{g}/2)\sim R_{g}/2$. Since using
the variable $\psi$ allows us to smoothly connect all the profiles from small
to large values of $R_{g}$, we also additionally replace all the appearances
of $R_{g}/2$ in the factorization formulae above by $\psi\tan(R/2)$. The
difference between the two is a power correction, and when $R_{g}$ is ${\cal
O}(1)$ it automatically results in the correct expressions. This includes all
the fixed-order results for factorization functions in App. B, the appearances
of $R_{g}$’s in the logarithms (which are simply the canonical scales
displayed above), the pre-factors in soft drop boundary cross section (see Eq.
(121)), in the ratio $\upsilon=(R_{g}/\theta_{g}^{\star})^{2+\beta}$ that
shows up in the resummation kernels in the large $R_{g}$ region (see Eqs.
(4.3) and (120)), in the weight functions in Eq. (147), and as well as any
non-logarithmic dependences on $R_{g}$ (such as in the collinear function in
the small $R_{g}$ region in Eq. (63)).

####  5.3.2 Profile functions for double differential cross section

We now turn to implementing the profile functions based on these canonical
scale choices and relations. Here we state results solely for $e^{+}e^{-}$
collisions with their generalizations to $pp$ case being straightforward. Our
goal is twofold: accounting for freezing of any perturbative scale that
approaches $\Lambda_{\rm QCD}$ and providing a way to estimate perturbative
uncertainty though scale variation. We consider profile variations in the next
section. We base the freezing behavior of all the scales on the the
$\mu_{cs_{g}}$ profile, defined as follows:

$\displaystyle\mu_{cs_{g}}(\psi)\equiv\mu_{gs}\,f_{\rm
run}\big{(}\psi(R_{g})^{1+\beta}\big{)}\,,$ (174)

where $f_{\rm run}(\psi)$ is defined by

$\displaystyle f_{\rm run}(\psi)$
$\displaystyle\equiv\left\\{\begin{array}[]{ll}y_{0}\Big{(}1+\frac{\psi^{2}}{4y_{0}^{2}}\Big{)}&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\psi\leq
2y_{0}\\\\[4.0pt] \psi&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2y_{0}<\psi\leq
1\end{array}\right.\;.$ (177)

In the resummation region $\mu_{cs_{g}}(\psi)\sim\mu_{gs}\psi^{1+\beta}$,
consistent with Eq. (152). The transition of the $\mu_{cs_{g}}$ scale into the
NP region is determined by

$\displaystyle y_{0}\equiv\frac{n_{0}}{(\mu_{gs}/1\,{\rm
GeV})}>\frac{\Lambda_{\rm QCD}}{Q_{\text{cut}}}\,,$ (178)

where $n_{0}$ is an ${\cal O}(1)$ number that controls the value for the
perturbative scale we freeze to which is $>\Lambda_{\rm QCD}$. Recall that it
is mandatory for the scales to be frozen in this manner, since the entire
formalism is based on perturbative expansions of things like the anomaous
dimensions. In the numerical analysis presented in Sec. 6 we use $n_{0}=0.75$
which from Eq. (177) corresponds to onset of the freezing of the coupling at
$\mu=1.5$ GeV.

Note that $\mu_{cs_{g}}$ has the lowest virtuality, so it is always the first
scale to enter the NP region when the jet mass grows smaller. To freeze in
turn the scales with larger virtuality as we venture into the NP region while
maintaining the canonical scaling relations, we now simply insert Eq. (174) in
the canonical relations derived above. First, as seen from Eq. (170), the
collinear-soft scale can be derived using $\mu_{cs_{g}}(\psi)$ by setting
$R_{g}=\theta_{g}^{\star}(m_{J}^{2},Q_{\rm cut},\beta)$. Thus we have

$\displaystyle\mu_{cs}(\xi)\equiv\mu_{gs}\,f_{\rm
run}\Big{(}\big{[}\psi^{\star}(\xi)\big{]}^{1+\beta}\Big{)}=\mu_{gs}\,f_{\rm
run}\bigg{(}\Big{(}\frac{\xi}{\xi_{0}}\Big{)}^{\frac{1+\beta}{2+\beta}}\bigg{)}\,,\qquad\xi<\xi_{0}\,.$
(179)

We note that, compared to the plain jet-mass soft scale in Eq. (148), the
collinear-soft scale should become nonperturbative at yet smaller jet masses
in the SDNP region defined by Eq. (17). This is correctly captured in Eq.
(179) because from Eq. (177) we see that the $\mu_{cs}(\xi)$ profile freezes
for $\xi\leq\xi_{\rm SDNP}$ where

$\displaystyle\xi_{\rm
SDNP}\equiv\xi_{0}(2y_{0})^{\frac{2+\beta}{1+\beta}}>\frac{(m_{J}^{2})_{\rm
SDNP}}{\mu_{N}^{2}}=\frac{\Lambda_{\rm QCD}}{\mu_{N}}\Big{(}\frac{\Lambda_{\rm
QCD}}{\mu_{gs}}\Big{)}^{\frac{1}{1+\beta}}\,,$ (180)

which is consistent with Eq. (17).

Having defined the $\mu_{cs}(\xi)$ and $\mu_{cs_{g}}(\psi)$ profiles we can
now use Eq. (153) to implement the $\mu_{cs_{m}}$ profile as follows:

$\displaystyle\mu_{cs_{m}}(\xi,\psi)$
$\displaystyle\equiv\mu_{cs}(\xi)\Bigg{[}\frac{\mu_{cs}(\xi)}{\mu_{cs_{g}}(\psi)}\Bigg{]}^{\frac{1}{1+\beta}}\,,\qquad\xi<\xi_{0}\,,$
(181)

while using Eqs. (149) and (5.3.1) we implement the profile for the jet scale
as follows:

$\displaystyle\mu_{J}(\xi)\equiv\Bigg{[}\mu_{N}\>\mu_{cs}(\xi)\>\bigg{(}\frac{\mu_{cs}(\xi)}{\mu_{gs}}\bigg{)}^{\frac{1}{1+\beta}}\Bigg{]}^{\frac{1}{2}}\,,\qquad\xi<\xi_{0}\,.$
(182)

Finally, using Eq. (155) the $\mu_{c}(\psi)$ profile is given by

$\displaystyle\mu_{c}(\psi)\equiv\mu_{N}\Bigg{(}\frac{\mu_{cs_{g}}(\psi)}{\mu_{gs}}\Bigg{)}^{\frac{1}{1+\beta}}\,.$
(183)

Figure 4: Profile scales $\mu_{i}$ as a function of the groomed radius at
fixed jet mass (top left) and as a function of the mass at fixed radius (top
right), for $\beta=1$. For more clarity the former plot uses a logarithmic
scale, while we omit the hard scale $\mu_{N}$ in the latter. At small groomed
radii (large jet masses) $\mu_{cs_{m}}$ and $\mu_{J}$ merge into $\mu_{C}$,
while at large radii (small masses) $\mu_{cs_{m}}$ and $\mu_{cs_{g}}$ become a
single soft-collinear scale $\mu_{cs}$, consistent with the mode picture.

We have so far defined the profile functions only in the groomed region for
$\xi<\xi_{0}$. A complete analysis of the distribution in the ungroomed region
is beyond the scope of this work. We will, however, extend our results into
the ungroomed region making use of the following extensions of the collinear-
soft profile scales:

$\displaystyle\text{Large $R_{g}$}:$
$\displaystyle\mu_{cs}=\mu_{gs}=\mu_{s}(\xi)=\mu_{N}\xi\,$
$\displaystyle(\xi>\xi_{0})\,,$ (184) $\displaystyle\text{Intermediate/Small
$R_{g}$}:$
$\displaystyle\mu_{cs_{g}}=\mu_{gs}\,,\quad\mu_{cs_{m}}=\mu_{s}(\xi)=\mu_{N}\xi$
$\displaystyle(\xi>\xi_{0})\,.$ (185)

The large $R_{g}$ transition occurs when we cross the dot-dashed green line in
Fig. 3 for values of $\psi$ that place us near the upper solid magenta line,
whereas the intermediate/small $R_{g}$ transitions occur when we cross the
dot-dashed green line when we are far from all solid lines, or close to the
lower blue line, respectively. Note that in the large $R_{g}$ regime the
resummation related to jet-grooming must be immediately turned off as
$\xi>\xi_{0}$. This is the same as the groomed to ungroomed profile transition
for the single differential jet mass case, which was discussed in Ref.
Chien:2019osu . In case of intermediate $R_{g}$ regime, however, this happens
more smoothly owing to its more factorized nature. Finally, the jet and the
collinear scales remain the same in the ungroomed region.

In Fig. 4 we show the profile scales resulting from the above formulae picking
$\beta=1$. In the left panel we show them as a function of the groomed jet
radius for a fixed jet mass. The scales satisfy the joining condition for the
end points of the $R_{g}$ spectrum: $\mu_{cs}$, $\mu_{cs_{g}}$, and
$\mu_{cs_{m}}$ merge at the end point
$R_{g}=\theta_{g}^{\star}(m_{J}^{2},Q_{\rm cut},\beta)$, whereas $\mu_{J}$ and
$\mu_{cs_{m}}$ merge into $\mu_{c}$ at the lower end point
$R_{g}=\theta_{c}(m_{J}^{2})$. In the right panel of Fig. 4 we fix the $R_{g}$
to a representative value of $R_{g}=0.3$ and plot the scales as a function of
the jet mass. Our choice of $R_{g}=0.3<R$ implies that the entire allowed mass
range lies within the groomed region. Once again we see the expected behavior
when the scales join. The profile functions for other values of $\beta$ look
qualitatively similar.

Figure 5:  Hybrid profile scales $\mu_{{\rm hyb},i}$ for $\beta=0$ and
$\beta=2$, shown as a function of the groomed radius $R_{g}$ at a fixed jet
mass. The bottom panels show the transition functions that give weight to the
various regimes. In the left panel we have a situation involving the 2-EFT
situation, whereas in the right panel we have a 3-EFT case.

As we discussed above in Sec. 5.1, the implementation of the matched
cumulative $R_{g}$ cross section involves constructing hybrid profiles
$\mu_{\rm hyb}$, that select the appropriate set of scales from each regime
following Eq. (138). These involve the weight functions
$w_{i}(m_{J}^{2},R_{g})$ that we constructed in Sec. 5.2. In Fig. 5 we show
the hybrid profiles for $\beta=0$ and $\beta=2$ for a given jet mass value in
the upper left and right panels respectively. In both these plots we show the
scales only in the region where they are valid (except for the global-soft
scale that is common to all). The $\mu_{c}$ scale splits off into $\mu_{J}$
and $\mu_{cs_{m}}$ as we transition out of the small $R_{g}$ regime, and the
$\mu_{cs_{m}}$ and $\mu_{cs_{g}}$ scales merge into the single collinear soft
scale $\mu_{cs}$ as we enter the large $R_{g}$ regime. In the bottom panels we
show the weight functions that correspond to that specific choice of jet mass
and and soft drop parameters. In the case of $\beta=0$ we transition directly
between the small and large $R_{g}$ regimes, thus providing an example of the
2-EFT scheme. In this case the scale $\mu_{cs_{m}}$ only serves to provide an
interpolation into the large $R_{g}$ region. On the other hand, we see that
for $\beta=2$ there is a significant region where $w_{\rm int}>0$ and the
intermediate $R_{g}$ resummation is active.

