# The soft drop momentum sharing fraction $z_{g}$ beyond leading-logarithmic
accuracy

Pedro Cal<EMAIL_ADDRESS>Institute for Theoretical Physics Amsterdam and
Delta Institute for Theoretical Physics, University of Amsterdam, Science Park
904, 1098 XH Amsterdam, The Netherlands  Nikhef, Theory Group, Science Park
105, 1098 XG, Amsterdam, The Netherlands  Kyle Lee<EMAIL_ADDRESS>Department
of Physics, University of California, Berkeley, CA 94720, USA  Nuclear Science
Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720,
USA  Felix Ringer<EMAIL_ADDRESS>Nuclear Science Division, Lawrence
Berkeley National Laboratory, Berkeley, California 94720, USA  Wouter J.
Waalewijn<EMAIL_ADDRESS>Institute for Theoretical Physics Amsterdam
and Delta Institute for Theoretical Physics, University of Amsterdam, Science
Park 904, 1098 XH Amsterdam, The Netherlands  Nikhef, Theory Group, Science
Park 105, 1098 XG, Amsterdam, The Netherlands

###### Abstract

Grooming techniques, such as soft drop, play a central role in reducing
sensitivity of jets to e.g. underlying event and hadronization at current
collider experiments. The momentum sharing fraction $z_{g}$, of the two
branches in a jet that pass the soft drop condition, is one of the most
important observables characterizing a collinear splitting inside the jet, and
directly probes the QCD splitting functions. In this work, we present a
factorization framework that enables a systematic calculation of the
corresponding cross section beyond leading-logarithmic (LL) accuracy, showing
that this measurement is not only sensitive to the QCD charge but also the
spin of the parton that initiates the jet. Our results at next-to-leading
logarithmic (NLL′) accuracy include non-global logarithms, and provide a first
meaningful assessment of the perturbative uncertainty. We present a comparison
to the available experimental data from ALICE, ATLAS, and STAR and find
excellent agreement.

Introduction. At high-energy collider experiments, jets and their substructure
play a central role in probing fundamental aspects of QCD and searching for
physics beyond the standard model Larkoski:2017jix ; Asquith:2018igt ;
Marzani:2019hun . Jet grooming techniques Krohn:2009th ; Ellis:2009me ;
Dasgupta:2013ihk ; Cacciari:2014gra ; Larkoski:2014wba  are designed to remove
soft radiation inside jets, crucially reducing contamination in the
complicated environment of hadron colliders. These techniques will become even
more important during the high-luminosity era of the LHC. Grooming algorithms
can also lead to significantly reduced nonperturbative (hadronization)
corrections, allowing for direct and precise comparisons between theory and
data, see e.g. Refs. Aaboud:2017qwh ; Sirunyan:2018xdh ; Aad:2019vyi .

In this letter, we study the soft drop grooming algorithm Larkoski:2014wba .
After reclustering a jet with the Cambridge/Aachen (C/A) Dokshitzer:1997in ;
Wobisch:1998wt  algorithm, it iteratively declusters the jet, at each step
removing the softer branch if its momentum fraction $z$ fails the soft drop
condition

$z>z_{\mathrm{cut}}\left(\Delta R_{12}/R\right)^{\beta}\,.$ (1)

Here, $\Delta R_{12}$ is the distance between the branches in the
$\eta$-$\phi$ plane, $R$ is the radius of the initial ungroomed jet, and
$z_{\rm cut},\beta$ are tuneable grooming parameters. (Soft drop grooming with
$\beta=0$ corresponds to the modified mass drop tagger Dasgupta:2013ihk .)
Once Eq. (1) is satisfied, the algorithm terminates and $z_{g}=z$ and
$R_{g}=\Delta R_{12}$, as illustrated in Fig. 1. Both $z_{g}$ and $R_{g}$ are
central to characterizing the two hard branches of the groomed jet.

