aainstitutetext: Institute for Advanced Study, Einstein Drive, Princeton, NJ
08540, USAbbinstitutetext: Department of Physics, Harvard University,
Cambridge, MA 02138, USAccinstitutetext: Deutsches Elektronen-Synchrotron
DESY, Notkestr. 85, 22607 Hamburg, Germanyddinstitutetext: University of
Manchester, School of Physics and Astronomy, Manchester, M13 9PL, United
Kingdomeeinstitutetext: University of Vienna, Faculty of Physics,
Boltzmanngasse 5, A-1090 Vienna, Austriaffinstitutetext: Center for
Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA
02139, USA

# Prospects for strong coupling measurement at hadron colliders using soft-
drop jet mass

Holmfridur S. Hannesdottir c,d,e Aditya Pathak b Matthew D. Schwartz f Iain W.
Stewart<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We compute the soft-drop jet-mass distribution from $pp$ collisions to NNLL
accuracy while including nonperturbative corrections through a field-theory
based formalism. Using these calculations, we assess the theoretical
uncertainties on an $\alpha_{s}$ precision measurement due to higher order
perturbative effects, nonperturbative corrections, and PDF uncertainty. We
identify which soft-drop parameters are well-suited for measuring
$\alpha_{s}$, and find that higher-logarithmic resummation has a qualitatively
important effect on the shape of the jet-mass distribution. We find that quark
jets and gluon jets have similar sensitivity to $\alpha_{s}$, and emphasize
that experimentally distinguishing quark and gluon jets is not required for an
$\alpha_{s}$ measurement. We conclude that measuring $\alpha_{s}$ to the 10%
level is feasible now, and with improvements in theory a 5% level measurement
is possible. Getting down to the 1% level to be competitive with other state-
of-the-art measurements will be challenging.

###### Keywords:

QCD, Colliders, Precision Physics

††preprint:  MIT–CTP 5461 DESY–22–156 UWThPh 2022–15

##  1 Introduction

The QCD strong coupling constant $\alpha_{s}$ is an essential ingredient in
theoretical predictions for the Large Hadron Collider (LHC). Using a variety
of high-energy event-shape measurements in scattering processes L3:1992nwf ;
SLD:1994idb ; DELPHI:1996oqw ; ALEPH:2003obs ; DELPHI:2004omy ; OPAL:2004wof ;
Dissertori:2007xa ; Becher:2008cf ; Davison:2009wzs ; Bethke:2009ehn ;
Dissertori:2009ik ; Abbate:2010xh ; Abbate:2012jh ; Hoang:2015hka ;
Hoang:2014wka ; CMS:2019oeb ; ATLAS:2020mee ; Aoki:2021kgd , the value of
$\alpha_{s}$ has been measured to a precision of $\lesssim 1$%
ParticleDataGroup:2018ovx ; ParticleDataGroup:2020ssz . However, there is
currently more than 3$\sigma$ tension between the most precise extractions,
which are from low-energy extractions in lattice QCD Aoki:2021kgd , and from
thrust and C-parameter measurements in $e^{+}e^{-}$ collisions at LEP
Abbate:2010xh ; Abbate:2012jh ; Hoang:2015hka . To resolve this discrepancy,
it is important to consider alternative observables for determining
$\alpha_{s}$. One such candidate is the soft-drop (SD) Larkoski:2014wba  jet-
mass cross section at hadron colliders. Soft-drop jet mass is an important
jet-substructure observable Larkoski:2017cqq ; Larkoski:2017iuy ;
Baron:2018nfz ; Kang:2018vgn ; Makris:2018npl ; Kardos:2018kth ; Chay:2018pvp
; Napoletano:2018ohv ; Lee:2019lge ; Hoang:2019ceu ; Gutierrez-Reyes:2019msa ;
Kardos:2019iwa ; Marzani:2019evv ; Mehtar-Tani:2019rrk ; Kardos:2020ppl ;
Larkoski:2020wgx ; Lifson:2020gua ; Caucal:2021cfb  because the soft-drop
grooming reduces pile-up contamination and sensitivity to hadronization while
still allowing for precise theoretical calculations. The perturbative state-
of-the-art of the soft-drop jet mass is next-to-next-to-leading logarithmic
(NNLL) accuracy for the general case Frye:2016aiz , and results are even known
to N3LL for $e^{+}e^{-}$ collisions with no angular weight to the grooming
Kardos:2020gty . Thus a precise extraction of $\alpha_{s}$ through soft-drop
observables is foreseeable.

The feasibility of using soft-drop thrust to measure $\alpha_{s}$ has already
been investigated in detail in Ref. Marzani:2019evv  for $e^{+}e^{-}$
colliders. Here our focus is on the use of soft-drop jet mass for $\alpha_{s}$
extraction in $pp$ colliders. Although extractions from $pp$ colliders are
more challenging than those at $e^{+}e^{-}$ colliders, due to the more
complicated environment, given the upcoming high-luminosity phase of the Large
Hadron Collider (LHC), it is timely to probe the prospects of this observable
for precision $\alpha_{s}$ determination. Points of reference for current
precision of hadron collider $\alpha_{s}$ measurements are the 2019 PDG value
of $\alpha_{s}(m_{Z})=0.1159\pm 0.0034$ ParticleDataGroup:2020ssz  which has
3% uncertainty, as well as the more recent TEEC measurement by ATLAS
ATLAS:2020mee , $\alpha_{s}(m_{Z})=0.1196\pm
0.004(\text{exp.}){}^{+0.0072}_{-0.0105}(\text{theo.})$ which has 9%
uncertainty. In this work, we will consider the cross-section differential in
the soft-drop jet mass for boosted jets with $p_{T}\geq 600$ GeV, with an eye
towards the opportunity for future higher precision experimental measurements
of soft-drop jet-mass cross sections.

The soft-drop procedure traces the clustering history of the jet, grooming
branches with soft radiation, until a sufficiently hard branch is found. More
precisely, the jet is first re-clustered using the Cambridge/Aachen algorithm,
and next, the groomer un-clusteres the jet, rejecting the softer subjet at
each step, unless the pair satisfies the soft-drop criteria given by

$\displaystyle\frac{\min(p_{T_{i}},p_{T_{j}})}{p_{T_{i}}+p_{T_{j}}}>z_{\mathrm{cut}}\Big{(}\frac{\Delta
R_{ij}}{R_{0}}\Big{)}^{\beta}\,,$ (1.1)

where $\Delta R_{ij}$ is the distance between the two branches and $z_{\rm
cut}$, $\beta$ and $R_{0}$ are parameters that control the strength of the
groomer. The first time a branch is found that satisfies Eq. (1.1), the
groomer stops, and all the remaining emissions are kept.

Compared to ungroomed observables, the soft-drop jet-mass spectrum is less
sensitive to pile-up and hadronization corrections, for a wide range of jet
masses. In addition, jet grooming allows for a more robust perturbative
prediction: in the region soft drop is active, the non-global logarithms (NGL)
arising from soft radiation crossing the jet boundary can only modify the
normalization Frye:2016aiz . The impact on the shape of the spectrum arises
only due to differences in the NGL-effect on the normalization in quark and
gluon-initiated jets, and hence is typically smaller compared to ungroomed
jets. Thus, the soft-drop jet mass is well-suited for hadron-collider
predictions and may provide a powerful cross check on the existing
$\alpha_{s}$ measurements based on ungroomed event shapes in $e^{+}e^{-}$
collisions.

Figure 1: Jet-mass spectrum for quark and gluon jets in Pythia8, at parton and
hadron levels and with the underlying event (modeled as multi-parton-
interaction (MPI)) with and without soft-drop grooming. We use the shorthand
$\xi=m_{J}^{2}/(p_{T}R)^{2}$. The green vertical dot-dashed line indicates the
jet masses below which soft-drop grooming becomes active and the groomed
spectrum deviates significantly from the corresponding ungroomed spectrum. To
the right of the black-dotted line, for $\xi\gtrsim\xi_{\rm FO}$ ,the fixed
order non-singular corrections to the spectrum become important. For the soft-
drop spectrum, the nonperturbative corrections become ${\cal O}(1)$ only to
the left of the brown dotted line given by $\xi=\xi_{\rm SDOE}$.

Some of the characteristic features of the soft-drop jet-mass distribution are
shown in Fig. 1. The figure shows the differential jet-mass cross section
simulated in Pythia8 for quark and gluon jets with $p_{T}\in[550,650]$ GeV and
$R_{\rm jet}=0.8$ without (upper panel) and with (lower panel) grooming,
choosing representative values of soft-drop parameters: $(z_{\rm
cut},\beta)=(0.1,1)$. The figure demonstrates that in the low jet-mass region,
the soft-drop jet mass has a much larger range that is dominated by
perturbation theory, compared to its ungroomed counterparts, allowing for fits
to $\alpha_{s}$ in a region with a larger cross section.

It is important to delineate the region where soft-drop grooming is active and
predictions for the spectrum are dominated by perturbation theory, with
nonperturbative effects giving small power-suppressed corrections. The
criterion that ensures that nonperturbative effects are power corrections is

$\displaystyle\frac{Q\Lambda_{\rm
QCD}}{m_{J}^{2}}\Bigl{(}\frac{m_{J}^{2}}{QQ_{\rm
cut}}\Bigr{)}^{\frac{1}{2+\beta}}\ll 1\qquad\Rightarrow\,\text{soft-drop
stopping emission is perturbative.}$ (1.2)

For jet masses $m_{J}$ that satisfy this condition, the transverse momentum
$k_{t}$ of the soft-drop stopping emission will be perturbative, i.e.
$k_{t}\gg\Lambda_{\rm QCD}$, and consequently the nonperturbative effects are
subleading. More precisely, we take $\Lambda_{\rm QCD}=1$ GeV and replace the
inequality with $\frac{Q\Lambda_{\rm
QCD}}{m_{J}^{2}}\Bigl{(}\frac{m_{J}^{2}}{QQ_{\rm
cut}}\Bigr{)}^{\frac{1}{2+\beta}}=\frac{1}{5}$. This transition point
demarcates the start of the soft drop operator expansion region (SDOE)
Hoang:2019ceu  and is shown in Fig. 1 by the brown dotted line at
$\xi=\xi_{\rm SDOE}$, where $\xi=m_{J}^{2}/(p_{T}R)^{2}$. This region is also
the one where resummation of logarithms of ratio of scales corresponding to
soft drop passing and failing emissions is important, and a rigorous
perturbative factorization theorem can be established. At even lower values of
the jet mass, when $k_{t}\sim\Lambda_{\rm QCD}$ for the emission that stops
soft drop, then we are in the soft-drop nonperturbative region (SDNP), i.e.
when $\frac{Q\Lambda_{\rm QCD}}{m_{J}^{2}}\Bigl{(}\frac{m_{J}^{2}}{QQ_{\rm
cut}}\Bigr{)}^{\frac{1}{2+\beta}}=1$. We define this region as $\xi<\xi_{\rm
SDNP}$, with a precise definition given in Eq. (4.1) below. Here
nonperturbative effects are ${\cal O}(1)$ and we will not attempt to compute
the cross section in this region. As the jet mass increases above the green
dot-dashed line at $\xi=\xi_{\rm 0}$ in Fig. 1, the soft drop grooming becomes
less effective. This occurs because for jet masses $\xi>\xi_{0}$ the very
first subjet seen upon declustering the jet is already energetic enough to
pass the groomer. The criterion for this to happen is

$\displaystyle\frac{m_{J}^{2}}{QQ_{\rm cut}}\gtrsim
1\qquad\Rightarrow\,\text{soft drop is less significant.}$ (1.3)

Hence, we refer to this region as the plain jet-mass resummation region. As
can be see in from Fig. 1 the parton, hadron and underlying event level
spectra look almost identical for $\xi>\xi_{0}$ for both groomed and ungroomed
cases. Finally, the region, for which $\xi>\xi_{\rm FO}$ is referred to as the
fixed-order region. In this analysis, we will not investigate fitting
$\alpha_{s}$ for $\xi>\xi_{0}$, since obtaining the same precision as in the
SDOE region would require a more dedicated study of the cusp region at
$\xi\sim\xi_{\rm 0}$. We discuss the various regions in detail and provide
precise definitions of these transition points in Sec. 2.1.

In our analysis we compute NNLL perturbative predictions, in addition to
employing the formalism of Ref. Hoang:2019ceu  to include the leading
nonperturbative power corrections. We also include soft-drop non-singular
fixed-order effects, such as the ones in Ref. Marzani:2019evv , but, as
discussed above, we do not include resummation in the cusp region as
investigated in Ref. Benkendorfer:2021unv . We also show in Sec. 2.7.2 below
that the effects of NGLs at LL and large $N_{c}$ limit is within the NNLL
perturbative uncertainty, and hence we do not directly study the effects of
NGL normalization effects on the $\alpha_{s}$-sensitivity of the groomed jet
mass. A more detailed comparison with previous work is summarized below in
Sec. 2.9.

As we can see from Fig. 1, the hadronization corrections are still non-
negligible in the perturbative region and must be taken into account in order
to achieve a percent-level precision measurement on $\alpha_{s}$. The
nonperturbative (NP) effects to the soft-drop jet mass were studied in Ref.
Hoang:2019ceu  using a field-theoretic method that is independent of any
hadronization model, and are given by

$\displaystyle\frac{1}{\sigma_{\kappa}}\frac{\textrm{d}\sigma_{\kappa}}{\textrm{d}m_{J}^{2}}=\frac{1}{\hat{\sigma}_{\kappa}}\frac{\textrm{d}\hat{\sigma}_{\kappa}}{\textrm{d}m_{J}^{2}}-Q\Omega_{1\kappa}^{\circ\\!\\!\circ}\frac{\textrm{d}}{\textrm{d}m_{J}^{2}}\bigg{(}C_{1}^{\kappa}(m_{J}^{2})\frac{1}{\hat{\sigma}_{\kappa}}\frac{\textrm{d}\hat{\sigma}_{\kappa}}{\textrm{d}m_{J}^{2}}\bigg{)}+\frac{Q\Upsilon_{1\kappa}^{\varocircle}(\beta)}{m_{J}^{2}}C_{2}^{\kappa}(m_{J}^{2})\frac{1}{\hat{\sigma}_{\kappa}}\frac{\textrm{d}\hat{\sigma}_{\kappa}}{\textrm{d}m_{J}^{2}}\,.$
(1.4)

In this expression, $\textrm{d}\hat{\sigma}_{\kappa}$ is the partonic cross
section, and the hadron-level cross section $\textrm{d}\sigma_{\kappa}$ is
obtained by including the NP power corrections represented by the two latter
terms. $\textrm{d}\sigma_{\kappa}$ represents the normalized cross section for
jets initiated by parton $\kappa$ and will be precisely defined below. The
functions $C_{1}^{\kappa}$ and $C_{2}^{\kappa}$ are parton-level objects that
describe how the power corrections vary as a function of jet-kinematic and
grooming parameters. Eq. (1.4) is valid at LL accuracy where a distinct two-
pronged geometry of the groomed jet can be defined. At higher orders there can
be subleading corrections that are distortions to this LL-approximation.
Nevertheless, as shown in Refs. Pathak:2020iue ; AP , a more accurate
computation of the Wilson coefficients leads to improved description of the
leading NP corrections. They have been computed to NNLL accuracy in Refs.
Pathak:2020iue ; AP , which we will employ in our analysis below. The power
corrections for each $\kappa=q,g$ involve six unknown nonperturbative
constants $\Omega_{1\kappa}^{\circ\\!\\!\circ}$,
$\Upsilon_{1,0\kappa}^{\varocircle}$ and $\Upsilon_{1,1\kappa}^{\varocircle}$,
with
$\Upsilon_{1\kappa}^{\varocircle}(\beta)=\Upsilon_{1,0\kappa}^{\varocircle}+\beta\Upsilon_{1,1\kappa}^{\varocircle}$,
where $\kappa$ indicates that the constants differ for quark and gluon jets.
These constants are universal in the sense that they are independent of the
jet kinematics and the grooming parameters. The superscripts signify specific
soft-drop related geometric constraints that we review in Sec. 3.

Since the impact of the power corrections is non-uniform across the SDOE
region, as can be seen in Fig. 1, the nonperturbative constants must ideally
be fit for along with $\alpha_{s}$. It will, however, be challenging to fit
the theoretical prediction to experimental data for 7 parameters: $\alpha_{s}$
along with the 6 constants. Instead of analyzing the prospects for such a fit,
we instead assess the uncertainty on a precision $\alpha_{s}$ measurement due
to the lack of knowledge of these NP parameters. This allows us to estimate
the ultimate precision on $\alpha_{s}$ that can be achieved in absence of any
perturbative uncertainty, while not attempting to constrain the NP power
corrections. In this work, we use values of these 6 constants from studies of
Monte Carlo hadronization models Ferdinand:2022xxx  for the estimate, and show
that the size of the power corrections turns out to be comparable to
perturbative uncertainty at NNLL accuracy.

Given the setup above, we will investigate the sensitivity of the cross
section to $\alpha_{s}$ in the SDOE region, and to what extent perturbative
variations and nonperturbative power corrections impact the uncertainty in
$\alpha_{s}$ determination. The two main features of the jet-mass spectrum
that can be exploited for a precision-$\alpha_{s}$ measurement are its shape
and the integral of the jet-mass curve. We will state results for two ways of
normalizing the cross section:

$\displaystyle\text{Normalize to inclusive cross section in the $p_{T}$-$\eta$
bin:}\quad\frac{1}{\sigma_{\rm
incl}}\frac{\textrm{d}^{3}\sigma}{\textrm{d}p_{T}\textrm{d}\eta\textrm{d}\xi}\,,$
(1.5) $\displaystyle\text{Normalize to cross section in a fixed jet-mass
range:}\quad\frac{1}{\sigma_{\rm fitrg}(\xi_{\rm SDOE},\xi_{\rm
0}^{\prime})}\frac{\textrm{d}^{3}\sigma}{\textrm{d}p_{T}\textrm{d}\eta\textrm{d}\xi}\,,$
(1.6)

where

$\displaystyle\sigma_{\rm incl}$
$\displaystyle=\frac{\textrm{d}^{2}\sigma}{\textrm{d}p_{T}\textrm{d}\eta}\,,$
$\displaystyle\sigma_{\rm fitrg}(\xi_{\rm SDOE},\xi_{\rm 0}^{\prime})$
$\displaystyle=\int_{\xi_{\rm
SDOE}}^{\xi_{0}^{\prime}}\textrm{d}\xi\>\frac{\textrm{d}^{3}\sigma}{\textrm{d}p_{T}\textrm{d}\eta\textrm{d}\xi}\,.$
(1.7)

Both normalization choices reduce sensitivity to the PDFs, non-global
logarithms and the luminosity. Here $\xi_{0}^{\prime}$ is the transition point
for the NNLL spectrum, that differs slightly from $\xi_{0}$, and is defined in
Eq. (2.9). Hence, the dominant region of interest for $\alpha_{s}$ fits is
taken to be $\xi\in[\xi_{\rm SDOE},\xi_{0}^{\prime}]$. It is customary to
normalize the spectrum by its integral in the SDOE region, i.e. the choice in
Eq. (1.6), as was done in the recent soft-drop jet-mass measurements by the
ATLAS collaboration ATLAS:2017zda ; ATLAS:2019mgf ; ATLAS:2021urs . While
normalizing the cross section in a fixed range reduces perturbative
uncertainty, we will show below that it loses almost all sensitivity to
$\alpha_{s}$, since the shape of the cross section in the SDOE region remains
largely invariant upon variations in $\alpha_{s}$. The sensitivity to
$\alpha_{s}$ therefore lies in the integral of the cross section in the
perturbative region (SDOE region and beyond). Thus, the fixed-range
normalization choice greatly reduces sensitivity to $\alpha_{s}$ variations
compared with the variations due to perturbative uncertainty and
nonperturbative corrections, and hence increases uncertainties in any
potential $\alpha_{s}$ measurements. This insensitivity is particularly severe
for quark jets where, at NNLL accuracy, the spectrum in the SDOE region
flattens and variations in $\alpha_{s}$ have almost no effect on the cross
section normalized to a fixed range. Based on this study, we will justify our
recommendation for the first choice in Eq. (1.5), and avoid self-normalizing
the cross section in the SDOE region.

