continuous grooming.

The Piranha grooming strategy has intricate geometric origins, but its goal is
simple: the optimal and continuous removal of contaminating low-energy
radiation. We hope that the simple strategies and examples of continuous
grooming discussed in this work may provide and inspire new tools for clear
communication between experimental results and theoretical predictions
regarding our microscopic universe.

###### Acknowledgements.

The authors would like to express their gratitude to Matthew LeBlanc, Jennifer
Roloff, and the ATLAS Jet Definitions subgroup for valuable discussions, and
to Nima Zardoshti for raising insightful questions during LHCP 2022. We would
also like to thank Pier Monni for helpful discussions regarding resummation,
and Sean Benevedes and Rikab Gambhir for engaging conversations and careful
readings of the manuscript. Finally, we extend our appreciation to an
anonymous referee whose thoughtful comments and feedback contributed to
improving the quality and clarity of this work and especially of Sec. 3. This
work was supported by the Office of Nuclear Physics of the U.S. Department of
Energy (DOE) under grant DE-SC-0011090, by the DOE Office of High Energy
Physics under grants DE-SC0012567 and DE-SC0019128, and by the National
Science Foundation under Cooperative Agreement PHY-2019786 (The NSF AI
Institute for Artificial Intelligence and Fundamental Interactions,
http://iaifi.org).

##  Appendix A Grooming in EMD Mode

In this appendix, we describe another route for developing Piranha groomers:
EMD mode. EMD-mode grooming produces a groomed event with a fixed EMD relative
to its ungroomed counterpart, and furnishes a complementary approach to
“$p_{T}$-mode” groomers. We provide a conceptual introduction to EMD-mode
grooming below, leaving an exploration of the phenomenological implications of
EMD mode to future work.

The Piranha groomers described above all subtract a fixed amount of transverse
momentum from the energy flow of an event. Let us call them $p_{T}$-mode
groomers. $p_{T}$-mode grooming gives us a great deal of control over the
amount of energy we remove from an event. When using Piranha groomers in
$p_{T}$ mode to remove additive contamination, however, this precise control
over $\Delta p_{T}$ leads to additional complications; since additive
contamination may add arbitrary amounts of contaminating soft radiation to an
event, removing a fixed amount of $p_{T}$ from the contaminated event may not
be the best strategy for reproducing the un-contaminated event. These
complications lead to important considerations when using Piranha for pileup
mitigation, as in Sec. 5.1 and App. B.2, and they are especially
disadvantageous when using Piranha to correct for the presence of the
underlying event, as in Sec. 5.2 and App. B.3.

Grooming in EMD mode is a complementary approach that allows Piranha groomers
to remove arbitrary amounts of soft radiation from an event. Piranha in EMD
mode therefore furnishes a conceptually interesting alternative for continuous
grooming.

First, let us briefly review how the $p_{T}$-mode algorithms we have presented
for P-AS, P-IVS, and P-RS all subtract a fixed amount of transverse momentum,
$\Delta p_{T}=\rho A_{\rm tot}$, or $\Delta p_{T}=z_{\rm cut}p_{T}$, from the
event under consideration.

  * •

P-AS subtracts the pre-specified $p_{T}$ all at once by finding the Apollonius
regions associated with the event, as in Sec. 2.3.

  * •

P-IVS subtracts the pre-specified $p_{T}$ by subtracting $p_{T}$ from each
particle in the event until it removes a particle, and then continues
recursively until the specified $p_{T}$ has been removed, as in Sec. 2.4.

  * •

P-RS subtracts the pre-specified $p_{T}$ by recursively assigning fractions of
the total $p_{T}$ to be removed to each branch of the jet tree until it
reaches the final-state particles of the jet, as in Sec. 2.5.

The approach for EMD-mode grooming is quite similar. In EMD mode, however,
each algorithm continues until a specific amount of EMD has been subtracted
from the event, often in an iterative procedure. As a concrete example, let us
construct the algorithm for P-IVS in EMD mode, or P-IVS(EMD):

  1. 1.

P-IVS(EMD) finds the Voronoi diagram for the event, as discussed in Sec. 2.4,
and indexes the steps of the algorithm by $n$, starting at $n=1$. We define
the energy flow of the algorithm after step $n$ is completed by
$\mathcal{E}_{n}$, where $\mathcal{E}_{0}$ is the energy flow of the original
event. We also define the EMD between $\mathcal{E}_{n-1}$ and
$\mathcal{E}_{n}$ as ${\rm EMD}_{n}$.

  2. 2.

P-IVS(EMD) attempts to modify the $p_{T}$ of every particle in the event in
the same way as P-IVS in $p_{T}$-mode:

$\displaystyle p_{T\,i}^{(n)}=p_{T\,i}^{(n-1)}-\rho^{(n)}A^{(n-1)}_{i},$ (31)

where the $p_{T\,i}^{(n-1)}$ and $A^{(n-1)}$ are the transverse
momenta/Voronoi areas after step $n-1$, and

$\displaystyle\rho^{(n)}=\min_{i}p^{(n)}_{T\,i}/A^{(n)}_{i},$ (32)

in analogy with Eq. (9). As in Sec. 2.4, $\rho^{(n)}$ describes the maximum
$p_{T}$ that may be subtracted from an event before one of the particles in
the event is groomed away entirely.

  3. 3.

P-IVS(EMD) next calculates ${\rm EMD}_{n}$ for this proposed groomed event,
and asks whether it has subtracted a total EMD of ${\rm EMD}_{\rm cut}$ from
the event. If $\sum_{k=1}^{n}{\rm EMD}_{n}<{\rm EMD}_{\rm cut}$, P-IVS(EMD)
has not yet subtracted the full ${\rm EMD}_{\rm cut}$ from the event. In this
case, we have more grooming to do: P-IVS(EMD) continues recursively, going
back to the first step of the algorithm.

  4. 4.

If $\sum_{k=1}^{n}{\rm EMD}_{k}>{\rm EMD}_{\rm cut}$, P-IVS(EMD) has
subtracted too much EMD from the event. In this case, P-IVS(EMD) revises its
proposed groomed event at this step in the algorithm by evaluating the value
of $\rho^{(n)}$ that would give $\sum_{k=1}^{n}{\rm EMD}_{k}>{\rm EMD}_{\rm
cut}$, and finding the associated groomed event by using Eq. (31). It returns
the resulting event as the final groomed event.

As in P-IVS, P-IVS(EMD) does not need to re-compute the Voronoi diagram from
scratch at each step of the algorithm. Similarly, our procedure for finding
the value of $\rho^{(n)}$ that subtracts the correct amount of total EMD in
the final step of the algorithm is computationally efficient, and does not
rely on a complete re-evaluation of the EMD at each step of the algorithm.

At present, we do not have an implementation of P-RS in EMD mode. That said,
we expect that the generalization of EMD-mode grooming to P-RS and P-RSF will
be similar: at step $n$ of the EMD mode algorithm, we perform grooming with
the maximum value of $z_{\rm cut}$ until a particle would be removed, and
define EMDn as the EMD between the energy flows at steps $n-1$ and $n$ of the
grooming. If the sum of the subtracted EMDs is greater than EMDcut,
$\sum_{k=1}^{n}{\rm EMD}_{k}>{\rm EMD}_{\rm cut}$, we must compute the value
of $z_{\rm cut}$ that fixes ${\rm EMD}_{n}>{\rm EMD}_{\rm
cut}-\sum_{k=1}^{n-1}{\rm EMD}_{k}$, groom with this value of $z_{\rm cut}$,
and return the resulting groomed event. Otherwise, if $\sum_{k=1}^{n}{\rm
EMD}_{k}<{\rm EMD}_{\rm cut}$, we would continue grooming recursively.

We hope that future exploration of EMD-mode grooming may offer additional
utility in the use of continuous grooming in the subtraction of additive
contamination, and in the mitigation of obfuscating radiation in general.

##  Appendix B Feeding Frenzy: Comparing a Plethora of Grooming Options

In this appendix, we present a brief collection of additional results
regarding the responses of Piranha and Soft Drop groomers to soft distortions
and additive contamination. We extend the results in the main text by
comparing P-RSFf=1/2, the focus of our phenomenological studies in the main
text, to three groups of grooming algorithms:

  * •

Hard-Cutoff Groomers: SDβ=0, SDβ=1, and SDβ=2;

  * •

Fully Continuous Groomers: P-AS and P-IVS;

  * •

Recursive Subtractors: P-RSFf=0, P-RSFf=1/2, P-RSFf=3/4, and P-RSFf=1.

In our pileup studies, we also count Constituent Subtraction (CS)
Berta:2014eza  among the continuous groomers due to its continuity properties
in the continuum limit. The comparisons of this appendix demonstrate that the
conclusions we drew in the main text by comparing P-RSFf=1/2 to SDβ=2 seem to
hold in greater generality: Piranha groomers tend to have more robust
responses to soft distortions and additive contamination than traditional
grooming methods. Furthermore, the responses of P-RSFf=1/2 are similar to the
responses of the fully continuous groomers P-AS and P-IVS (and CS, in the case
of pileup mitigation); each of the Piranha algorithms we explore in this
appendix are available on GitHub piranhagithub .