The momentum sharing fraction $z_{g}$ has received a lot of attention by both
the theoretical and experimental particle and nuclear physics communities in
the past years. The main reason is that it allows for the most direct
measurement of the QCD (Altarelli-Parisi) splitting functions Altarelli:1977zs
, providing a glimpse into fundamental splittings at parton level. The cross
section differential in $z_{g}$ was measured by the ALICE Acharya:2019djg ;
Mulligan:2020cnp , ATLAS Aad:2019vyi , CMS Sirunyan:2017bsd ; Sirunyan:2018asm
collaborations at the LHC and by STAR Adam:2020kug  at RHIC, which we compare
to in this work. In addition, the $z_{g}$ distribution was also extracted from
CMS open data Larkoski:2017bvj ; Tripathee:2017ybi .

Our calculation reveals that this measurement not only probes the color charge
but also the spin of the parton initiating the jet, and our precision is
essential to have sensitivity to this effect in interpreting experimental
results. The momentum sharing fraction is also of great interest for heavy-ion
collisions, as it probes modifications of hard-collinear splittings in the
quark-gluon plasma. For recent theoretical results, see Refs. Mehtar-
Tani:2016aco ; Chien:2016led ; Chang:2017gkt ; Li:2017wwc ; Milhano:2017nzm ;
KunnawalkamElayavalli:2017hxo ; Casalderrey-Solana:2019ubu ; Caucal:2019uvr .
We also expect $z_{g}$ to be of great phenomenological importance at the
future Electron-Ion Collider (EIC) AbdulKhalek:2021gbh .

Figure 1: Illustration of the clustering tree of a jet: Branches are groomed
away (grey) until the soft drop condition in Eq. (1) is first satisfied. This
splitting sets the observables $z_{g}$ and $R_{g}$, and subsequent splittings
(green) are kept.

The observable $z_{g}$ was first introduced in Ref. Larkoski:2015lea , where a
calculation of the corresponding cross section at leading-logarithmic (LL)
accuracy was performed. It was found that $z_{g}$ is Infrared-Collinear (IRC)
safe only for $\beta<0$. For $\beta\geq 0$, $z_{g}$ is IRC unsafe but
calculable: the IRC divergence is tamed by accounting for the Sudakov
suppression, making $z_{g}$ a Sudakov safe observable Larkoski:2013paa . The
cross section can thus be calculated by performing the joint resummation of
the logarithms of $z_{g}$ and the groomed radius $R_{g}$. Alternatively, one
can impose a cut on $R_{g}$, but this case also requires resummation. Here we
extend the work of Ref. Larkoski:2015lea  by setting up a factorization
framework within Soft Collinear Effective Theory (SCET) Bauer:2000ew ;
Bauer:2000yr ; Bauer:2001yt ; Bauer:2002nz ; Beneke:2002ph , which allows for
the systematic extension beyond LL. We obtain results at next-to-leading
logarithmic (NLL′) accuracy, accounting for non-global logarithms (NGLs)
Dasgupta:2001sh . Our work also provides the first meaningful assessment of
perturbative uncertainties, and opens the door to future calculations at
higher perturbative accuracy and for a systematic treatment of nonperturbative
effects Hoang:2019ceu ; Pathak:2020iue . Theoretical calculations of other
groomed observables, such as the groomed jet mass, can be found in Refs.
Frye:2016aiz ; Marzani:2017mva ; Larkoski:2017cqq ; Kang:2018jwa ;
Kardos:2018kth ; Lee:2019lge ; Chien:2019osu ; Larkoski:2020wgx ;
Kardos:2020gty ; Anderle:2020mxj ; Mehtar-Tani:2020oux ; Baron:2020xoi ;
Makris:2021drz ; Caucal:2021bae ; Cal:2019gxa ; Cal:2020flh ; Caletti:2021oor
.

Figure 2: Lund diagram for the cross section differential in $z_{g}$ and
$\theta_{g}$. The green dot corresponds to the emission passing the soft drop
condition, and emissions in the red area are vetoed. The modes that appear in
the corresponding SCET calculation are indicated by green and red dots.