The outline of the paper is as follows: In Sec. 2 we review the details of
resummation of the soft-drop jet-mass spectrum using soft-collinear effective
field theory (SCET) Bauer:2000ew ; Bauer:2000yr ; Bauer:2001ct ; Bauer:2001yt
. The framework for the NP power corrections is discussed in Sec. 3. Finally,
in Sec. 4 we employ this setup in testing the sensitivity to $\alpha_{s}$ and
impact of the NP power corrections via the procedure outlined above. We
conclude in Sec. 5. The appendices provide details of derivations and
supplementary material on the numerical analysis.

##  2 Perturbative predictions

The various regions shown in Fig. 1 are distinguished by specific logarithmic
or power corrections involving ratios of different energy scales that become
significant. We implement the resummation of these logarithms and the
treatment of nonperturbative power corrections using SCET.

###  2.1 Kinematic features

We first review the main kinematic features of the soft-drop spectrum.
Although we are primarily concerned with $\alpha_{s}$ determination at the
hadron colliders, our presentation of the formalism is done such that it is
equally applicable to $e^{+}e^{-}$ collisions. In the case of $pp$ collisions,
we will primarily focus on jets identified inclusively with $R/2\ll 1$. The
results for $e^{+}e^{-}$ remain valid for large $R$ jets and reduce to
familiar expressions for hemisphere jets when $R=\pi/2$. We refer the reader
to Refs. Frye:2016aiz ; Hoang:2019ceu ; Pathak:2020iue  for a more thorough
discussion and derivation of some of the results stated here.

The large momentum associated with the jet is given by

$\displaystyle Q=p_{T}R$ $\displaystyle(\textrm{$pp$ case})\,,$ $\displaystyle
Q=2E_{J}\tan\frac{R}{2}$ $\displaystyle(\textrm{$e^{+}e^{-}$ case})\,.$ (2.1)

We will be concerned with the region of the jet-mass spectrum that is
dominated by soft and collinear emissions. We define variables $z_{i}$, the
$p_{T}$ or energy fraction of a soft subjet, given by

$\displaystyle z_{i}=\frac{p_{T_{i}}}{p_{T_{J}}}$ $\displaystyle(\textrm{$pp$
case})\,,$ $\displaystyle z_{i}=\frac{E_{i}}{E_{J}}$
$\displaystyle(\textrm{$e^{+}e^{-}$ case})\,,$ (2.2)

and angle $\Delta R_{iJ}$ or $\theta_{iJ}$ that it makes with the jet axis.
The soft-drop criteria in Eq. (1.1) for passing for soft collinear limit
$z_{i}\rightarrow 0$ and $\theta_{i}\rightarrow 0$ can be compactly expressed
as

$\displaystyle z_{i}>z_{\rm cut}^{\prime}\Delta R_{iJ}^{\beta}$
$\displaystyle(\textrm{$pp$ case})\,,$ $\displaystyle z_{i}>\tilde{z}_{\rm
cut}\Big{(}2\sin\frac{\theta_{iJ}}{2}\Big{)}^{\beta}$
$\displaystyle(\textrm{$e^{+}e^{-}$ case})\,,$ (2.3)

where in defining $z_{\rm cut}^{\prime}$ and $\tilde{z}_{\rm cut}$ we have
absorbed the parameter $R_{0}$:

$\displaystyle z_{\rm cut}^{\prime}\equiv z_{\rm
cut}\big{(}R_{0}^{(pp)}\big{)}^{-\beta}$ $\displaystyle(\textrm{$pp$
case})\,,$ $\displaystyle\tilde{z}_{\rm cut}\equiv z_{\rm
cut}\bigg{[}\sqrt{2}\sin\Big{(}\frac{R^{e^{+}e^{-}}_{0}}{2}\Big{)}\bigg{]}^{-\beta}$
$\displaystyle(\textrm{$e^{+}e^{-}$ case})\,.$ (2.4)

With this definition, we can identify the energy scale $Q_{\rm cut}$
associated with soft drop:

$\displaystyle Q_{\rm cut}=p_{T}z_{\rm cut}^{\prime}R^{1+\beta}$
$\displaystyle\text{($pp$ case)}\,,$ $\displaystyle Q_{\rm
cut}=\Big{(}2\tan\frac{R}{2}\Big{)}^{1+\beta}E_{J}\tilde{z}_{\rm cut}$
$\displaystyle(\textrm{$e^{+}e^{-}$ case})\,.$ (2.5)

Finally, we will find it convenient to work with a dimensionless combination
of the jet mass and the large momentum $Q$

$\displaystyle\xi\equiv\frac{m_{J}^{2}}{Q^{2}}\,.$ (2.6)

As shown in Fig. 1, the soft-drop jet-mass spectrum can be divided into four
distinct regions. The regions encountered as we reduce $\xi$ from its maximum
value $\xi=1$ are as follows:

  1. 1.

The fixed-order region: This region corresponds to groomed jet masses close to
the end point of the spectrum for $\xi\lesssim 1$. In this region the
logarithms $\textrm{ln}(\xi)$ generated at each perturbative order are ${\cal
O}(1)$ and the region can be adequately described in fixed-order perturbation
theory. In practice, at a given order in $\alpha_{s}$ expansion, only a finite
number of particles contribute to the jet mass, and hence the end point is
seen at somewhat smaller values. For leading-order (LO) analysis, $\xi_{\rm
LO}=0.25$, and for next-to-leading-order (NLO) we have $\xi_{\rm NLO}\simeq
0.4$ Marzani:2017kqd . In our analysis we will treat this region to ${\cal
O}(\alpha_{s})$, the LO accuracy. We also retain all the finite $z_{\rm cut}$
terms in this region.

  2. 2.

Plain jet-mass resummation region: This region is dominated by soft and
collinear emissions and corresponds to $\xi_{0}<\xi\ll 1$, and the logarithms,
$\alpha_{s}^{n}\textrm{ln}^{m}(\xi)$ with $m\leq 2n$ become large and must be
resummed. Here

$\displaystyle\xi_{0}=z_{\rm cut}^{\prime}R^{\beta}$ $\displaystyle(\text{$pp$
case})\,,$ $\displaystyle\xi_{0}=\tilde{z}_{\rm
cut}\Big{(}2\tan\frac{R}{2}\Big{)}^{\beta}$ $\displaystyle(\text{$e^{+}e^{-}$
case})\,.$ (2.7)

From Eqs. (2.1) and (2.5), we see that

$\displaystyle Q_{\rm cut}=Q\xi_{0}\,.$ (2.8)

As discussed below in Sec. 2.4.2, power corrections of ${\cal O}(\xi)$ are
small and contributions from soft and collinear modes in this region
factorize. However, soft drop does not significantly groom the jet in this
region, as one of the first few de-clusterings pass the soft-drop condition.
Consequently, the jet mass after grooming is only slightly smaller than its
value prior to grooming. Thus, in this region the effect of soft drop alone
can be treated in fixed-order perturbation theory. Below the $\xi_{0}$
transition point this situation changes, and resummation must take into
account a change in the logarithmic structure due to soft drop. Note, however,
that close to the cusp, the soft emissions passing soft drop are at angles
$\Delta R_{ij}\sim R$ and hence one must employ the version of soft drop
without the collinear approximation $\Delta R_{ij}\rightarrow 0$. Doing so
leads to the “soft wide-angle” transition point $\xi_{0}^{\prime}$ differing
slightly from $\xi_{0}$ Baron:2018nfz :

$\displaystyle\xi_{0}^{\prime}\equiv\frac{\xi_{0}}{(1+\zeta^{2})^{\frac{2+\beta}{2}}}\,.$
(2.9)

where $\zeta$ for the two cases is given by:

$\displaystyle\zeta\equiv\tan\frac{R}{2}\quad(\text{$e^{+}e^{-}$
case})\,,\qquad\zeta\equiv\frac{R}{2\cosh\eta_{J}}\quad(\text{$pp$ case})\,.$
(2.10)

The value of $\xi_{0}^{\prime}$ corresponds to the location of the distinct
soft-drop cusp. Defining the parameter $\zeta$ will prove useful in
universally keeping track of wide-angle soft effects.

  3. 3.

Soft-drop resummation region: As we move past $\xi_{0}$ to yet smaller jet
masses $\xi<\xi_{0}$, the application of soft drop removes a significant
amount of soft, wide-angle radiation. The hierarchy $\xi\ll\xi_{0}$ results in
large logarithms $\textrm{ln}(\xi/\xi_{0})$, and a different resummation from
the previous case is required. At the same time, perturbative power
corrections of ${\cal O}(\xi/\xi_{0})$ are small and can be neglected. We will
also restrict ourselves to the case $\xi_{0}\ll 1$, such that power suppressed
terms in $\xi_{0}$, or equivalently $z_{\rm cut}$, can be ignored and the
leading power factorization derived in Ref. Frye:2016aiz  can be employed.
This enables us to further factorize the soft function from the plain jet-mass
region into contributions that fail and pass soft drop, such that the large
logarithms $\alpha_{s}^{n}\textrm{ln}^{m}(\xi/\xi_{0})$ can be resummed. (For
the special case of $\beta=0$ the leading logarithms are single logs with
$m=n$.) From the point of view of nonperturbative power corrections, as long
as the first subjet that stops soft-drop is perturbative (with
$k_{t}\gg\Lambda_{\rm QCD}$), which amounts to satisfying the criterion in Eq.
(1.2), the perturbative prediction dominates. This is the SDOE region where
nonperturbative power corrections can be incorporated through constant
parameters that appear in a systematic expansion. In terms of $\xi$, the
condition in Eq. (1.2) for being in the SDOE region can be expressed as

$\displaystyle\Big{(}\frac{\xi_{\rm
SDNP}}{\xi}\Big{)}^{\frac{1+\beta}{2+\beta}}\ll 1\,,\qquad\text{(SDOE
region)}\,,$ (2.11)

where the parameter $\xi_{\rm SDNP}$, defined by Eq. (2.12) below,
characterizes the region below which nonperturbative corrections become ${\cal
O}(1)$.

  4. 4.

Soft-drop nonperturbative region: Here the nonperturbative effects are ${\cal
O}(1)$. This happens for jet masses

$\displaystyle\xi\gtrsim\xi_{\rm SDNP}\equiv\xi_{0}\bigg{(}\frac{1\,{\rm
GeV}}{Q_{\rm cut}}\bigg{)}^{\frac{2+\beta}{1+\beta}}\,.$ (2.12)

In the SDNP region, the perturbative resummation of large logarithms in SDOE
region still carries over. However, the collinear-soft function describing
dynamics of the emissions that pass the groomer becomes nonperturbative. This
region is equivalent to the shape-function region in ungroomed jets, however
the shape function that appears here is not normalized to one and involves
convolution in fractional power $k_{\rm NP}^{\frac{2+\beta}{1+\beta}}$ of the
nonperturbative momentum $k_{\rm NP}$ Hoang:2019ceu . It remains to be
explored how the nonperturbative power corrections in the SDOE region
(reviewed in Sec. 3) are related to the shape function in the SDNP region.

Further below we will review the perturbative resummation that is necessary in
the plain jet-mass and the soft-drop resummation region. We note that close to
the soft-drop cusp, a more refined treatment along the lines of Ref.
Benkendorfer:2021unv  can be performed by identifying a resolved soft emission
that stops the groomer. Our treatment nonetheless fully captures the dominant
leading order effect at the soft-drop cusp including all the perturbative
power corrections related to the jet mass and soft drop at ${\cal
O}(\alpha_{s})$, and we estimate that the impact from the lack of cusp
resummation is smaller than all other uncertainties for the large values of
$p_{T}$ considered for analysis of $\alpha_{s}$ in $pp$ collisions. We leave
to future work a more detailed integration of the cusp region resummation with
the setup presented here.

###  2.2 Quark/gluon fraction

The quark gluon fraction is a very useful theoretical concept, regardless of
whether there is a direct experimental way to measure the number of quark jets
and gluon jets. To compute the soft-drop jet mass we do not even need to know
whether the two classes of jets can be distinguished. What we compute is the
total jet-mass distribution, inclusive over quarks and gluons. To do so, we
compute the individual jet-mass distribution for quark jets and for gluon jets
separately in SCET, add them together using perturbative predictions for the
quark and gluon jet cross sections or equivalently the quark/gluon fraction,
and then match the sum to full QCD results to include additional fixed-order
corrections.

At tree level, the quark/gluon cross sections are identical in SCET and QCD.
Beyond leading order, the quark/gluon cross sections are ambiguous in QCD but
not in SCET. In SCET, whether a jet is quark or gluon is determined by the
quantum numbers of the collinear fields in the hard scattering operator. Thus,
in SCET the quark/gluon fraction is well-defined and systematically calculable
order-by-order in perturbation theory. Whatever ambiguity there is for the
quark/gluon jet definition in QCD is immaterial, since after matching to the
NLO inclusive (over quarks and gluons) jet-mass distribution we no longer need
to discuss quark/gluon fractions. So at NLO and beyond the quark/gluon
fraction is not meaningful in QCD, but it does not have to be. The quark/gluon
fraction is nevertheless a useful concept because the leading-order prediction
gets small corrections at NLO in SCET and provides valuable intuition for what
the full matched distribution might look like.

Note that the prediction for the cross section also depends on PDFs, as can be
seen from Eqs. (2.17) and (2.66). It has been argued in Ref. Forte:2020pyp
that for observables that are sensitive to PDFs, a consistent determination of
$\alpha_{s}$ cannot be carried out unless PDFs are simultaneously determined.
The arguments in Ref. Forte:2020pyp  apply to inclusive cross-section
measurements, while here we study jet substructure measurements. We remove the
dependence on the inclusive cross section by appropriately normalizing the
cross section as shown in Eq. (1.5), so that our observables become mostly
insensitive to PDFs. The remaining small uncertainty is propagated into our
result through the quark/gluon fractions, but we will show that the resulting
uncertainties on an $\alpha_{s}$ measurement are a subdominant effect compared
with the uncertainties due to scale-variations of the perturbative cross
section, as well as those due to nonperturbative effects.

####  2.2.1 Impact of PDF variations on quark/gluon fractions

To compute the quark gluon fraction we begin at leading order by computing the
rates for dijet production using the different partonic process $q\bar{q}\to
gg$, $qq^{\prime}\to qq^{\prime}$, etc. The quark fraction at leading order is
then

$x_{q}=\frac{2\sigma(pp\to qq)+\sigma(pp\to gq)}{2\sigma(pp\to{\rm
dijet})}\,,$ (2.13)

where $q$ heuristically refers to quarks or antiquarks, identical or distinct.
The gluon fraction is $x_{g}=1-x_{q}$.

We show in Table 1 the tree-level quark/gluon fraction computed with various
PDFs. These fractions were computed with Madgraph Alwall:2014hca  for $pp$
collisions at 13 TeV with a $p_{T}$ cut of 600 GeV on the jets. We show
results for a representative sample from among various available PDF sets. We
see that there is surprisingly little variation among the PDFs with a typical
quark gluon fraction of $x_{q}=0.51$ with about a 2% uncertainty.

PDF |  $\alpha_{s}$ used | $x_{q}$ | % change
---|---|---|---
NNPDF 23 LO | 0.119 | 0.479 | -6.0
NNPDF 23 NLO | 0.119 | 0.517 | 1.3
NNPDF 23 NNLO | 0.119 | 0.523 | 2.5
NNPDF 23 NNLO | 0.120 | 0.514 | 0.84
CT18NLO$\\_$as$\\_$0119 | 0.119 | 0.514 | 0.87
CT18NNLO$\\_$as$\\_$0119 | 0.119 | 0.507 | -0.49
MSTW2008nlo68cl | 0.120 | 0.510 | 0.063
MSTW2008nlo68cl | 0.117 | 0.514 | 0.87
mean | – | 0.510 | 1.6
Table 1: Quark/Gluon fraction for different parton distribution functions
which average to $x_{q}=0.510$ with a standard deviation of $0.013$. The %
changes are listed relative to this mean and the average % change is the
average of the absolute value of these %’s.

To systematically improve the computation for the quark gluon fraction, we
include perturbative corrections using SCET factorization, as discussed below
in Sec. 2.6. In particular, we provide expressions for quark/gluon fractions
for exclusive soft drop groomed dijets identified via a $p_{T}^{\rm veto}$ on
additional jets. Taking the result from there (in a schematic form) we have

$\displaystyle
x_{q}=\frac{1}{N}\sum_{\\{\kappa_{i}\\}}\int\\!\\!\textrm{d}\xi_{a}\textrm{d}\xi_{b}\>f_{\kappa_{a}}(\xi_{a},\mu)f_{\kappa_{b}}(\xi_{b},\mu)\otimes\big{(}{\cal
I}_{\kappa_{a}\kappa_{a^{\prime}}}{\cal
I}_{\kappa_{b}\kappa_{b^{\prime}}}\big{)}\ {\rm
tr}\Big{[}{\widehat{H}}_{\kappa_{a^{\prime}}\kappa_{b^{\prime}}}^{\kappa_{1}\kappa_{2}}(\Phi_{\rm
born})\otimes{\widehat{S}}_{G,a^{\prime}b^{\prime}}^{\kappa_{1}\kappa_{2}}\Big{]}\,,$
(2.14)

where here $\kappa_{a^{\prime}}\kappa_{b^{\prime}}\to\kappa_{1}\kappa_{2}$ is
the partonic exclusive dijet hard scattering process, and
$\sum_{\\{\kappa_{i}\\}}$ indicates a sum over all channels for each
$\kappa_{i}$ variable, except that at least one of the final-state jet labels,
$\kappa_{1}$ or $\kappa_{2}$, is fixed to be a quark. Here $f_{\kappa}$ is a
PDF, ${\cal I}_{\kappa\kappa^{\prime}}$ accounts for initial-state radiation
which depends on the exclusive jet veto and that may or may not change the
identity of the parton participating in the hard scattering, $\widehat{H}$
encodes the hard scattering including virtual corrections, and
${\widehat{S}}_{G}$ are global soft functions. $N$ is a normalization factor.

We would like to determine the impact on $x_{q}$ from using PDFs obtained at
different orders in perturbation theory, which also requires systematically
including higher order terms in other parts of the factorization theorem.
Since we are interested in an estimate for these effects in a context that is
more general than exclusive jets, we choose to ignore the jet veto dependence
as well as higher order corrections from the global soft functions, and hence
work with tree level functions ${\cal I}_{\kappa\kappa^{\prime}}$ and
${\widehat{S}}_{G}$. Thus, we focus on including the perturbative corrections
to $H$ in a manner that is consistent with the use of higher order PDFs (in
particular, for example, with the same choice of factorization scale $\mu$ in
the $f$s and $H$ so that the $\mu$ dependence cancels to the order being
considered). At this order the soft functions are just matrices, for example
in the $qq\to qq$ channel

$S_{IJ}^{0}=\begin{pmatrix}N^{2}&0\\\ 0&\frac{C_{F}N}{2}\end{pmatrix}\,.$
(2.15)

This and the other tree-level soft functions can be found in Broggio:2014hoa .
The hard functions have been computed to NLO Kelley:2010fn  and NNLO
Broggio:2014hoa .