As in the main text, we first extend our results involving hadronization
corrections (Sec. 4.2) and all-charged corrections (Sec. 4.3) to explore how
several choices of grooming and grooming parameters may respond to soft
distortions. We then begin our discussion of additive contamination by
providing more detail regarding our pileup studies: we explain in greater
detail our procedure for pileup mitigation used in Sec. 5.1 and extend our
results to several groomers and values of $\langle n_{\rm PU}\rangle$.
Finally, we similarly explore how different groomers respond to the underlying
event, extending the results of Sec. 5.2.

(a)  (b)  (c)

(d)  (e)  (f)

(g)  (h)  (i)

Figure 13:  Per-jet hadronization responses of Piranha and traditional
groomers for (top row) EMD, (middle row) $\Delta p_{T}$, and (bottom row)
$C_{1}^{(2)}$; we compare P-RSFf=1/2 to (left column) hard-cutoff groomers,
(middle column) fully continuous groomers, and (right column) recursive
subtractors. For brevity in this appendix, we focus on the variance of the
shifts in each observable due to hadronization in groomed jets. The red arrows
indicate the direction corresponding to better performance.

(a)  (b)  (c)

(d)  (e)  (f)

(g)  (h)  (i)

Figure 14:  Same as Fig. 13, but for jet responses to the exclusion of neutral
particles.

###  B.1 Soft Distortions

In the main text, we made the argument that continuity provides more
predictable responses to soft distortions; we therefore focus on the
fluctuations induced by soft distortions by studying the standard deviations
of the groomed EMD and $C_{1}^{(2)}$ as a function of $z_{\rm cut}$.

Fig. 13 shows the fluctuations in the parton-hadron groomed EMD and the
parton-hadron $\Delta C_{1}^{(2)}$, as discussed in Sec. 4.2, for a variety of
grooming options. The first column of Fig. 13 demonstrates that P-RSFf=1/2
groomed jets tend to have significantly smaller fluctuations due to
hadronization, echoing the conclusions of Sec. 4.2. We note that SDβ=2 is
significantly more robust to hadronization than SDβ=0 when considering
extensive observables (EMD and $p_{T}$). The second two columns of Fig. 13
show that the Piranha groomers we consider all have similar hadronization
responses in the considered range of $z_{\rm cut}$.

Fig. 14 presents the all-charged groomed EMD and $\Delta C_{1}^{(2)}$ as
discussed in Sec. 4.3. The first column again shows that P-RSFf=1/2
experiences significantly smaller fluctuations due to the exclusion of neutral
particles than Soft Drop, and the second two columns show similar responses
for all Piranha groomers.

###  B.2 Pileup

We now provide more detail regarding our PU mitigation procedure. As discussed
in Sec. 5.1, we use the dijet and minimum bias events of Ref. Soyez:2018opl
to study the effects of PU. We use the GridMedianBackgroundEstimator (GMBE)
method of FastJet Cacciari:2011ma  to provide a rough estimate, $\rho_{\rm
GMBE}$, of the energy density of additive contamination due to pileup in each
jet we would like to groom. However, since GMBE is designed to provide an
estimate of the contaminating energy in an entire event,222222 Note that
FastJet also supports Jet Median Background Estimation (JMBE), which is more
suited to the task of estimating the contaminating energy density in each jet.
We were unable to find Python support for JMBE and used the method detailed
here as a replacement.  we improve upon the GMBE procedure by optimizing a
correction factor that scales the GMBE estimate for each groomer. We optimize
the correction factor for each groomer by considering its action on 100,000
events, throwing away events where the presence of PU changes which jet within
the dijet has more energy, and implementing the following procedure:

  1. 1.

For each epoch $i$ during our optimization, estimate the contaminating energy
density due to PU within a jet,

$\displaystyle\rho_{\rm
PU}[\mathcal{E}]=\frac{p_{T}^{\rm(PU~{}in~{}jet)}}{A_{\rm jet}},$ (33)

that needs to be applied by a groomer $g$ to accurately remove PU
contamination as

$\displaystyle\rho^{{\rm(epoch}\,i)}_{\rm est.}[\mathcal{E};g)=\rho_{\rm
GMBE}\,[\mathcal{E}]f_{i}(g),$ (34)

where we use parentheses to indicate that $\rho_{\rm est.}$ and $f$ are
functions of the groomer $g$, and square brackets to indicate that the energy
densities $\rho$ are functionals of the energy flow $\mathcal{E}$. Since we
are using jets clustered using the anti-$k_{t}$ algorithm, we may take the jet
area to be $\pi R_{\rm jet}^{2}$ Cacciari:2008gp .

  2. 2.

Introduce a new epoch every 1,000 events we consider, by adjusting our
proposed correction factor $f_{i}$ to

$\displaystyle f_{i+1}=f_{i}+r_{\rm
learn}\left\langle\frac{\Delta\rho}{\rho}\right\rangle_{i},$ (35)

where the angular brackets indicate an average over the jets considered during
epoch $i$, $\Delta\rho$ indicates the difference in $p_{T}$ density in a
pileup subtracted event and a hard event,

$\displaystyle\Delta\rho=\frac{\Delta p_{T}}{A_{\rm
jet}}=\frac{p_{T}^{\rm(groomed~{}PU)}-p_{T}^{\rm(hard)}}{A_{\rm jet}},$ (36)

and $r_{\rm learn}$ is a learning rate which we set to 1/3.

  3. 3.

At the end of the final epoch, save the final correction factor $f_{\rm
learned}(g)$ for each groomer.

  4. 4.

Perform a new analysis of the ability of each groomer to remove PU by
estimating the optimal $\rho$ required for PU-subtraction as

$\displaystyle\rho_{\rm est.}[\mathcal{E};g)=\rho_{\rm
GMBE}\,[\mathcal{E}]f_{\rm learned}(g).$ (37)

We recall for completeness that $z_{\rm cut}=\rho_{\rm est.}A_{\rm
jet}/p_{T}^{(\rm jet)}$.

(a)  (b)  (c)

(d)  (e)  (f)

(g)  (h)  (i)

Figure 15:  Average per-jet pileup-induced shifts in (top row) EMD, (middle
row) $\Delta p_{T}$, and (bottom row) $C_{1}^{(2)}$; we compare P-RSFf=1/2 to
(left column) hard-cutoff groomers, (middle column) fully continuous groomers,
and (right column) recursive subtractors, as discussed in Sec. 5.1 and App. B.
The red arrows indicate the direction corresponding to better performance.

(a)  (b)  (c)

(d)  (e)  (f)

(g)  (h)  (i)

Figure 16:  Same as Fig. 15, but for the standard deviation of the pileup-
induced shifts.

In Figs. 15 and 16, we present results on PU mitigation using the optimization
procedure described above for a variety of groomers, as an extension of the
discussion in Sec. 5.1. We focus on the PU-subtracted EMD, the PU-subtracted
$\Delta p_{T}$, and the PU-subtracted $\Delta C_{1}^{(2)}$ as a function of
the average number of PU events $\langle n_{\rm PU}\rangle$.

The Piranha groomers displayed in Fig. 15, together with CS, tend to perform
only slightly better than the traditional groomers, on average, in removing
pileup; both Piranha and traditional groomers have been tuned so that
$\langle\Delta p_{T}\rangle=0$, and this tuning is reflected by the relatively
small changes to the extensive properties and even the substructure of the PU-
subtracted jets. The notable exception is P-RSFf=0. Since P-RSFf=0
preferentially grooms away hard radiation, it is poorly suited to grooming
away the low-energy additive contamination that is contributed by pileup, and
it performs more poorly than other Piranha groomers in reproducing both
extensive and substructure properties of PU-subtracted jets.

However, Fig. 16 shows that Piranha groomers and CS are more reliable PU
mitigators than traditional groomers: the standard deviations in the shifts of
PU-subtracted extensive and substructure properties tend to be significantly
smaller for Piranha groomers than for traditional groomers. Again, P-RSFf=0 is
a noticeable exception and is less reliable as a PU mitigator than either
Piranha or traditional grooming methods.

(a)  (b)

Figure 17:  Runtimes for Piranha algorithms, and comparison to some
traditional grooming and pileup mitigation algorithms. (a) shows the
probability density for the logarithm of the runtime for each algorithm when
$\langle n_{\rm PU}=50\rangle$, where $n_{\rm PU}$ is the average number of
pileup events. (b) shows the average runtime for each algorithm as a function
of $\langle n_{\rm PU}\rangle$. Runtimes are shown for the Pythia 8.185
datasets of Ref. Soyez:2018opl , and we simulate pileup by layering a Poisson-
distributed number of minimum-bias events over a hard dijet event.