Theoretical framework. We now describe how we calculate the cross section
differential in the jet’s transverse momentum $p_{T}$ and rapidity $\eta$, as
well as the groomed jet substructure observables $z_{g}$ and $\theta_{g}\equiv
R_{g}/R$. For small jet radii, we can separate the production of the inclusive
jet sample from the jet substructure measurement using collinear
factorization,

$\displaystyle\frac{{\rm d}\sigma}{{\rm d}p_{T}\,{\rm d}\eta\,{\rm
d}z_{g}\,{\rm d}\theta_{g}}=$ $\displaystyle\sum_{i}f_{i}(p_{T},\eta,R,\mu)$
$\displaystyle\times\tilde{\cal G}_{i}(z_{g},\theta_{g},p_{T}R,z_{\rm
cut},\beta,\mu)\,.$ (2)

The jet production is summarized by quark/gluon fractions $f_{i=q,g}$, which
account for parton distribution functions, the hard-scattering, and semi-
inclusive jet functions. The jet functions $\tilde{\cal G}_{i}$ (using the
notation of Ref. Cal:2020flh ) encode the substructure measurement. See Refs.
Aversa:1988vb ; Jager:2002xm ; Mukherjee:2012uz ; Dasgupta:2014yra ;
Kaufmann:2015hma ; Kang:2016mcy ; Dai:2016hzf ; Cal:2020flh  for more details
on this first step of the factorization.

Next, we calculate the jet function $\tilde{\cal G}_{i}$, which we will need
to describe the region where $z_{g}$ is order one. We find at leading order
(LO) for $z_{g},\theta_{g}>0$

$\displaystyle\tilde{\cal G}^{(1)}_{q}$ $\displaystyle=\Theta(1/2>z_{g}>z_{\rm
cut}\theta_{g}^{\beta})\,\Theta(\theta_{g}<1)$
$\displaystyle\quad\frac{\alpha_{s}}{\pi}\frac{1}{\theta_{g}}\bigl{[}P_{qq}(z_{g})+P_{gq}(z_{g})\bigr{]}\,,$
(3)

and similarly for $\tilde{\cal G}^{(1)}_{g}$. Here $P_{ij}$ are the leading-
order QCD splitting functions, showing that the measurement of $z_{g}$ probes
these. Since $\tilde{\cal G}^{(1)}_{i}\sim 1/\theta_{g}$, and there is no
lower bound on $\theta_{g}$ for $\beta\geq 0$, we cannot integrate out the
dependence on $\theta_{g}$ in this case. The integration over $\theta_{g}$ is
only possible after taking into account the Sudakov suppression through
resummation.

To achieve this, we perform the resummation of large logarithmic corrections
of $z_{g},\theta_{g}$ and $z_{\rm cut}$. We start with the result at LL
accuracy, which provides physical intuition and helps identify the relevant
modes in SCET. It is described by strongly-ordered emission of gluons in the
collinear and soft limit. The Lund diagram in Fig. 2 shows the phase space of
such emissions in terms of their energy fraction $z$ and angle $\theta$, with
dashed lines indicating the soft drop condition in Eq. (1) and the measurement
of $\theta_{g}$ and $z_{g}$. The emission at the green dot sets $z_{g}$ and
$\theta_{g}$. Emissions in the red region are not allowed, and the
corresponding area enters in the Sudakov exponent:

$\displaystyle\tilde{\cal G}_{i}$ $\displaystyle=\Theta(1/2>z_{g}>z_{\rm
cut}\theta_{g}^{\beta})\frac{2\alpha_{s}C_{i}}{\pi}\frac{1}{z_{g}\,\theta_{g}}$
$\displaystyle\quad\times\exp\bigg{(}-\frac{\alpha_{s}C_{i}}{\pi}(\beta\ln^{2}\theta_{g}+2\ln
z_{\rm cut}\ln\theta_{g})\bigg{)}\,,$ (4)

Here $C_{i=F,A}$ denotes the appropriate color factor for quarks and gluons,
which is the only dependence on the initial parton at LL. As Eq. (The soft
drop momentum sharing fraction $z_{g}$ beyond leading-logarithmic accuracy)
indicates, it is now safe to integrate over $\theta_{g}$ due to the Sudakov
suppression, and the resulting expression agrees with Eq. (14) of Ref.
Larkoski:2015lea .