Using the CTEQ NNLO PDFs with $\alpha_{s}=0.119$ we find the LO, NLO and NNLO
quark/gluon fractions to be

$x_{q}^{\text{LO}}=0.507,\quad x_{q}^{\text{NLO}}=0.530,\quad
x_{q}^{\text{NNLO}}=0.538,\quad$ (2.16)

corresponding to an 4.5% increase in the fraction at NLO and a further 1.5%
increase at NNLO. Thus, we see that the impact of variations of PDF induces at
most 2% uncertainty on quark gluon fractions. As we will see in Section 4.2,
this translates to around a 1% uncertainty on $\alpha_{s}$, which we will show
is quite subdominant compared to other uncertainties in an extraction of
$\alpha_{s}$ from soft-drop jet-mass spectra.

####  2.2.2 Measurements of inclusive jets

We will be concerned with jets produced in hadron collisions with an
underlying process such as $pp\rightarrow\text{dijets}+X$ or $pp\rightarrow
Z+j_{\kappa}+X$ with the jet initiated by the parton $\kappa=q,g$. We will
derive the factorization in the context of inclusive jets measurement but our
conclusions will also hold for the exclusive case. We begin by considering the
cross section, differential in $p_{T}$ and $\eta$, for inclusive jets
Ellis:1996mzs ; Kang:2018jwa :

$\displaystyle\frac{\textrm{d}^{3}\sigma}{\textrm{d}p_{T}\textrm{d}\eta\textrm{d}\xi}=\sum_{abc}\int\frac{\textrm{d}x_{a}\textrm{d}x_{b}\textrm{d}z}{x_{a}x_{b}z}f_{a}(x_{a},\mu)f_{b}(x_{b},\mu)H_{ab}^{c}\Big{(}x_{a},x_{b},\eta,\frac{p_{T}}{z},\mu\Big{)}{\cal
G}_{c}(z,\xi,p_{T},R,\mu)\,,$ (2.17)

where $f_{a,b}$ are the parton distribution functions, $H_{ab}^{c}$ is the
hard function describing inclusive production of parton $c$ with transverse
momentum ${\cal O}(p_{T})$, which initiates a signal jet carrying momentum
fraction $z$. The inclusive jet function Kang:2016mcy  ${\cal G}_{c}$
describes the formation of jet of radius $R$ initiated by a parton $c$ and the
jet-mass measurement $\xi$. The factorization is valid for $R/2\ll 1$, which
implies that the relevant modes describing the jet dynamics are the hard-
collinear modes with transverse momentum $k_{t}\sim p_{T}R$ (but with energy
$\sim p_{T}$), which is parametrically smaller than the scale $p_{T}$
characterizing the jet production. Numerically, however, it has been shown to
work very well for $R/2\lesssim 0.4$ Dasgupta:2014yra ; Becher:2015hka ;
Chien:2015cka ; Hornig:2016ahz ; Dasgupta:2016bnd ; Kolodrubetz:2016dzb . The
$\mu$ evolution in Eq. (2.17) resums the large (single) logarithms
$\alpha_{s}^{n}\textrm{ln}^{k}(p_{T}R/p_{T})$, with $k\leq n$:

$\displaystyle\mu\frac{\textrm{d}}{\textrm{d}\mu}{\cal
G}_{i}\big{(}z,\xi,p_{T}R,\mu\big{)}=\frac{\alpha_{s}(\mu)}{\pi}\sum_{j}\int_{z}^{1}\frac{\textrm{d}z^{\prime}}{z^{\prime}}\>P_{ji}\bigg{(}\frac{z}{z^{\prime}},\mu\bigg{)}{\cal
G}_{j}\big{(}z^{\prime},\xi,p_{T},R,\mu\big{)}\,,$ (2.18)

where $P_{ji}$ are the time-like splitting functions. As the DGLAP evolution
between the hard and the hard collinear scale $p_{T}R$ involves branching of
the original parton leading to formation of multiple jets. See also Ref.
Neill:2021std  for an extension of this formalism to leading jets.

While the quark/gluon fraction is internal to the entire machinery for a
theory prediction, it is nevertheless useful to pull them out such that we can
define the intermediate objects corresponding to quark- and gluon-initiated
jets, and analyze their properties separately. The integral over the jet
function in Eq. (2.17) defines the event averaged number of jets for quarks
and gluons Neill:2021std ,

$\displaystyle\int_{0}^{1}\\!\textrm{d}z\,\textrm{d}\xi\>{\cal
G}_{\kappa}(z,\xi,p_{T},R,\mu)=\langle N_{\kappa,\rm
jets}\rangle\,,\qquad\kappa=q,g\,.$ (2.19)

The number density is generated dynamically via QCD fragmentation process and
depends on the center of mass energy and the jet algorithm. As such it is not
straightforward to identify quark and gluon fractions associated with
inclusive jet measurement.

Our goal, however, is to analyze jets in the region of small jet masses,
where, as we describe below, the hard collinear modes at the scale $p_{T}R$
factorize from the collinear and soft dynamics governed by the jet-mass
measurement. The additional constraint of the small jet-mass measurement
enables us to speak of quark- and gluon-initiated jets. When we deem a jet to
be a quark or a gluon jet in perturbation theory, we are not referring to the
color charge of all the partons inside but instead the color charge of the
only collinear sector that is singled out due to the small jet-mass constraint
at leading power, i.e. with corrections suppressed at ${\cal O}(\xi)$. In
general, there will be non-global emissions of soft gluons from outside of the
jet but they do not modify this aspect. In the limit of small jet mass, the
inclusive jet function in Eq. (2.17) factorizes as

$\displaystyle{\cal G}_{c}\big{(}z,\xi,p_{T},R,\mu\big{)}=\sum_{i}{\cal
H}_{c\rightarrow i}\big{(}z,p_{T}R,\mu\big{)}\>{\cal
J}_{i}(\xi,p_{T},\eta,R,\mu)\,,\qquad\xi\ll 1\,.$ (2.20)

The ${\cal H}_{c\rightarrow i}$ function involves hard-collinear modes
integrated out at scales $p_{T}R$, while ${\cal J}_{i}$ is still a multiscale
function whose factorization depends on the jet-mass region, which will be
treated in subsequent sections. The ${\cal H}_{c\rightarrow i}$ obeys the
following evolution equation,

$\displaystyle\mu\frac{\textrm{d}}{\textrm{d}\mu}{\cal H}_{i\rightarrow
j}\big{(}z,p_{T}R,\mu\big{)}=\sum_{k}\int_{z}^{1}\frac{\textrm{d}z^{\prime}}{z^{\prime}}\>\gamma_{ik}\bigg{(}\frac{z}{z^{\prime}},p_{T}R,\mu\bigg{)}{\cal
H}_{k\rightarrow j}\big{(}z^{\prime},p_{T}R,\mu\big{)}\,,$ (2.21)

where the anomalous dimension $\gamma_{ij}$ is given by

$\displaystyle\gamma_{ij}\big{(}z,p_{T}R,\mu\big{)}=\delta_{ij}\delta(1-z)\Gamma_{i}\big{(}p_{T}R,\mu\big{)}+\frac{\alpha_{s}}{\pi}P_{ji}(z)\,.$
(2.22)

We see that the same splitting function appears that is tied to the $z$
dependence in ${\cal H}_{i\rightarrow j}$. The functions $\Gamma_{i}$ are
responsible for Sudakov double logarithmic evolution between the hard-
collinear, collinear and soft modes in the small jet-mass region. This
diagonal evolution is consistent with the RG evolution of soft and collinear
modes in ${\cal J}_{i}$ in Eq. (2.20). It is helpful to separate the purely
diagonal piece at all orders by writing Cal:2019hjc

$\displaystyle{\cal H}_{i\rightarrow
j}\big{(}z,p_{T}R,\mu\big{)}=J_{ij}\big{(}z,p_{T}R,\mu\big{)}N_{\rm
incl}^{j}\big{(}p_{T}R,\mu\big{)}\,,$ (2.23)

where the functions $J_{ij}(z,p_{T}R,\mu)$ also track the flavor of the jet
which the original parton fragments into and describes the dependence of their
corresponding energy fraction. They satisfy the same DGLAP evolution equation
as the inclusive jet function in Eq. (2.18), whereas the functions $N_{\rm
incl}^{j}$ satisfy multiplicative RG equations:

$\displaystyle\mu\frac{\textrm{d}}{\textrm{d}\mu}\textrm{ln}N^{i}_{\rm
incl}\big{(}p_{T}R,\mu\big{)}=\Gamma_{i}\big{(}p_{T}R,\mu\big{)}\,.$ (2.24)

The evolution equations for $N^{i}_{\rm incl}$ and $J_{ij}$ alone do not fix
their all-orders expressions, and hence there is some freedom in separating
the constant, non-logarithmic pieces of ${\cal H}_{i\rightarrow j}$ between
the two functions. At each order, we define $J_{ij}$ such that summing over
the flavor index $j$ leads to the inclusive jet function:

$\displaystyle\sum_{j}J_{ij}\big{(}z,p_{T}R,\mu\big{)}=J_{i}\big{(}z,p_{T}R,\mu\big{)}\,.$
(2.25)

This also fixes the $N_{\rm incl}^{\kappa}$.

This enables us to write an all-orders definition of quark and gluon fractions
in this small jet-mass region:111We thank Kyle Lee for discussions on this.

$x_{\kappa}\big{(}p_{T}R,\eta,\mu)\equiv\frac{\sum_{a,b,c}f_{a}\otimes
f_{b}\otimes H_{ab}^{c}\otimes J_{c\kappa}}{\sigma_{\rm incl}(p_{T},\eta)}\,,$
(2.26)

with

$\sigma_{\rm
incl}(p_{T},\eta)\equiv\frac{\textrm{d}^{2}\sigma}{\textrm{d}p_{T}\textrm{d}\eta}=\sum_{a,b,c,d}f_{a}\otimes
f_{b}\otimes H_{ab}^{c}\otimes J_{cd}\,,$

where the convolutions denoted by $\otimes$ are written out as in Eq. (2.17).
In the denominator we have summed over all the indices resulting in the
inclusive cross section. Thus, the normalized jet-mass cross section can be
written as

$\displaystyle\frac{1}{\sigma_{\rm
incl}(p_{T},\eta)}\frac{\textrm{d}^{3}\sigma}{\textrm{d}p_{T}\textrm{d}\eta\textrm{d}\xi}=x_{q}\,\tilde{\cal
G}_{q}\big{(}\xi,p_{T}R,\mu\big{)}+x_{g}\,\tilde{\cal
G}_{g}\big{(}\xi,p_{T}R,\mu\big{)}\,,$ (2.27)

where we have defined Cal:2020flh

$\displaystyle\tilde{\cal
G}_{\kappa}\big{(}\xi,p_{T}R,\mu\big{)}\equiv\frac{1}{\sigma_{\kappa}^{\rm
incl}}\frac{\textrm{d}\sigma_{\kappa}}{\textrm{d}\xi}(p_{T},\eta)=N_{\rm
incl}^{\kappa}(p_{T}R,\mu){\cal J}_{\kappa}(\xi,p_{T},\eta,R,\mu)\,.$ (2.28)

In the expression above we have suppressed the effects of non-global
logarithms for now, and we discuss them in detail below.

###  2.3 The fixed-order region

Using Eqs. (2.25) and (2.28), at ${\cal O}(\alpha_{s})$ we can write Eq.
(2.20) as

$\displaystyle{\cal G}_{\kappa}(z,\xi,p_{T}R,\mu)$
$\displaystyle=\delta(1-z)\big{[}\delta(\xi)+\tilde{\cal
G}^{[1]}_{\kappa}(\xi,\alpha_{s}(\mu))\big{]}+\delta(\xi)\sum_{j}J_{\kappa
j}(z,p_{T}R,\mu)$ $\displaystyle=\delta(1-z)\big{[}\delta(\xi)+\tilde{\cal
G}^{[1]}_{\kappa}(\xi,\alpha_{s}(\mu))\big{]}+\delta(\xi)J_{\kappa}(z,p_{T}R,\mu)\,.$
(2.29)

In the second line various pieces $J_{\kappa j}$ have been summed to yield the
semi-inclusive jet function $J_{\kappa}$ with $\tilde{\cal G}^{[1]}_{\kappa}$
being the ${\cal O}(\alpha_{s})$ term. Furthermore, our extension of
$\tilde{\cal G}_{\kappa}$ to include the large $\xi$ fixed-order region is
defined in such a way that

$\displaystyle\int_{0}^{1}\textrm{d}\xi\>\tilde{\cal
G}_{\kappa}(\xi,p_{T}R,\mu)=1\,.$ (2.30)

We first state the results for the ungroomed cross section which will prove
useful for the groomed case. We evaluated the one-loop, fixed-order results
for $\tilde{\cal G}_{\kappa}$ in Eq. (2.28) for ungroomed jet-mass
distribution for quark and gluon jets clustered with $k_{T}$-type algorithm:

$\displaystyle\tilde{\cal G}^{\rm FO}_{q,{\rm
No\,sd}}(\xi,\alpha_{s}(\mu))=\delta(\xi)+\frac{\alpha_{s}(\mu)C_{F}}{2\pi}\Bigg{\\{}\delta(\xi)\bigg{(}-3+\frac{\pi^{2}}{3}\bigg{)}$
(2.31) $\displaystyle\qquad+\Theta(1-4\xi)\Bigg{[}-2{\cal L}_{1}(\xi)+{\cal
L}_{0}(\xi)\bigg{(}4\log\Big{(}\frac{1+\sqrt{1-4\xi}}{2}\Big{)}-\frac{3}{2}\sqrt{1-4\xi}\bigg{)}\Bigg{]}\Bigg{\\}}\,,$
$\displaystyle\tilde{\cal G}^{\rm FO}_{g,{\rm
No\,sd}}(\xi,\alpha_{s}(\mu))=\delta(\xi)+\frac{\alpha_{s}(\mu)}{2\pi}\Biggl{\\{}\delta(\xi)\bigg{(}C_{A}\Big{(}\frac{1}{4}+\frac{\pi^{2}}{3}\Big{)}-\frac{13}{12}\beta_{0}\bigg{)}+\Theta(1-4\xi)\Bigg{[}-2C_{A}{\cal
L}_{1}(\xi)$ $\displaystyle\qquad+{\cal
L}_{0}(\xi)\bigg{(}4C_{A}\log\Big{(}\frac{1+\sqrt{1-4\xi}}{2}\Big{)}-\frac{\beta_{0}}{2}\sqrt{1-4\xi}+\frac{C_{A}-2n_{f}T_{F}}{3}4\xi\sqrt{1-4\xi}\bigg{)}\Bigg{]}\Biggr{\\}}\,,$

with $\mathcal{L}_{n}(x)\equiv[\Theta(x)\textrm{ln}^{n}x/x]_{+}$. We derived
these results using one-loop calculations for quark and gluon collinear
functions that appears in the soft-drop double-differential jet-mass and
groomed jet-radius factorization Pathak:2020iue , discussed below in Sec.
2.7.2. See App. A.2 for more details. We see that at LO the end point of the
jet-mass distribution is at $\xi=0.25$. It is straightforward to check that
these expressions are normalized to 1, such that the ${\cal O}(\alpha_{s})$
piece upon integration over $\xi$ vanishes:

$\displaystyle\int_{0}^{1}\textrm{d}\xi\>\tilde{\cal G}^{\rm FO}_{\kappa,{\rm
No\,sd}}(\xi,\alpha_{s}(\mu))=1\,.$ (2.32)

As we show in App. A.3, the one-loop calculation for the groomed jet-mass
spectrum can be expressed in terms of the results for the ungroomed spectrum
stated above, plus an additional correction:

$\displaystyle\tilde{\cal G}^{\rm FO}_{\kappa,\rm sd}(\xi,\tilde{z}_{\rm
cut},\beta,R,\alpha_{s}(\mu))$ $\displaystyle=\tilde{\cal G}^{\rm
FO}_{\kappa,{\rm No\,sd}}\big{(}\xi,\alpha_{s}(\mu)\big{)}+\Delta\tilde{\cal
G}^{\rm FO}_{\kappa,{\rm sd}}(\xi,\tilde{z}_{\rm
cut},\beta,R,\alpha_{s}(\mu))\,.$ (2.33)

The groomed cross section $\tilde{\cal G}^{\rm FO}_{\kappa,\rm sd}$ is
similarly normalized, such that all the effects of grooming in the cumulative
jet-mass cross section vanish at the end point. The evaluation of the
remainder $\Delta{\cal G}^{\rm FO}_{\kappa,\rm sd}$ is detailed in App. A.3.

###  2.4 The plain jet-mass region

We now turn to plain jet-mass region with $\xi\ll 1$ where the large logs of
$\textrm{ln}(\xi)$ in $\tilde{\cal G}_{\kappa}$ necessitate factorization and
resummation. We first review the results for jet-mass measurement without
grooming and include the soft-drop related pieces in the subsequent section.

####  2.4.1 The ungroomed jet-mass spectrum

We first set up some useful notation. For a jet of radius $R$ emitted in the
direction $\vec{n}$ at rapidity $\eta_{J}$, we define the light-like vectors

$\displaystyle
n^{\mu}\equiv\zeta^{-1}(1,\vec{n})\,,\qquad{\bar{n}}^{\mu}\equiv\zeta(1,-\vec{n})\,,$
(2.34)

where we have performed a large boost in the direction of the jet using the
parameter $\zeta$ defined above in Eq. (2.10). Doing so allows us to work with
hemisphere-jets like coordinates and all the intermediate factors of jet
radius appearing in the factorization formulae are easily accounted for. In
terms of these light-like vectors, we define the light-cone decomposition of a
momentum $q^{\mu}$ as

$\displaystyle
q^{\mu}=q^{+}\frac{{\bar{n}}^{\mu}}{2}+q^{-}\frac{n^{\mu}}{2}+q_{\perp}^{\mu}\,.$
(2.35)

The factorization of the ${\cal G}_{\kappa}$ function in Eq. (2.17) for
ungroomed jet-mass measurement on inclusive jets involves double logarithms
and NGLs due to soft emissions:

$\displaystyle{\cal G}_{\kappa,\rm No\,sd}^{\rm plain}(z,\xi,Q,\mu)$
$\displaystyle=\frac{\textrm{d}}{\textrm{d}\xi}\Bigg{[}\sum_{\kappa^{\prime}}\sum_{j}\widehat{\cal
H}_{\kappa^{\prime}\rightarrow\kappa}^{j}(z,Q,\mu)\otimes_{\Omega}\widehat{\Sigma}_{\kappa}^{j}(\xi,Q,\mu)\Bigg{]}\,.$
(2.36)

The hard collinear scale $Q$ was defined above in Eq. (2.1). This
factorization formula is valid to all orders in $\alpha_{s}$ and at leading
power in the power counting set by $\xi$, such that its relation to the full
theory result is given by

$\displaystyle{\cal G}_{\kappa,\rm No\,sd}(z,\xi,Q,\alpha_{s}(\mu))={\cal
G}_{\kappa,\rm No\,sd}^{\rm plain}(z,\xi,Q,\mu)\big{(}1+{\cal
O}(\xi)\big{)}\,.$ (2.37)

In Eq. (2.36) $\widehat{\Sigma}_{\kappa}^{j}$ is the cumulative jet-mass cross
section that accounts for the contributions from the soft and collinear modes.
The index $j$ runs over the number of hard partons outside of the jet that
contribute to soft radiation everywhere, part of which is clustered with the
jet. Such configurations lead to non-global logarithms (NGL) and the
$\otimes_{\Omega}$ denotes angular integrals between hard collinear and soft
radiation. The cumulative jet-mass cross section is given by convolution in
the $+$ component between the soft and collinear contributions:

$\displaystyle\widehat{\Sigma}^{j}_{\kappa}(\xi,Q,\mu)$
$\displaystyle=\int_{0}^{\infty}\textrm{d}s\int_{0}^{\infty}\textrm{d}\ell_{c}^{+}\>J_{\kappa}\big{(}s,\mu\big{)}\>\widehat{\mathscr{S}}_{c_{m}}^{\kappa,j}\big{(}\ell_{c}^{+},\mu\big{)}\>\delta\bigg{(}\xi-\frac{s}{Q^{2}}-\frac{\ell_{c}^{+}}{Q}\bigg{)}\,.$
(2.38)

Here $J_{\kappa}(s,\mu)$ is the standard inclusive jet function discussed
below in App. B.2 and $\widehat{\mathscr{S}}_{c_{m}}^{\kappa,j}$ is the
cumulative ungroomed soft function discussed below. For simplicity, we will
treat the NGLs in the convolution in Eq. (2.36) at NLL accuracy (and at LL and
large-$N_{c}$ in our numerical studies below) where they can be included
multiplicatively Kang:2019prh :

$\displaystyle\sum_{j}\widehat{\cal
H}_{\kappa^{\prime}\rightarrow\kappa}^{j}(z,Q,\mu)\otimes_{\Omega}\widehat{\Sigma}_{\kappa}^{j}(\xi,Q,\mu)\rightarrow\langle\widehat{\cal
H}_{\kappa^{\prime}\rightarrow\kappa}(z,Q,\mu)\rangle\,{\cal S}_{\rm
NGL}^{\kappa}\big{(}t[\xi
Q,Q]\big{)}\,\langle\widehat{\Sigma}_{\kappa}(\xi,Q,\mu)\rangle\,,$ (2.39)

where defining
$\mathscr{S}_{c_{m}}^{\kappa}\big{(}\ell_{c}^{+},\mu\big{)}=\bigl{\langle}\widehat{\mathscr{S}}_{c_{m}}^{\kappa}\big{(}\ell_{c}^{+},\mu\big{)}\bigr{\rangle}$
we have

$\displaystyle\langle\widehat{\Sigma}_{\kappa}(\xi,Q,\mu)\rangle=\int_{0}^{\infty}\textrm{d}s\int_{0}^{\infty}\textrm{d}\ell_{c}^{+}\>J_{\kappa}\big{(}s,\mu\big{)}\>\mathscr{S}_{c_{m}}^{\kappa}\big{(}\ell_{c}^{+},\mu\big{)}\>\delta\bigg{(}\xi-\frac{s}{Q^{2}}-\frac{\ell_{c}^{+}}{Q}\bigg{)}\,.$
(2.40)

Here $\langle\ldots\rangle$ denotes angular averaging after solid angle
integration and

$\displaystyle
t[\mu_{0},\mu_{1}]\equiv\frac{1}{2\pi}\int_{\mu_{0}}^{\mu_{1}}\frac{d\mu^{\prime}}{\mu^{\prime}}\>\alpha_{s}(\mu^{\prime})\,.$
(2.41)

Similarly ${\cal
H}_{\kappa^{\prime}\rightarrow\kappa}(z,Q,\mu)=\langle\widehat{\cal
H}_{\kappa^{\prime}\rightarrow\kappa}(z,Q,\mu)\rangle$, where ${\cal
H}_{\kappa^{\prime}\rightarrow\kappa}(z,Q,\mu)$ is the same function appearing
above in Eq. (2.20). At the lowest order for jets clustered with anti-$k_{T}$
algorithm, the non-global piece ${\cal S}_{\rm NGL}^{\kappa}$ is given by

$\displaystyle{\cal S}_{\rm
NGL}^{\kappa}\big{(}t[\mu_{0},\mu_{1}]\big{)}=1-C_{\kappa}C_{A}\Big{(}\frac{\alpha_{s}}{2\pi}\Big{)}^{2}\frac{\pi^{2}}{3}\log^{2}\Big{(}\frac{\mu_{1}}{\mu_{0}}\Big{)}+\ldots\,.$
(2.42)

The function $\mathscr{S}^{\kappa}_{c_{m}}$ now no longer depends on the
angles of hard partons, and can be expressed as a cumulative integral of the
differential soft function:

$\displaystyle\mathscr{S}^{\kappa}_{c_{m}}\big{(}\ell_{c}^{+},\mu\big{)}=\int_{0}^{\ell_{c}^{+}}d\ell^{+}S^{\kappa}_{c_{m}}\big{(}\ell^{+},\mu\big{)}\,.$
(2.43)

The one-loop result is given in Eq. (B.20).

From Eqs. (2.28) and (2.36) the differential cross section stripped of its
DGLAP evolution at ${\cal O}(\alpha_{s})$ can be written as

$\displaystyle\tilde{\cal G}^{\rm plain}_{\kappa,{\rm
No\,sd}}(\xi,Q,\mu)=N^{\kappa}_{\rm incl}(Q,\mu){\cal J}^{\rm
plain}_{\kappa,{\rm No\,sd}}(\xi,Q,\mu)\bigg{(}1+{\cal O}(\xi)\bigg{)}\,,$
(2.44)

The ${\cal O}(\alpha_{s})$ expressions of $N_{\rm incl}^{\kappa}$ are given
below in Eq. (B.1).

From Eq. (2.39) we have

$\displaystyle{\cal J}^{\rm plain}_{\kappa,{\rm
No\,sd}}(\xi,Q,\mu)=\frac{\textrm{d}}{\textrm{d}\xi}\Big{(}{\cal S}_{\rm
NGL}^{\kappa}\big{(}t_{S}\big{)}\,\Sigma_{\kappa}(\xi,Q,\mu)\Big{)}\,,\qquad
t_{S}\equiv t[Q\xi,Q]\,.$ (2.45)

With the RG evolution made explicit, Eq. (2.45) reads

$\displaystyle{\cal J}^{\rm plain}_{\kappa,{\rm No\,sd}}(\xi,Q,\mu)$
$\displaystyle=\Bigg{(}{\cal S}_{\rm NGL}^{\kappa}\big{(}t_{S}\big{)}{\cal
J}^{\kappa}_{\rm
plain}[\partial_{\Omega};\xi,Q,\mu]\frac{e^{\gamma_{E}\Omega}}{\Gamma(-\Omega)}$
(2.46)
$\displaystyle\quad+\Big{(}\frac{\textrm{d}}{\textrm{d}\textrm{ln}\xi}{\cal
S}_{\rm NGL}^{\kappa}\big{(}t_{S}\big{)}\Big{)}{\cal J}^{\kappa}_{\rm
plain}[\partial_{\Omega};\xi,Q,\mu]\frac{e^{\gamma_{E}\Omega}}{\Gamma(1-\Omega)}\Bigg{)}\Bigg{|}_{\Omega=\tilde{\omega}(\mu_{s},\mu_{J})}\,,$

where the function of the derivative operator ${\cal J}^{\kappa}_{\rm
plain}[\partial_{\Omega};\xi,Q,\mu]$ is defined in terms of Laplace transforms
of the jet and the soft functions and the explicit expression is provided in
Eq. (C).

Next, we define the non-singular piece that is needed to recover the full
plain jet-mass one-loop cross section in Eq. (2.31) including power suppressed
terms. It is obtained by subtracting the plain contribution from the full
theory result,

$\displaystyle{\cal G}_{\kappa,\rm No\,sd}^{\rm
n.s.}(z,\xi,Q,\alpha_{s}(\mu))={\cal G}^{\rm FO}_{\kappa,\rm
No\,sd}(z,\xi,Q,\alpha_{s}(\mu))-{\cal G}_{\kappa,\rm No\,sd}^{\rm
plain}(z,\xi,Q,\mu)\,,$ (2.47)

which at ${\cal O}(\alpha_{s})$ yields

$\displaystyle\tilde{\cal G}_{\kappa,{\rm No\,sd}}^{\rm
n.s.[1]}(\xi,\alpha_{s}(\mu))$ $\displaystyle=\tilde{\cal G}^{\rm
FO[1]}_{\kappa,{\rm No\,sd}}(\xi,\alpha_{s}(\mu))$ (2.48)
$\displaystyle\quad-\Theta(1-4\xi)\bigg{(}\delta(\xi)N_{\rm
incl}^{\kappa[1]}(Q,\mu)+Q^{2}J_{\kappa}^{[1]}(\xi
Q^{2},\mu)+QS_{c_{m}}^{\kappa[1]}(Q\xi,\mu)\bigg{)}\,.$

Hence we have

$\displaystyle\tilde{\cal G}_{q,{\rm No\,sd}}^{\rm
n.s.[1]}(\xi,\alpha_{s}(\mu))$
$\displaystyle=\frac{\alpha_{s}C_{F}}{2\pi}\Theta(1-4\xi){\cal
L}_{0}(\xi)\Bigg{(}4\log\bigg{(}\frac{1+\sqrt{1-4\xi}}{2}\bigg{)}-\frac{3}{2}\big{(}\sqrt{1-4\xi}-1\big{)}\Bigg{)}\,,$
$\displaystyle\tilde{\cal G}_{g,{\rm No\,sd}}^{\rm
n.s.[1]}(\xi,\alpha_{s}(\mu))$
$\displaystyle=\frac{\alpha_{s}}{2\pi}\Theta(1-4\xi){\cal
L}_{0}(\xi)\bigg{(}4C_{A}\log\Big{(}\frac{1+\sqrt{1-4\xi}}{2}\Big{)}-\frac{\beta_{0}}{2}\big{(}\sqrt{1-4\xi}-1\big{)}$
$\displaystyle\qquad\qquad+\frac{C_{A}-2n_{f}T_{F}}{3}4\xi\sqrt{1-4\xi}\bigg{)}\,.$
(2.49)

Thus the total resummed ungroomed jet-mass cross section including ${\cal
O}(\alpha_{s})$ power suppressed terms reads

$\displaystyle\frac{1}{\sigma_{\kappa}}\frac{\textrm{d}\sigma^{\rm
No\,sd}_{\kappa}}{\textrm{d}\xi}=N^{\kappa}_{\rm incl}(Q,\mu){\cal J}^{\rm
plain}_{\kappa,{\rm No\,sd}}(\xi,Q,\mu)+\tilde{\cal G}^{\rm
n.s.[1]}_{\kappa,{\rm No\,sd}}(\xi,\alpha_{s})\,.$ (2.50)

####  2.4.2 The groomed jet-mass spectrum

In the plain jet-mass region the modifications due to soft drop can be treated
in fixed-order perturbation theory. Because up to ${\cal O}(z_{\rm cut})$ the
jet grooming primarily only affects the measurement on the soft radiation,
this amounts to making in Eq. (2.39) the replacement

$\displaystyle\mathscr{S}_{c_{m}}^{\kappa}\big{(}\ell_{c}^{+},\mu\big{)}\rightarrow\mathscr{S}_{\rm
sd}^{\kappa}\big{(}\ell_{c}^{+},Q\xi_{0},\beta,\zeta,\mu\big{)}\equiv\mathscr{S}_{c_{m}}^{\kappa}\big{(}\ell_{c}^{+},\mu\big{)}+\int_{0}^{\ell_{c}^{+}}d\ell^{+}\>\Delta
S_{\rm
sd}^{\kappa}\big{(}\ell^{+},Q\xi_{0}^{\prime},\beta,\zeta,\alpha_{s}(\mu)\big{)}\,,$
(2.51)

where $\Delta S_{\rm sd}^{\kappa}$ is the fixed-order soft-drop related
correction. At ${\cal O}(\alpha_{s})$ we have

$\displaystyle\Delta S^{\kappa[1]}_{\rm
sd}\big{(}\ell^{+},Q\xi_{0}^{\prime},\beta,\zeta,\alpha_{s}(\mu)\big{)}$
(2.52)
$\displaystyle\qquad=-\frac{\alpha_{s}(\mu)C_{\kappa}}{\pi}\Biggl{[}\frac{\Theta(\ell^{+})}{\ell^{+}}\Theta\bigg{(}\Big{(}\frac{Q\xi_{0}^{\prime}}{\ell^{+}}\Big{)}^{\frac{2}{2+\beta}}-1\bigg{)}\textrm{ln}\bigg{(}\Big{(}\frac{Q\xi_{0}}{\ell^{+}}\Big{)}^{\frac{2}{2+\beta}}-\zeta^{2}\bigg{)}\Biggr{]}^{[Q\xi_{0}^{\prime}]}_{+}\,,$

where $\zeta$ was defined above in Eq. (2.10). Here, the plus function is
defined as

$\displaystyle\big{[}\Theta(x)q(x)\big{]}_{+}^{[x_{0}]}\equiv\lim_{\epsilon\rightarrow
0}\Big{(}\Theta(x-\epsilon)q(x)-\delta(x-\epsilon)\int_{\epsilon}^{x_{0}}\textrm{d}x^{\prime}\>q(x^{\prime})\Big{)}\,.$
(2.53)

Note that because the soft modes defining this function are at wide angles,
the end-point of the plus function appears at the soft wide-angle transition
point $\xi_{0}^{\prime}$ given in Eq. (2.9). We derive this one-loop
expression in App. A.3.

This replacement allows us to write down the factorized formula for soft-drop
jet-mass cross section in the plain jet-mass region:

$\displaystyle{\cal G}_{\kappa,\rm sd}^{\rm plain}(z,\xi,Q,\mu)$
$\displaystyle=\frac{\textrm{d}}{\textrm{d}\xi}\Bigg{[}\sum_{\kappa^{\prime}}{\cal
H}_{\kappa^{\prime}\rightarrow\kappa}(z,Q,\mu)\,\Sigma_{\kappa,\rm
sd}(\xi,Q,\mu)\Bigg{]}\,,$ (2.54)

where $\Sigma_{\kappa,\rm sd}$ is defined analogously to Eq. (2.38) with
additional soft-drop related fixed-order pieces in the soft function in Eq.
(2.51). Its relation to the full theory cross section is given by

$\displaystyle{\cal G}_{\kappa,\rm sd}\big{(}z,\xi,Q,\alpha_{s}(\mu)\big{)}$
$\displaystyle={\cal G}_{\kappa,\rm sd}^{\rm plain}(z,\xi,Q,\mu)\Big{(}1+{\cal
O}(\xi,\xi_{0})\Big{)}$ (2.55)

Thus, analogous to Eq. (2.44) the resummed cross section (to NNLL accuracy) is
given by

$\displaystyle\tilde{\cal G}_{\kappa,\rm sd}^{\rm plain}\big{(}\xi,z_{\rm
cut},\beta,R,\mu\big{)}=N_{\rm incl}^{\kappa}(Q,\mu){\cal J}_{\kappa,\rm
sd}^{\rm plain}(\xi,Q,\mu)\,,$ (2.56)

where

$\displaystyle{\cal J}_{\kappa,\rm sd}^{\rm plain}(\xi,Q,\mu)={\cal S}_{\rm
NGL}^{\kappa}\big{(}t_{S}\big{)}{\cal J}^{\kappa}_{\rm
plain}[\partial_{\Omega};\xi,Q,\mu]\bigg{(}\frac{e^{\gamma_{E}\Omega}}{\Gamma(-\Omega)}+{\cal
Q}_{\rm
sd}^{\kappa}\bigg{[}\Omega,\frac{\xi}{\xi_{0}},\beta,R,\mu\bigg{]}\bigg{)}\Bigg{|}_{\Omega=\tilde{\omega}(\mu_{s},\mu_{J})}$
$\displaystyle\quad+\Big{(}\frac{\textrm{d}}{\textrm{d}\textrm{ln}\xi}{\cal
S}_{\rm NGL}^{\kappa}\big{(}t_{S}\big{)}\Big{)}{\cal J}^{\kappa}_{\rm
plain}[\partial_{\Omega};\xi,Q,\mu]\bigg{(}\frac{e^{\gamma_{E}\Omega}}{\Gamma(1-\Omega)}+{\cal
Q}_{\rm
sd}^{\kappa}\bigg{[}\Omega-1,\frac{\xi}{\xi_{0}},\beta,R,\mu\bigg{]}\bigg{)}\Bigg{|}_{\Omega=\tilde{\omega}(\mu_{s},\mu_{J})}\,.$
(2.57)

The kernel ${\cal Q}_{\rm sd}^{\kappa}$ accounts for the effects of soft drop,
including resummation between the jet and soft scales. The explicit expression
is given in Eq. (C.13).

The additional ${\cal O}(\xi_{0})$ power corrections dropped in Eq. (2.56)
include finite $z_{\rm cut}$ terms and are captured by the following non-
singular term:

$\displaystyle\Delta\tilde{\cal G}^{\rm n.s.[1]}_{\kappa,\rm
sd}=\Delta\tilde{\cal G}_{\kappa,\rm sd}^{\rm FO[1]}(\xi,\tilde{z}_{\rm
cut},\beta,R,\alpha_{s}(\mu))-Q\Delta S_{\rm
sd}^{[1]}\big{(}Q\xi,Q\xi_{0}^{\prime},\beta,\zeta,\alpha_{s}(\mu)\big{)}\,.$
(2.58)

where $\Delta\tilde{\cal G}_{\kappa,\rm sd}^{\rm FO}$ appeared in Eq. (2.33).
At one-loop the factor of $Q$ cancels against a $1/(Q\xi)$ from $\Delta S_{\rm
sd}^{[1]}$ such that $\Delta\tilde{\cal G}^{\rm n.s.[1]}_{\kappa,\rm sd}$ is
independent of $Q$. The $\Delta S_{\rm sd}^{[1]}$ accounts for all the soft
singularities which allows us to evaluate this non-singular piece via simple
numerical integration. Including the non-singular corrections in Eqs. (2.4.1)
and (2.56) accounts for all perturbative power corrections at ${\cal
O}(\alpha_{s})$ in the plain jet-mass region:

$\displaystyle\tilde{\cal G}_{\kappa,\rm sd}=N_{\rm incl}^{\kappa}(Q,\mu){\cal
J}_{\kappa,\rm sd}^{\rm plain}(\xi,\mu)+{\cal G}_{\kappa,\rm No\,sd}^{\rm
n.s.[1]}\big{(}\xi,\alpha_{s}(\mu)\big{)}+\Delta\tilde{\cal G}^{\rm
n.s.[1]}_{\kappa,\rm sd}(\xi,\tilde{z}_{\rm cut},\beta,R,\alpha_{s}(\mu))\,,$
(2.59)

where we have dropped some of the function arguments for simplicity. In App.
A.3 we will show that $\Delta\tilde{\cal G}^{\rm n.s.[1]}_{\kappa,\rm sd}$
results in a numerically small contribution which can be ignored in comparison
to perturbative uncertainty from scale variation at NNLL. Hence, we will drop
$\Delta\tilde{\cal G}^{\rm n.s.[1]}_{\kappa,\rm sd}$ in our
$\alpha_{s}$-sensitivity analysis.