###  5.4 Perturbative uncertainty

We now briefly discuss our prescription to assess the perturbative uncertainty
of the EFT results by means of scale variations and varying the unknown ${\cal
O}(\alpha_{s}^{2})$ non-logarithmic terms necessary for NLL′ resummation in
the large and small $R_{g}$ regimes through the parameters $a_{20}^{\cal C}$,
$a_{20}^{S_{c}}$, and $a_{20,\varepsilon}^{S_{c}}$ that appear in Eqs. (63),
(81) and (113) respectively. Note that we will not consider variations in the
ungroomed region, leaving a more detailed analysis of this transition to
future work, and hence we often cutoff our plots with uncertainty bands prior
to hitting this transition.

####  5.4.1 Scale variations

In this analysis we will consider the following three types of scale
variations:

  1. 1.

Vary the hard and global-soft scales by a multiplicative factor, and let this
variation propagate to other scales through the canonical relations:

$\displaystyle\mu_{N}^{\rm vary}(e_{gs})$ $\displaystyle\equiv
e_{gs}\mu_{N}^{\rm can.}\,,\qquad\mu_{gs}^{\rm vary}(e_{gs})\equiv
e_{gs}\mu_{gs}^{\rm can.}\,.$ (186)

Here we will vary $e_{gs}$ from the default value of $e_{gs}=1$ to
$e_{gs}=1/2$ and $2$. Note that we vary the $\mu_{N}$ and $\mu_{gs}$ scales
simultaneously. This is to ensure that the variation does not affect the
groomed to ungroomed transition point $\xi_{0}$ defined in Eq. (2). This
variation will affect all the scales in Eqs. (174), (179), (181), (182), and
(183).

  2. 2.

A variation of the jet mass and $R_{g}$ dependent scales from their canonical
values in Eqs. (148), (150), (152), and (154) while maintaining the various
see-saw relations in Eqs. (149), (5.3.1), (153), and (155). This is done such
that the variations are frozen to the default value at the end point. More
specifically, we first define the variation of the $\mu_{cs_{g}}$ scale as

$\displaystyle\mu_{cs_{g}}^{\rm vary}(\psi,\alpha)$
$\displaystyle\equiv\mu_{gs}\Big{[}f_{\rm
vary}\big{(}\psi^{1+\beta}\big{)}\Big{]}^{\alpha}\,f_{\rm
run}\big{(}\psi^{1+\beta}\big{)}\,$ (187)

and use it to derive variations of other scales. Here we make use of the
following trumpet function $f_{\rm vary}(\psi)$ to turn off the variation at
the end point $\psi=1$:

$f_{\rm vary}(\psi)=\left\\{\begin{array}[]{l
r}2(1-\psi^{2})\,,&\qquad\psi<0.5\\\ 1+2(1-\psi)^{2}\,,&\qquad 0.5\leq\psi\leq
1\end{array}\right..$ (188)

The choice $\alpha=0$ returns the default profile, and varying $\alpha=\pm 1$
allows for variation in the resummation region up to a factor of $2$. In
accordance with Eq. (179), the variation of $\mu_{cs}$ scale is given by

$\displaystyle\mu_{cs}^{\rm vary}(\xi,\alpha)$
$\displaystyle\equiv\mu_{gs}\Bigg{[}f_{\rm
vary}\bigg{(}\Big{(}\frac{\xi}{\xi_{0}}\Big{)}^{\frac{1+\beta}{2+\beta}}\bigg{)}\Bigg{]}^{\alpha}\,f_{\rm
run}\bigg{(}\Big{(}\frac{\xi}{\xi_{0}}\Big{)}^{\frac{1+\beta}{2+\beta}}\bigg{)}\,,\qquad\xi<\xi_{0}\,.$

Having defined $\alpha$-variations of $\mu_{cs_{g}}$ and $\mu_{cs}$, the
corresponding variations of $\mu_{J}$, $\mu_{cs_{m}}$ and $\mu_{c}$ scales are
derived using Eqs. (182), (181) and (183) by replacing
$\mu_{cs_{g},cs}\rightarrow\mu_{cs_{g},cs}^{\rm vary}$.

  3. 3.

A variation that relaxes the canonical see-saw relations that we used to
derive the $\mu_{cs_{m}}$, $\mu_{J}$, and $\mu_{c}$ scales. The two sets of
canonical relations, Eqs. (5.3.1) and (153), and Eqs. (149) and (156), are a
consequence of soft drop and cumulative-$R_{g}$ constraints respectively. The
two sets are equivalent when $R_{g}\rightarrow R$, where both constraints are
lifted. For convenience, we define an auxiliary plain-jet mass soft scale
$\mu_{s}(\xi,\alpha)$ using Eq. (5.3.1):

$\displaystyle\widetilde{\mu}_{s}(\xi)\equiv\mu_{cs}(\xi)\Bigg{[}\frac{\mu_{cs}(\xi)}{\mu_{gs}}\Bigg{]}^{\frac{1}{1+\beta}}\,,\qquad\xi<\xi_{0}\,.$
(189)

We now parameterize the deviations in the $\mu_{cs_{m}}$ scale upon modifying
the soft-drop see-saw relations in Eqs. (5.3.1) and (153) with a small
exponent $\rho$:

$\displaystyle\Big{[}\widetilde{\mu}_{cs}(\xi,\rho)\Big{]}^{2+\beta}=\Big{[}\widetilde{\mu}_{s}(\xi)\Big{]}^{1+\beta+\rho}\big{[}\mu_{gs}\big{]}^{1-\rho}$
$\displaystyle=\Big{[}\mu^{\rm
vary}_{cs_{m}}(\xi,\psi,\rho)\Big{]}^{1+\beta+\rho}\Big{[}\mu_{cs_{g}}\Big{]}^{1-\rho}\,.$
(190)

Solving for $\mu_{cs_{m}}$ yields the variation

$\displaystyle\mu^{\rm vary}_{cs_{m}}(\xi,\psi,\rho)$
$\displaystyle\equiv\mu_{cs}(\xi)\Bigg{[}\frac{\mu_{cs}(\xi)}{\mu_{gs}}\Bigg{]}^{\frac{1}{1+\beta}}\Bigg{[}\frac{\mu_{cs_{g}}(\psi)}{\mu_{gs}}\Bigg{]}^{\frac{-(1-\rho)}{1+\rho+\beta}}\,.$
(191)

Next, we apply the same strategy to the $\mu_{J}$ and $\mu_{cs_{m}}$ scales by
entering deviations in Eqs. (149) and (156) through a small exponent $\gamma$:

$\displaystyle\big{[}\mu^{\rm vary}_{J}(\xi,\gamma)\big{]}^{2}$
$\displaystyle=\big{[}\mu_{N}\big{]}^{1+\gamma}\Big{[}\widetilde{\mu}_{s}(\xi)\Big{]}^{1-\gamma}$
(192) $\displaystyle=\Big{[}\mu^{\rm
vary}_{c}(\psi,\gamma)\Big{]}^{1+\gamma}\Big{[}\mu_{cs_{m}}(\xi,\psi)\Big{]}^{1-\gamma}\,.$

Note that we have set $\rho=0$ in Eq. (191) as these two variations are
independent. Solving for the scales $\mu_{J}$ and $\mu_{c}$ yields the
variations

$\displaystyle\mu_{J}^{\rm vary}(\xi,\gamma)$
$\displaystyle\equiv\big{[}\mu_{N}\big{]}^{\frac{1}{2}+\gamma}\Bigg{[}\mu_{cs}(\xi)\>\bigg{(}\frac{\mu_{cs}(\xi)}{\mu_{gs}}\bigg{)}^{\frac{1}{1+\beta}}\Bigg{]}^{\frac{1}{2}-\gamma}\,,$
(193) $\displaystyle\mu^{\rm vary}_{c}(\psi,\gamma)$
$\displaystyle\equiv\mu_{N}\>\Bigg{[}\frac{\mu_{cs_{g}}(\psi)}{\mu_{gs}}\Bigg{]}^{\frac{1-\gamma}{1+\gamma}\frac{1}{1+\beta}}\,.$

####  5.4.2 Parameterizing variation of the ${\cal O}(\alpha_{s}^{2})$ non-
logarithmic terms

We now consider the variations of the two-loop parameters $a_{20}^{\cal C}$,
$a_{20}^{S_{c}}$, and $a_{20,\varepsilon}^{S_{c}}$ introduced in Eqs. (63),
(81) and (113) respectively, in order to assess uncertainty of our lack of
knowledge of these terms. We first note that in combining the cross section
through the recipe in Eq. (137) we are relying on the fact that the
intermediate $R_{g}$ cross section reproduces the EFT result on either end of
the $R_{g}$ spectrum up to power corrections. This is guaranteed to hold from
Eqs. (50) and (54) when all the functions are expanded to a given order. Given
our complete knowledge of ${\cal O}(\alpha_{s})$ pieces, we can use the
construction in Eq. (137) to NLL accuracy. However, when we include ${\cal
O}(\alpha_{s}^{2})$ pieces in the small and large $R_{g}$ regimes to account
for NLL′ resummation, the missing non-logarithmic terms parameterized by
$a_{20}^{i}$ spoil this delicate cancellation when they are extrapolated into
the intermediate regime. We solve this issue by explicitly multiplying the
${\cal O}(\alpha_{s}^{2})$ fixed-order pieces in the large and small $R_{g}$
regimes by the weight factors $w_{i}(m_{J}^{2},R_{g})$ so that they are set to
zero when extrapolated into other regimes:

$\displaystyle d\Sigma_{\rm small}$ $\displaystyle\equiv d\Sigma^{[1]}_{\rm
small}(a_{10}^{\cal C})+w_{\rm small}(m_{J}^{2},R_{g})\,d\Sigma^{[2]}_{\rm
small}(a_{20}^{\cal C})\,\qquad(m_{J}>0)\,,$ (194) $\displaystyle d\Sigma_{\rm
large}$ $\displaystyle\equiv d\Sigma^{[0]}_{\rm large}+d\Sigma^{[1]}_{\rm
large}(a_{10}^{S_{c}})+w_{\rm large}(m_{J}^{2},R_{g})\,d\Sigma^{[2]}_{\rm
large}(a_{20}^{S_{c}})\,,$

where the terms $d\Sigma^{[i]}$ are ${\cal O}(\alpha_{s}^{i})$. The
$d\Sigma^{[0]}_{\rm large}$ term in the large $R_{g}$ regime is simply the
single differential jet mass cross section, and does not contribute to the
differential $R_{g}$ measurement. By demanding $m_{J}>0$, the ${\cal
O}(\alpha_{s}^{0})$ $\delta$-function piece in the small $R_{g}$ regime does
not contribute. The ${\cal O}(\alpha_{s})$ terms depend on the LO expressions
in Eqs. (64) and (93) in the large and small $R_{g}$ regimes respectively. The
${\cal O}(\alpha_{s}^{2})$ terms are the additional pieces required to achieve
NLL′ accuracy which include the cross terms shown in Tab. 1, as well as the
parameters $a_{20}^{\cal C}$ and $a_{20}^{S_{c}}$ that are included for
uncertainty estimation. Thus, we achieve the reasonable outcome that the
variations induced by these parameters will be limited to the region where the
corresponding weight function $w_{i}(m_{J}^{2},R_{g})$ is non-zero.