Figure 3: Comparison of our numerical results for the momentum sharing
fraction $z_{g}$ at NLL′+LO to Pythia 8 simulations Sjostrand:2014zea  at
parton and hadron level, for representative jet kinematics and three choices
of soft drop grooming parameters.

We can extend this result to NLL′ by identifying the relevant modes within
SCET, for which the scaling of the momentum components can be read off from
the location of the points in the Lund diagram in Fig. 2. We find

$\displaystyle\tilde{\cal G}_{i}$ $\displaystyle=\Theta(1/2>z_{g}>z_{\rm
cut}\theta_{g}^{\beta})\tilde{H}_{i}(p_{T}R,\mu)\,C_{i}^{\in{\rm
gr}}(\theta_{g}p_{T}R,\mu)$ $\displaystyle\quad\times S_{i}^{\notin{\rm
gr}}(z_{\rm cut}p_{T}R,\beta,\mu)\,\tilde{\cal S}_{G}(z_{\rm
cut}\theta_{g}^{1+\beta}p_{T}R,\beta,\mu)\ $ $\displaystyle\quad\times
S_{i}^{\rm NG}(z_{\rm cut})\bigg{[}\frac{\rm d}{{\rm d}z_{g}}\frac{\rm d}{{\rm
d}\theta_{g}}\tilde{\cal S}_{z_{g}}(z_{g}\theta_{g}p_{T}R,\mu)$
$\displaystyle\quad+\tilde{\cal S}^{\prime\rm
NG}_{i,1}(z_{g}\theta_{g},z_{g})+\tilde{\cal S}^{\prime\rm
NG}_{i,2}\Big{(}z_{g}\theta_{g},\frac{z_{g}}{z_{\rm
cut}\theta_{g}^{\beta}}\Big{)}\bigg{]}\,.$ (5)

The resummation of logarithms of $z_{g},\theta_{g}$, and $z_{\rm cut}$ is
achieved by evaluating each ingredient (except those describing NGLs) at its
natural scale, which can be read off from their first argument, and using the
renormalization group equations (RGEs) to evolve them to a common scale $\mu$.
The modes associated with the red points in Fig. 2 also appeared in the
factorization of the groomed jet radius Kang:2019prh , and the expressions for
the corresponding functions can be found there. The mode corresponding to the
green point is rather different because it describes the _single_ emission
that passes the soft drop condition. We set the $\mu$ scales for the cross
section differential in $\theta_{g}$ and cumulative in $z_{g}$, and therefore
present the order $\alpha_{s}$ expression and RGE for the new ingredient
$\tilde{\cal S}_{z_{g}}$ differential in $\theta_{g}$:

$\displaystyle\frac{{\rm d}}{{\rm d}\theta_{g}}\tilde{\cal
S}_{z_{g}}(z_{g}\theta_{g}p_{T}R,\mu)$
$\displaystyle=-\frac{2\alpha_{s}C_{i}}{\pi}\frac{1}{\theta_{g}}\ln\frac{\mu}{z_{g}\theta_{g}p_{T}R}\,,$
$\displaystyle\frac{{\rm d}}{{\rm d}\ln\mu}\,\frac{{\rm d}}{{\rm
d}\theta_{g}}\tilde{\cal S}_{z_{g}}$
$\displaystyle=-\frac{2\alpha_{s}C_{i}}{\pi}\frac{1}{\theta_{g}}\,.$ (6)

A similarly unusual RGE was encountered in Refs. Cal:2019gxa ; Cal:2020flh ,
to which we refer the reader for details.

There are three types of non-global logarithms Dasgupta:2001sh ; Banfi:2002hw
; Hornig:2011iu ; Kelley:2011ng ; Schwartz:2014wha ; Caron-Huot:2015bja ;
Hagiwara:2015bia ; Larkoski:2015zka ; Becher:2015hka ; Balsiger:2018ezi ;
Banfi:2021owj  in Eq. (The soft drop momentum sharing fraction $z_{g}$ beyond
leading-logarithmic accuracy): First, the NGLs described by $S_{i}^{\rm NG}$
are similar to the usual hemisphere case, and arise due to correlations
between the unconstrained emissions in the region outside the jet and the
radiation inside the jet that fails the grooming condition. Second, the NGLs
${\cal S}_{i,1,2}^{\prime\rm NG}$ arise from correlated emissions, where one
sets $z_{g}$ and $\theta_{g}$ (green dot) and the other is either outside the
groomed radius and fails the grooming condition or inside the groomed radius
and is unconstrained (corresponding to one of the two red dots on the vertical
dashed line $\theta=\theta_{g}R$ in Fig. 2). These NGLs are sensitive to C/A
clustering effects Appleby:2002ke ; Delenda:2006nf ; Neill:2018yet . We
include their leading contribution at order $\alpha_{s}^{2}$,