We also briefly present results for the runtimes of the grooming algorithms
discussed in this paper, including the Constituent Subtraction (CS) pileup
mitigation algorithm, as a function of the amount of additive contamination in
events with PU. In particular, Fig. 17 displays the properties of runtimes for
grooming algorithms functioning as pileup mitigators as we vary the average
number of layered minimum bias events, $\langle n_{\rm PU}\rangle$. The tree-
based grooming strategies, such as Soft Drop and P-RS, are the fastest. P-IVS
is an order of magnitude slower. Finally, P-AS and CS are the slowest, with
runtimes of $\mathcal{O}({\rm 1s})$ even for a reasonable number of pileup
events. The combination of high speed and high performance of balanced P-RS
provides simple preliminary evidence for their utility in the removal of
pileup.

(a)

(b)  (c)

(d)

(e)  (f)

(g)

(h)  (i)

Figure 18:  Wasserstein distance between jet $C_{1}^{(2)}$ distributions (top
row) without UE correction and (middle row) with UE correction, and (bottom
row) the optimal value of $\delta z_{\rm cut}$ for UE correction.

###  B.3 Underlying Event

For the underlying event, we present results for the Wasserstein distances
between substructure distributions with and without UE, discussed in Sec. 5.2,
as well as the additional grooming $\delta z_{\rm cut}$ required for optimal
UE correction. We collect these results in Fig. 18.

In each of the plots, a lower value along the $y$-axis – of either the
Wasserstein distance or the optimal additional $\delta z_{\rm cut}$ –
indicates a desirable property for UE correction. A lower Wasserstein distance
without UE correction indicates inherent robustness of the groomer to the
presence of UE. A lower UE-corrected Wasserstein distance indicates that the
groomer is able to better correct for the presence of UE. Finally, a lower
$\delta z_{\rm cut}$ indicates greater robustness against UE: less additional
grooming must be applied for UE correction.

Notably, Fig. 18d shows that the UE-corrected Wasserstein distance of P-RSFf=1
is smaller than that of Soft Drop for several values of $\beta_{\rm SD}$ and a
range of $z_{\rm cut}^{\rm(base)}$, indicating that P-RSFf=1 is a more
delicate tool for the removal of additive contamination due to UE. However,
P-IVS and P-AS perform even better at UE removal, as shown in Fig. 18e. It may
be that simple improvements to P-RS that emulate the features of P-IVS and
P-AS, such as the “Hard-Balanced Recursive Subtractors” mentioned briefly in
Sec. 3.4, may lead to even better performance in UE removal. Finally, while
the P-RSF groomers generally exhibit similar performance in UE correction,
P-RSFf=0 performs poorly at UE correction and is extremely sensitive to the
effects of UE; this is unsurprising, as P-RSFf=0 deliberately grooms away hard
radiation and leaves in soft radiation, such as the radiation contributed by
UE.

Since the procedure we use to study the ability of each groomer to remove UE
is based on a computationally expensive optimization of $\delta z_{\rm cut}$,
the plots we show in Fig. 18g have some unphysical, jagged behavior,
especially in the second two rows. We expect that with a dedicated study with
better statistics that the jagged behavior of these plots will be smoothed,
but that the qualitative nature of our conclusions will remain unchanged.

##  Appendix C Towards Resummed Recursive Subtraction

We would like to gain theoretical insight, where possible, into the properties
of Piranha-groomed jets. To move towards a perturbative understanding of
Piranha, we begin by studying the behavior of P-RSF at leading order (LO),
deriving the results for P-RSF substructure presented in Sec. 3.3, and
describe why this is sufficient for generic Piranha groomers at LO. Beyond LO,
we find difficulties in using existing resummation techniques because of the
intricate correlations between subtractions of different emissions.
Nonetheless, we can study the specific example of P-RSFf=1 to obtain analytic
resummed results up to leading logarithmic accuracy (LL) and numerical results
beyond LL.

The angular-ordered, tree-based structure of P-RSF is an important feature
that helps facilitate perturbative calculation. In particular, we recall that
the branching structure of an angular-ordered tree of emissions is comparable
to the history of coherent parton branchings in the parton model
Collins:2011zzd , and that the philosophy of local parton-hadron duality
Azimov:1984np ; Andersson:1989ww ; Shifman:2003de ; Neill:2020tzl  asserts
that energy flows are not dramatically changed in the presence of
hadronization. We may therefore approximate the probability of finding a
physical particle within an angular-ordered jet tree by the probability of
finding an unphysical, unconfined QCD parton with the same energy and angle.
Each partonic emission in a jet is accompanied by a factor of the strong
coupling $\alpha_{s}$, enabling a perturbative analysis of the behavior of
groomed observables.

We focus on quantifying the substructure of Piranha-groomed jets using the
two-prong generalized energy correlation functions (ECFs) of Ref.
Larkoski:2013eya . These were described by Eqs. (16) and (28), and we repeat
their approximate form in the central ($y=0$) and narrow ($R_{0}\ll 1$) limit
here for convenience:

$C_{1}^{(\varsigma)}\simeq\frac{1}{2}\sum_{i=1}^{M}\sum_{j=1}^{M}z_{i}z_{j}\left(\frac{\theta_{ij}}{R_{0}}\right)^{\varsigma}.$
(38)

As before, $z_{i}$ indicates the energy fraction of particle $i$,
$\theta_{ij}$ indicates the opening angle between particles $i$ and $j$, and
$R_{0}$ is the jet radius.

The $C_{1}^{(\varsigma)}$ furnish a set of interesting but relatively simple
observables to study in the context of Piranha. The behavior of the Piranha
groomed jet $p_{T}$, for example, is relatively uninteresting: since the
Piranha groomers discussed in this paper operate by subtracting a fixed amount
of transverse momentum

$\displaystyle\Delta p_{T}=z_{\rm cut}\,p_{T}=\rho\,A_{\rm jet},$ (39)

the changes to the transverse momentum of a jet due to the P-RSF grooming
procedure are set trivially by the parameter $z_{\rm cut}$. However, as
discussed in Sec. 3.3, refining calculations of observables involving the
energy fraction $z_{g}$ or angle $r_{g}$ of the first emission to pass the
grooming is a nuanced task that we will defer to future work.

We discuss the leading behavior of the distributions for P-RSF groomed ECFs in
App. C.1, with a brief discussion of the LO behavior of Piranha groomers in
general. We discuss the behavior of P-RSF1 groomed ECFs at resummed accuracy,
comparing results obtained with perturbative QCD, leading logarithmic parton
showers, and Pythia 8.244, in App. C.3. We introduce the mathematical
technology we use for calculations in perturbative QCD in App. C.4, and our
leading logarithmic parton shower in App. C.5.

###  C.1 Substructure at Fixed Order

For convenience, we first briefly repeat some comments made in Sec. 3.3 while
discussing the behavior of P-RSF at lowest order in $\alpha_{s}$ (LO). At LO,
the emission of a single parton with energy fraction $z$ and angle $\theta$ is
described approximately by the pseudo-probability distribution

$\displaystyle\frac{\alpha_{s}}{\pi}p_{i}(z)\frac{1}{\theta}\approx\frac{2C_{R_{i}}\alpha_{s}}{\pi}~{}\frac{1}{z}~{}\frac{1}{\theta},$
(40)

where $p_{i}(z)$ is a Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP)
splitting function Gribov:1972ri ; Dokshitzer:1977sg ; Altarelli:1977zs  and
$C_{R_{i}}$ is the quadratic Casimir for the SU(3) color representation
$R_{i}$ of the mother parton, with
$C_{F}=\frac{N_{C}^{2}-1}{2N_{C}}=\frac{4}{3}$ for quarks and $C_{A}=N_{C}=3$
for gluons.232323 By focusing on the singular pieces of the splitting function
in Eq. (40), we implicitly focus on the probability that a gluon is emitted
from the mother parton. For example, a quark may emit a gluon, so that the
resulting final state consists of both a gluon and a quark, while a gluon may
split into two gluons. In the language of splitting functions, a gluon may in
principle split into two quarks; however, the associated splitting function
does not have any singularities as $z\to 0$ and is therefore sub-leading to
the singular behavior reflected in Eq. (40).  We use reduced splitting
functions,

$\overline{p}_{i}(z)=p_{i}(z)+p_{i}(1-z),$ (41)

with $z\in(0,1/2)$, which describe the probability distribution of the softest
parton produced in the splitting.

The singular nature of the DGLAP splitting functions in the limits
$z,\,\theta\to 0$ indicates that the phase space of partonic emissions is
dominated by emissions with small energy fraction $z$ and angle $\theta$. The
problems posed by these singularities are addressed systematically by
resummation, and we review the resummation method we apply here (often called
the CAESAR method Banfi:2001bz ; Banfi:2004yd ) in App. C.4. The non-singular
pieces of these splitting functions are written explicitly in Eqs. (57) and
(58).