###  2.5 The soft-drop resummation region

In the soft-drop resummation region for $\xi\ll\xi_{0}$, the (differential)
soft function $S_{\rm sd}^{\kappa}$ in Eq. (2.51) can be factorized into
pieces that correspond to soft-drop passing and failing soft radiation, such
that

$\displaystyle S^{\kappa}_{c_{m}}\big{(}\ell^{+},\mu\big{)}+\Delta
S^{\kappa}_{\rm sd}\big{(}\ell^{+},Q\xi_{0}^{\prime},\beta,\zeta,\mu\big{)}$
(2.60) $\displaystyle\qquad=S_{G}^{\kappa}\big{(}Q_{\rm
cut},\beta,\zeta,\mu\big{)}Q_{\rm
cut}^{\frac{1}{1+\beta}}S_{c}^{\kappa}\big{(}\ell^{+}Q_{\rm
cut}^{\frac{1}{1+\beta}},\beta,\mu\big{)}\Bigg{[}1+{\cal
O}\bigg{(}\Big{(}\frac{\ell^{+}}{Q_{\rm
cut}}\Big{)}^{\frac{2}{2+\beta}}\bigg{)}\Bigg{]}\,,$

where the term in the square brackets indicates the size of the power
corrections. The $S_{G}^{\kappa}$ and $S_{c}^{\kappa}$ are the global soft and
collinear soft functions Frye:2016aiz  that respectively describe physics of
soft modes that fail and pass the soft-drop test. $Q_{\rm cut}$, defined in
Eqs. (2.5) and (2.8), defines the scale of the global soft function. The one-
loop expressions are provided in Eqs. (B.39) and (B.29). Here we have made use
of a dimension $\big{(}\frac{2+\beta}{1+\beta}\big{)}$ variable in the
argument of $S_{c}^{\kappa}$ following Hoang:2019ceu . We show in App. B.4
that the collinear-soft function is independent of $Q_{\rm cut}$, as $Q_{\rm
cut}$ is a high scale from the perspective of collinear-soft emissions
Frye:2016aiz . It is also independent of the jet radius $R$, since collinear-
soft modes are at parametrically smaller angles than the jet radius. Thus,
from Eq. (2.60) we have the following factorization formula in this region:

$\displaystyle{\cal G}^{\rm resum}_{\kappa,\rm sd}$
$\displaystyle\big{(}z,\xi,Q,z_{\rm
cut},\beta,\mu\big{)}=\sum_{\kappa^{\prime}}{\cal
H}_{\kappa^{\prime}\rightarrow\kappa}(z,Q,\mu)S_{G}^{\kappa}\big{(}Q_{\rm
cut},\beta,\zeta,\mu\big{)}{\cal S}_{\rm NGL}^{\kappa}\big{(}t_{gs}\big{)}$
(2.61)
$\displaystyle\qquad\times\int\textrm{d}\tilde{k}\int\textrm{d}s\>J_{\kappa}\big{(}s,\mu\big{)}S_{c}^{\kappa}\big{(}\tilde{k},\beta,\mu\big{)}\delta\bigg{(}\xi-\frac{s}{Q^{2}}-\frac{\tilde{k}(Q_{\rm
cut})^{\frac{-1}{1+\beta}}}{Q}\bigg{)}\,.$

The relation between this factorized cross section and the full theory result
is given by

$\displaystyle{\cal G}_{\kappa,\rm sd}\big{(}z,\xi,Q,z_{\rm
cut},\beta,\alpha_{s}(\mu)\big{)}={\cal G}^{\rm resum}_{\kappa,\rm
sd}\big{(}z,\xi,Q,z_{\rm cut},\beta,\mu\big{)}\Bigg{[}1+{\cal
O}\bigg{(}\xi,\xi_{0},\Big{(}\frac{\xi}{\xi_{0}}\Big{)}^{\frac{2}{2+\beta}}\bigg{)}\Bigg{]}\,,$
(2.62)

where we have displayed all the power corrections that are small in the soft-
drop resummation region. We discuss these corrections in detail in the next
section. Next, we note that since the wide-angle modes are groomed away in
this region, the NGLs give a contribution that is independent of the jet mass.
The argument of the NGL piece is given by

$\displaystyle t_{gs}\equiv t\big{[}Q_{\rm cut},Q\big{]}\,.$ (2.63)

Comparing with Eq. (2.45) we see that the evolution of the NGLs is frozen to
the fixed scale $Q_{\rm cut}$.

Finally, at one-loop we can remove the DGLAP piece and write

$\displaystyle\tilde{\cal G}^{\rm resum}_{\kappa,\rm sd}(\xi,Q,Q_{\rm
cut},\beta,R,\mu)=N_{\rm incl}^{\kappa}(Q,\mu){\cal J}_{\kappa}^{\rm
sd\,resum}(\xi,Q,Q_{\rm cut},\beta,R,\mu)\,,\qquad(\xi<\xi_{0})\,,$ (2.64)

where

$\displaystyle{\cal J}_{\kappa}^{\rm sd\,resum}$
$\displaystyle=S_{G}^{\kappa}\big{(}Q_{\rm cut},\beta,\zeta,\mu\big{)}{\cal
S}_{\rm NGL}^{\kappa}\big{(}t_{gs}\big{)}{\cal J}^{\kappa}_{\rm
sd}[\partial_{\Omega};\xi,Q,\mu]\frac{e^{\gamma_{E}\Omega}}{\Gamma(-\Omega)}\Bigg{|}_{\Omega=\tilde{\omega}(\mu_{cs},\mu_{J})}\,.$
(2.65)

The function ${\cal J}^{\kappa}_{\rm sd}[\partial_{\Omega};\xi,Q,\mu]$ now
involves Laplace transforms of the jet and the collinear-soft functions, and
is given in Eq. (C.18). Because the NGLs are independent of $\xi$, the terms
in the second line in Eq. (2.4.2) are not present in the soft-drop resummation
region.

###  2.6 Soft drop on exclusive jets

Another way of isolating jets is via an exclusive measurement which involves
vetoing additional jets. For example, for $pp\rightarrow$ dijets with a veto
on transverse momentum of additional jets $p_{T_{i}}<p_{T}^{\rm veto}$, one
can derive a factorization formula with a $p_{T}$ jet veto Stewart:2010tn ;
Berger:2010xi ; Banfi:2012yh ; Becher:2012qa ; Dasgupta:2012hg ; Banfi:2012jm
; Chien:2012ur ; Jouttenus:2013hs ; Becher:2013xia ; Stewart:2013faa ;
Stewart:2014nna ; Stewart:2015waa ; Banfi:2015pju  for a soft drop groomed
jet-mass measurement

$\displaystyle\frac{\textrm{d}\sigma(p_{T}^{\rm
veto})}{\textrm{d}\xi_{1}\textrm{d}\xi_{2}\textrm{d}\Phi_{\rm born}}$
$\displaystyle=\sum_{\\{\kappa_{1},\kappa_{2},\kappa_{a},\kappa_{b}\\}}{\cal
B}_{n}^{\kappa_{a}}\big{(}x_{a},Q,p_{T}^{\rm veto},R^{2},\mu,\nu\big{)}{\cal
B}_{\bar{n}}^{\kappa_{b}}\big{(}x_{b},Q,p_{T}^{\rm veto},R^{2},\mu,\nu\big{)}$
(2.66)
$\displaystyle\qquad\times\textrm{tr}\Bigg{[}\widehat{H}_{ab}^{\kappa_{1}\kappa_{2}}\big{(}Q_{i},\Phi_{\rm
born},\mu\big{)}\otimes_{\Omega}\widehat{S}_{G,ab}^{\kappa_{1}\kappa_{2}}\big{(}p_{T}^{\rm
veto},R,Q_{1\rm cut},Q_{2\rm cut},\beta,\mu,\nu\big{)}\Bigg{]}$
$\displaystyle\qquad\times\int\textrm{d}s_{1}\,\textrm{d}\tilde{k}_{1}\>J_{\kappa_{1}}\big{(}s_{1},\mu\big{)}\,S_{c}^{\kappa_{1}}(\tilde{k}_{1},\beta,\mu)\delta\bigg{(}\xi_{1}-\frac{s_{1}+Q_{1}Q_{1\rm
cut}^{-\frac{1}{1+\beta}}\tilde{k}_{1}}{Q_{1}^{2}}\bigg{)}$
$\displaystyle\qquad\times\int\textrm{d}s_{2}\,\textrm{d}\tilde{k}_{2}\>J_{\kappa_{2}}\big{(}s_{2},\mu\big{)}S_{c}^{\kappa_{2}}\big{(}\tilde{k}_{2},\beta,\mu\big{)}\delta\bigg{(}\xi_{2}-\frac{s_{2}+Q_{2}Q_{2\rm
cut}^{-\frac{1}{1+\beta}}\tilde{k}_{2}}{Q_{2}^{2}}\bigg{)}\,.$

Here $Q_{1}$, $Q_{2}$ and $Q$ are appropriate large momenta associated with
the two jets and the beam functions respectively, and $Q_{i\,\rm cut}$ are the
global soft scales:

$\displaystyle Q_{i}=p_{T_{i}}\cosh\eta_{J_{i}}\,,\qquad Q_{i\rm
cut}=(2\cosh\eta_{J_{i}})^{1+\beta}p_{T_{i}}z_{\rm cut}\,.$ (2.67)

The momentum fractions $x_{a,b}$ are constrained by the born kinematics. The
born kinematics phase space $\Phi_{\rm born}$ is captured in the hard function
$\widehat{H}_{ab}^{\kappa_{1}\kappa_{2}}$. The sum runs over various partonic
channels $\kappa_{a}\kappa_{b}\rightarrow\kappa_{1}\kappa_{2}$ which lead to
the production of two jets. Color is traced over, and there are two color
channels for $\kappa_{a}$-$\kappa_{b}=$ quark-quark, three for quark-gluon,
and nine for gluon-gluon. The parton distribution function dependence appears
through the beam functions ${\cal B}_{n}^{\kappa_{a}}$ and ${\cal
B}_{\bar{n}}^{\kappa_{b}}$, via a further factorization between perturbative
(${\cal I}$) and nonperturbative ($f$) contributions in the direction of the
beams:

$\displaystyle{\cal B}^{\kappa}\big{(}x,Q,p_{T}^{\rm
veto},R^{2},\mu,\nu\big{)}$
$\displaystyle=\sum_{\kappa^{\prime}}\int_{x}^{1}\frac{\textrm{d}\xi}{\xi}\>{\cal
I}_{\kappa\kappa^{\prime}}(x/\xi,Q,p_{T}^{\rm
veto},R^{2},\mu,\nu)f_{\kappa^{\prime}}(\xi,\mu)\,.$ (2.68)

The global soft function $\widehat{S}_{G,ab}^{\kappa_{1}\kappa_{2}}$ describes
the dynamics of soft particles emitted off the initial state partons as well
as those that are clustered in the final state jets but groomed away. As a
consequence of jet grooming the jet-mass spectra of the two jets are
decoupled.

It is useful to separate Eq. (2.66) into quark-gluon fractions that are
independent of the jet-mass measurement, times a function that depends on the
jet mass. To make this meaningful we must carry out the separation in a
$\mu$-independent manner. To this end we define

$\displaystyle x^{\rm excl}_{\kappa_{1}\kappa_{2}}(p_{T}^{\rm veto},\Phi_{\rm
born})$ $\displaystyle=\sum_{\\{\kappa_{a},\kappa_{b}\\}}{\cal
B}_{n}^{\kappa_{a}}\big{(}x_{a},Q,p_{T}^{\rm veto},R^{2},\mu,\nu\big{)}{\cal
B}_{\bar{n}}^{\kappa_{b}}\big{(}x_{b},Q,p_{T}^{\rm veto},R^{2},\mu,\nu\big{)}$
(2.69)
$\displaystyle\quad\times\frac{\textrm{tr}\Big{[}\widehat{H}_{ab}^{\kappa_{1}\kappa_{2}}\big{(}Q_{i},\Phi_{\rm
born},\mu\big{)}\otimes_{\Omega}\widehat{S}_{G,ab}^{\kappa_{1}\kappa_{2}}\big{(}p_{T}^{\rm
veto},R,Q_{1\rm cut},Q_{2\rm cut},\beta,\mu,\nu\big{)}\Big{]}}{N_{\rm
incl}^{\kappa_{1}}(Q_{1},\mu)S_{G}^{\kappa_{1}}(Q_{1\rm cut},\beta,R,\mu)\
N_{\rm incl}^{\kappa_{2}}(Q_{2},\mu)S_{G}^{\kappa_{2}}(Q_{2\rm
cut},\beta,R,\mu)}\,,$

such that the exclusive jet cross section in the soft drop resummation region
is given by

$\displaystyle\frac{\textrm{d}\sigma(p_{T}^{\rm
veto})}{\textrm{d}\xi_{1}\textrm{d}\xi_{2}\textrm{d}\Phi_{\rm born}}$ (2.70)
$\displaystyle\quad=\sum_{\\{\kappa_{1},\kappa_{2}\\}}x^{\rm
excl}_{\kappa_{1}\kappa_{2}}(p_{T}^{\rm veto},\Phi_{\rm born})\>\tilde{\cal
G}^{\rm resum}_{\kappa_{1},\rm sd}\big{(}\xi_{1},Q_{1},Q_{1\rm
cut},\beta,R,\mu\big{)}\,\tilde{\cal G}^{\rm resum}_{\kappa_{2},\rm
sd}\big{(}\xi_{2},Q_{2},Q_{2\rm cut},\beta,R,\mu\big{)}\,.$

This equation is analogous to Eq. (2.27) for the inclusive case, except that
here we have fractions $x_{\kappa_{1}\kappa_{2}}$ that depend on the quark-
gluon identities $\kappa_{i}$ of each of the dijets. The jet-mass shape
dependence is described by the same partonic cross section $\tilde{\cal
G}_{\kappa,\rm sd}$ for flavor $\kappa$ defined in Eq. (2.28), whose resummed
version $\tilde{\cal G}_{\kappa,\rm sd}^{\rm resum}$ was defined above in Eq.
(2.64). The definition of $x_{\kappa_{1}\kappa_{2}}^{\rm excl}$ in Eq. (2.69)
is RG invariant since the $\mu$-dependence arising from the $J_{\kappa_{i}}$
and $S_{c}^{\kappa_{i}}$ functions associated with the final state jets in Eq.
(2.66), is equal but opposite to that of the factors in the denominator of Eq.
(2.69).

In Sec. 2.2 we made use of the exclusive dijet factorization to determine the
impact that variations in the PDFs cause to the quark-gluon fractions. Since
there the fraction for a single quark jet was examined, we can make the
connection with the fractions defined here using

$\displaystyle x_{q}^{\rm excl}$ $\displaystyle=\frac{2x_{qq}^{\rm
excl}+x^{\rm excl}_{qg}}{2(x_{qq}^{\rm excl}+x_{gg}^{\rm excl}+x_{qg}^{\rm
excl})}\,,$ $\displaystyle x_{g}^{\rm excl}$ $\displaystyle=\frac{x^{\rm
excl}_{qg}+2x^{\rm excl}_{gg}}{2(x_{qq}^{\rm excl}+x_{gg}^{\rm
excl}+x_{qg}^{\rm excl})}\,.$ (2.71)

Here a subscript $q$ indicates a sum over all quark and antiquark flavor
channels. Note that by construction $x_{q}^{\rm excl}+x_{g}^{\rm excl}=1$.

###  2.7 Power corrections in the soft-drop resummation region

We notice from Eq. (2.62) that there are three kinds of power corrections
appearing on top of the soft-drop resummed result. The ${\cal O}(\xi)$ is
related to jet-mass resummation, the ${\cal O}(\xi_{0})$ is related to finite
$z_{\rm cut}$ effects that are ignored in the soft approximation, and lastly
the ${\cal O}((\xi/\xi_{0})^{\frac{2}{2+\beta}})$ results from collinear-soft
approximation of the soft-drop passing subjet. We investigate the ${\cal
O}(\xi_{0})$ power correction in Eq. (2.58) in resummation region at ${\cal
O}(\alpha_{s})$. Thus, we can refine Eq. (2.62) at ${\cal O}(\alpha_{s})$ as

$\displaystyle\tilde{\cal G}_{\kappa,\rm sd}\big{(}\xi,Q,z_{\rm
cut},\beta,\alpha_{s}(\mu)\big{)}$ $\displaystyle=\tilde{\cal G}^{\rm
resum}_{\kappa,\rm sd}\big{(}\xi,Q,z_{\rm cut},\beta,\mu\big{)}\Bigg{[}1+{\cal
O}\bigg{(}\xi,\Big{(}\frac{\xi}{\xi_{0}}\Big{)}^{\frac{2}{2+\beta}}\bigg{)}\Bigg{]}$
$\displaystyle\quad+\Delta\tilde{\cal G}^{\rm n.s.[1]}_{\kappa,\rm
sd}(\xi,z_{\rm cut},\beta,R,\alpha_{s}(\mu))\,.$ (2.72)

As remarked above, a fixed-order analysis of $\Delta\tilde{\cal G}^{\rm
n.s.[1]}_{\kappa,\rm sd}$ in App. A.3 shows that this is a small correction.
Thus, we will suppress the corrections from $\Delta\tilde{\cal G}^{\rm
n.s.[1]}_{\kappa,\rm sd}$ in the analysis below.

The remaining two power corrections, however, require a careful treatment.
Unlike Eqs. (2.4.1) and (2.56) where the power corrections were of purely
fixed-order origin, we cannot simply set all the scales in Eq. (2.62) to be
equal as these power corrections are related to turning off one, but not
every, resummation. We now describe our strategy to include these power
corrections using profile functions.

####  2.7.1 Soft-drop power corrections

Let us first consider the ${\cal
O}\big{(}(\xi/\xi_{0})^{\frac{2}{2+\beta}}\big{)}$ power correction. Eq.
(2.60) prescribes calculating this power correction by setting scales of the
soft pieces in the difference of two factorization formulae in Eqs. (2.56) and
(2.64). On the other hand, the jet scale must be left where it should be in
the two cases222The canonical jet scale is the same in the two cases but can
differ in the precise implementation through profile functions as we show
below., as we would still like to retain the jet-mass resummation in the two
cases. This can be straightforwardly implemented via the following set of
$\xi$-dependent scales

$\displaystyle\text{Plain jet-mass profiles}:$ $\displaystyle\mu_{\rm plain}$
$\displaystyle\equiv\\{\mu_{J}(\xi),\mu_{s}(\xi),\mu_{N}\\}\,,$ (2.73)
$\displaystyle\text{soft-drop jet-mass profiles}:$ $\displaystyle\mu_{\rm sd}$
$\displaystyle\equiv\\{\mu_{J}(\xi),\mu_{gs},\mu_{cs}(\xi),\mu_{N}\\}\,,$
$\displaystyle\text{soft-drop to plain jet-mass transition}:$
$\displaystyle\mu_{\rm sd\rightarrow plain}$
$\displaystyle\equiv\\{\mu_{J}(\xi),\mu_{gs},\mu_{cs\rightarrow
s}(\xi),\mu_{N}\\}\,.$

They are chosen to individually minimize logarithms in the momentum space
expressions of the factorization functions. We describe their implementation
in detail in Sec. 2.8, but for now it suffices to note that in the $\mu_{\rm
plain}$ profile the $\mu_{cs}$ and $\mu_{gs}$ scales merge into a single soft
scale $\mu_{s}$ Chien:2019osu , turning off any soft-drop related resummation.
This, however, continues to retain resummation between the jet and the soft
scales. We have also defined a profile $\mu_{\rm sd\rightarrow plain}$ that
merges the collinear and global soft scales for $\xi\geq\xi_{0}$.

Using these three profiles, the soft-drop power corrections can be captured by
considering the combination:

$\displaystyle\tilde{\cal G}^{\rm matched}_{\kappa,\rm
sd}(\xi)\equiv\tilde{\cal G}^{\rm resum}_{\kappa,\rm sd}\big{(}\xi,\mu_{\rm
sd\rightarrow plain}\big{)}+\tilde{\cal G}^{\rm plain}_{\kappa,\rm
sd}\big{(}\xi,\mu_{\rm plain}\big{)}-\tilde{\cal G}_{\rm sd}^{\rm
resum}\big{(}\xi,\mu_{\rm plain}\big{)}+\ldots\,,$ (2.74)

where the arguments $\mu_{\rm plain}$ etc. in these expressions are understood
as inserting each scale from the set in Eq. (2.73) into the respective
factorization function. For simplicity we have suppressed other arguments. The
behavior of $\mu_{\rm sd\rightarrow plain}$ ensures that in the plain jet-mass
region the first and the third term cancel, leaving the fixed-order soft-drop
cross section. Note that the choice $\mu_{cs\rightarrow s}(\xi)$ in $\mu_{\rm
sd\rightarrow plain}$ is arbitrary as long as the same profiles are used in
the first and the third term in the $\xi>\xi_{0}$ region. This matching also
ensures that the NGL pieces cancel with each other in precisely the same
fashion. Additionally, in the implementation below, the $\mu_{\rm plain}$
profile is designed to saturate to $\mu_{\rm N}$ for $\xi>\xi_{0}$, but ahead
of the LO jet-mass end point $\xi=1/4$, such that close to the endpoint all
the resummations are turned off. Finally, the terms ‘$\ldots$’ in Eq. (2.74)
correspond to jet-mass dependent power corrections which we now address.