Having constructed the matched cumulative $R_{g}$ cross section, we will use
it directly to evaluate $C_{1}^{q}(m_{J}^{2})$. The uncertainties resulting
from scale variations and 2-loop parameter variations will then be directly
translated to uncertainties in predictions for $C_{1}^{q}(m_{J}^{2})$. For
computing $C_{2}^{q}(m_{J}^{2})$ we will, however, only make use of the large
$R_{g}$ boundary cross section at the differential level. Since we have no
contribution from the small and intermediate $R_{g}$ regime, following Eq.
(194) we adopt the prescription

$\displaystyle\frac{d}{d\varepsilon}\frac{d\sigma(\varepsilon)}{dm_{J}^{2}d\theta_{g}}\bigg{|}_{\rm
large}\equiv w_{\rm
large}(m_{J}^{2},\theta_{g})\frac{d}{d\varepsilon}\,\Bigg{(}\frac{d\sigma^{[1]}(\varepsilon)}{dm_{J}^{2}d\theta_{g}}\bigg{|}_{\theta_{g}\sim\theta_{g}^{\star}}+\frac{d\sigma^{[2]}(\varepsilon)}{dm_{J}^{2}d\theta_{g}}\bigg{|}_{\theta_{g}\sim\theta_{g}^{\star}}\Bigg{)}\,.$
(195)

From this equation we see explicitly that the cutoff on the large $R_{g}$
cross section appearing in the determination of $C_{2}^{q}$ is directly
determined by $w_{\rm large}$.

##  6 Numerical results and comparison with Monte Carlo simulations

Having set up the formalism, provided the method to match across different
regimes and described our prescription for quantifying the perturbative
uncertainty, we are now in position to present the numerical results for the
matched cumulative-$R_{g}$ cross section and the moments
$C_{1}^{q}(m_{J}^{2})$ and $C_{2}^{q}(m_{J}^{2})$. We build our c++ code based
on the core of the SCETlib package scetlib , extending it for soft drop
observables, and cross check the numerical results using Mathematica. We first
present the results for the matched double differential cross section in Sec.
6.1, and then use it to calculate $C_{1}^{q}(m_{J}^{2})$ in Sec. 6.2 and
$C_{2}^{q}(m_{J}^{2})$ in Sec. 6.3 in the manner described in earlier
sections. For definiteness, we fix $z_{\rm cut}=0.1$, $E_{J}=500$ GeV and
$R=1$ throughout the whole analysis, but we sample different values of the
soft drop angular weight $\beta$. Along with the numerical results for the
moments we will also present a comparison with parton shower Monte Carlo
predictions.

Figure 6:  The matching prescription smoothly interpolating between the three
EFT regimes is shown for the distribution cumulative in $R_{g}$, at NLL′
accuracy, for different values of $\beta$ and a fixed value of $m_{J}$. The
large $R_{g}$ (red dashed) and small $R_{g}$ (green dashed) predictions
include fixed-order corrections that are relevant in the respective plotted
range; the profile-improved intermediate prediction (blue, dot-dashed) does
not yet include these corrections while resumming large logarithms in the
central region. We obtain the matched result (solid black) by weighting the
three curves with the exponents $w_{i}(R_{g})$ shown in the bottom panels, and
subtracting the overlap (grey) as described in the text.

###  6.1 Results for the matched cross section

Following the strategy outlined in Sec. 5.1, we now utilize the profile and
transition functions described in Sec. 5.2 and Sec. 5.3 to smoothly match the
cross section across all our EFT regimes. Here we consider explicitly the
cross section cumulative in $R_{g}$ which determines $C^{q}_{1}$ as well as
the normalization factor $C_{0}^{q}$ it depends on, whereas $C^{q}_{2}$ will
be calculated at the differential level using the prescription in Eq. (195).
Since our focus is on making predictions for $C_{1,2}^{q}$, where NGL effects
in the global soft sector do not contribute, we will not include the function
$\Xi_{G}^{q}$ in our numerical results for the double differential cross
sections presented in this section (despite the fact that the resummation of
these NGLs may be important to determine the overall normalization). However,
we do include the NGLs described by $\Xi_{S}^{q}$, and $\Xi_{C}^{q}$, which
affect both the shape and normalization of the distribution.

In Fig. 6 we show the various components that enter the calculation of the
matched cross section for $\beta=0,1,$ and $2$ plotted as a function of
$R_{g}$ for a representative jet mass value giving
$\log_{10}(m_{J}^{2}/E_{J}^{2})=-1.7$, which with our choice for $E_{J}$ is
$m_{J}=70.6\,{\rm GeV}$. In order to explain the legend and the meaning of
each curve, consider the first panel with $\beta=0$. The solid black line is
the matched cross section that is evaluated after combining the cross sections
in the three regimes according to Eq. (137). The results for cross section in
the large $R_{g}$ EFT, $d\Sigma_{\rm large}|_{\mu_{\rm large}}$, and in the
small $R_{g}$ EFT, $d\Sigma_{\rm small}|_{\mu_{\rm small}}$, evaluated with
their corresponding profiles are shown by the red dashed and green dashed
curves respectively. The overlap of the intermediate $R_{g}$ with either of
these regimes are shown in gray-solid for $d\Sigma_{\rm int}|_{\mu_{\rm
large}}$ (labeled as “Large $R_{g}$ w/o ${\cal
O}\big{(}(R_{g}/\theta_{g}^{\star})^{2+\beta}\big{)}$”), and gray-dotted for
$d\Sigma_{\rm int}|_{\mu_{\rm small}}$ (labeled as “Small $R_{g}$ w/o ${\cal
O}\big{(}\theta_{c}/R_{g})$”). We see that the $d\Sigma_{\rm int}|_{\mu_{\rm
large}}$ term matches with the $d\Sigma_{\rm large}|_{\mu_{\rm large}}$ term
as $R_{g}\rightarrow 0$, such that the power correction ${\cal
O}\big{(}(R_{g}/\theta_{g}^{\star})^{2+\beta}\big{)}$ becomes small. Likewise,
the $d\Sigma_{\rm int}|_{\mu_{\rm small}}$ cross section merges with the small
$R_{g}$ result as $R_{g}\rightarrow\theta_{g}^{\star}$, up to the ${\cal
O}\big{(}\theta_{c}^{2}/R_{g}^{2})$ power correction. Finally, the blue dot-
dashed curve that spans the entire range of $R_{g}$ values is the
interpolation $\Sigma_{\rm int}|_{\mu_{\rm hyb}}$ that makes use of the hybrid
profiles described in Eq. (138). The hybrid profiles are constructed to
reproduce the large-$R_{g}$ and small-$R_{g}$ profiles in their respective
regions. Thus, we see that $\Sigma_{\rm int}|_{\mu_{\rm hyb}}$ matches exactly
with the $d\Sigma_{\rm int}|_{\mu_{\rm large}}$ (gray-solid) in the large
$R_{g}$ regime and $d\Sigma_{\rm int}|_{\mu_{\rm small}}$ (gray-dotted) in the
small-$R_{g}$ regime. The weight functions $w_{i}(m_{J}^{2},R_{g})$ given in
Eq. (147) demarcate the boundary of each region and govern the behavior of the
hybrid profiles, and are shown below each cross section panel.

Figure 7: Theoretical uncertainty on the cross section cumulant in $R_{g}$,
for $\log_{10}(m_{J}^{2}/E_{J}^{2})=-1.7$ and different values of $\beta$. The
central curve (black dashed) is obtained with the matching prescription
described in the text, while the colored bands probe respectively scale
variation (magenta), and variation of the fixed order two-loop constants
$a_{20}^{S_{c}}$ and $a_{20}^{\cal C}$, collectively denoted as $a^{i}_{20}$
variation (green). In the top right plot, the envelope of scale variations for
$\beta=0$ is broken down into various components: overall variation by a
common factor (red), variation by a trumpet factor preserving canonical
relations (yellow), breaking of canonical relations (green).

As mentioned in Sec. 5.2, for the cases $\beta=0,1$ the intermediate $R_{g}$
resummation is not present (for the jet mass value shown in Fig. 6) as there
is simply no room for this EFT to be valid according to Eq. (144), whereas for
$\beta=2$ there is a significant region where the intermediate $R_{g}$ EFT is
valid. For each of the three $\beta$ values that we show, we see that our
choice of the locations where the weight functions turn themselves on and off
agrees with the locations where the power corrections become visible. This can
be seen from the difference of the large and small $R_{g}$ cross sections with
the corresponding intermediate $R_{g}$ overlaps.

Next we show in Fig. 7 our estimate of perturbative uncertainty on the NLL′
cumulant cross section, again for $\beta=0$, $1$, and $2$. The uncertainty
bands in the top left plot are obtained by varying the profile scales as
described in Sec. 5.4 (magenta) and by varying the quantities $a_{20}^{{\cal
C}}$ and $a_{20}^{S_{c}}$ that parametrize our ignorance of the two-loop
constants terms in respectively the small $R_{g}$ and large $R_{g}$ EFTs
(green). Specifically, we vary both $a_{20}^{S_{c}}$ and $a_{20}^{{\cal C}}$
in the range $[-2\pi,2\pi]$, and take the envelope of the two independent
variations. We notice that scale variation is the dominant source of
uncertainty at large $R_{g}$, while at small $R_{g}$ the perturbative
uncertainty is mainly set by the $a_{20}^{{\cal C}}$ term. As discussed below
Eq. (64), the lower endpoint of the distribution receives perturbative
corrections starting at NLO. It is clear from Eq. (63) that this effect is
captured by variations of $a_{20}^{{\cal C}}$, resulting in a non-vanishing
uncertainty band below the endpoint of the central value. This effect has no
equivalent in the scale variations, where by construction we set
$a_{20}^{{\cal C}}=0$ and use the same LO endpoint as for the central value.

The top right plot in Fig. 7 shows for $\beta=0$ the breakdown of the scale
uncertainty into the three independent variations listed in Sec. 5.4: overall
scale uncertainty obtained by simultaneously varying the hard and soft scales
by a factor 2 (red band); variations of collinear-soft scales by a trumpet
factor, setting $\alpha=\pm 1$ in Eqs. (187) and (2) (yellow band); and
uncertainty due to relaxing the canonical relations between scales, achieved
by varying $\rho,\gamma\in[-0.1,0.1]$ in Eqs. (190) and (192) (green band). We
see that the three sources of uncertainty have similar size, with the trumpet
factor playing the most relevant role. A similar qualitative behavior occurs
for $\beta=1,2$, so for simplicity we do not plot the breakdown for these
cases, and only show the envelope of scale variations and the fixed-order
uncertainty due to the parameters $\\{a^{i}_{20}\\}$ in the lower panels of
Fig. 7. In these plots we do not show the relative impact of the NGLs we have
included in our cross section, but we note that here their contribution is
much smaller than the theoretical uncertainty across the whole range in
$R_{g}$.