Because each emission is suppressed by a factor of $\alpha_{s}$, the behavior
of the P-RSF groomed ECFs of Eq. (38), away from $C_{1}^{(\varsigma)}=0$, is
determined by the phase space distribution of two-parton jets whose particles
survive the P-RSF grooming procedure, up to corrections proportional to
$\alpha_{s}^{2}$. If the two partons survive the grooming procedure, they can
be described with the modified energy fractions

$\displaystyle z_{\rm soft}^{\prime}$ $\displaystyle=z-f\,z_{\rm cut}>0,$
(42a) $\displaystyle z_{\rm hard}^{\prime}$ $\displaystyle=1-z-(1-f)z_{\rm
cut}>0,$ (42b)

where $z$ is the original energy fraction of the softer parton. The groomed
jet ECF then takes the form

$C_{1}^{(\varsigma)}(z,\theta)=\left(z-f\,z_{\rm
cut}\right)\left(\frac{\theta}{R_{0}}\right)^{\varsigma}\frac{\left(1-z-(1-f)z_{\rm
cut}\right)}{\left(1-z_{\rm cut}\right)^{2}}\approx\left(z-f\,z_{\rm
cut}\right)\left(\frac{\theta}{R_{0}}\right)^{\varsigma}\,\frac{\left(1-(1-f)z_{\rm
cut}\right)}{\left(1-z_{\rm cut}\right)^{2}}.$ (43)

In the last step, we use that the phase space of the emission is dominated by
regions with $z,\,\theta\ll 1$. The factors of $(1-z_{\rm cut})^{2}$ that
appear in the denominators of Eq. (43) emerge from the normalization of the
groomed $C_{1}^{(\varsigma)}$ by the groomed jet $p_{T}$, as in the definition
of Eq. (38).

We also note that if either of the partons in our two-parton configuration at
LO is entirely groomed away, the groomed value of $C_{1}^{(\varsigma)}$ is
zero. The region of phase space where one parton is groomed away at LO
conspires with virtual contributions and the one-parton configuration with no
splittings to ensure that the $C_{1}^{(\varsigma)}$ distribution integrates to
one. These contributions may quickly be calculated once we find the
distribution away from the origin, $C_{1}^{(\varsigma)}>0$, and we include
them at the end of the following discussion.

In the DGLAP splitting functions, the energy fraction $z$ takes values from
$0$ to $1$. Since the $C_{1}^{(\varsigma)}$ are sensitive to the softest
emission in the jet, we are instead interested in restricting $z$ to take
values from $0$ to $1/2$. We are therefore interested in the reduced splitting
functions, $\bar{p}(z)=p(z)+p(1-z)$; these have the same singular behavior as
the unmodified DGLAP splitting functions near $z=0$, but are only used in the
range $z\in(0,1/2)$. From these considerations, the fixed-order distribution
of $C_{1}^{(\varsigma)}$ away from zero becomes

$\displaystyle\rho^{\rm(LO)}_{i,~{}\varsigma}(C>0)=\frac{\alpha_{s}}{\pi}\int_{0}^{R_{0}}\frac{\differential\theta}{\theta}\int_{0}^{1/2}\differential
z\,\bar{p}_{i}(z)\delta\left(C-C_{1}^{(\varsigma)}(z,\theta)\right),$ (44)

where $C_{1}^{(\varsigma)}(z,\theta)$ is the functional form of the groomed
ECF given in Eq. (43).

Up to terms that are power-suppressed in $C$, $z_{\rm cut}$, or both, the
fixed-order distribution for the groomed observable away from $C=0$ is

$\displaystyle\rho^{\rm(LO)}_{i,~{}\varsigma}(C>0)$
$\displaystyle=\frac{2\alpha_{s}C_{R_{i}}}{\varsigma~{}\pi}\frac{1}{C}\left(2\tanh^{-1}\left(1-2\tilde{C}-2\,f\,z_{\rm
cut}\right)+B_{i}+\mathcal{O}\left(C,\,z_{\rm cut}\right)\right)$ (45a)
$\displaystyle=\frac{2\alpha_{s}C_{R_{i}}}{\varsigma~{}\pi}\frac{1}{C}\left(-\log\left(\tilde{C}+f\,z_{\rm
cut}\right)+B_{i}+\mathcal{O}\left(C,\,z_{\rm cut}\right)\right)$ (45b)

where $\tilde{C}=C\times(1-z_{\rm cut})^{2}/(1-(1-f)z_{\rm cut})$, $C_{R_{i}}$
is the quadratic Casimir of the color representation of the initiating parton,
and we have defined the factor $B_{i}$ to capture behavior from non-singular
components of the splitting functions: $B_{q}=-3/4$ for quark initiated jets
and $B_{g}=-11/12+n_{f}/(6C_{A})$ for gluon initiated jets.

Including the $\mathcal{O}(\alpha_{s}^{0})$ jet configuration with a single
parton for which $C_{1}^{(\varsigma)}=0$, the two-parton contributions for
which $z<fz_{\rm cut}$ and the softer emission is completely eliminated, and
the virtual contributions at $\mathcal{O}(\alpha_{s}^{1})$, we can write the
full LO probability distribution for $C_{1}^{(\varsigma)}$ as

$\displaystyle\rho^{\rm(LO)}_{i,~{}\varsigma}(C)$
$\displaystyle\approx\delta(C)+\frac{2\alpha_{s}C_{R_{i}}}{\varsigma~{}\pi}\left[\frac{1}{C}\left(-\log\left(\tilde{C}+f\,z_{\rm
cut}\right)+B_{i}+\mathcal{O}\left(C,\,z_{\rm
cut}\right)\right)\right]^{(C_{\rm max})}_{+},$ (46)

where ${C_{\rm max}}=(1/2-fz_{\rm cut})(1-(1-f)z_{\rm cut})/(1-z_{\rm
cut})^{2}$ is the maximum value of $C_{1}^{(\varsigma)}$, and we have fixed
the normalization of the distribution with plus-function regularization.242424
The plus-function regulation of the function $q(x)$ is a distribution
$[q(x)]^{(x_{0})}_{+}$ defined by its integral against any function $f(x)$ in
a suitable space of test functions,
$\displaystyle\int_{0}^{x_{0}}\differential x[q(x)]^{(x_{0})}_{+}f(x)$
$\displaystyle=\int_{0}^{x_{0}}\differential x~{}q(x)\left(f(x)-f(0)\right).$
(47) See Appendix B of Ref. Ligeti:2008ac  and the presentation of Ref.
Ebert:2016gcn  for satisfying discussions.  The plus-function regularized
expression of Eq. (46) may be derived by replacing $\overline{p}(z)$ in Eq.
(44) with its plus-function analog. Alternatively, one can recognize that the
full distribution for $C_{1}^{(\varsigma)}$ must integrate to one even at LO,
and that additional contributions to the LO distribution in Eq. (45) can only
come from the regions of phase space where $C_{1}^{(\varsigma)}=0$; this is
enough to uniquely specify the result of Eq. (46) at LO.

Unlike the LO distributions for traditionally groomed jet ECFs, such as those
of Soft Drop Larkoski:2014wba , the LO distributions for P-RSFf=1/2 and P-RSF
groomed $C_{1}^{(\varsigma)}$ do not exhibit piece-wise behavior. While the
piece-wise behavior of Soft Drop observable distributions is smoothed out by
all-orders effects Benkendorfer:2021unv , the smoothness of Piranha-groomed
distributions at LO is a manifestation of their continuity.

###  C.2 The Groomed Energy Fraction for a Single Emission

Next, we discuss the groomed energy fraction, $z_{g}$, of the first emission
to survive the grooming procedure. In Soft Drop, $z_{g}$ is the energy
fraction of the first emission to pass the Soft Drop criterion $z>z_{\rm
cut}\theta^{\beta_{\rm SD}}$. In P-RSF, $z_{g}$ is the groomed energy fraction
of the first emission that is not entirely removed by the grooming, which we
call the critical emission.