####  2.7.2 Jet-mass power corrections

The jet-mass power corrections need careful treatment. We cannot simply
extrapolate the formula for the non-singular term in Eq. (2.47) into the
groomed region as it lacks resummation related to any soft-drop factorization
that we would like to retain. Eq. (2.47) is actually connected to a soft-
collinear factorization where the soft modes live at the boundary of the jet.
This is perfectly fine for large jet masses beyond the soft-drop cusp and Eq.
(2.47) remains legitimate in the ungroomed region. However, in the region soft
drop is active, the soft modes cannot lie at angles close to the jet radius,
but must be confined within the maximum kinematically allowed groomed jet
radius. We define

$\displaystyle r_{g}\equiv\frac{R_{g}}{R}\,,\qquad(\text{$pp$ case})\,,\qquad
r_{g}\equiv\frac{\tan\frac{R_{g}}{2}}{\tan\frac{R}{2}}\,,\qquad(\text{$e^{+}e^{-}$
case})\,,$ (2.75)

such that the groomed jet radius is constrained by the jet mass as
Pathak:2020iue

$\displaystyle r_{g}<r_{g}^{\rm max}(\xi)$
$\displaystyle=\min\Bigg{\\{}\bigg{(}\frac{m_{J}^{2}}{QQ_{\rm
cut}}\bigg{)}^{\frac{1}{2+\beta}},1\Bigg{\\}}$ (2.76)
$\displaystyle=\min\bigg{\\{}\Big{(}\frac{\xi}{\xi_{0}}\Big{)}^{\frac{1}{2+\beta}},1\bigg{\\}}\,.$

Because of this restriction, there is a non-trivial resummation between the
global soft and collinear soft scales. The second argument of min is valid
when $\xi>\xi_{0}$, the ungroomed region.

To properly evaluate these mass-dependent power corrections, we instead need
to consider the cross section, differential in groomed jet mass, and
simultaneously cumulative in the groomed jet radius, as worked out in Ref.
Pathak:2020iue . The jet-mass measurement constrains the range of groomed jet
measurement, with the maximum allowed value given by Eq. (2.76). When
$r_{g}\lesssim r_{g}^{\rm max}$, one has a distinct collinear-soft subjet that
stops the soft drop. Eq. (2.61) is in fact a special case when the
(cumulative) $r_{g}=r_{g}^{\rm max}$, such that all groomed jet radius values
are allowed. On the other hand, for a given $\xi$ measurement, the groomed jet
radius can be as small as

$\displaystyle r_{g}\gtrsim r_{g}^{\rm min}(\xi)=\sqrt{\xi}\,.$ (2.77)

This corresponds to the situation where there is a single hard collinear mode
that fills up the entire jet, and the soft drop is stopped by a haze of soft
radiation at the angles $R_{g}$ to the jet. In this region, the jet-mass
measurement is not factorized, but groomed jet radius is, with the
corresponding factorization formula given by

$\displaystyle{\cal G}^{\rm min}_{\kappa,\rm sd}\big{(}z,\xi,r_{g},z_{\rm
cut},\beta,\mu\big{)}$ $\displaystyle=\sum_{\kappa^{\prime}}{\cal
H}_{\kappa^{\prime}\rightarrow\kappa}(z,Q,\mu)S_{G}^{\kappa}\big{(}Q_{\rm
cut},\beta,\zeta,\mu\big{)}{\cal S}_{\rm NGL}^{\kappa}\big{(}t_{gs}\big{)}$
(2.78) $\displaystyle\quad\times S_{c_{g}}^{\kappa}\big{(}Q_{\rm
cut}r_{g}^{1+\beta},\beta,\mu\big{)}{\cal S}_{\rm
NGL}^{\kappa}\big{(}t_{c}\big{)}\frac{1}{r_{g}^{2}}{\cal
C}^{\kappa}\bigg{(}\frac{\xi}{r_{g}^{2}},Qr_{g},\mu\bigg{)}\,.$

The soft function $S_{c_{g}}^{\kappa}$ describes dynamics of soft modes at the
groomed jet boundary. The soft-drop resummation is included through an RG
evolution between the global soft and $S_{c_{g}}$ scales. Here ${\cal
C}^{\kappa}$ is the collinear function which, as can be seen from Eqs. (A.11)
and (A.2), is precisely the fixed-order result for ungroomed cross section in
Eq. (2.31) with a jet radius $r_{g}R$, differing only in the $\delta(\xi)$
pieces at one-loop to satisfy RG consistency (which are moved into the DGLAP
evolution piece). Next, in addition to the non-global logarithms present at
the jet boundary, probing the groomed jet radius results in additional NGL at
the $R_{g}$ boundary Kang:2019prh , with the argument of this NGL piece being

$\displaystyle t_{c}\equiv t\big{[}Q_{\rm
cut}r_{g}^{1+\beta},Qr_{g}\big{]}\,.$ (2.79)

The two arguments are canonical scales of the $S_{c_{g}}^{\kappa}$ and ${\cal
C}^{\kappa}$ functions. Since we are only interested in capturing the ${\cal
O}(\alpha_{s})$ jet-mass related power suppressed terms we will turn off this
NGL piece in the numerical analysis.

For our purpose, it is important to note that the jet-mass dependence in
collinear function factorizes as in the case of an ungroomed jet with the jet
radius $R\rightarrow r_{g}R$:

$\displaystyle\frac{1}{r_{g}^{2}}{\cal
C}^{\kappa}\Big{(}\frac{\xi}{r_{g}^{2}}\Big{)}$
$\displaystyle=\int\textrm{d}s\int\textrm{d}\ell^{+}\>J_{\kappa}\big{(}s,\mu\big{)}\,S^{\kappa}_{c_{m}}(\ell^{+},\mu)\,\delta\bigg{(}\frac{\xi}{r_{g}^{2}}-\frac{s}{Q^{2}r_{g}^{2}}-\frac{\ell^{+}}{Qr_{g}}\bigg{)}\bigg{(}1+{\cal
O}\Big{(}\frac{\xi}{r_{g}^{2}}\Big{)}\bigg{)}$
$\displaystyle=\Sigma_{\kappa}\bigg{(}\frac{\xi}{r_{g}^{2}},Qr_{g},\mu\bigg{)}\bigg{(}1+{\cal
O}\Big{(}\frac{\xi}{r_{g}^{2}}\Big{)}\bigg{)}\,,$ (2.80)

where $\Sigma_{\kappa}$ was defined above in Eq. (2.38). However, the
collinear function is not RG invariant alone, but only in combination with the
other pieces in Eq. (2.78). The cross section resulting from Eq. (2.7.2)
defines the “intermediate-$R_{g}$” factorization:

$\displaystyle{\cal G}^{\rm int}_{\kappa,\rm sd}(z,\xi,r_{g},z_{\rm
cut},\beta,\mu)$
$\displaystyle=\frac{\textrm{d}}{\textrm{d}\xi}\Bigg{[}\sum_{\kappa^{\prime}}{\cal
H}_{\kappa^{\prime}\rightarrow\kappa}(z,Q,\mu)S_{G}^{\kappa}\big{(}Q_{\rm
cut},\beta,\zeta,\mu\big{)}{\cal S}_{\rm NGL}^{\kappa}(t_{gs})$
$\displaystyle\quad\times S_{c_{g}}^{\kappa}\big{(}Q_{\rm
cut}r_{g}^{1+\beta},\beta,\mu\big{)}{\cal S}_{\rm
NGL}^{\kappa}(t_{cs})\Sigma_{\kappa}\bigg{(}\frac{\xi}{r_{g}^{2}},Qr_{g},\mu\bigg{)}\Bigg{]}\,,$
(2.81)

where, as before, we have NGLs associated with the groomed jet boundary, and
this time the arguments are

$\displaystyle t_{cs}\equiv t\bigg{[}Q_{\rm
cut}r_{g}^{1+\beta},\frac{Q\xi}{r_{g}}\bigg{]}\,.$ (2.82)

Since we are not constraining the groomed jet radius, we set it to its maximum
allowed value in Eq. (2.76) in these expressions. Thus, from Eq. (2.7.2) we
see that all the jet-mass dependent power corrections in the soft-drop
resummation region are given by the difference of two factorized cross
sections in Eqs. (2.78) and (2.7.2) at $r_{g}=r_{g}^{\rm max}(\xi)$ by setting
the scale of jet and collinear soft functions to be the same. The scale of the
global soft function must be left to its canonical value to retain soft-drop
resummation. To this end, we define the profile:

Min-$R_{g}$ factorization profile: $\displaystyle\mu_{\rm
min}\equiv\\{\mu_{\cal C}(\xi),\mu_{cs}(\xi),\mu_{gs},\mu_{N}\\}\,,$ (2.83)

such that the jet-mass dependent power corrections in the soft-drop
resummation region are given by

$\displaystyle\text{${\cal O}(\xi)$ p.c. for $\xi<\xi_{0}$}:$
$\displaystyle\tilde{\cal G}^{\rm min}_{\kappa,\rm sd}\big{(}\xi,r_{g}^{\rm
max}(\xi),\mu_{\rm min}\big{)}-\tilde{\cal G}_{\rm sd}^{\rm
int}\big{(}\xi,r_{g}^{\rm max}(\xi),\mu_{\rm min}\big{)}\,,$ (2.84)

where we defined the maximum groomed jet radius in Eq. (2.76).

As a consequence of setting $r_{g}=r_{g}^{\rm max}(\xi)$, the same collinear
soft scale $\mu_{cs}(\xi)$ appears in the $S_{c_{g}}^{\kappa}$ function in
Eqs. (2.78) and (2.7.2). The new scale $\mu_{\cal C}(\xi)$ minimizes
logarithms in the collinear function and is defined below in Sec. 2.8.
Following Eq. (2.7.2), we ought to use this same scale in the $J_{\kappa}$ and
$S_{c_{m}}$ functions in the intermediate $R_{g}$ cross section. We give
explicit formulae for evaluating these power corrections in App. D.

We would like to connect this formula with the jet-mass power corrections in
the ungroomed region in Eq. (2.47). This is quite straightforward as the
profile $\mu_{\cal C}(\xi)$ automatically becomes $\mu_{N}$ when $r_{g}^{\rm
max}=1$ in the ungroomed region. We only need to ensure that $\mu_{cs}(\xi)$
is switched to $\mu_{gs}$ for $\xi>\xi_{0}$ to turn off any soft-drop
resummation in this region. Thus, we will define a $\mu_{\rm min\rightarrow
plain}$ profile, which includes this behavior of the collinear soft profile,

Min-$R_{g}$ to plain jet-mass transition: $\displaystyle\mu_{\rm
min\rightarrow plain}\equiv\\{\mu_{\cal C}(\xi),\mu_{cs\rightarrow
gs}(\xi),\mu_{gs},\mu_{N}\\}\,.$ (2.85)

Using this result we arrive at the complete expression for the matched cross
section in Eq. (2.74):

$\displaystyle\tilde{\cal G}^{\rm matched}_{\kappa,\rm sd}(\xi)$
$\displaystyle\equiv{\color[rgb]{1,0.39,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.39,0.13}\pgfsys@color@cmyk@stroke{0}{0.61}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>G}^{\rm resum}_{\kappa,\rm sd}\big{(}\xi,\mu_{\rm sd\rightarrow
plain}\big{)}}+{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}\tilde{\cal
G}^{\rm plain}_{\kappa,\rm sd}\big{(}\xi,\mu_{\rm
plain}\big{)}}-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@cmyk@stroke{0}{1}{1}{0}\pgfsys@color@cmyk@fill{0}{1}{1}{0}\tilde{\cal
G}_{\rm sd}^{\rm resum}\big{(}\xi,\mu_{\rm plain}\big{)}}$ (2.86)
$\displaystyle\quad+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@cmyk@stroke{1}{1}{0}{0}\pgfsys@color@cmyk@fill{1}{1}{0}{0}\tilde{\cal
G}^{\rm min}_{\kappa,\rm sd}\big{(}\xi,r_{g}^{\rm max}(\xi),\mu_{\rm
min\rightarrow
plain}\big{)}}-{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\tilde{\cal
G}_{\rm sd}^{\rm int}\big{(}\xi,r_{g}^{\rm max}(\xi),\mu_{\rm min\rightarrow
plain}\big{)}}\,.$

Figure 2: The matched curves at NNLL for quark jets (left) and gluon jets
(right). The dotted curves correspond to overlap terms that are subtracted in
the matched cross section.

The second line now captures all the jet-mass dependent power corrections. We
display in Fig. 2 how this works in practice. Recall from Eq. (2.28) that
$\tilde{\cal G}_{\kappa}$ is defined as the normalized cross section
$\frac{1}{\sigma_{\kappa}^{\text{incl}}}\frac{\textrm{d}\sigma_{\kappa}}{\textrm{d}\xi}$,
which is what we plot in Fig. 2. For now we have turned off all the NGL terms,
which we discuss separately below. We have colored each term to match the
corresponding curve in Fig. 2. We see from Fig. 2 that the two power
corrections have opposite signs and their size can depend on the kinematic and
grooming parameters, as well as the flavor of the jet.333In the figure we
follow a slightly different implementation than written in Eqs. (2.73) and
(2.86) by letting the collinear soft scale merge with global soft scale for
both $\tilde{\cal G}^{\rm resum}_{\kappa,\rm sd}$ pieces. This still has the
same effect of the two pieces canceling out each other entirely in the
ungroomed resummation region, as can be seen in Fig. 2 for
$\log_{10}\xi>\log_{10}\xi_{0}\sim-1$.  Next, we see that the pieces
$\tilde{\cal G}^{\rm plain}_{\kappa,\rm sd}$ and $\tilde{\cal G}^{\rm
int}_{\rm sd}$ pieces cancel each other in the same region. This is
straightforward to see from Eq. (2.7.2) where $r_{g}=1$ results in complete
(internal) cancellation of $S_{c_{g}}^{\kappa}$ and $S_{G}^{\kappa}$ functions
as well as canceling of $\Sigma_{\kappa}(\xi)$ between the two cross sections.

Figure 3: Impact of non-global logarithms on the matched groomed jet-mass
spectrum. Here we treat NGLs at LL and large $N_{c}$ limit. The differences
between the solid and dashed curves are within NNLL perturbative uncertainty.

We now turn to the NGL pieces. In the implementation of the NGL, we make use
of the canonical scales of the set of profiles used above. This allows us to
employ the same matching scheme in Eq. (2.86) to merge NGLs that differ in
each of the terms. In this work we will limit ourselves to analyzing the
impact of NGLs at LL in the large $N_{c}$ limit using the parameterization
provided in Ref. Dasgupta:2001sh . In Fig. 3 we show the NNLL spectrum for
quark and gluon jets for $\beta\in\\{0,1,2\\}$. We find that the impact of NGL
on the quark jets is very small, while the difference for gluon jets is
somewhat larger. As mentioned earlier, despite the fact that the NGLs appear
only in the normalization in the SDOE region, these differences in their
relative impact on quark and gluon jets can lead to modification of the shape.
However, we will see that the differences in Fig. 3 are well within the NNLL
perturbative uncertainty due to scale variation. (The only borderline case is
$\beta=2$ gluon jets.) Thus for simplicity we not include $\alpha_{s}$
variations in these NGL pieces in our analysis. We leave a more detailed
exploration of NGL effects on groomed jet mass to future work when higher
precision results become available for $pp$ collisions, such ${\cal
O}(\alpha_{s}^{2})$ terms or the full N3LL resummation.

###  2.8 Profile functions

In this section we describe the implementation of profile functions
Ligeti:2008ac ; Abbate:2010xh  as well as the scheme for varying them to
estimate perturbative uncertainty. We will employ the profile functions
described in Ref. Pathak:2020iue  and review the essential ingredients here.
We start with the two scales $\mu_{N}$ and $\mu_{gs}$, from which other
profile functions will be derived:

$\displaystyle\mu_{N}=e_{N}Q\,,\qquad\mu_{gs}=e_{gs}Q_{\rm cut}\,,$ (2.87)

where the parameters $e_{N}$ and $e_{gs}$ give a handle for varying these
scales, as described in the next subsection. For central profiles, we take
$e_{N}=e_{gs}=1$.

####  2.8.1 Soft-drop resummation region

We first consider the soft-drop profiles valid in the region $\xi<\xi_{0}$.
The value $\xi_{0}$, defined in Eq. (2.7), is the point at which the
collinear-soft and global-soft scales merge. To define the jet-mass dependent
soft and collinear scales we first introduce the auxiliary function $f_{\rm
run}(r_{g})$ defined by

$\displaystyle f_{\rm run}(r_{g})$
$\displaystyle\equiv\left\\{\begin{array}[]{ll}y_{0}\Big{(}1+\frac{r_{g}^{2}}{4y_{0}^{2}}\Big{)}&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}r_{g}\leq
2y_{0}\\\\[4.0pt] r_{g}&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2y_{0}<r_{g}\leq
1\end{array}\right.\;.$ (2.90)

Here

$\displaystyle y_{0}\equiv\frac{n_{0}}{(\mu_{gs}/1\,{\rm
GeV})}>\frac{\Lambda_{\rm QCD}}{Q_{\rm cut}}\,,$ (2.91)

and $n_{0}$ is an ${\cal O}(1)$ number. We take $n_{0}=1$ GeV as the central
value for our numerical analysis. This number governs the point $\xi_{\rm
SDNP}$ at which the jet-mass dependent scales will freeze,

$\displaystyle\xi_{\rm
SDNP}\equiv\xi_{0}\,y_{0}^{\frac{2+\beta}{1+\beta}}=\frac{n_{0}}{\mu_{N}}\Big{(}\frac{n_{0}}{\mu_{gs}}\Big{)}^{\frac{1}{1+\beta}}\,.$
(2.92)

Using the auxiliary function, the central collinear-soft profile
$\mu_{cs}(\xi)$ is given by

$\displaystyle\mu_{cs}(\xi)\equiv\mu_{gs}\,f_{\rm
run}\bigg{(}\Big{(}\frac{\xi}{\xi_{0}}\Big{)}^{\frac{1+\beta}{2+\beta}}\bigg{)}\,,\qquad\quad\xi<\xi_{0}\,.$
(2.93)

To derive the jet function scale $\mu_{J}$ from $\mu_{cs}$, we first derive an
auxiliary ungroomed soft scale

$\displaystyle\tilde{\mu}_{s}(\xi)\equiv\mu_{cs}(\xi)\>\bigg{(}\frac{\mu_{cs}(\xi)}{\mu_{gs}}\bigg{)}^{\frac{1}{1+\beta}}\,,\qquad\quad\xi<\xi_{0}\,,$
(2.94)

such that the jet function scale is given by

$\displaystyle\mu_{J}(\xi)$
$\displaystyle\equiv\Big{[}\mu_{N}\tilde{\mu}_{s}(\xi)\Big{]}^{\frac{1}{2}}=\Bigg{[}\mu_{N}\>\mu_{cs}(\xi)\>\bigg{(}\frac{\mu_{cs}(\xi)}{\mu_{gs}}\bigg{)}^{\frac{1}{1+\beta}}\Bigg{]}^{\frac{1}{2}}\,,\qquad\quad\xi<\xi_{0}\,.$
(2.95)

Finally the collinear function scale $\mu_{\cal C}$ is defined as

$\displaystyle\mu_{\cal
C}(\xi)=\mu_{N}\Big{(}\frac{\mu_{cs}(\xi)}{\mu_{gs}}\Big{)}^{\frac{1}{1+\beta}}\,.$
(2.96)

Thus $\mu_{\cal C}$ inherits its variations from $\mu_{cs}$ and $\mu_{gs}$.