Before moving on to the numerical results for $C_{1}^{q}$ and $C_{2}^{q}$, we
anticipate that the theoretical uncertainty for these two observables will
look rather different than the one just shown for the double differential
cumulative cross section. In particular, the part of uncertainty that only
depends on the jet mass but not on $R_{g}$ will cancel in the ratio of the
angular moments in Eq. (5) and the cumulant distribution, since effectively
this means they are normalized point by point in the $m_{J}^{2}$ spectrum. We
anticipate that this will lead to smaller uncertainties for both the
$C_{1}^{q}$ and $C_{2}^{q}$ coefficients.

###  6.2 Results for $C_{1}^{q}$

We now show predictions for $C_{1}^{q}$, based on the NLL′ expressions for
$M_{1}^{q}$ obtained by evaluating Eq. (4) on the resummed cross section
described in Sec. 4 and shown in Sec. 6.1. We then compare these EFT results
against Monte Carlo simulations and numerical results from coherent branching.
We explicitly present predictions for quark jets, expecting gluon jets to
follow the same qualitative behavior. We set again $E_{J}=500$ GeV, $R=1$, and
$z_{\rm cut}=0.1$, and explore the three cases $\beta=0,1,2$ across the mass
range $-3.5<\log_{10}(m_{J}^{2}/E_{J}^{2})<-0.5$. At low jet masses, the
double differential distribution on which we base our predictions becomes
nonperturbative. However, these NP effect should not be included in
$C^{q}_{1}$, which is by definition a purely perturbative coefficient. What
portion of the jet mass range is significantly affected by NP corrections
crucially depends on the value of $\beta$. Our method of calculating
$C_{1}^{q}$ automatically deals with this as discussed in Sec. 5.3.

Figure 8: Comparison between the LL (blue dashed) and NLL′ (red) effective
theory predictions for the perturbative coefficient $C_{1}$, the LL
predictions from coherent branching formalism (black dot-dashed) and Monte
Carlo simulations (colored markers), for different values of $\beta$. The
vertical dotted magenta line signals the transition from the SDNP region at
small $m_{J}$ to the perturbative SDOE region at larger $m_{J}$.

In Fig. 8 we present central values for our EFT results for $C_{1}^{q}$ at
both LL and NLL′ orders. These results are compared to results from coherent
branching and from Monte Carlo simulations. For the latter, we consider
predictions from Pythia 8.2 Sjostrand:2014zea , Vincia 2.2 Fischer:2016vfv
and Herwig 7.1 Bellm:2015jjp , where the jet reconstruction and soft drop are
performed using FastJet 3.3 Cacciari:2011ma . We signal the region where NP
effects are relevant by showing the transition from SDNP to SDOE region by a
vertical dotted magenta line, as defined in Eq. (1). Note that Pythia, Herwig
and Vincia return very similar results for $C_{1}^{q}$ Hoang:2019ceu . At
intermediate and large masses the NLL′ predictions differ significantly from
the ones from coherent branching, in better agreement with the Monte Carlo,
while the LL-EFT prediction lies in-between these two results. In the SDNP
region the NLL′ and LL coherent branching curves agree very well with each
other, while the LL-EFT result shows some deviation. The fact that our LL-EFT
predictions differ from the LL coherent branching derived in Hoang:2019ceu  is
not too surprising: as we discussed in Sec. 4.5, the EFT description is more
refined in that it involves matching the three regimes via profile scales
which includes a resummation of additional large logarithms, but on the other
hand the LL-EFT result lacks some of the power corrections that are present in
the splitting kernel $p_{gq}(z)$. These power corrections have a small effect
for $C_{1}^{q}$, thus the difference seen in Fig. 8 is driven by the
additional resummation included in the LL-EFT results.

Figure 9: Uncertainty estimate for the NLL′ and LL effective theory
predictions for the perturbative coefficient $C_{1}^{q}$. We do not estimate
LL uncertainty in the ungroomed region, where the EFT description loses
validity. NLL′ uncertainty bands are tiny and examined in more detail in Fig.
10.

In Fig. 9 we reconsider the EFT predictions, now including also our estimate
of their perturbative uncertainties (repeating the Monte Carlo simulation
results for reference). While the LL uncertainty band (blue) is obtained
purely from scale variation, the NLL′ uncertainty band (orange) is obtained
from the envelope of scale variation and variation of the two-loop fixed-order
constants $\\{a^{i}_{20}\\}$. Since our estimates for the perturbative
uncertainty does not cover the ungroomed region, we exclude values
$m_{J}>m_{0}^{ee}$ in the plotted range. We observe good convergence between
the two orders, with only the $\beta=2$ results being slightly outside the LL
uncertainty band. The most striking feature is the very small size of the NLL′
uncertainty band, whose width becomes noticeable only for $\beta=2$ (low
panel). Given that for the NLL′ cumulant cross section the variations are
rather large (see Fig. 7), the tiny uncertainty bands for $C_{1}^{q}$ are due
to cancellations between numerator and denominator of Eq. (57). Indeed, we
know that the leading double logarithms cancel in the ratio to all orders in
perturbation theory.

To further study the small NLL′ theoretical uncertainty, in Fig. 10 we show
the relative difference of the variations to the central curve. As we did for
the cumulant in Fig. 7, we separate scale variation (magenta) and the
variation of the two-loop constants (green). Again, we do not show the
perturbative uncertainty beyond the groomed region. We first notice that in
the SDOE region the uncertainty, of a few percent, is dominated by the unknown
two-loop term. Scale variation gradually takes over in the SDNP region, which
becomes larger as $\beta$ increases. For $\beta=1,2$ the enlarged uncertainty
at high jet mass comes from varying the two-loop constant $a_{20}^{\cal C}$ in
the small $R_{g}$ region, whereas variations of the large $R_{g}$ term
$a_{20}^{S_{c}}$ are roughly constant across the mass range, and effectively
negligible. The situation is different for $\beta=0$, where varying
$a_{20}^{S_{c}}$ gives the dominant contribution. That the variations from the
small $R_{g}$ region are less relevant for this $\beta$ was already clear from
Fig. 7. In the bottom right plot of Fig. 10 we further break down the scale
variations for the $\beta=2$ case into the three different sources, where the
same descriptions made for Fig. 7 apply. A similar picture holds for the other
values of $\beta$, except that the overall scale variation (light red) becomes
less and less relevant as $\beta$ decreases.

In Fig. 10 we also investigate the relative contribution of NGLs to
$C_{1}^{q}$, by showing the shift to the central curve obtained when omitting
the leading NGLs and abelian clustering logarithms (black dashed). For
$\beta=0$ they are entirely negligible, while for larger $\beta$ their
contribution is small but comparable to the theoretical uncertainty. This
further justifies including the leading NGLs in our central value predictions
for $C_{1}^{q}$ with the method described above. Finally, we tested the
dependence of our results on the choice of weight functions discussed in Sec.
5.2. Specifically, we modified the inequality in Eq. (144) from
$\leq\frac{1}{3}$ to $\leq\frac{1}{2}$ and $\leq\frac{1}{4}$ (thus varying the
range where the intermediate $R_{g}$ EFT is included) and changed the
transition rate $r_{t}$ between EFTs in Eq. (146) by a factor 2. We find that
in both cases the variations lie within our uncertainty bands.

Figure 10: Theoretical uncertainly on the NLL′ predictions for $C_{1}$,
estimated through scale variation (magenta) and by varying the two-loop fixed-
order constant (green). When the jet mass is perturbative, both methods return
very small uncertainty bands, thus we normalize variations to the central
value for clarity. The black dashed curve shows the shift to the central value
when NGLs are ignored. The bottom right panel shows the breakdown of the
envelope of scale variation for $\beta=2$, labeled as in Fig. 7.

###  6.3 Results for $C_{2}^{q}$

Figure 11: Comparison between the NLL′/ LL effective theory predictions for
the perturbative coefficient $C_{2}$ (red/blue dashed), the analogous
predictions from coherent branching formalism (black dotdashed) and Monte
Carlo simulations (colored markers). The last panel shows the impact of the
additional anomalous dimension $\gamma_{0}(\varepsilon,z_{\rm cut})$ present
in the boundary distribution in the case $\beta=0$, where we zoom on the range
of validity of our EFT.

Based on the definition in Eq. (5), we obtain numerical results for
$C_{2}^{q}(m_{J}^{2})$ by using only the large $R_{g}$ EFT, setting up the
calculation at the differential level, weighting the large $R_{g}$ cross
section according to Eq. (195) and following the explicit construction given
in App. B.4. Note that since the large $R_{g}$ regime is free of NGLs in
contributions modifying the spectrum, so are our predictions for $C^{q}_{2}$.
We consider the same kinematical values as for $C_{1}^{q}$, and in Fig. 11
plot the comparison between our LL and NLL′ EFT results, the LL coherent
branching prediction, and MC data. For this observable the spread between
different MC results is much larger, with Herwig predictions being
systematically lower than Pythia and Vincia. We see that our calculations are
in good agreement with the MC in the region of validity of the EFT, with a
preference for the Herwig results for larger $\beta$. Unlike the treatment in
Hoang:2019ceu , here we do not make the small jet radius approximation in the
coherent branching LL predictions, i.e. we distinguish $\tan(R_{g}/2)\neq
R_{g}/2$. This ensures that the coherent branching and EFT calculations both
have their transitions to the ungroomed region at the same value of the jet
mass. Although the impact of these corrections for $C_{1}^{q}$ were minimal,
here they induce considerable corrections to $C_{2}^{q}$ at large mass,
shifting the peak of the distribution towards larger values and improving the
agreement with MC data. This happens because the boundary cross section, on
which the calculation of $C_{2}^{q}$ is based, is highly peaked at larger
$R_{g}$, where the $R_{g}/2\ll 1$ approximation is less justified. We stress,
however, that a more accurate description of this region would involve a an
extension of our EFT to account for power corrections ${\cal
O}(m_{J}^{2}/(QQ_{\rm cut}))$ that become relevant at the groomed-ungroomed
transition point, $m_{0}^{(ee)}$ defined in Eq. (15), which we do not attempt
here. We further notice in Fig. 11 that the LL coherent branching result is
systematically larger than the LL-EFT result. Compared to NLL′, the LL
coherent branching result is similar for $\beta=0$ but systematically larger
for $\beta=1,2$. As noted in Sec. 4.5, our LL-EFT predictions include extra
resummation in Laplace space, which is absent in the LL coherent branching
results, while the latter include some additional power corrections through
the splitting function $p_{gq}$. We find that at the large jet masses
($\log_{10}(m_{J}^{2}/E_{J}^{2})\gtrsim-1.5$) the disagreement is almost
entirely due to the power corrections, while at smaller jet masses the
additional evolution in the EFT becomes more important. Finally, in the bottom
right plot we show the effect of including the extra anomalous dimension
$\gamma_{0}(\varepsilon,z_{\rm cut})$ from Eq. (109), which is present only in
the case $\beta=0$. The NLL′ result is shown before (dashed magenta) and after
(red) including the fixed-order logarithms and additional running induced by
$\gamma_{0}$. For more clarity, we zoom in on the range $[-3.5,-1.5]$ in the
logarithmic jet mass variable, leaving out the ungroomed region. We observe
that the $\gamma_{0}$ contribution is important across the whole plotted
range, and the improved agreement with MC predictions shows that the parton
showers considered here likely do incorporate this NLL effect.