The behavior of $z_{g}$ for Soft Drop was derived to double-logarithmic
accuracy by Ref. Larkoski:2015lea  with the technology of Sudakov safety.
Expanding their double-logarithmic result to $\mathcal{O}(\alpha_{s})$, the
$z_{g}$ distribution for Soft Drop takes the approximate form

$\displaystyle\rho_{i,\,\,\rm SD}(z_{g};\beta_{\rm
SD})=\begin{dcases}\frac{2\alpha_{s}C_{R_{i}}}{\pi|\beta|}\,\overline{p}_{i}(z_{g})\,\log(z_{g}/z_{c})\,\Theta(z_{\rm
cut}<z_{g})+\mathcal{O}(\alpha_{s}^{2}),&\beta_{\rm SD}<0;\\\
\frac{\overline{p}_{i}(z_{g})}{\int_{z_{\rm cut}}^{1/2}\differential
z\,\,\overline{p}_{i}(z)}\,\Theta(z_{\rm cut}<z_{g}),&\beta_{\rm SD}=0;\\\
\sqrt{\frac{\alpha_{s}C_{R_{i}}}{\beta}}\,\,\overline{p}_{i}(z_{g})+\frac{2\alpha_{s}C_{R_{i}}}{\beta}\,\,\overline{p}_{i}(z_{g})\log\min(2z_{\rm
cut},2z_{g})+\mathcal{O}(\alpha_{s}^{3/2}),&\beta_{\rm SD}>0,\end{dcases}$
(48)

where $i$ again indicates the flavor (quark or gluon) of the initiating
parton, and we note that the expression for $\beta_{\rm SD}=0$ receives no
$\mathcal{O}(\alpha_{s})$ corrections at double-logarithmic accuracy. If
$\beta_{\rm SD}\leq 0$, the probability distribution has no support when
$z_{g}<z_{\rm cut}$ because $z_{g}$ is determined by the Soft Drop criterion;
the vanishing of the $z_{g}$ distribution when $z_{g}<z_{\rm cut}$ for
$\beta_{\rm SD}\leq 0$ therefore reflects the soft discontinuous behavior of
Soft Drop. If $\beta_{\rm SD}>0$ there is instead a kink at $z_{g}=z_{\rm
cut}$ that similarly reflects Soft Drop’s soft discontinuity.

The behavior of the P-RSF $z_{g}$ distribution is more subtle, for reasons we
discuss below. At the level of a single emission, however, the $z_{g}$
distribution for Soft Drop may be translated directly to P-RSF because of the
similarities between Soft Drop with $\beta=0$ and P-RSF:252525 In Eq. (49), we
choose $z_{g}$ to be the groomed energy fraction of what had been the softer
emission before grooming, regardless of which emission is softer after
grooming. This avoids additional corrections if $f_{\rm soft}<1/2$, where more
grooming is applied to the harder emission than the softer emission at each
branch and it becomes possible that the harder of two ungroomed emissions
becomes the softer of the two emissions after grooming. If we instead define
$z_{g}$ to be the softer groomed emission, there are corrections to this
formula for $z_{g}>1/2-\mathcal{O}(z_{\rm cut})$ when $f_{\rm soft}<1/2$. We
also assume that $z_{\rm cut}<1/2$; if $f_{\rm soft}<1/2$ and $z_{\rm
cut}>1/2$, there are additional $\mathcal{O}(z_{\rm cut})$ corrections to Eq.
(49) due to the constraint that the harder ungroomed emission is not entirely
groomed away.

$\displaystyle\rho_{i,\,\,\text{P-RSF}}(z_{g};f)=(1-z_{\rm
cut})\frac{\overline{p}_{i}\left[\left(1-z_{\rm cut}\right)z_{g}\,+\,fz_{\rm
cut}\right]}{\int_{fz_{\rm cut}}^{1/2}\differential
z^{\prime}\,\,\overline{p}(z^{\prime})}+\mathcal{O}(\alpha_{s}).$ (49)

$\rho_{i,\,\,\text{P-RSF}}(z_{g};f)$ has support in the domain
$z_{g}\in\left(0,\,(1/2-fz_{\rm cut})\,/\,(1-z_{\rm cut})\right)$ at this
order of accuracy. The additional factors of $(1-z_{\rm cut})$ relative to Eq.
(48) capture the fact that $z_{g}$ is normalized further after grooming:
$z_{g}=(z_{0}-fz_{\rm cut})/(1-z_{\rm cut})$, where $z_{0}$ denotes the
ungroomed softer energy fraction. The lower limit $fz_{\rm cut}$ on the
integration variable $z^{\prime}$ reflects that we consider configurations for
which the softer emission survives the grooming procedure.

Much like Soft Drop with $\beta_{\rm SD}=0$, the LO behavior of the P-RSF
$z_{g}$ distribution is determined by the fact that the first surviving
emission must have an energy fraction greater than the assigned grooming. At
LO, there is a single emission, and therefore the assigned grooming is simply
$fz_{\rm cut}$. Unlike Soft Drop, however, the $z_{g}$ distribution for P-RSF
has support down to $z_{g}=0$: a two-parton event with $z=fz_{\rm
cut}+\delta$, $\delta\ll 1$, is mapped to a groomed two-parton event with
$z_{g}=\delta$, while the two-parton event with $z=fz_{\rm cut}-\delta$ is
mapped to a one-parton event; though $z_{g}$ is undefined in the latter case,
where the groomed jet consists of only a single parton, the fact that $z_{g}$
can become infinitesimally small is a mark of the continuity of P-RSF.

Improving the accuracy of the calculation of $z_{g}$ for P-RSF will require a
more detailed analysis which we leave to future work. An important
complication emerges because there may be emissions before the critical
emission that can soak up some of the grooming; the amount by which the
critical emission is groomed is therefore not $z_{\rm cut}$ but a smaller
effective value, $z_{\rm cut}^{\rm(eff)}=z_{\rm cut}-\Delta z_{\rm cut}$,
where $\Delta z_{\rm cut}$ depends on the energy fractions of earlier
emissions. This leads to problems when following the approach of Ref.
Larkoski:2015lea , suited to the calculation of $z_{g}$ for Soft Drop. Ref.
Larkoski:2015lea  calculates the distribution of $z_{g}$ in the framework of
Sudakov safety by first marginalizing over the distribution of the angle of
the critical emission, $r_{g}$. However, the distribution of $r_{g}$ depends
on the amount by which the critical emission is groomed, $z_{\rm cut}^{(\rm
eff)}$, while a leading logarithmic calculation of $z_{\rm cut}^{(\rm eff)}$
depends on the value of $r_{g}$. The circular dependence of the distributions
of $r_{g}$ and $z_{\rm cut}^{\rm(eff)}$ prevents us from applying the
formalism of Ref. Larkoski:2015lea  without more careful modification.

###  C.3 Substructure at Leading Logarithmic Accuracy and Beyond

The global nature of Piranha grooming algorithms leads to intricate
correlations between the grooming of different final-state particles, and thus
to complications in perturbative calculations beyond LO. For P-RSFf=1/2,
P-IVS, and P-AS, every particle in a jet is groomed by some amount, and the
grooming of any particular final-state particle is correlated with the
grooming of every other. This subtlety renders existing techniques for
multiple emissions calculations less effective. Piranha-groomed observables
depend on global information, and a more detailed understanding of Piranha-
groomed correlators will require techniques that can elucidate global
information associated with a jet.

While we do not get around this subtlety for the Piranha grooming procedures
presented in this paper, we may use P-RSF to at least gain some intuition for
the resummed behavior of recursive subtractors. In particular, we navigate
around the subtleties associated with Piranha by studying the substructure of
P-RSF with $f_{\rm soft}=1$, or P-RSFf=1. Crucial to our analysis is that
P-RSFf=1 eventually terminates, such that it does not necessarily affect every
particle within an event.262626 P-RSFf=0 is another subtraction algorithm that
eventually terminates, but it subtracts away hard radiation instead of soft
radiation. Therefore, we focus on P-RSFf=1 in the discussion below. We leave a
more detailed survey of the analytic properties of P-RSFf=1/2, P-RSFf=0, and
Recursive Subtraction more generally to future work.  Due to this additional
simplicity, we can study P-RSFf=1 at leading logarithmic (LL) accuracy in
slightly more detail, and find analytic expressions for observables related to
the first emission to survive the grooming procedure, or critical emission.

The calculation of these resummed distributions is discussed in more detail in
App. C.4, whose methods may be used to show that the LL distribution for the
energy fraction $z$ and the angle $\theta$ of the first emission to survive
the P-RSF1 procedure is

$\rho^{\rm(LL)}_{\rm crit}(z_{\rm crit},\theta_{\rm
crit})=\frac{2C_{R}\alpha_{s}}{\pi}\frac{1}{z_{\rm crit}}\frac{1}{\theta_{\rm
crit}}\exp\left[-\frac{2C_{R}\alpha_{s}}{\pi}\log\theta_{\rm crit}\log 2z_{\rm
cut}\right].$ (50)

In Eq. (50) and in the remainder of our discussion, we suppress the index $i$
denoting the flavor of the initiating parton because our LL results depend
unambiguously on the representation of the initiating parton through $C_{R}$.