####  2.8.2 Ungroomed region

At the value $\xi_{0}$ the global-soft and collinear-soft scales merge, i.e.
$\mu_{cs}|_{\xi_{0}}=\mu_{gs}|_{\xi_{0}}$. Beyond this value, we enter the
ungroomed resummation region. We first derive the profile for the ungroomed
soft function, $\mu_{s}$. Following Ref. Lustermans:2019plv  its profile can
be written as:

$\displaystyle\mu_{s}(\xi)\equiv\mu_{N}\,f_{\rm run}^{\rm plain}(\xi)\,,$
(2.97)

where $f_{\rm run}^{m_{J}}(\xi)$ is given by

$\displaystyle f_{\rm run}^{\rm plain}(\xi)$
$\displaystyle\equiv\left\\{\begin{array}[]{ll}x_{0}\big{(}1+\frac{\xi^{2}}{4x_{0}^{2}}\big{)}&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\xi\leq
2x_{0}\\\\[2.0pt] \xi&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}2x_{0}<\xi\leq
x_{1}\\\\[2.0pt] \xi+\frac{(2-x_{2}-x_{3})(\xi-
x_{1})^{2}}{2(x_{2}-x_{1})(x_{3}-x_{1})}&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}x_{1}<\xi\leq
x_{2}\\\\[2.0pt] 1-\frac{(2-x_{1}-x_{2})(\xi-
x_{3})^{2}}{2(x_{3}-x_{1})(x_{3}-x_{2})}&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}x_{2}<\xi\leq
x_{3}\\\\[2.0pt] 1&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}x_{3}<\xi\leq 1\\\\[2.0pt]
\end{array}\right.\;.$ (2.103)

Here $x_{0}$ corresponds to the point where the scale is frozen in the
nonperturbative region. To be consistent with Eq. (2.91), we set it to

$\displaystyle x_{0}=\frac{n_{0}}{(\mu_{N}/1{\rm GeV})}\,.$ (2.104)

The purpose of this function is to give a smooth transition between the linear
dependence on $\xi$ for $2x_{0}<\xi<x_{1}$, and a constant value for
$\xi>x_{3}$. To this end, we add and subtract a quadradic function in $\xi$ in
the regions $[x_{1},x_{2}]$, and $[x_{2},x_{3}]$, respectively. When choosing
values for $x_{1}$, $x_{2}$ and $x_{3}$, we must ensure that the transition to
the fixed-order region, whenever possible, starts only after the soft drop
turns off, but in addition to finishing before the jet-mass endpoint at
$\xi=0.25$. To this end, we take $x_{1}=1.15\xi_{0}$, $x_{3}=0.2$, and
$x_{2}=(x_{1}+x_{3})/2$.444We deal with cases with aggressive grooming, such
as $z_{\rm cut}=0.2$ and $\beta=0$, where the soft-drop transition point can
appear in the fixed-order region with $0.2<\xi_{0}<0.25$ by setting
$x_{3}=(0.25+\xi_{0})/2$. For yet higher $z_{\rm cut}$, when $\xi_{0}>0.25$,
we set $x_{3}=\min\\{\xi_{0}+0.1,1\\}$. In each of these two cases we set
$x_{1}=1.15\xi_{0}<x_{3}$ or $x_{1}=\xi_{0}<x_{3}$, and
$x_{2}=(x_{1}+x_{3})/2$.

Analogously to Eq. (2.95), the jet scale for ungroomed profiles is derived
from $\mu_{s}(\xi)$ profile

$\displaystyle\mu_{J}^{\rm
plain}(\xi)\equiv\Big{[}\mu_{N}\>\mu_{s}\Big{]}^{\frac{1}{2}}\,,$ (2.105)

such that the jet scale also has the plain jet-mass transition points. In the
implementation of the set of profiles $\mu_{\rm sd\rightarrow plain}$ the jet
scale in Eq. (2.95) will be replaced by the profile in Eq. (2.105) once
$\xi>\xi_{0}$ and the global and collinear-soft scales by the ungroomed soft
scale in Eq. (2.97).

Figure 4: Profile functions for $pp$ collisions at $p_{T}=600$ GeV. Top panel
shows profiles used for the soft-drop resummed and plain jet-mass cross
sections. The bottom panel shows the profile variations with $\alpha$ (left
panel) and with $\gamma$ (right panel) in the soft-drop resummation region.

####  2.8.3 Profile variation

We now describe the profile variations used to assess perturbative
uncertainties. They are implemented through a set of parameters introduced
above,

$\\{e_{i},\alpha,\lambda,\gamma,\rho,n_{0}\\}\,.$ (2.106)

The order in which the parameters are introduced in the section also
corresponds with how they are implemented in the code: the variation of
parameters introduced later are impacted by simultaneous variation of
parameters introduced earlier, but not vice versa. We now turn to describing
the effects of varying each parameter.

First, the hard and global soft scales are varied by varying the $e_{i}\equiv
e_{N}=e_{gs}$, where we have set $e_{N}=e_{gs}$ to prevent the soft-drop cusp
point $\xi_{0}$ from varying. These variations affect all the other scales
derived from these scales, as shown in the previous section, and below.

Second, we implement an independent variation of the collinear-soft scale, by
making use of a trumpet function $f_{\rm vary}(r_{g})$:

$f_{\rm vary}(r_{g})=\left\\{\begin{array}[]{l r}2(1-r_{g}^{2})\,,&\qquad
r_{g}<0.5\\\ 1+2(1-r_{g})^{2}\,,&\qquad 0.5\leq r_{g}\leq
1\end{array}\right.,$ (2.107)

which implements a variation for $r_{g}<1$. Following Eq. (2.93), the
variation of $\mu_{cs}$ scale is given by

$\displaystyle\tilde{\mu}_{cs}^{\rm vary}(\xi,\alpha)$
$\displaystyle\equiv\mu_{gs}\Bigg{[}f_{\rm
vary}\bigg{(}\Big{(}\frac{\xi}{\xi_{0}}\Big{)}^{\frac{1+\beta}{2+\beta}}\bigg{)}\Bigg{]}^{\alpha}\,f_{\rm
run}\bigg{(}\Big{(}\frac{\xi}{\xi_{0}}\Big{)}^{\frac{1+\beta}{2+\beta}}\bigg{)}\,,\qquad\xi<\xi_{0}\,.$
(2.108)

The choice $\alpha=0$ returns the default profile, and varying $\alpha=\pm
0.5$ allows for variation in the resummation region up to a factor of
$\sqrt{2}$. The tilde indicates that this is not quite the final scale. This
is because, as written, the $\alpha$ variation will drive this scale below the
nonperturbative scale $n_{0}$ in Eq. (2.91). To prevent that, we re-freeze
this scale in the nonperturbative region:

$\displaystyle\mu_{cs}^{\rm vary}(\xi,\alpha)\equiv f_{\rm
freeze}\big{[}\tilde{\mu}_{cs}^{\rm vary}(\xi,\alpha)\big{]}$ (2.109)

where

$f_{\rm freeze}[\mu]\equiv\left\\{\begin{array}[]{l r}\mu\,,&\qquad\mu\geq
2n_{0}\\\
n_{0}\Big{(}1+\frac{\mu^{2}}{4n_{0}^{2}}\Big{)}\,,&\qquad\mu<2n_{0}\end{array}\right..$
(2.110)

Having defined $\alpha$-variation of $\mu_{cs}$, the corresponding variations
of $\mu_{J}$ is derived using Eq. (2.95) by replacing
$\mu_{cs}\rightarrow\mu_{cs}^{\rm vary}$. As mentioned above, these two scales
inherit the $e_{gs}$ variation through the $\mu_{gs}$ scale.

Third, we consider the variation that breaks the following canonical relation
between the soft scales: $\mu_{cs}^{2+\beta}=\mu_{s}^{1+\beta}\mu_{gs}$. It is
implemented via a parameter $\rho$ as follows:

$\displaystyle\tilde{\mu}_{cs}^{\rm vary}(\xi,\alpha,\rho)$
$\displaystyle\equiv\Bigg{[}f_{\rm
vary}\bigg{(}\Big{(}\frac{\xi}{\xi_{0}}\Big{)}^{\frac{1+\beta}{2+\beta}}\bigg{)}\Bigg{]}^{\alpha}\big{(}\tilde{\mu}_{s}(\xi)\big{)}^{\frac{1+\beta+\rho}{2+\beta}}\big{(}\mu_{gs}\big{)}^{\frac{1-\rho}{2+\beta}}\,,$
$\displaystyle\mu_{cs}^{\rm vary}(\xi,\alpha,\rho)$ $\displaystyle\equiv
f_{\rm freeze}\big{[}\tilde{\mu}_{cs}^{\rm vary}(\xi,\alpha,\rho)\big{]}\,.$
(2.111)

Here the auxiliary scale $\tilde{\mu}_{s}$ is defined in Eq. (2.94), i.e.
without any $\alpha$ variation.

As a final variation in the resummation region, we break the canonical
relation between the jet and the hard scale in Eq. (2.95), via parameter
$\gamma$:

$\displaystyle\mu_{J}^{\rm vary}(\xi,\alpha,\rho,\gamma)=\big{(}\mu_{N}^{\rm
vary}\big{)}^{\frac{1}{2}+\gamma}\big{(}\tilde{\mu}^{\rm
vary}_{s}(\xi,\alpha,\rho)\big{)}^{\frac{1}{2}-\gamma}\,,$ (2.112)

where, analogously to before, the auxiliary scale $\tilde{\mu}^{\rm
vary}_{s}(\xi,\alpha,\rho)$ inherits the $\alpha$ and $\rho$ variation from
being defined by substituting $\mu_{cs}(\xi)$ in Eq. (2.94) by $\mu_{cs}^{\rm
vary}(\xi,\alpha,\rho)$ defined in Eq. (2.8.3):

$\displaystyle\tilde{\mu}_{s}^{\rm vary}(\xi,\alpha,\rho)\equiv\mu_{cs}^{\rm
vary}\Big{(}\frac{\mu_{cs}^{\rm vary}}{\mu_{gs}^{\rm
vary}}\Big{)}^{\frac{1}{1+\beta}}\,.$ (2.113)

Next, we consider profile variation in the ungroomed region. The hard scale is
varied as before using the parameter $e_{i}$. The trumpet variation of the
$\mu_{s}(\xi)$ scale, defined in Eq. (2.97), is implemented only in the
ungroomed resummation region $\xi_{0}\leq\xi\leq x_{3}$, where the parameter
$x_{3}$ appears in $f_{\rm run}^{\rm plain}$ defined in Eq. (2.103). To this
end, we first define

$\displaystyle f^{\rm plain}_{\rm vary}(\xi)\equiv\left\\{\begin{array}[]{l
l}1&\qquad 0<\xi\leq\xi_{0}\\\ \zeta\big{(}\xi,\xi_{0},x_{\rm
mid};1,2\big{)}\,,&\qquad\xi_{0}\leq\xi<x_{\rm mid}\\\ \zeta\big{(}\xi,x_{\rm
mid},x_{3};2,1\big{)}\,,&\qquad x_{\rm mid}\leq\xi\leq x_{3}\\\ 1&\qquad
x_{3}<\xi\leq 1\end{array}\right..$ (2.118)

where

$\displaystyle x_{\rm mid}\equiv\frac{\xi_{0}+x_{3}}{2}\,,$ (2.119)

and

$\displaystyle\zeta\big{(}\xi,x_{\rm start},x_{\rm
end};a_{1},a_{2}\big{)}\equiv\left\\{\begin{array}[]{l l}a_{1}&\qquad\xi\leq
x_{\rm start}\\\ a_{1}+\frac{2(a_{2}-a_{1})(\xi-x_{\rm start})^{2}}{(x_{\rm
end}-x_{\rm start})^{2}}\,,&\qquad x_{\rm start}\leq\xi<\frac{x_{\rm
start}+x_{\rm end}}{2}\\\ a_{2}-\frac{2(a_{2}-a_{1})(x_{\rm
end}-\xi)^{2}}{(x_{\rm end}-x_{\rm start})^{2}}\,,&\qquad\frac{x_{\rm
start}+x_{\rm end}}{2}\leq\xi\leq x_{\rm end}\\\ a_{2}&\qquad x_{\rm
end}<\xi\end{array}\right..$ (2.124)

Thus, the profile $\mu_{s}(\xi)$ is varied as

$\displaystyle\tilde{\mu}^{\rm
vary}_{s}(\xi,\lambda)\equiv\mu_{N}\Big{[}f_{\rm vary}^{\rm
plain}(\xi)\Big{]}^{\lambda}f_{\rm run}^{\rm plain}(\xi)\,,$ (2.125)

but since the variation with $\lambda$ may spoil the freezing in the
nonperturbative region, we must re-freeze the soft scale by taking

$\mu^{\rm vary}_{s}(\xi,\lambda)=f_{\rm freeze}[\tilde{\mu}^{\rm
vary}_{s}(\xi,\lambda)]\,.$ (2.126)

Finally, analogously to Eq. (2.112), the variation of the scale $\mu_{J}^{\rm
plain}$ is given by

$\displaystyle\mu_{J}^{\rm
plain,vary}(\xi,\lambda,\gamma)\equiv\mu_{N}^{\frac{1}{2}+\gamma}\big{(}\mu^{\rm
vary}_{s}(\xi,\lambda)\big{)}^{\frac{1}{2}-\gamma}\,.$ (2.127)

In our analysis, we choose the following profile variation ranges for the six
different parameters:

$\displaystyle e_{i}\in[0.5,2]\,,$ $\displaystyle\alpha\in[-1,1]\,,$
$\displaystyle\lambda\in[-1,0.3]\,,$ (2.128)
$\displaystyle\gamma\in[-0.1,0.1]\,,$ $\displaystyle\rho\in[-0.1,0.1]\,,$
$\displaystyle n_{0}\in[0.75,1.25]\,.$

We have adopted the same range of variations as in Ref. Pathak:2020iue , with
the exception of $\lambda$ parameter which did not appear there. A key
criterion in deciding the range of variation is to ensure that the hierarchies
among various scales is maintained. We also ensure that variation does not
interrupt the monotonic behavior of the scales or introduce unphysical kinks.
These profiles with the variations are displayed in Fig. 4. Since it is a
priori not clear how the different variations might affect each other, we vary
the profiles randomly in all six parameters, and take an envelope for our
uncertainty.

Figure 5: Variations for each of the six different profile-variation
parameters individually, for quark and gluon jets at $p_{T}=600$ GeV, with
$\beta=1$. Other values of $\beta$ are shown in App. E.

Fig. 5 shows impact of profile variations for each of the parameter for the
LL, NLL and NNLL jet-mass spectrum of quarks and gluons. Note that at LL and
NLL, the shape and the transition points are governed solely by profile
functions entering the RG evolution kernels. The soft drop cusp for LL and NLL
cross section is thus located at $\xi_{0}$, whereas it appears at
$\xi_{0}^{\prime}$ for NNLL once $\mathcal{O}(\alpha_{s})$ terms are included.
The end-point of the jet-mass spectrum for LL and NLL is located at
$\xi=x_{3}$, where $x_{3}$ was defined above in Eq. (2.103). The NNLL curve,
being matched to LO non-singular cross section, has the end point at
$\xi=0.25$. The transition to the nonperturbative region, however, happens for
the same jet mass for all of the three orders. Next, since we are normalizing
to the inclusive cross section according to Eq. (2.86), and indeed the
integral over the curve for the NLL and NNLL spectra is close to 1, although
the integral of the LL curve is around a factor of 20% smaller. We find that
the dominant variation is induced by the overall scale variation due to
$e_{i}$, the trumpet variation via $\alpha$, and the variation due to breaking
of jet, soft and hard collinear canonical relation in Eqs. (2.112) and (2.127)
through $\gamma$. On the other hand, the variation of the plain soft function
scale in the ungroomed region via $\lambda$, the breaking of the soft drop
canonical relation via $\rho$ in Eq. (2.8.3), and the nonperturbative scale
variation via $n_{0}$ lead to subdominant effects. We also notice here a more
significant dependence on the perturbative order for the slope of the spectrum
in the soft drop resummation region for quark jets compared to gluon jets. We
will discuss the impact of this effect on $\alpha_{s}$ below. The analogous
profile variation figures for $\beta=0$ and $\beta=2$ are provided in App. E.

###  2.9 Comparison with previous work

We now make a comparison with earlier calculations of the groomed jet-mass
spectrum and list the various improvements achieved in this paper.

  1. 1.

Resummation and matching: The leading power factorization in the small $z_{\rm
cut}$ limit was formulated in SCET in Ref. Frye:2016aiz . In Refs.
Marzani:2017mva ; Marzani:2017kqd  the analysis was extended by including
finite $z_{\rm cut}$ terms in framework of diagrammatic QCD for $\beta=0$ soft
drop at NLL (single-logarithmic) accuracy and matched to NLO fixed-order
results. The finite $z_{\rm cut}$ pieces were incorporated both at the level
of fixed-order perturbation theory as well as by modifying the resummation
kernels in the Sudakov double log exponent. Using the $1\rightarrow 3$
splitting functions in the collinear limit, the calculation in this framework
was improved to NNLL accuracy in Ref. Anderle:2020mxj  providing a cross check
for results in Ref. Frye:2016aiz . In Ref. Kang:2018jwa  the groomed and
ungroomed jet-mass spectrum for inclusive jets in $pp$ collisions was
calculated at NLL accuracy, and matched to LO fixed-order results. Formulating
the problem in terms of inclusive jets enabled determination of the absolute
normalization up to NGLs.

In our analysis we have performed the state-of-the-art NNLL resummation with
the ingredients available to-date. In order to extend the analysis to N3LL
accuracy, we need two loop constant pieces of the collinear-soft
$S_{c}^{\kappa}$ and global soft $S_{G}^{\kappa}$ functions for quarks and
gluon jets with generic jet radius $R<\pi/2$, as well as the corresponding
3-loop non-cusp anomalous dimension. This is partially achieved through the
calculation for hemisphere quark jets for $\beta=0$ in Ref. Kardos:2020ppl .
However, the 2-loop global soft constant term calculation for non-hemisphere
jets is still lacking.

Next, while our matching to fixed-order results is performed at LO, we have
done so by computing fully analytic expressions for the jet-mass dependent
pieces $\tilde{\cal G}_{\kappa,\rm No\,sd}$ and $\tilde{\cal G}_{\kappa,\rm
sd}$ of the inclusive jet function, as defined in Eq. (2.28). In the groomed
region, we have analytically captured the jet-mass related power corrections
using the doubly differential groomed jet mass and groomed jet radius result
in the $r_{g}\gtrsim r_{g}^{\rm min}(\xi)$ regime. We have also checked that
the finite $z_{\rm cut}$ power corrections in the $\Delta\tilde{\cal G}^{\rm
n.s.[1]}_{\kappa,\rm sd}$ term defined in Eq. (2.58) are numerically
insignificant for our analysis. This is consistent with the observation in
Ref. Larkoski:2020wgx .

  2. 2.