$\displaystyle\tilde{\cal S}_{i,1}^{\prime{\rm
NG},(2)}(z_{g}\theta_{g},z_{g})$
$\displaystyle=2.58C_{i}C_{A}\Big{(}\frac{\alpha_{s}}{2\pi}\Big{)}^{2}\frac{1}{z_{g}\theta_{g}}\ln
z_{g}\,,$ (7)

and the form of $\tilde{\cal S}_{i,2}^{\prime{\rm NG},(2)}$ is the same at
this order. In our numerical implementation these NGLs are multiplied by the
global Sudakov suppression factor, see Eq. (The soft drop momentum sharing
fraction $z_{g}$ beyond leading-logarithmic accuracy), making their numerical
size small (percent level).

Numerical results and comparison to data. Throughout this section we consider
(ungroomed) jets which are reconstructed with the anti-$k_{T}$ algorithm
Cacciari:2008gp , as in the measurement of the experimental collaborations. We
use the parton distribution functions of Ref. Dulat:2015mca .

Figure 4: Relative difference between the $z_{g}$ distribution for quark and
gluon-initiated jets. The difference between the green and blue curve
indicates the size of the nonsingular part of the QCD splitting function,
encoding the spin-dependence.  Figure 5: Comparison of our results to ATLAS
Aad:2019vyi , ALICE Mulligan:2020cnp  and STAR Adam:2020kug  data for
$\beta=0$.

We start by comparing our numerical results at NLL′+LO accuracy to Pythia 8
simulations Sjostrand:2014zea . The comparison for exemplary jet kinematics is
shown in Fig. 3. We choose three representative values of the grooming
parameter $\beta$ and impose a cutoff on the groomed jet radius of
$\theta_{g}^{\rm cut}>0.25$ to reduce the sensitivity to nonperturbative
physics. The QCD scale uncertainty bands in the figures shown here are
obtained by independently varying all relevant scales in Eqs. (The soft drop
momentum sharing fraction $z_{g}$ beyond leading-logarithmic accuracy) and
(The soft drop momentum sharing fraction $z_{g}$ beyond leading-logarithmic
accuracy) by a factor of 2 around the central scale choice. In addition, we
smoothly freeze all scales at ${\cal O}(1~{}\text{GeV})$ Ligeti:2008ac . The
hadronization corrections for the chosen kinematics are very small which can
be seen by comparing Pythia results at parton and hadron level.

Overall, we observe very good agreement of our results with Pythia. In
addition to the improved precision, there are notable qualitative differences
with the earlier results of Ref. Larkoski:2015lea : The shape of our
distribution for $\beta<0$ is rather different, and we find a smooth
transition to $\beta=0$ for $z_{g}>z_{\rm cut}$, whether we approach from
negative or positive $\beta$. For large $z_{g}$ the matching to the LO in Eq.
(The soft drop momentum sharing fraction $z_{g}$ beyond leading-logarithmic
accuracy) is essential, which is done multiplicatively because of the common
singularity in $\theta_{g}$. Indeed, the non-singular terms added by the
matching are important to achieve good agreement with Pythia, as illustrated
in the left panel of Fig. 3 by the black NLL′ curve (that does not include the
matching). This demonstrates the sensitivity of $z_{g}$ distribution to the
full splitting function beyond the leading $1/z_{g}$ behavior in the singular
limit. The size of these corrections, which encode the spin-dependence of the
splitting function, are visualized in Fig. 4. Since their size can be up to
order 10%, we expect that this can be probed experimentally by for example
comparing inclusive vs. photon-tagged jets or jets with different rapidities.