Figure 12: Uncertainty estimate for the NLL′ (orange band) and LL (blue band)
effective theory predictions for the perturbative coefficient $C_{2}^{q}$, at
different values of $\beta$. The right top panel shows the breakdown of the LL
envelope into the various sources of variation, showing that the large
uncertainty is induced by the trumpet variation (yellow) rather than by
breaking the canonical relations (green). The same panel shows that at NLL′
varying the two-loop constant (blue) dominates over the scale uncertainty
(red).

We also show the LL and NLL′ EFT results with theoretical uncertainties in
figure Fig. 12, again restricting to the region $m_{J}<m_{0}^{ee}$ where
grooming is active. The NLL′ uncertainties are estimated again by combining
scale variation with variations of the two-loop non-logarithmic term via
$a_{20}^{S_{c},\varepsilon}$ introduced in Eq. (113). We see that the
uncertainty bands shrink from LL to NLL′, but unlike $C_{1}^{q}$, their size
is still significant and visible at NLL′. For this reason we do not show plots
of relative variations here. For $\beta=0$ the NLL′ curves have excellent
agreement with the MC data, while $\beta=1$ and $\beta=2$ show some tension at
larger jet masses, especially with Pythia and Vincia. This could be due to
power corrections that we have not added to the EFT results, as inferred from
the comparison with the LL coherent branching results for $\beta=1,2$ in Fig.
11. Note that since the $C_{2}^{q}$ calculation is carried out with the large
$R_{g}$ cross section, that the NGL and clustering effects present for the
corresponding double differential cross section only modify the normalization,
and hence cancel out in the ratio of moments that determine $C_{2}^{q}$.

While the NLL′ uncertainty in Fig. 12 has roughly the same size for all
$\beta$, the LL band becomes larger as $\beta$ decreases. To investigate this
further, we show the breakdown of variations for $\beta=0$ in the top right
panel of Fig. 12. The LL uncertainty, entirely determined by scale variation,
is once more decomposed into the variation of the collinear-soft scale by a
trumpet factor (yellow) and the breaking of the canonical relations (green).
We find that the overall variation of the hard and soft scales is irrelevant
for $C_{2}^{q}$, and hence have not displayed it here. In the same plot, the
NLL band is decomposed into scale variation (red) and variation of the two-
loop constant $a_{20}^{S_{c},\varepsilon}$ (blue), where we vary
$a_{20}^{S_{c},\varepsilon}\in[-\pi,\pi]$. We see that the variation of the
two-loop constant dominates over scale variation across the whole range in the
jet mass. This is because the denominator in Eq. (5) does not involve
$a_{20}^{S_{c},\varepsilon}$ (which parametrizes uncertainty in the boundary
cross section), and hence unlike other variations does not have a cancellation
in the ratio of moments used to compute $C_{2}^{q}$. We also checked that
variations of the weight functions through the parameters in Eqs. (144) and
(146) have a small impact compared to the theoretical uncertainty. (Note that
even though $C_{2}^{q}$ uses only the large $R_{g}$ regime, it still involves
the weight function as in Eq. (195).)

##  7 Conclusion

In this paper we have set up an effective field theory description of
processes involving jets which are groomed using soft drop, where the jet mass
and groomed jet radius are simultaneously measured. We presented factorization
formulae formulated using SCET in order to resum large logarithms in the cross
section which is differential in these two variables. We also combined the
three EFT regimes that span the phase space of the double differential
measurement to obtain a coherent description of the double differential cross
section across the whole range of kinematics where grooming is active. We
performed a detailed numerical analysis of results for $e^{+}e^{-}$ collisions
in dijet limit.

As a first application of this formalism, we computed two angular moments of
the double differential cross section: the average value of the groomed jet
radius, $M_{1}^{\kappa}(m_{J}^{2})$, and the average value of the inverse
groomed jet radius for emissions at the soft drop boundary,
$M_{-1}^{\kappa\varocircle}(m_{J}^{2})$. Apart from being interesting
observables in their own right, they are closely related to the Wilson
coefficients $C_{1}^{\kappa}(m_{J}^{2})$ and $C_{2}^{\kappa}(m_{J}^{2})$ which
describe the leading nonperturbative corrections to the soft drop jet mass
cross section Hoang:2019ceu . By making the correspondence
$C_{1}^{\kappa}\simeq M_{1}^{\kappa}$ and $C_{2}^{\kappa}\simeq
M_{-1}^{\kappa\varocircle}$, we use our NLL′ calculation of the moments to
determine these Wilson coefficients with higher precision than previously
available. This leads in turn to a more accurate description of the leading
nonperturbative corrections to the groomed jet mass.

We determined the regions of validity of three different EFT regimes based on
their value of the groomed jet radius $R_{g}$, combined them using profile
scales, and estimated the theoretical uncertainties on the double differential
cross section and its angular moments via scale variation. The NLL′ improved
computation involved, in certain regimes, fixed order ingredients at ${\cal
O}(\alpha_{s}^{2})$ accuracy. We parameterized unknown 2-loop terms that are
not predicted by the renormalization group equations, and included them in our
estimate of perturbative uncertainty. We also compared our predictions with
parton shower Monte Carlo simulations and found that the two agree within
uncertainty in the region of validity of the formalism. Although we did not
resum the nonglobal logarithms in our factorization formulae, we showed that
they cancel in $C_{2}^{q}$. We computed the leading effects on $C_{1}^{q}$
finding them to be not larger than other missing ${\cal O}(\alpha_{s}^{2})$
effects included in our uncertainty estimate.

While we have focused on the application of simultaneous measurement of the
groomed jet radius and the jet mass, we expect the EFTs explored in this work
to have a much wider applicability, and our work to pave the way to analogous
computations for other groomed jet observables. Single differential jet based
observables, such as the jet mass and jet angularities, have exciting
prospects for measurements of standard model parameters such as the strong
coupling constant $\alpha_{s}$ and the top mass $m_{t}$, with grooming playing
a key role in reducing contamination in jets. Having a double differential
cross section prediction at hand offers complementary ways of extracting these
parameters.

The double differential cross section includes a resummation of large
logarithms of the groomed jet radius, and this resummation remains relevant
even when $m_{J}$ is taken to be in the ungroomed region. The study of power
corrections for the multi-differential cross section in this region are also
rather interesting. In the future work, we envision extending our double
differential predictions to the ungroomed region, to achieve a more complete
description of the double differential cross section.

One of the greatest advantages of jet grooming is the decrease to the impact
of nonperturbative corrections compared to those in ungroomed jets. Grooming
also significantly reduces contamination from multi-parton interactions, and
is thus an invaluable tool for carrying out phenomenology with jets, and for
analyzing heavy ion collisions.

In hadronic collisions, measuring observables on groomed jets also allows for
much cleaner access to the initial state hadron structure, which is one of the
principal goals of nuclear physics. By bringing under control the small
nonperturbative effects from final state radiation, we are one step closer to
the accurate extraction of hadron structure functions such as the transverse
momentum dependent parton distributions. Additional natural areas for future
explorations of multi-differential groomed jets include for transverse
momentum observables and for the phenomenology of jets in heavy ion
collisions.

###### Acknowledgements.

This work was supported in part by FWF Austrian Science Fund under the Project
No. P28535-N27 and by the Office of Nuclear Physics of the U.S. Department of
Energy under the Grant No. DE-SCD011090. A.P. is a member of the Lancaster-
Manchester-Sheffield Consortium for Fundamental Physics, which is supported by
the UK Science and Technology Facilities Council (STFC) under grant number
ST/T001038/1. I.S. was also supported by the Simons Foundation through the
Investigator grant 327942. L.Z. was supported in part by the ERC grant ERC-
STG-2015-677323 and in part by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) – Research Unit FOR 2926, grant number 409651613.
A.P. also thanks FWF Austrian Science Fund under the Doctoral Program
Particles and Interactions No.W1252-N27 for partial support. L.Z. would like
to thank the Center for Theoretical Physics of MIT for hospitality during the
early stage of this project. Figures are made using the Mathematica package
PLHot plhot .

##  Appendix A Formulae

###  A.1 Laplace transforms and RG Evolution

Here we present the details of the resummation of large logarithms via RG
evolution. The earliest papers exploring evolution equations, with the type of
momentum space convolutions encountered here, were Refs. Korchemsky:1993uz ;
Balzereit:1998yf , and further advances to the approach were made in Ref.
Neubert:2005nt  which presented formulas valid to all orders in perturbation
theory. These rely on the equations being multiplicative in Laplace or Fourier
space. Solutions to these equations can also be obtained functionally to all
orders, both directly in momentum space by formulas combining plus functions
Fleming:2007xt ; Ligeti:2008ac , or with derivatives to compactly transform
logarithms from Laplace to momentum space Becher:2006nr ; Becher:2006mr . Ref.
Almeida_2014  provides a review and we follow the notation used there.

We consider a generic $\overline{\rm MS}$ renormalized SCET matrix element
${\cal F}(q,\mu)$, with $j_{\cal F}$ the mass dimensions of $q$. When ${\cal
F}(q,\mu)$ enters factorization theorems in a Laplace convolution over the
momentum space variable $q$, its renormalization group equation (RGE) reads

$\displaystyle\mu\frac{d}{d\mu}{\cal F}(q,\mu)=\int dq^{\prime}\,\gamma_{\cal
F}(q-q^{\prime},\mu)\,{\cal F}(q,\mu)\,,$ (196)

where $\gamma_{\cal F}$ is the anomalous dimension in momentum space. In SCET
such functions are ubiquitous and result from the fact that UV renormalization
of soft and collinear matrix elements accounts for double poles in the
dimensional regulator, that are in turn reminiscent of the double IR poles of
the corresponding QCD matrix elements. The presence of double poles in
counterterms, and the convolution nature of the factorization theorem, lead to
nontrivial running such as the one shown in Eq. (196). To solve of Eq. (196),
we consider the Laplace transform of ${\cal F}$ given by

$\displaystyle\widetilde{\cal F}(x,\mu)\equiv\int_{0}^{\infty}dq\,e^{-qx}{\cal
F}(q,\mu)\,,$ (197)

such that $x$ has mass dimensions $-j_{\cal F}$. The formula for inverse
Laplace transform reads

$\displaystyle{\cal F}(q,\mu)=\int_{c-i\infty}^{c+i\infty}\frac{dx}{2\pi
i}\,e^{qx}\widetilde{\cal F}(x,\mu)\,,$ (198)

where the real number $c$ determines a vertical contour in the complex plane
that leaves any pole of $\widetilde{\cal F}$ to its left. In position space,
the RGE is simply multiplicative:

$\displaystyle\mu\frac{d}{d\mu}\widetilde{\cal
F}(x,\mu)=\widetilde{\gamma}_{\cal F}(x,\mu)\,\widetilde{{\cal
F}}(x,\mu)\equiv\Big{[}\Gamma_{\cal
F}[\alpha_{s}]\,\textrm{ln}(e^{\gamma_{E}}x\mu^{j_{\cal F}})+\gamma_{\cal
F}[\alpha_{s}]\Big{]}\,\widetilde{{\cal F}}(x,\mu)\,,$ (199)

where the anomalous dimension999Had there been higher powers of the logarithm
in Eq. (199) the factorization into soft and collinear matrix elements would
not have been possible. This has been shown to be true explicitly up to four
loops Becher:2019avh . $\widetilde{\gamma}_{\cal F}(x,\mu)$ consists of a cusp
piece $\Gamma_{\cal F}[\alpha_{s}]$, responsible for resumming double
logarithms, and a non-cusp part $\gamma_{\cal F}[\alpha_{s}]$. The cusp part,
$\Gamma_{\cal F}[\alpha_{s}]$, is proportional to the cusp-anomalous dimension
such that $\Gamma_{\cal F}[\alpha_{s}]=\kappa_{\cal F}C_{\kappa}\Gamma^{\rm
cusp}[\alpha_{s}]$, where $C_{\kappa}=C_{F}$ or $C_{A}$. Equation (199) has a
formal solution

$\displaystyle{\cal F}(q,\mu)=\int\\!dq^{\prime}\,U_{\cal
F}(q-q^{\prime},\mu,\mu_{0})\,{\cal
F}(q^{\prime},\mu_{0})=\int_{c-i\infty}^{c+i\infty}\\!\frac{dx}{2\pi
i}\,e^{qx}\,\widetilde{U}_{\cal F}(x,\mu,\mu_{0})\,\widetilde{\cal
F}(x,\mu_{0})\,,$ (200)

in terms of the evolution operator

$\displaystyle\widetilde{U}_{\cal F}(x,\mu,\mu_{0})=e^{K_{\cal
F}(\mu,\mu_{0})}\,(e^{\gamma_{E}}x\mu_{0}^{j_{\cal{F}}})^{\omega_{\cal
F}(\mu,\mu_{0})}\,.$ (201)

The evolution kernels $K_{\cal F}$ and $\omega_{\cal F}$ are obtained from
integrals of the anomalous dimensions,

$\displaystyle K_{\cal F}(\mu,\mu_{0})$ $\displaystyle=K^{\Gamma}_{\cal
F}(\mu,\mu_{0})+K^{\gamma}_{\cal F}(\mu,\mu_{0})$ (202) $\displaystyle
K^{\Gamma}_{\cal F}(\mu,\mu_{0})$
$\displaystyle=j_{\cal{F}}\int_{\alpha_{s}(\mu_{0})}^{\alpha_{s}(\mu)}\frac{d\alpha}{\beta[\alpha]}\Gamma_{\cal
F}[\alpha]\int_{\alpha_{s}(\mu_{0})}^{\alpha}\frac{d\alpha^{\prime}}{\beta[\alpha^{\prime}]}$
$\displaystyle K^{\gamma}_{\cal F}(\mu,\mu_{0})$
$\displaystyle=\int_{\alpha_{s}(\mu_{0})}^{\alpha_{s}(\mu)}\frac{d\alpha}{\beta[\alpha]}\gamma_{\cal
F}[\alpha]\,,$ $\displaystyle\omega_{\cal F}(\mu,\mu_{0})$
$\displaystyle=\int_{\alpha_{s}(\mu_{0})}^{\alpha_{s}(\mu)}\frac{d\alpha}{\beta[\alpha]}\Gamma_{\cal
F}[\alpha]\,,$

involving the QCD beta function $\beta[\alpha_{s}]$, whose perturbative
expansion we give in Eq. (208). We also state the results for functions
$F(\mu,\mu_{F})$ that enter the factorized cross section as multiplicative
factors (and not through convolutions) with $\mu_{F}$ being the energy scale
specific to $F$. Here the RGE is directly given by

$\displaystyle\mu\frac{\textrm{d}}{\textrm{d}\mu}F(\mu,\mu_{F})=\gamma_{F}(\mu,\mu_{F})F(\mu,\mu_{F})=\Big{[}\Gamma_{F}[\alpha_{s}]\log\Big{(}\frac{\mu}{\mu_{F}}\Big{)}+\gamma_{F}(\alpha_{s})\Big{]}F(\mu,\mu_{0})\,,$
(203)

with the solution for evolution from scale $\mu_{0}$ to $\mu$ being

$\displaystyle F(\mu,\mu_{F})$
$\displaystyle=U_{F}(\mu,\mu_{0};\mu_{F})F(\mu_{0},\mu_{F})\equiv
e^{K_{F}(\mu,\mu_{0})}\,\Big{(}\frac{\mu_{0}}{\mu_{F}}\Big{)}^{\omega_{F}(\mu,\mu_{F})}F(\mu_{0},\mu_{F})\,,$
(204)

where the evolution kernels $K_{F}$ and $\omega_{F}$ are again given by Eq.
(202). The kernels in Eq. (202) depend on the specific function $\cal{F}$ (or
$F$) only through the anomalous dimensions: we can thus rewrite them in terms
of two universal kernels

$\displaystyle K^{\Gamma}_{\cal F}(\mu,\mu_{0})=j_{\cal{F}}K(\Gamma_{\cal
F}[\alpha_{s}],\mu,\mu_{0})\,,\quad K^{\gamma}_{\cal
F}(\mu,\mu_{0})=\eta(\gamma_{\cal
F}[\alpha_{s}],\mu,\mu_{0})\,,\quad\omega_{\cal
F}(\mu,\mu_{0})=\eta(\Gamma_{\cal F}[\alpha_{s}],\mu,\mu_{0})\,,$ (205)

whose explicit NLL expressions are

$\displaystyle\eta(\Gamma,\mu,\mu_{0})$
$\displaystyle=-\frac{\Gamma_{0}}{2\beta_{0}}\bigg{[}\textrm{ln}r+\frac{\alpha_{s}(\mu_{0})}{4\pi}\bigg{(}\frac{\Gamma_{1}}{\Gamma_{0}}-\frac{\beta_{1}}{\beta_{0}}\bigg{)}(r-1)\bigg{]}\,,$
(206) $\displaystyle K(\Gamma,\mu,\mu_{0})$
$\displaystyle=\frac{\Gamma_{0}}{4\beta_{0}^{2}}\Bigg{[}\frac{4\pi}{\alpha_{s}(\mu_{0})}\Big{(}\textrm{ln}r+\frac{1}{r}-1\Big{)}+\bigg{(}\frac{\Gamma_{1}}{\Gamma_{0}}-\frac{\beta_{1}}{\beta_{0}}\bigg{)}(r-1-\textrm{ln}r)-\frac{\beta_{1}}{2\beta_{0}}\textrm{ln}^{2}r\Bigg{]}\,,$
(207)

where $r=\alpha_{s}(\mu)/\alpha_{s}(\mu_{0})$. Here the perturbative expansion
of a generic anomalous dimension $\Gamma[\alpha_{s}]$ and $\beta[\alpha_{s}]$
is written as

$\displaystyle\Gamma[\alpha_{s}]=\sum_{n=0}^{\infty}\Gamma_{n}\Big{(}\frac{\alpha_{s}}{4\pi}\Big{)}^{n+1}\,,\qquad\beta[\alpha_{s}]=-2\,\alpha_{s}\,\sum_{n=0}^{\infty}\beta_{n}\Big{(}\frac{\alpha_{s}}{4\pi}\Big{)}^{n+1}\,$
(208)

and the terms relevant for NLL resummation read

$\displaystyle\Gamma_{0}^{\rm cusp}$ $\displaystyle=4\,,\qquad\Gamma^{\rm
cusp}_{1}=4C_{A}\bigg{(}\frac{67}{9}-\frac{\pi^{2}}{3}\bigg{)}-\frac{80n_{f}T_{R}}{9}\,,$
(209) $\displaystyle\beta_{0}$
$\displaystyle=\frac{11}{3}C_{A}-\frac{4}{3}T_{R}n_{f}\,,\qquad\beta_{1}=\frac{34}{3}C_{A}^{2}-4T_{R}n_{f}\bigg{(}C_{F}+\frac{5}{3}C_{A}\bigg{)}\,.$
(210)

###  A.2 One-loop anomalous dimensions

We state here one-loop results for the anomalous dimensions of the various
functions appearing in Eq. (71) with the cusp and noncusp pieces defined as in
Eqs. (199) and (203). The RGEs of these functions are given by

$\displaystyle\mu\frac{d}{d\mu}\log N^{\kappa}(Q,R,\mu)$
$\displaystyle=\Gamma_{N^{\kappa}}[\alpha_{s}]\log\Big{(}\frac{\mu}{Q\tan\frac{R}{2}}\Big{)}+\gamma_{N^{\kappa}}[\alpha_{s}]\,,$
(211) $\displaystyle\mu\frac{d}{d\mu}\log{\cal
C}^{\kappa}\bigg{(}\frac{m_{J}^{2}}{Q^{2}R_{g}^{2}},QR_{g},\mu\bigg{)}$
$\displaystyle=\Gamma_{{\cal
C}^{\kappa}}[\alpha_{s}]\log\Big{(}\frac{\mu}{Q\frac{R_{g}}{2}}\Big{)}+\gamma_{{\cal
C}^{\kappa}}[\alpha_{s}]\,,$
$\displaystyle\mu\frac{d}{d\mu}\log\tilde{J}_{\kappa}(x,\mu)$
$\displaystyle=\Gamma_{J_{\kappa}}[\alpha_{s}]\log\big{(}e^{\gamma_{E}}x\mu^{2}\big{)}+\gamma_{J_{\kappa}}[\alpha_{s}]\,,$
$\displaystyle\mu\frac{d}{d\mu}\log\tilde{S}^{\kappa}_{c_{m}}(y,\mu)$
$\displaystyle=\Gamma_{S^{\kappa}_{c_{m}}}[\alpha_{s}]\log\big{(}e^{\gamma_{E}}y\mu\big{)}+\gamma_{S^{\kappa}_{c_{m}}}[\alpha_{s}]\,,$
$\displaystyle\mu\frac{d}{d\mu}\log\tilde{S}^{\kappa}_{c}(z,\beta,\mu)$
$\displaystyle=\Gamma_{S^{\kappa}_{c}}[\alpha_{s}]\log\big{(}e^{\gamma_{E}}z\mu^{\frac{2+\beta}{1+\beta}}\big{)}+\gamma_{S^{\kappa}_{c}}(\beta)[\alpha_{s}]\,,$
$\displaystyle\mu\frac{d}{d\mu}\log S_{G}^{\kappa}\Big{(}Q_{\rm
cut}\tan^{1+\beta}\frac{R}{2},R,\beta,\mu\Big{)}$
$\displaystyle=\frac{1}{1+\beta}\Gamma_{S^{\kappa}_{G}}[\alpha_{s}]\log\Big{(}\frac{\mu}{Q_{\rm
cut}\tan^{1+\beta}\frac{R}{2}}\Big{)}+\gamma_{S^{\kappa}_{G}}(\beta)[\alpha_{s}]\,,$
$\displaystyle\mu\frac{d}{d\mu}\log
S^{\kappa}_{c_{g}}\Big{(}\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta,\mu\Big{)}$
$\displaystyle=\frac{1}{1+\beta}\Gamma_{S^{\kappa}_{c_{g}}}[\alpha_{s}]\log\bigg{[}\bigg{(}\frac{\mu}{Q_{\rm
cut}\big{(}\frac{R_{g}}{2}\big{)}^{1+\beta}}\bigg{)}\bigg{]}+\gamma_{S^{\kappa}_{c_{g}}}(\beta)[\alpha_{s}]\,.$

and the inverse Laplace transforms for $J_{\kappa}$, $S_{c_{m}}^{\kappa}$ and
$S_{c}^{\kappa}$ are defined as follows,