Perturbative uncertainty: Typically, the perturbative uncertainty has been
estimated by varying resummation scales up and down by a factor of two around
their canonical values. In Ref. Kang:2018jwa  the jet scale and the global
soft scales were defined as a function of the collinear soft (or ungroomed
soft) scales, and hence their variations were analogously constrained. In
contrast, we have parameterized our variation in terms of parameters that each
probe a different physical effect associated with resummation. Unlike Ref.
Kang:2018jwa  we have also included profile variations that probe breaking of
the canonical relations related to the jet-mass and soft-drop measurements. In
fact, breaking the jet, hard collinear and soft scale relation via parameter
$\gamma$ leads to a variation comparable to other standard variations, and is
the main determinant in the overall perturbative uncertainty in $\beta=0$ soft
drop (see Fig. 17). Furthermore, to avoid being biased by correlations when
varying scales one at a time, we consider a more comprehensive variation by
randomly choosing points from the specified range of profile parameters. We
have, however, chosen not to consider variations in the cusp location as in
Ref. Marzani:2019evv , since this is a kinematically calculable endpoint.

  3. 3.

Transition into the ungroomed region: The problem of consistently combining
the results in groomed and ungroomed regions around the soft-drop cusp has
been considered in the previous literature. In Refs. Frye:2016aiz  this
matching was performed using numerical fixed-order code, and thus lacked
resummation in the ungroomed region. Refs. Marzani:2017mva ; Marzani:2017kqd
did include resummation in the ungroomed region and performed the matching to
NLL ungroomed resummed result. Note that no subtlety in the analysis of the
transition point arises at this order since the entire shape of jet-mass
distribution is described via resummation kernels and the transition can be
straightforwardly implemented using collinear-soft transition point $\xi_{0}$.
The same applies to the NLL analysis of Ref. Kang:2018jwa ; Marzani:2019evv .
Ref. Marzani:2019evv  however, did take into account the uncertainty induced
due to differences between the collinear-soft transition point $\xi_{0}$ and
the soft wide-angle transition point $\xi_{0}^{\prime}$. Finally, in Ref.
Baron:2018nfz , the transition region was more closely analyzed by computing
the cumulative of the fixed-order soft-drop correction $\Delta S_{\rm
sd}^{\kappa[1]}$ in Eq. (2.52). Their implementation of matching between
groomed and ungroomed region involved directly using the (cumulative of)
$\Delta S_{\rm sd}^{\kappa[1]}$ correction augmented with running coupling
corrections. However, this partially resummed result could not remove in
discontinuity in the matched groomed result at ${\cal O}(\alpha_{s}^{2})$.

As shown above, our treatment of the cusp using profile functions results in a
seamless transition between the groomed and ungroomed regions, such that the
transition happens consistently at the soft wide-angle transition point
$\xi_{0}^{\prime}$ defined in Eq. (2.9) in the matched result. This was
possible only after including the resummation between the soft and the jet
scales which was not accounted for in Ref. Baron:2018nfz  (which can also be
phrased as neglecting the multiple emission contributions). Furthermore, our
results include complete NNLL resummation of Sudakov logarithms across the
entire spectrum.

  4. 4.

Resummation at the soft-drop cusp: The resummation of logarithms of $\xi$ and
$\xi_{0}$ in the cusp region becomes subtle because, similar to the
consideration of NGL in ungroomed jet mass, there can be arbitrary number of
widely separated soft subjets whose angular location must be tracked to
determine the outcome of grooming. In Ref. Benkendorfer:2021unv  an all-orders
resummation was performed for the case of single resolved soft subjet at “NLL
accuracy”. Note that because the groomed jet-mass measurement is inclusive
over the kinematics and the number of resolved soft subjets, the calculation
with a single resolved soft subjet alone does not capture the entire NLL tower
of $\alpha_{s}^{n}\textrm{ln}^{n}\xi$ terms. Nevertheless, this calculation
showed that the effect of resummation is to shift the location of the cusp and
cause small variations in the normalization.

Because Ref. Benkendorfer:2021unv  did not match the cusp-resummation results
to the cross sections in the ungroomed jet-mass region above and the groomed
jet-mass resummation region with collinear-soft modes below, it is not clear
how much the impact of resummation on the location of the cusp really is. For
jets of $p_{T}>600$ GeV that we consider in this analysis, Fig. 6 of Ref.
Benkendorfer:2021unv  suggests a positive shift in $\xi_{0}$ of about 20%. A
proper matching to results for larger and smaller jet masses may render the
shift smaller. Furthermore, a 20% shift may already be ruled out by the ATLAS
data when compared to NNLL theory. Nevertheless, we find that a 20% shift to
$\xi_{0}$ is only about a 5% modification to the range that we use for
analyzing sensitivity to the $\alpha_{s}$. Thus our conclusions derived below
while ignoring resummation of the cusp are expected to continue to hold,
though more investigation of this is warranted.

##  3 Nonperturbative corrections

It was shown in Ref. Hoang:2019ceu  that nonperturbative power corrections in
the soft drop resummation region take the form in Eq. (1.4). It is helpful to
rewrite the NP power corrections as Pathak:2020iue

$\displaystyle\frac{1}{\sigma_{\kappa}}\frac{\textrm{d}\sigma^{\rm
had}_{\kappa}}{\textrm{d}\xi}$
$\displaystyle=\frac{1}{\hat{\sigma}_{\kappa}}\frac{\textrm{d}\hat{\sigma}_{\kappa}}{\textrm{d}\xi}-\frac{\Omega_{1\kappa}^{\circ\\!\\!\circ}}{Q}\frac{\textrm{d}}{\textrm{d}\xi}\bigg{(}\int_{r_{g}^{\rm
min}(\xi)}^{r_{g}^{\mathrm{max}}(\xi)}\textrm{d}r_{g}\>r_{g}\,\frac{1}{\hat{\sigma}_{\kappa}}\frac{\textrm{d}\hat{\sigma}_{\kappa}}{\textrm{d}\xi\textrm{d}r_{g}}\bigg{)}$
(3.1)
$\displaystyle\quad+\frac{\Upsilon_{1,0\kappa}^{\varocircle}+\beta\Upsilon_{1,1\kappa}^{\varocircle}}{Q_{\rm
cut}}\int_{r_{g}^{\rm
min}(\xi)}^{r_{g}^{\mathrm{max}}(\xi)}\textrm{d}r_{g}\>\frac{w_{\rm
max}(\xi,r_{g})}{r_{g}}\,\frac{\textrm{d}}{\textrm{d}\varepsilon}\bigg{(}\frac{1}{\hat{\sigma}_{\kappa}(\varepsilon)}\frac{\textrm{d}\hat{\sigma}_{\kappa}(\varepsilon)}{\textrm{d}\xi\textrm{d}r_{g}}\bigg{)}\bigg{|}_{\varepsilon\rightarrow
0}+\ldots\,,$

where $\textrm{d}\sigma^{\rm had}_{\kappa}$ is the hadron-level jet-mass cross
section for flavor $\kappa$ as defined above in Eq. (2.28) and
$\\{\Omega_{1\kappa}^{\circ\\!\\!\circ},\Upsilon_{1,0\kappa}^{\varocircle},\Upsilon_{1,1\kappa}^{\varocircle}\\}$
are ${\cal O}(\Lambda_{\rm{QCD}})$ constants that multiply $r_{g}$ moments of
doubly differential, parton level cross sections
$\textrm{d}\hat{\sigma}_{\kappa}/(\textrm{d}\xi\textrm{d}r_{g})$. Here
$r_{g}=R_{g}/R$ defined above in Eq. (2.75). The meaning of the parameter
$\varepsilon$ and the function $w_{\rm max}$ is explained below. Comparing
with Eq. (1.4) we see how functions $C_{1,2}^{\kappa}$ are defined in terms of
these moments. The result in Eq. (3.1) is a model independent statement
derived from an effective field theory analysis, and a places strong
constraint on any hadronization model. The leading hadronization corrections
are described by the three ${\cal O}(\Lambda_{\rm{QCD}})$ constants that only
depend on the quark or gluon flavor of the jet, but are independent of $z_{\rm
cut}$, $\beta$, $R$, $p_{T}$ and $\eta$. This implies that one can in
principle constrain the parameters
$\\{\Omega_{1\kappa}^{\circ\\!\\!\circ},\Upsilon_{1,0\kappa}^{\varocircle},\Upsilon_{1,1\kappa}^{\varocircle}\\}$
by comparing experimental data on groomed jet mass and theoretical
predictions.

The statement of Eq. (3.1) is that all the hadronization power corrections can
be descried in terms of these constants, whereas the $r_{g}$-weighted cross
sections multiplying these numbers are perturbatively calculable and account
for the dependence of these power corrections on grooming and kinematic
parameters. The nature of power corrections is clearly more involved than
compared to the ungroomed jets. This mainly results from the dynamical nature
of the groomed jet radius $R_{g}$ in contrast to a fixed jet radius $R$ in the
ungroomed jets, and the requirement to pass the soft drop test at each stage
of de-clustering. As a consequence, the entire power correction has both
perturbative and nonperturbative components. The perturbative component,
involving doubly differential cross sections, is responsible for describing
the dynamics of the $R_{g}$ distribution, whereas the nonperturbative
component is parameterized by three unknown ${\cal O}(\Lambda_{\rm{QCD}})$
constants at leading order for each of $\kappa=q,g$. The two terms that are
added to the parton level cross section in Eq. (3.1) represent the “shift” and
“boundary” nonperturbative corrections resulting from modification in the jet-
mass measurement as well as change in the outcome of the soft drop test due to
hadronization.

The above factorization is valid in the LL strong ordering limit of
perturbative collinear-soft (c-soft) emissions. This captures the dominant
“two-pronged” configuration of the collinear and c-soft subjet that stop the
soft drop and determine the groomed jet radius $R_{g}$. The ‘$\ldots$’ in Eq.
(3.1) represent further subleading power corrections which include effects
that are suppressed by higher powers in $\Lambda_{\rm{QCD}}$, and corrections
at ${\cal O}(\Lambda_{\rm{QCD}})$ where strong ordering is not obeyed and
higher-pronged configurations of c-soft subjets and collinear subjet
contribute. The latter is thus an NLL effect. Nevertheless, similar to the
expansion in number of resolved and ordered soft subjets in resummation of
non-global logarithms Larkoski:2015zka , the two-pronged configuration of the
“LL factorization” captures the leading nonperturbative power corrections, and
is sufficient for the precision studies here. As we will show below, the sizes
of these hadronization corrections are comparable to NNLL perturbative
uncertainty, which impacts the precision to which these nonperturbative
effects can be constrained. Hence, any further subleading effects not
accounted in Eq. (3.1) are well beyond our current sensitivity. We also note
that typically in analytical models of hadronization corrections, such as in
Ref. Dasgupta:2007wa , effects of hadron masses are ignored and any “universal
nonperturbative parameters” are understood with additional ${\cal O}(1)$ fudge
factors. In our case, however, the factorization in Eq. (3.1) is more general,
and the form of the NP corrections shown above remains the same even if hadron
masses are explicitly taken into consideration because scaling related to
$R_{g}$ is governed solely by perturbative dynamics.555However, the hadron
mass effects do complicate the relation between the NP constants that appear
in Eq. (3.1) from those in other groomed observables, for example,
angularities. See Ref. Mateu:2012nk  for how hadron mass effects are
incorporated in operator-based analyses.

The two terms in Eq. (3.1) capture dependence on the groomed jet radius
analogous to how the jet radius $R$ appears in the expressions for
nonperturbative power corrections to (ungroomed) jet mass and jet $p_{T}$
respectively Dasgupta:2007wa ; Stewart:2014nna . For ungroomed jets, the
nonperturbative contribution to the jet mass $\Omega_{1\kappa}(R)$ is
proportional to $R$, which for groomed jets is now replaced by $r_{g}$ times
$\Omega_{1\kappa}^{\circ\\!\\!\circ}$ in Eq. (3.1). The reason a term
analogous to jet $p_{T}$ power correction is present is because for groomed
jets, the recoil of the collinear soft subjet due to hadronization can change
the outcome of the soft drop test relative to a reference parton level
configuration, and is of the same order as the effect of shift in the jet
mass. The recoil includes both shift in the c-soft subjet $p_{T,cs}$ (ignoring
the subleading effect on the jet $p_{T}$) as well as a change in its direction
relative to the collinear subjet, the parton level $r_{g}=R_{g}/R$. These two
aspects of c-soft subjet recoil are accounted by the parameters
$\Upsilon_{1,0\kappa}^{\varocircle}$ and $\Upsilon_{1,1\kappa}^{\varocircle}$
respectively. Thus, the boundary correction includes a $1/r_{g}$ dependence
familiar from the $1/R$ correction appearing in the jet $p_{T}$ shift due to
hadronization.

Moreover, the boundary effect only appears for c-soft subjets that barely pass
or fail soft drop. As a result, the change in the parton level $r_{g}$ due to
c-soft subjet recoil captured by $\Upsilon_{1,1\kappa}^{\varocircle}$ is
relevant only for $\beta>0$. The boundary effect can be probed by replacing
the inequality in the soft drop condition by equality. To implement this, we
evaluate the double differential cross section using a shifted version of the
soft drop condition (for $pp$ collisions):

$\displaystyle\overline{\Theta}_{\rm
sd}(\varepsilon)\equiv\Theta\bigg{(}\min\\{z_{i},z_{j}\\}-\xi_{0}\Big{(}\frac{\Delta
R_{ij}}{R}\Big{)}^{\beta}+\xi_{0}\varepsilon\bigg{)}\,.$ (3.2)

Taking the $\varepsilon$ derivative then turns this into a $\delta$-function
constraint. The weight function $w_{\rm max}(\xi,r_{g})=1$ in the region where
the $r_{g}\lesssim r_{g}^{\rm max}(\xi)$ defined in Eq. (2.76). This condition
ensures that there is a distinct collinear-soft subjet that stops the groomer.
The contribution from the region where $r_{g}\ll r_{g}^{\rm max}(\xi)$ is a
subleading NLL effect, which must be, however, suppressed by hand using this
weight function because of the $1/r_{g}$ enhancement.

Figure 6: Weighted cross sections for shift and boundary power corrections.
Vertical lines denote the extent of the SDOE region.

These moments being perturbative quantities can be in principle computed to
arbitrarily high accuracy. Despite the fact that beyond LL factorization in
Eq. (3.1) there are subleading corrections arising from higher pronged
configurations, a more precise computation of these weighted cross section
allows one to accurately capture the dominant geometrical effect of soft drop,
and account for some of the higher order terms. These moments were first
computed in Ref. Hoang:2019ceu  at LL in the coherent branching framework. In
Ref. Pathak:2020iue  the problem of computing the $r_{g}$ moments as a
function of jet mass was cast into a computation of the double differential
cross section. Ref. Pathak:2020iue  developed the effective field theory for
doubly differential cross section and used it to compute these moments at NLL′
accuracy. We have already reviewed and utilized some aspects of this EFT setup
in Sec. 2.7.2. The calculation was further improved in Ref. AP  by matching
the doubly differential cross section to the ungroomed region, including an
improved treatment of the cusp region along the lines of the analysis
presented in Sec. 2.7, and extending the resummation to NNLL accuracy. In what
follows we will make use of results from Ref. AP .

To assess the jet-mass dependence and the size of the NP power corrections in
Eq. (3.1), we write

$\displaystyle\frac{1}{\sigma_{\kappa}}\frac{\textrm{d}\sigma_{\kappa}}{\textrm{d}\xi}=\frac{1}{\hat{\sigma}_{\kappa}}\frac{\textrm{d}\hat{\sigma}_{\kappa}}{\textrm{d}\xi}+\frac{\Omega_{1\kappa}^{\circ\\!\\!\circ}}{1\,{\rm
GeV}}\frac{1}{\hat{\sigma}_{\kappa}}\frac{\textrm{d}\hat{\sigma}_{1\kappa}}{\textrm{d}\xi}+\frac{\Upsilon_{1,0\kappa}^{\varocircle}+\beta\Upsilon_{1,1\kappa}^{\varocircle}}{1\,{\rm
GeV}}\frac{1}{\hat{\sigma}_{\kappa}}\frac{\textrm{d}\hat{\sigma}_{2\kappa}}{\textrm{d}\xi}\,,$
(3.3)

and the coefficients of the NP parameters,
$(1/\hat{\sigma}_{\kappa}){\textrm{d}\hat{\sigma}_{i\kappa}}/{\textrm{d}\xi}$
for $i=1,2$, are defined by the functions that appeared in Eq. (3.1). The
weighted cross sections for the two effects are shown in Fig. 6 for quark and
gluon jets. The orange and blue bands denote the perturbative uncertainty
associated with the NNLL calculation, ignoring subdominant NGL effects. The
vertical lines mark the extent of the SDOE region given in Eq. (4.1).
Comparing with the results in Fig. 5 we see that these weighted cross sections
in the SDOE region are about factor of 10 smaller than the leading parton
level cross section. Thus, as expected, for ${\cal O}(1\,{\rm GeV})$ values of
these parameters, these corrections are small in the SDOE region.

##  4 Sensitivity of matched jet-mass cross section to $\bm{\alpha_{s}}$

In this section, we consider how the various theoretical uncertainties
translate into uncertainty on the extraction of $\alpha_{s}$. We will state
results for two ways of normalizing the cross section, as shown in Eq. (1.5),
namely either using the inclusive cross section $\sigma_{\rm
incl}(p_{T},\eta)$ or the cross section obtained by integrating over the fit
range of jet-mass values $\sigma_{\rm fitrg}(p_{T},\eta,\xi_{\rm
SDOE},\xi_{\rm 0}^{\prime})$. The fit range that we consider for $\alpha_{s}$
analysis is given by

$\displaystyle\xi\in\big{[}\xi_{\rm SDOE},\xi_{0}^{\prime}\big{]}\,,$
$\displaystyle\xi_{\rm SDOE}=\xi_{\rm SDNP}\,5^{\frac{2+\beta}{1+\beta}}\,,$
(4.1)

where $\xi_{\rm SDNP}$ was defined in Eq. (2.92) and $\xi_{0}^{\prime}$ in Eq.
(2.9). Here the factor of 5 has replaced the strong inequality in Eq. (2.11).
The choice of 5 is motivated by analyzing the impact of perturbative
uncertainty in the shift and boundary corrections shown in Fig. 6. For
$\xi<\xi_{\rm SDOE}$ defined in Eq. (4.1), the perturbative uncertainty grows
and leads to poor description of these two effects in the SDOE region.

Since it will be useful to consider the individual quark and gluon components
of the cross section for these two choices of normalization, we also utilize
Eq. (2.27) and write

$\displaystyle\frac{1}{\sigma_{\rm
incl}}\>\frac{\textrm{d}^{3}\sigma}{\textrm{d}p_{T}\textrm{d}\eta\textrm{d}\xi}$
$\displaystyle=\frac{x_{q}}{\sigma_{q}^{\rm
incl}}\frac{\textrm{d}\sigma_{q}}{\textrm{d}\xi}+\frac{x_{g}}{\sigma_{g}^{\rm
incl}}\frac{\textrm{d}\sigma_{g}}{\textrm{d}\xi}\,,$ (4.2)
$\displaystyle\frac{1}{\sigma_{\rm fitrg}(\xi_{\rm SDOE},\xi_{\rm
0}^{\prime})}\>\frac{\textrm{d}^{3}\sigma}{\textrm{d}p_{T}\textrm{d}\eta\textrm{d}\xi}$
$\displaystyle=\frac{x_{q}}{\sigma_{q}^{\rm
fitrg}}\frac{\textrm{d}\sigma_{q}}{\textrm{d}\xi}+\frac{x_{g}}{\sigma_{g}^{\rm
fitrg}}\frac{\textrm{d}\sigma_{g}}{\textrm{d}\xi}\,,$ (4.3)

where $\sigma_{\rm incl}$ and $\sigma_{\rm fitrg}$ were defined in Eq.(1.7),
the $\sigma_{\kappa}^{\rm incl}$ are defined such that

$\int d\xi\frac{1}{\sigma_{\kappa}^{\rm