Using this resummed distribution, the singular pieces of the splitting
function, and dropping subleading terms in $z_{\rm cut}$, we use the
cumulative analog of Eq. (44) at LL to find the cumulative distribution
$\Sigma^{\rm(LL)}_{\varsigma,~{}{\rm crit}}(C)$ for the contribution of the
critical emission to the groomed $C_{1}^{(\varsigma)}$ for general $f_{\rm
soft}=f$:

$\displaystyle\Sigma^{\rm(LL)}_{\varsigma,~{}{\rm crit}}(C)$
$\displaystyle=C^{\eta_{f}}+\frac{2C_{R}\alpha_{s}}{\varsigma\pi}\left(\frac{C}{f\,z_{\rm
cut}}\right)^{\eta_{f}}\frac{1}{\eta_{f}(\eta_{f}-1)}$ (51)
$\displaystyle~{}~{}~{}~{}~{}~{}\times\Bigg{[}\left(\frac{f\,z_{\rm
cut}}{C}\right)^{\eta_{f}-1}{}_{~{}~{}2}F_{1}\left(1,1-\eta_{f},2-\eta_{f},-\frac{C}{f\,z_{\rm
cut}}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\left(\frac{f\,z_{\rm
cut}}{C}\right)^{\eta_{f}}\left(\eta_{f}-1\right)\log\left(1+\frac{C}{f\,z_{\rm
cut}}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\left(f\,z_{\rm
cut}\right)^{\eta_{f}-1}{}_{~{}~{}2}F_{1}\left(1,1-\eta_{f},2-\eta_{f},1-1/(2\,f\,z_{\rm
cut})\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\left(f\,z_{\rm
cut}\right)^{\eta_{f}}\left(\eta_{f}-1\right)\log\left(1/(2\,f\,z_{\rm
cut})\right)\Bigg{]},$

where we have defined

$\displaystyle\eta_{f}$
$\displaystyle=-\frac{2C_{R}\alpha_{s}}{\varsigma\pi}\log(2\,f\,z_{\rm cut}).$
(52)

To gain some intuition for this intimidating expression, we may also expand
this to first order in our fixed coupling $\alpha_{s}$ to retrieve

$\displaystyle\Sigma^{\rm(LL)}_{\varsigma~{}{\rm crit}}(C)$
$\displaystyle\approx
1+\frac{2C_{R}\alpha_{s}}{\varsigma\pi}\left(-\log(2\,f\,z_{\rm cut})\log
C+{\rm Li}_{2}\left(-\frac{C}{f\,z_{\rm cut}}\right)-{\rm
Li}_{2}(1-1/(2\,f\,z_{\rm cut}))\right).$ (53)

Taking a derivative gives

$\rho^{\rm(LL)}_{\varsigma~{}{\rm
crit}}(C\,>\,0)=\frac{\differential}{\differential
C}\Sigma^{\rm(LL)}_{\varsigma~{}{\rm
crit}}(C\,>\,0)\approx-\frac{2C_{R}\alpha_{s}}{\varsigma~{}\pi}\,\frac{1}{C}\,\log\left(2C+2fz_{\rm
cut}\right),$ (54)

which more clearly reveals the LL contribution of the critical emission to the
groomed value of $C_{1}^{(\varsigma)}$. Eq. (54) differs from the LO result of
Eq. (45) by a factor of $2$ inside the logarithm simply because we have
integrated $z$ from $0$ to $1/2$ and used only the singular parts of the
splitting function in computing the LL result, while the LO result includes
non-singular pieces of the splitting function. By noting that the probability
distribution must integrate to 1, one can also produce an expression with end-
point contributions at $C=0$ which can then be compared to Eq. (46).

Even for P-RSF with $f_{\rm soft}=1$, however, the subtractive nature of the
grooming procedure leads to additional subtleties. Usually, the distribution
of partonic emissions in the $\log z$–$\log\theta$ plane is approximately
uniform, so that a particular emission will tend to have an exponentially
higher contribution to the two-pronged substructure of a jet than any other.
Since the first emission to survive an angular ordered hard-cutoff grooming
procedure such as Soft Drop will have a non-negligible energy fraction
$z>z_{\rm cut}$ and a large angle, it is this first surviving emission, the
critical emission, that contributes dominantly to $C_{1}^{(\varsigma)}$. This
trick made the original calculation of Soft Drop groomed substructure quite
simple Larkoski:2014wba .

In the case of P-RSFf=1, however, the critical emission is partially groomed
and may have a groomed energy fraction $z_{\text{P-RSF}_{1}}\sim z-z_{\rm
cut}\ll 1$. It is therefore possible for other emissions – the ungroomed
subsequent emissions which are narrower than the critical emission and
therefore untouched by the P-RSFf=1 algorithm – to contribute greater values
to $C_{1}^{(\varsigma)}$. The ungroomed subsequent emissions after the
P-RSFf=1 algorithm terminates can lead to larger contributions to substructure
observables than they would in the case of Soft Drop.

(a)

(b)

(c)

(d)

Figure 19:  Distributions of the observable $C_{1}^{(2)}\approx
m^{2}/p_{T}^{2}$ of Eq. (38) for quark jets groomed with SDβ=0 (top row) and
P-RSFf=1 (bottom row), with and without multiple emissions, at (left column)
LL accuracy and (right column) MLL accuracy. For each plot, we display the
results of the convolution method of Eq. (55) using numerical integration and
perturbative QCD with $10^{6}$ phase space samples as well as the parton
shower algorithm of App. C.5. At LL, we compare these results to the analytic
prediction of Eq. (51); at MLL, we compare them to parton-level and hadron-
level results obtained with Pythia 8.244 using the anti-$k_{t}$ algorithm. The
figures demonstrate that P-RSFf=1 is less sensitive to hadronization
corrections in the large observable regime, near $C_{1}^{(2)}\approx 10^{-2}$,
and that P-RSFf=1 groomed substructure is more sensitive to the effects of
multiple emissions than that of Soft Drop.

As a result, performing resummed calculations of Soft Drop groomed
substructure distributions that use only the critical emissions are more
accurate than calculations for P-RSFf=1 that only use the critical emission.
The relative accuracy of the LL, single-emission result for Soft Drop and
P-RSFf=1 is demonstrated in Figs. 19a and 19c, which compares the fixed-order
behavior of $C_{1}^{(2)}$ distributions for jets groomed with Soft Drop or
P-RSFf=1, respectively, using either the critical emission or all emissions in
the jet. Our parton shower procedure is inspired by Ref. Larkoski:2013paa  and
is described in more detail in App. C.5. Including multiple emissions does not
dramatically change the Soft Drop substructure distribution, but leads to non-
negligible changes in P-RSFf=1 substructure distributions.

We may include effects from the running of the strong coupling constant and
multiple emissions within the ungroomed jet to determine the behavior of the
cumulative distribution of $C_{1}^{(\varsigma)}$ at an even higher level of
accuracy, modified leading logarithmic (MLL) accuracy, as described in App.
C.4. While including these features in our calculation does not fully achieve
next-to-leading-logarithmic (NLL) accuracy, the MLL calculation is an
important piece of the full NLL result. To consider the effects of several
emissions on the distribution of the groomed ECFs, we convolve the probability
distributions of several emissions that contribute to the total observable
$C_{1}^{(\varsigma)}$:

$\displaystyle\rho^{\rm(MLL)}_{\varsigma}(C_{\rm tot})$
$\displaystyle=\int_{f\,z_{\rm cut}}^{1/2}\differential z_{\rm
crit}\int_{0}^{1}\differential\theta_{\rm crit}~{}\rho_{\rm crit}(\theta_{\rm
crit})\rho_{\rm crit}(z_{\rm crit}|z_{\rm pre})$ (55)
$\displaystyle~{}~{}~{}\int\differential z_{\rm pre}~{}\rho_{\rm pre}(z_{\rm
pre}|\theta_{\rm crit})\int\differential c_{\rm sub}~{}\rho_{\rm sub}(c_{\rm
sub}|z_{\rm crit},\theta_{\rm crit})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\delta\left(C_{\rm
tot}-C^{(\varsigma)}_{1,~{}\rm crit}(\theta_{\rm crit},z_{\rm crit},z_{\rm
pre})-c_{\rm sub}\right).$

Each factor of $\rho$ in Eq. (55) indicates a resummed probability
distribution corresponding to a particular emission, discussed in greater
detail in App. C.4. At the MLL level of accuracy, these resummed probability
distributions are all calculated with a running coupling $\alpha_{s}$. We
model non-perturbative effects by freezing the value of $\alpha_{s}$ below the
non-perturbative scale of $\mu_{\rm freeze}=1$ GeV, as in Eq. (67). Variations
in our choice of non-perturbative scale lead only to small changes to our
results.

The expressions that describe multiple emissions within the groomed jet are:

  * •

$\rho_{\rm crit}(\theta_{\rm crit})\rho(z_{\rm crit}|z_{\rm pre})$ represents
the probability distribution for the emission angle $\theta_{\rm crit}$ and
energy fraction $z_{\rm crit}$ of the critical emission, or the first emission
to survive the P-RSFf=1/2 procedure. This distribution is conditioned on the
presence of earlier emissions that modify the grooming procedure.