$\displaystyle J_{\kappa}(m_{J}^{2},\mu)$
$\displaystyle=\int_{c-i\infty}^{c+i\infty}\frac{dx}{2\pi
i}\>e^{x\,m_{J}^{2}}\tilde{J}_{\kappa}(x,\mu)\,,$ (212) $\displaystyle
S^{\kappa}_{c_{m}}\Big{(}\frac{\ell^{+}}{R_{g}/2},\mu\Big{)}$
$\displaystyle=\int_{c-i\infty}^{c+i\infty}\frac{dy}{2\pi
i}\>e^{y\,\frac{\ell^{+}}{R_{g}/2}}\>\tilde{S}_{c_{m}}^{\kappa}(y,\mu)\,,$
$\displaystyle S_{c}^{\kappa}\big{(}\ell^{+}Q_{\rm
cut}^{\frac{1}{1+\beta}},\mu\big{)}$
$\displaystyle=\int_{c-i\infty}^{c+i\infty}\frac{dz}{2\pi
i}\>e^{z\,\ell^{+}Q_{\rm
cut}^{\frac{1}{1+\beta}}}\>\tilde{S}_{c}^{\kappa}(z,\mu)\,,$

such that the mass dimensions of $x$, $y$, and $z$ are respectively

$\displaystyle-j_{J_{\kappa}}=-2\,,\qquad-j_{S^{\kappa}_{c_{m}}}=-1\,,\qquad-
j_{S_{c}^{\kappa}}=-(2+\beta)/(1+\beta)\,.$ (213)

The cusp pieces in Eq. (211) are multiples of the universal cusp anomalous
dimension $\Gamma^{\rm cusp}[\alpha_{s}]$ and are given by

$\displaystyle\Gamma_{N^{\kappa}}[\alpha_{s}]$
$\displaystyle=-\Gamma_{S_{G}^{\kappa}}[\alpha_{s}]=-\Gamma_{J_{\kappa}}[\alpha_{s}]=-\Gamma_{{\cal
C}^{\kappa}}[\alpha_{s}]=\Gamma_{S^{\kappa}_{c_{m}}}[\alpha_{s}]=\Gamma_{S_{c}^{\kappa}}[\alpha_{s}]=\Gamma_{S_{c_{g}}^{\kappa}}[\alpha_{s}]=-2C_{\kappa}\,\Gamma^{\rm
cusp}[\alpha_{s}]\,,$ (214)

and the one-loop noncusp pieces are

$\displaystyle\gamma^{N^{q}}_{0}$
$\displaystyle=-\gamma^{J_{q}}_{0}=-\gamma^{{\cal
C}^{q}}_{0}=-6C_{F}\,,\quad\gamma^{N^{g}}_{0}=-\gamma^{J_{g}}_{0}=-\gamma^{{\cal
C}^{g}}_{0}=-2\beta_{0}\,,\quad\gamma^{S^{\kappa}_{c_{m}}}_{0}=\gamma^{S_{c}^{\kappa}}_{0}=\gamma^{S_{G}^{\kappa}}_{0}=\gamma^{S_{c_{g}}^{\kappa}}_{0}=0\,.$
(215)

We can now check the RG consistency relation for the factorized cross section
for doubly differential $m_{J}^{2}$ and $R_{g}$ measurements. Written in
position space, the dijet cross section for jet masses is given in the
intermediate regime by Eq. (4.2). By differentiating with respect to the
renormalization scale, it is immediate to show that the sum of the cusp terms
must vanish,

$\displaystyle\Gamma_{N^{\kappa}}[\alpha_{s}]\log\Big{(}\frac{\mu}{Q\tan\frac{R}{2}}\Big{)}+\frac{1}{2}\frac{1}{1+\beta}\Gamma_{S^{\kappa}_{G}}[\alpha_{s}]\log\Big{(}\frac{\mu}{Q_{\rm
cut}\tan^{1+\beta}\frac{R}{2}}\Big{)}+\Gamma_{J_{\kappa}}[\alpha_{s}]\log\big{(}e^{\gamma_{E}}x\mu^{2}\big{)}$
(216)
$\displaystyle\qquad+\Gamma_{S^{\kappa}_{c_{g}}}[\alpha_{s}]\log\bigg{[}\Big{(}\frac{\mu}{Q_{\rm
cut}}\Big{)}^{\frac{1}{1+\beta}}\frac{2}{R_{g}}\bigg{]}+\Gamma_{S^{\kappa}_{c_{m}}}[\alpha_{s}]\log\Big{(}e^{\gamma_{E}}x\frac{QR_{g_{L}}}{2}\mu\Big{)}=0\,,$

and that the noncusp pieces must also add to zero from RG consistency,

$\displaystyle\gamma_{N^{\kappa}}[\alpha_{s}]+\gamma_{J_{\kappa}}[\alpha_{s}]+\gamma_{S^{\kappa}_{G}}[\alpha_{s}]+\gamma_{S^{\kappa}_{c_{m}}}[\alpha_{s}]+\gamma_{S^{\kappa}_{c_{g}}}[\alpha_{s}]=0\,,$
(217)

as can be easily verified at one loop from Eq. (215). This puts nontrivial
constraint on the $\beta$ dependence of the global-soft and collinear-soft non
cusp anomalous dimensions. As a consequence of the refactorization in Eq. (50)
in the regime of large groomed jet radius, the relation

$\displaystyle\Gamma_{S^{\kappa}_{c_{g}}}[\alpha_{s}]\log\bigg{(}\Big{(}\frac{\mu}{Q_{\rm
cut}}\Big{)}^{\frac{1}{1+\beta}}\frac{2}{R_{g}}\bigg{)}+\Gamma_{S^{\kappa}_{c_{m}}}[\alpha_{s}]\log\Big{(}e^{\gamma_{E}}x\frac{QR_{g}}{2}\mu\Big{)}=\Gamma_{S^{\kappa}_{c}}[\alpha_{s}]\log\big{(}e^{\gamma_{E}}xQQ_{\rm
cut}^{\frac{-1}{1+\beta}}\mu^{\frac{2+\beta}{1+\beta}}\big{)}$ (218)

is also enforced by RG consistency.

##  Appendix B One-loop results

In this appendix we collect the one-loop results for the ingredients appearing
in the double differential cross section. We start from the functions that
enter the factorized expression for the cross section cumulant in $R_{g}$ and
describe their evolution, then we move on to describing the ingredients of the
boundary cross section that enters the definition of $C_{2}^{q}$.

###  B.1 Results for the cumulant distribution

####  B.1.1 Global-soft function

The global soft function accounts for the soft radiation which is groomed
away, namely lies within the original jet but fails the soft drop test. The
one loop correction for $e^{+}e^{-}$ collisions requires computing

$\displaystyle S_{G}^{\kappa[1],\rm bare}\bigl{(}Q_{\rm
cut},R,\beta,\mu\bigr{)}=\frac{\alpha_{s}C_{\kappa}}{\pi}\frac{(\mu^{2}e^{\gamma_{E}})^{\epsilon}}{\Gamma(1-\epsilon)}\int\frac{dp^{+}dp^{-}}{(p^{+}p^{-})^{1+\epsilon}}\,\Theta_{\rm
sd}\,\overline{\Theta}_{R}\,,$ (219)

where $C_{q}=C_{F}$ and $C_{g}=C_{A}$, the condition for failing soft drop is

$\displaystyle\Theta_{\rm sd}$ $\displaystyle\equiv\Theta\bigg{(}z_{\rm
cut}\Big{(}2\sin\frac{\theta_{p}}{2}\Big{)}^{\beta}-\frac{p^{+}+p^{-}}{Q}\bigg{)}\,,$
(220)

with $\sin^{2}(\theta_{p}/2)=p^{+}/(p^{+}+p^{-})$, and $\overline{\Theta}_{R}$
constrains radiation within a jet of radius $R$,

$\displaystyle\overline{\Theta}_{R}\equiv\Theta\bigg{(}\tan\frac{R}{2}-\sqrt{\frac{p^{+}}{p^{-}}}\bigg{)}\,.$
(221)

Solving the integration in Eq. (219) we get

$\displaystyle S_{G}^{\kappa}\Big{(}Q_{\rm
cut}\tan^{1+\beta}\frac{R}{2},R,\beta,\mu\Big{)}=1+\frac{\alpha_{s}(\mu)C_{\kappa}}{\pi}$
$\displaystyle\bigg{[}\frac{1}{(1+\beta)}\log^{2}\Big{(}\frac{\mu}{Q_{\rm
cut}\tan^{1+\beta}\frac{R}{2}}\Big{)}-\frac{\pi^{2}}{24}\Big{(}\frac{1}{1+\beta}\Big{)}$
(222) $\displaystyle-\frac{(2+\beta)}{4}\Big{(}2{\rm
Li}_{2}\Big{[}\sin^{2}\Big{(}\frac{R}{2}\Big{)}\Big{]}+\log^{2}\Big{[}\cos^{2}\frac{R}{2}\Big{]}\Big{)}\bigg{]}\,.$

For $pp$ collisions with small jet radius, $R\ll 1$, one retrieves the
familiar expression

$\displaystyle S_{G}^{\kappa}\bigl{(}p_{T}z_{\rm
cut}^{\prime}R^{1+\beta},\beta,\mu\bigr{)}=1+\frac{\alpha_{s}(\mu)C_{\kappa}}{\pi}\frac{1}{1+\beta}$
$\displaystyle\bigg{[}\log^{2}\Big{(}\frac{\mu}{p_{T}z_{\rm
cut}^{\prime}R^{1+\beta}}\Big{)}-\frac{\pi^{2}}{24}\bigg{]}\,.$ (223)

####  B.1.2 Jet function

The one-loop result for the jet function is well known Bauer:2003pi ;
Bosch:2004th  and given by