Figure 6: Comparison to ATLAS results Aad:2019vyi  for $\beta=1$.  Figure 7:
Prediction for the Electron-Ion Collider.

Next, we compare to the experimental results from ATLAS Aad:2019vyi , ALICE
Mulligan:2020cnp  and STAR Adam:2020kug  for $\beta=0$ in Fig. 5. We normalize
our results to the data 111The ALICE normalization is not normalized by a few
percent, because of their treatment of jets that never pass the soft drop
condition. and impose the same cut on $\theta_{g}$. The hadronization effects
in Pythia for ALICE and STAR kinematics (not shown) are much more sizable than
in Fig. 3, in accord with the larger perturbative uncertainties. Nevertheless,
we find very good agreement even for these relatively low jet transverse
momenta. We note that the CMS result of Ref. Sirunyan:2017bsd  is not
unfolded, prohibiting a direct comparison. As a representative example, we
show the LL QCD scale uncertainty band in the left panel of Fig. 5 which is
significantly larger than at NLL′. This implies that the NLL′ accuracy
achieved in this work is needed to match the current experimental precision.

Next, we compare in Fig. 6 our results to ATLAS measurement Aad:2019vyi  for
$\beta=1$, as an example. We normalize our results to the data in the region
to the right of the dotted line, as our prediction for the left most data
point are very sensitive to nonperturbative effects. Note that this was not
needed for $\beta=0$, where $z_{\rm cut}$ provides a lower bound on $z_{g}$.
We observe excellent agreement! In this case our NLL′+LO prediction is
substantially better than the LL result.

Lastly, we present predictions for jet kinematics at the future EIC in Fig. 7.
We consider jets reconstructed in the laboratory frame $ep\to e+{\rm jet}+X$
for typical EIC kinematics Arratia:2019vju ; Page:2019gbf  with cuts on the
photon virtuality $Q^{2}$ and the inelasticity $y$ as indicated in the figure.
The clean environment at the EIC will allow studies of hadronization effects,
and $z_{g}$ measurements in single- and di-jet events can help improve our
understanding of quark/gluon differences (see also Fig. 4).

Conclusions. In this work we have presented a calculation of the soft drop
groomed momentum sharing fraction $z_{g}$ at NLL′+LO accuracy. This Sudakov-
safe jet substructure observable, which probes the hard branching process
inside the jet, constitutes the most direct measurement of the QCD splitting
function. Our framework allows for a systematic extension beyond the
previously achieved LL accuracy, yielding qualitatively different results for
$\beta<0$ than in an earlier study, and provides the first meaningful
assessment of theoretical uncertainties. We also show that $z_{g}$ probes the
QCD splitting function beyond the leading $1/z_{g}$ dependence, indicating
sensitivity to the spin of the particle that initiates the jet. Our
calculations indicate that this effect is sufficiently large to be probed
experimentally. The momentum sharing fraction $z_{g}$ is one of the hallmark
observables in the field of jet substructure and has been measured by several
experimental collaborations at the LHC and RHIC. We compared to the available
experimental data and found very good agreement with our purely perturbative
calculation. In addition, we provided predictions for the future Electron-Ion
Collider. Our precise calculations reduced uncertainties and changed the shape
of the prediction, bringing it in excellent agreement with the data.

Acknowledgements. We thank Yi Chen, Raghav Elayavalli, Matt LeBlanc, Yen-Jie
Lee, Ezra Lesser, James Mulligan, Ben Nachman, Mateusz Ploskon, Jennifer
Roloff, and Marta Verweij for helpful discussions. We also thank Andrew
Larkoski and Jesse Thaler for feedback on our manuscript. PC and WW are
supported by the ERC grant ERC-STG-2015-677323, the NWO projectruimte
680-91-122 and the D-ITP consortium, a program of NWO that is funded by the
Dutch Ministry of Education, Culture and Science (OCW). KL is supported by the
US Department of Energy, Office of Nuclear Physics. FR is supported by the US
Department of Energy under Contract No. DE-AC02-05CH11231 and the LDRD Program
of LBNL.

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