  * •

$\rho_{\rm pre}(z_{\rm pre}|\theta_{\rm crit})$ encodes the probability that
emissions that are wider than the critical emission, which we call pre-
critical emissions, have an energy fraction $z_{\rm pre}$. Any non-zero value
of $z_{\rm pre}$ mitigates the grooming of the critical emission.

  * •

$\rho_{\rm sub}(c_{\rm sub}|z_{\rm crit},\theta_{\rm crit})$ is the
probability distribution for emissions narrower than the critical emission to
contribute a value $c_{\rm sub}$ to the overall observable
$C^{(\varsigma)}_{1}$. We call these emissions subsequent emissions, and they
are unmodified by the grooming procedure.

  * •

Finally, $C^{(\varsigma)}_{1,~{}\rm crit}(\theta_{\rm crit},z_{\rm
crit},z_{\rm pre})$ is the contribution of the critical emission to
$C^{(\varsigma)}_{1,~{}\rm tot}$, in the presence of pre-critical emissions
with energy fraction $z_{\rm pre}$.

We evaluate the convolution of Eq. (55) through numerical integration and
compare the resulting distributions to the results from running coupling
parton showers and Pythia 8.244 in Fig. 19. In particular, Figs. 19b and 19d
compare our numerical MLL results to hadron-level results for anti-$k_{t}$
jets in Pythia 8.244 and demonstrate that this trend of comparatively large
multiple-emissions effects in P-RSFf=1 continues at MLL.

First, let us note the difference between our MLL results obtained with Monte
Carlo integration, described in App. C.4, and via parton showers, described in
App. C.5. Our Monte Carlo results use a frozen coupling below a non-
perturbative scale $\mu_{\rm freeze}=1$ GeV (see, for example, Eq. (67)). Our
parton shower results both freeze the coupling at $\mu_{\rm freeze}=1$ GeV and
stop the shower below a non-perturbative cutoff $\mu_{\rm cutoff}=1$ MeV,
tuned by hand to match results from Pythia 8.244. The shower cutoff
effectively sets $\alpha_{s}$ to zero below the scale set by $\mu_{\rm
cutoff}$. Therefore, the plots in Figs. 19b and 19d effectively show the
results obtained with three different models of non-perturbative effects, with
Pythia 8.244 the most accurate. Our MLL parton shower results in Figs. 19b and
19d are still normalized, but the normalization is not immediately evident due
to the presence of zero bins that are not shown in the logarithmically-scaled
figures. In passing from our rough parton shower model to the more accurate
hadronization model of Pythia, we expect that it is these zero bins that
contribute to the additional hadron-level peak shown in the Pythia data.

We also point out that in Fig. 19b, the parton-level Pythia 8.244 distribution
for $C_{1}^{(2)}$ also has an additional peak near the discontinuity in the
parton-level LL result, in the regime of relatively large $C_{1}^{(2)}\approx
10^{-1}$. There is a comparable additional peak in the hadron-level Pythia
distribution for $C_{1}^{(2)}$ at this relatively large value for the
observable.

We also note that, while our numerical results for P-RSFf=1 in the MLL figures
agree roughly with Pythia 8.244 in the regime of large $C_{1}^{(2)}$, the
hadron-level Pythia distributions have an additional peak near
$C_{1}^{(2)}\approx 3\times 10^{-6}$ due to hadronization corrections to the
observable at that scale. There is a similar additional peak in the hadron-
level Pythia distribution for mMDT (SDβ=0) groomed $C_{1}^{(2)}$.

###  C.4 Summary of Resummation: Convolution

We now review the resummation procedure we apply in our LL and MLL results. In
particular, we introduce the basics of the CAESAR formalism, named due to its
use in the context of the Computer Automated Expert Semi-Analytical Resummer
(CAESAR) program Banfi:2004yd .272727 The results of our presentation below
are also a special case of the more general, observable-independent formalism
of jet calculus Konishi:1979cb  – closely connected to the resummation
procedure used by parton showers – for which pedagogical introductions may be
found in e.g. Refs. Dokshitzer:1991wu ; vanBeekveld:2023lsa .  To the best of
our knowledge, this formalism was first introduced in the context of semi-
analytic resummation of event shapes Banfi:2001bz . While we include an
introduction here for completeness, a presentation of the concepts that appear
in the following discussion can be found in more elegant or complete forms
throughout existing literature Larkoski:2017fip ; Larkoski:2021aav ;
Cohen:2020afv ; Banfi:2001bz ; Banfi:2004yd ; Banfi:2004nk ; Banfi:2010xy ;
Luisoni:2015xha ; Bauer:2018svx ; Baron:2020xoi .

The distribution of energy fraction and angle of a partonic emission in a jet
is described at fixed order by

$\displaystyle\rho_{i,~{}\rm f.o.}(z,\theta)$
$\displaystyle=\frac{\alpha_{s}}{\pi}\bar{p}_{i}(z)\frac{1}{\theta},$ (56)

where $z\in(0,1/2)$ is the energy fraction of the softer branch of the
emission and $\bar{p}(z)=p(z)+p(1-z)$ indicates a reduced splitting function,
modified from the splitting function $p(z)$ to describe the softer branch. We
would like to regulate the singularities of this distribution by using
techniques of resummation in perturbative QCD and thus gain greater predictive
power over groomed jet observables. In this appendix, we discuss the resummed
distributions used in the text and give some quick derivations.

To begin, we provide explicit expressions for the DGLAP splitting functions,
which describe the probability for the emission of partonic radiation from
quarks and gluons, at one loop accuracy:

$\displaystyle p_{q\to qg}(z)=C_{F}\frac{1+(1-z)^{2}}{z},$ (57) $\displaystyle
p_{g\to gg}(z)=2C_{A}\frac{1-z}{z}+C_{A}z(1-z)+T_{F}n_{f}(z^{2}+(1-z)^{2}).$
(58)

Here, $z$ is the energy fraction of an emitted gluon relative to the total
energy of the emitted gluon and an initial parton. $p_{q}$ describes the
distribution of $z$ in the case that a gluon emitted by a quark, $q\to qg$,
and $p_{g}$ describes the distribution of $z$ in the case $g\to gg$.
$T_{F}=\frac{1}{2}$ is the Dynkin index of the fundamental representation of
the standard model SU(3) color gauge group, and $n_{f}$ indicates the number
of quark flavors. In the following analysis, we take $n_{f}=5$, including all
quark flavors except for the top quark. As in the text, $C_{R}$ is the
quadratic Casimir for the representation $R$ of SU(3).
$C_{R}=C_{F}=\frac{4}{3}$ for quarks and $C_{R}=C_{A}=3$ for gluons.

We also introduce some graphical notation to represent the probability of a
splitting whose angle $\theta$ and softer energy fraction $z$ lie within some
region of phase space. In particular, we describe the probability of finding a
splitting in some region in the $\log(z^{-1})$-$\log(\theta^{-1})$ plane, or
Lund plane, with a diagram of the Lund plane with the corresponding region
filled:

$\displaystyle\iint_{\leavevmode\hbox to11.7pt{\vbox
to9.72pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
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}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
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}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.06}{0.0}{0.0}{0.06}{0.0pt}{0.0pt}\pgfsys@invoke{
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to162pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{ }
\pgfsys@beginscope\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope \pgfsys@beginscope\pgfsys@invoke{ }
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}{}{} {}
{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{194.9997pt}{0.0pt}\pgfsys@lineto{194.9997pt}{162.00256pt}\pgfsys@lineto{0.0pt}{162.00256pt}\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
} {{}{}}{{}{}}{{}{}}{}{}{}{}{} \pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@setlinewidth{9.0pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.3}\pgfsys@invoke{
}{}\pgfsys@moveto{97.49985pt}{139.3222pt}\pgfsys@curveto{123.34706pt}{148.26947pt}{144.29977pt}{129.4117pt}{144.29977pt}{97.20154pt}\pgfsys@curveto{144.29977pt}{64.99138pt}{123.34706pt}{31.62762pt}{97.49985pt}{22.68036pt}\pgfsys@curveto{71.65263pt}{13.7331pt}{50.69992pt}{32.59087pt}{50.69992pt}{64.80103pt}\pgfsys@curveto{50.69992pt}{97.01118pt}{71.65263pt}{130.37494pt}{97.49985pt}{139.3222pt}\pgfsys@closepath\pgfsys@moveto{97.49985pt}{81.00128pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}{}}{}{{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope }} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\frac{\alpha_{s}}{\pi}~{}\frac{\differential\theta}{\theta}~{}\differential
z~{}[\bar{p}_{i}(z)]^{(1/2)}_{+}~{}~{}~{}\triangleq~{}~{}~{}\leavevmode\hbox
to60.01pt{\vbox to50.47pt{\pgfpicture\makeatletter\hbox{\hskip
1.06201pt\lower-1.41602pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{189.09966pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{194.9997pt}{0.0pt}\pgfsys@lineto{194.9997pt}{162.00256pt}\pgfsys@lineto{0.0pt}{162.00256pt}\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
} {{}{}}{{}{}}{{}{}}{}{}{}{}{} \pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{35.09995pt}{81.00128pt}\pgfsys@curveto{50.17747pt}{89.94855pt}{65.89203pt}{84.14615pt}{70.19989pt}{68.04108pt}\pgfsys@curveto{74.50775pt}{51.936pt}{65.77745pt}{31.62762pt}{50.69992pt}{22.68036pt}\pgfsys@curveto{35.62239pt}{13.7331pt}{19.90784pt}{19.53549pt}{15.59998pt}{35.64056pt}\pgfsys@curveto{11.29211pt}{51.74564pt}{20.02242pt}{72.05402pt}{35.09995pt}{81.00128pt}\pgfsys@closepath\pgfsys@moveto{42.89993pt}{51.84082pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}{}}{}{{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{{}{{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}{}{}{}{ {}{}} {}{{}}{}}}{{}{}{}{}{}{ {}{}} {}{{}}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{116.67491pt}{9.783pt}\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scalebox{2.5}{$\log(\theta^{-1})$}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{{}{{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}
{}{}{}{ {}{}} {}{{}}{}}}{{}{} {}{}{}{ {}{}} {}{{}}{}} {{}{{}}}{{}{}}{}{{}{}} {
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scalebox{2.5}{$\log(z^{-1})$}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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to59.4pt{\vbox to49.5pt{\pgfpicture\makeatletter\hbox{\hskip
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
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}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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}\pgfsys@endscope }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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to165pt{\pgfpicture\makeatletter\hbox{\hskip 1.5pt\lower-1.5pt\hbox
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{ }
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{189.09966pt}{0.0pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{156.10252pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{1.0}{-1.0}{0.0}{0.0pt}{156.10252pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{194.9997pt}{0.0pt}\pgfsys@lineto{194.9997pt}{162.00256pt}\pgfsys@lineto{0.0pt}{162.00256pt}\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
} {{}{}}{{}{}}{{}{}}{}{}{}{}{} \pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@setlinewidth{3.0pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.3}\pgfsys@invoke{
}{}\pgfsys@moveto{35.09995pt}{81.00128pt}\pgfsys@curveto{50.17747pt}{89.94855pt}{65.89203pt}{84.14615pt}{70.19989pt}{68.04108pt}\pgfsys@curveto{74.50775pt}{51.936pt}{65.77745pt}{31.62762pt}{50.69992pt}{22.68036pt}\pgfsys@curveto{35.62239pt}{13.7331pt}{19.90784pt}{19.53549pt}{15.59998pt}{35.64056pt}\pgfsys@curveto{11.29211pt}{51.74564pt}{20.02242pt}{72.05402pt}{35.09995pt}{81.00128pt}\pgfsys@closepath\pgfsys@moveto{42.89993pt}{51.84082pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}{}}{}{{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope }} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$ (59)

where the shaded oval in the Lund plane above represents an arbitrary shape,
and $[\bar{p}(z)]^{(1/2)}_{+}$ is a plus-function regulated reduced splitting
function, $\bar{p}_{i}(z)=p_{i}(z)+p_{i}(1-z)$, using the definition of plus-
function regularization of Eq. (47).282828We may use
$[\bar{p}(z)]^{(1/2)}_{+}$ for the applications presented in this work because
we are interested in observables whose dependence on energy is captured
entirely by the energy fraction of the softer parton at any branching. If we
would like to deal with observables that require more information than just
the softer energy fraction, such as the flavor of the emitted partons, we
would need to work with a plus-function regulated splitting function
$[p(z)]^{(1)}_{+}$.  It follows, for example, that

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}\pgfsys<EMAIL_ADDRESS>(60)

As depicted above, we may replace any vertical line – representing an integral
over $z$ at fixed $\theta$ – or any vertical strip – representing an integral
over $z$ and some range of $\theta$ – with zero, since the integral of a plus-
function regulated distribution is zero.

In this paper, we are concerned with the jet ECFs $C_{1}^{(\varsigma)}$, which
can be roughly calculated by adding contributions from all of the emissions
within a jet, such that the contribution from each emission increases as $z$
or $\theta$ increases and vanishes in the limit that $z$ or $\theta$ vanishes.
We call such observables additive observables, and we may treat the
$C_{1}^{(\varsigma)}$ as additive observables in our leading order
computations.

The calculation of distributions of additive observables is simplified by the
approximately uniform distribution of partonic emissions within the Lund
plane. In particular, the uniformity of emissions in the Lund plane implies
that there is a single emission with splitting parameters $z$ and $\theta$
that we expect to be exponentially higher than the values of $z$ and $\theta$
of other emissions. Additive observables are therefore generically dominated
by the contribution from a single such emission, and we may approximate the
probability that an observable $\mathcal{O}$ is less than the sum of
contributions from different emissions,
$\mathbb{P}(\mathcal{O}<\sum_{i}x_{i})$, by the probability that it is less
than the dominant value, $\mathbb{P}(\mathcal{O}<X)$, with $X=\max_{j}x_{j}$.
We refer to this as the leading logarithmic (LL) approximation.

Mathematically, the LL approximation for an additive observable $\mathcal{O}$,
with contributions $x_{i}>0$ from emissions $i$, asserts that there is a
dominant contribution $X=\max_{i}x_{i}$ such that
$\mathcal{O}=\sum_{i}x_{i}\approx X$. This approximation holds up to effects
that are sub-leading in $\alpha_{s}$ and logarithms of the observable. In the
calculation of cumulative distributions for $\mathcal{O}$, we may write
without approximation that
$\Theta(\mathcal{O}-\sum_{i}x_{i})\leq\Theta(\mathcal{O}-X)$, where equality
emerges when we have at most a single non-zero $x_{j}=X$. The LL approximation
notes that $x_{i}/X$ is exponentially small, and therefore that
$\Theta(\mathcal{O}-\sum_{i}x_{i})\approx\Theta(\mathcal{O}-X)$. We should
also note that typically, calculations applying the LL approximation will also
drop terms that are suppressed by powers of $X$ since these are sub-dominant
to the leading logarithmic terms in observable distributions.

We therefore define the LL cumulative probability distribution
$\Sigma^{\rm(LL)}_{\mathcal{O}}(X)$ of an additive observable $\mathcal{O}$ as
the probability that no emission contributes a value greater than $X$ to
$\mathcal{O}$, up to sub-leading behavior in $\alpha_{s}$ and logarithms or
powers of the observable. The region of phase space in which an observable
contributes a value greater than $X$ to the observable is called a veto
region. We sum and integrate over all possible final-state emissions outside
the veto region when calculating $\Sigma^{\rm(LL)}_{\mathcal{O}}(x)$.

As an example, the approximate mass of a jet due to a single emission is given
by $m^{2}=Q^{2}z\theta^{2}$, where $Q$ is the total energy of the jet.
Therefore, the jet mass distribution is given at leading logarithmic accuracy
by

$\displaystyle\Sigma^{\rm(LL)}_{m^{2}}(m^{2})$
$\displaystyle=\sum_{n=0}^{\infty}\frac{1}{n!}\left(\frac{\alpha_{s}}{\pi}\int_{0}^{1}\frac{\differential\theta}{\theta}\int_{0}^{\frac{1}{2}}\differential
z\,[\bar{p}_{i}(z)]_{+}^{[1/2]}\Theta\left(\frac{m^{2}}{Q^{2}}-z\theta^{2}\right)\right)^{n}$
(61)
$\displaystyle=\exp\left[-\frac{\alpha_{s}}{\pi}\int_{0}^{1}\frac{\differential\theta}{\theta}\int_{0}^{\frac{1}{2}}\differential
z\,\bar{p}_{i}(z)\Theta\left(z\theta^{2}-\frac{m^{2}}{Q^{2}}\right)\right].$

In graphical notation, the analogous derivation is

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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}{}}{}{{}}{}{{}{}} {}{}
\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}{{}{}}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}}{}{}{}{}{} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.64279}{-0.76604}{0.76604}{0.64279}{5.7665pt}{40.44273pt}\pgfsys@invoke{
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}\hbox{{${\scriptstyle\frac{m^{2}}{Q^{2}}<z\theta^{2}}$}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}{}}{}{{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}{{}{}{}{}{{}}{}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.59999pt}{0.0pt}\pgfsys@invoke{
}\hbox{}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope }} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{}~{}\right],$ (62)

where in the final equality, we see that the LL cumulative distribution is one
over the exponent of the veto region area. The area of the veto region for an
observable is also called the radiator $R_{\mathcal{O}}$ for the observable.

A similar analysis yields the cumulative probability distribution function of
the angle $\theta$ for the first emission in the P-RSFf=1/2 procedure to
satisfy $z>z_{\rm cut}$, which we call the critical emission:

$\displaystyle\Sigma^{\rm(LL)}_{{\rm crit},\theta}(\theta_{\rm crit})$
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