$\displaystyle
J_{\kappa}(m_{J}^{2},\mu)\equiv\delta(m_{J}^{2})+\frac{\alpha_{s}(\mu)}{\pi}\Bigg{[}\frac{C_{\kappa}}{\mu^{2}}{\cal
L}_{1}\Big{(}\frac{m_{J}^{2}}{\mu^{2}}\Big{)}-\frac{\gamma_{0}^{J_{\kappa}}}{8}\frac{1}{\mu^{2}}{\cal
L}_{0}\Big{(}\frac{m_{J}^{2}}{\mu^{2}}\Big{)}-\delta(m_{J}^{2})\Big{(}C_{\kappa}\frac{\pi^{2}}{4}-c_{1}^{\kappa}\Big{)}\Bigg{]}\,.$
(224)

The Laplace transform is defined as (see Eq. (212))

$\displaystyle\tilde{J}_{\kappa}(x,\mu)$
$\displaystyle=\int_{0}^{\infty}dm_{J}^{2}\>e^{-x\,m_{J}^{2}}J_{\kappa}(m_{J}^{2},\mu)\,,$
(225)

such that Eq. (224) gives

$\displaystyle\tilde{J}_{\kappa}\bigl{(}x,\mu\bigr{)}$ $\displaystyle\equiv
1+\frac{\alpha_{s}(\mu)}{\pi}\bigg{[}\frac{C_{\kappa}}{2}\log^{2}\big{(}e^{\gamma_{E}}x\mu^{2}\big{)}+\frac{\gamma_{0}^{J_{\kappa}}}{8}\log\big{(}e^{\gamma_{E}}x\mu^{2}\big{)}-C_{\kappa}\frac{\pi^{2}}{6}+c_{1}^{\kappa}\bigg{]}\,,$
(226)

where the one-loop non-cusp anomalous dimension for the quark and gluon jet
functions are

$\displaystyle\gamma_{0}^{J_{q}}$
$\displaystyle=6C_{F}\,,\qquad\gamma_{0}^{J_{g}}=2\beta_{0}\,,$ (227)

the one-loop constant terms being

$\displaystyle c_{1}^{q}=\frac{7}{4}\,,\qquad c_{1}^{g}=\frac{\Gamma^{\rm
cusp}_{1}}{8}=\frac{C_{A}}{2}\bigg{(}\frac{67}{9}-\frac{\pi^{2}}{3}\bigg{)}-\frac{10n_{f}T_{R}}{9}\,.$
(228)

Equivalently, following to the alternative notation in Eq. (4.2), we write Eq.
(226) as

$\displaystyle\tilde{J}_{\kappa}\big{[}\ell_{J},\alpha_{s}(\mu_{J})\big{]}$
$\displaystyle=1+\frac{\alpha_{s}(\mu_{J})}{\pi}\bigg{[}\frac{C_{\kappa}}{2}\ell_{J}^{2}+\frac{\gamma_{0}^{J_{\kappa}}}{8}\ell_{J}-C_{\kappa}\frac{\pi^{2}}{6}+c_{1}^{\kappa}\bigg{]}\,.$
(229)

####  B.1.3 Collinear-soft functions at intermediate $R_{g}$

The CSg function describes emissions that live at the groomed jet boundary and
pass soft drop. At one loop, it requires computing

$\displaystyle S_{c_{g}}^{\kappa[1],\rm bare}\Big{(}\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta,\mu\Big{)}=\frac{\alpha_{s}C_{\kappa}}{\pi}\frac{(\mu^{2}e^{\gamma_{E}})^{\epsilon}}{\Gamma(1-\epsilon)}\int\frac{dp^{+}dp^{-}}{(p^{+}p^{-})^{1+\epsilon}}\,\overline{\,}\Theta\bigg{(}\frac{R_{g}}{2}-\sqrt{\frac{p_{+}}{p_{-}}}\bigg{)}\overline{\Theta}_{\rm
sd}\,,$ (230)

with

$\displaystyle\overline{\Theta}_{\rm sd}\,=\,1-\Theta_{\rm
sd}\,=\,\Theta\bigg{(}\frac{p^{-}}{Q}-z_{\rm
cut}\Big{(}\frac{p^{+}}{p^{-}}\Big{)}^{\beta/2}\bigg{)}\,,$ (231)

where the last equality uses that for collinear-soft radiation $p^{+}\ll
p^{-}$. This leads to the $e^{+}e^{-}$ equivalent of the one-loop result of
Kang:2019prh ,

$\displaystyle S_{c_{g}}^{\kappa}\Big{(}\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta,\mu\Big{)}$
$\displaystyle=1-\frac{\alpha_{s}C_{\kappa}}{\pi}\frac{1}{1+\beta}\bigg{[}\log^{2}\bigg{(}\frac{\mu}{Q_{\rm
cut}(R_{g}/2)^{(1+\beta)}}\bigg{)}-\frac{\pi^{2}}{24}\bigg{]}\,,$ (232)

The CSm function describes emissions at the groomed jet boundary that set the
jet mass (therefore, they automatically pass soft drop in this regime). From
the bare one-loop expression

$\displaystyle S_{c_{m}}^{\kappa[1],\rm bare}\Big{(}\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta,\mu\Big{)}=\frac{\alpha_{s}C_{\kappa}}{\pi}\frac{(\mu^{2}e^{\gamma_{E}})^{\epsilon}}{\Gamma(1-\epsilon)}\int\frac{dp^{+}dp^{-}}{(p^{+}p^{-})^{1+\epsilon}}\,\overline{\,}\Theta\bigg{(}\frac{R_{g}}{2}-\sqrt{\frac{p^{+}}{p^{-}}}\bigg{)}\delta\Big{(}\frac{\ell^{+}}{p^{+}}p^{-}-1\Big{)}\,,$
(233)

we get

$\displaystyle S_{c_{m}}^{\kappa}\Big{(}\frac{2\ell^{+}}{R_{g}},\mu\Big{)}$
$\displaystyle=\delta\Big{(}\frac{2\ell^{+}}{R_{g}}\Big{)}+\frac{\alpha_{s}C_{\kappa}}{\pi}\bigg{[}\frac{-2}{\mu}{\cal
L}_{1}\bigg{(}\frac{1}{\mu}\frac{2\ell^{+}}{R_{g}}\bigg{)}+\frac{\pi^{2}}{24}\delta\Big{(}\frac{2\ell^{+}}{R_{g}}\Big{)}\bigg{]}\,,$
(234)

which is the equivalent to the result for the soft function for ungroomed jets
of radius $R\rightarrow R_{g}$ computed in Ref. Ellis:2010rwa  (up to a
different choice in normalization). Following Eq. (212) the Laplace transform
of $S_{c_{m}}$ is defined as

$\displaystyle\tilde{S}_{c_{m}}^{\kappa}\bigl{(}u,\mu\bigr{)}=\int\frac{d\ell^{+}}{R_{g}/2}e^{-u\frac{\ell^{+}}{R_{g}/2}}S_{c_{m}}^{\kappa}\Bigl{[}\frac{\ell^{+}}{R_{g}/2},\mu\Bigr{]}\,$
$\displaystyle(\text{$e^{+}e^{-}$ case})\,,$ (235)
$\displaystyle\tilde{S}_{c_{m}}^{\kappa}\bigl{(}u,\mu\bigr{)}=\int\frac{dr^{+}}{R_{g}}e^{-u\frac{r^{+}}{R_{g}}}S_{c_{m}}^{\kappa}\Bigl{[}\frac{r^{+}}{R_{g}},\mu\Bigr{]}\,$
$\displaystyle(\text{$pp$ case})\,,$

such that at one loop (for either scenario)

$\displaystyle\tilde{S}_{c_{m}}^{\kappa}\bigl{(}u,\mu\bigr{)}$
$\displaystyle=1+\frac{\alpha_{s}C_{\kappa}}{\pi}\bigg{[}-\log^{2}\big{(}ue^{\gamma_{E}}\mu\big{)}-\frac{\pi^{2}}{8}\bigg{]}\,.$
(236)

In the notation of Eq. (4.2), useful in our final expressions for the resummed
cross sections, we get

$\displaystyle\tilde{S}_{c_{m}}^{\kappa}\bigl{[}L_{cs_{m}},\alpha_{s}(\mu_{cs_{m}})\bigr{]}$
$\displaystyle=1+\frac{\alpha_{s}(\mu_{cs_{m}})C_{\kappa}}{\pi}\bigg{[}-L_{cs_{m}}^{2}-\frac{\pi^{2}}{8}\bigg{]}\,.$
(237)

####  B.1.4 Collinear-soft functions at large $R_{g}$

In the large $R_{g}$ case, the collinear-soft function knows about both the
mass measurement and the soft-drop condition. At one loop, we need

$\displaystyle S_{c}^{\kappa[1],{\rm bare}}\Big{[}\ell^{+}Q_{\rm
cut}^{\frac{1}{1+\beta}},\frac{R_{g}}{2}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta,\mu\Big{]}=\frac{\alpha_{s}C_{\kappa}}{\pi}\frac{(\mu^{2}e^{\gamma_{E}})^{\epsilon}}{\Gamma(1-\epsilon)}\\!\int\\!\\!\frac{dp^{+}dp^{-}}{(p^{+}p^{-})^{1+\epsilon}}\,\overline{\,}\Theta\bigg{(}\frac{R^{2}_{g}}{4}-\frac{p^{+}}{p^{-}}\bigg{)}\delta\Big{(}\frac{\ell^{+}p^{-}}{p^{+}}-1\Big{)}\overline{\Theta}_{\rm
sd}\,,$ (238)

yielding the result presented in Eq. (34). Given our multiplicative treatment
of this function, see Eq. (78), we need to separate the terms that contribute
to RG evolution from the ones that depend on $R_{g}$. The $R_{g}$ independent
function coincides with the collinear-soft function for single differential
jet mass distribution. Its one-loop result in the $\overline{\rm MS}$ scheme
is

$\displaystyle S_{c}^{\kappa}\Bigl{[}\ell^{+}\,Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta,\mu\Bigr{]}$
$\displaystyle=\delta\Bigl{(}\ell^{+}Q_{\rm cut}^{\frac{1}{1+\beta}}\Bigr{)}$
(239)
$\displaystyle+\frac{C_{\kappa}\alpha_{s}(\mu)}{\pi}\Biggl{\\{}\\!\frac{-2(1+\beta)}{(2\\!+\\!\beta)\mu^{\frac{2+\beta}{1+\beta}}}\,{\cal
L}_{1}\\!\biggl{[}\frac{\ell^{+}Q_{\rm
cut}^{\frac{1}{1+\beta}}}{\mu^{\frac{2+\beta}{1+\beta}}}\biggr{]}+\frac{\pi^{2}}{24}\frac{2\\!+\\!\beta}{1\\!+\\!\beta}\,\delta\Bigl{(}\ell^{+}Q_{\rm
cut}^{\frac{1}{1+\beta}}\Bigr{)}\\!\Biggr{\\}},$

where ${\cal L}_{1}(x)=\big{[}\Theta(x)\frac{\textrm{ln}x}{x}\big{]}_{+}$ is
the standard plus function which integrates to zero on $x\in[0,1]$. Following
the convention for Laplace transform in Eq. (82) we have at one loop

$\displaystyle\tilde{S}_{c}^{\kappa}\bigl{(}s,\beta,\mu\bigr{)}$