#  $J/\psi$ production within a jet in high-energy proton-proton and nucleus-
nucleus collisions

Shan-Liang Zhang  Department of Physics, Hubei University, Wuhan 430062, China
Key Laboratory of Atomic and Subatomic Structure and Quantum Control (MOE),
Guangdong Basic Research Center of Excellence for Structure and Fundamental
Interactions of Matter, Institute of Quantum Matter, South China Normal
University, Guangzhou 510006, China  Guangdong-Hong Kong Joint Laboratory of
Quantum Matter, Guangdong Provincial Key Laboratory of Nuclear Science,
Southern Nuclear Science Computing Center, South China Normal University,
Guangzhou 510006, China  Hongxi Xing<EMAIL_ADDRESS>Key Laboratory of
Atomic and Subatomic Structure and Quantum Control (MOE), Guangdong Basic
Research Center of Excellence for Structure and Fundamental Interactions of
Matter, Institute of Quantum Matter, South China Normal University, Guangzhou
510006, China  Guangdong-Hong Kong Joint Laboratory of Quantum Matter,
Guangdong Provincial Key Laboratory of Nuclear Science, Southern Nuclear
Science Computing Center, South China Normal University, Guangzhou 510006,
China  Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern
Physics, Chinese Academy of Sciences, Huizhou 516000, China

( )

###### Abstract

Within the framework of leading power factorization formalism of
nonrelativistic quantum chromodynamics, we calculate the jet fragmentation
function for $J/\psi$ production in proton-proton (pp) collisions ranging from
$\sqrt{s}=500$ GeV to $13$ TeV. The reasonable agreements between theory and
experimental data indicate that $J/\psi$ production within a jet is mainly
dominated by gluon fragmentation. Such a mechanism can be further tested by
the predicted jet transverse momentum and radius dependence of jet
fragmentation function. Based on the satisfying description of pp baseline, we
carry out the first theoretical investigation on medium modification on
$J/\psi$ production within jet in heavy-ion collisions at the Large Hadron
Collider, using a linear Boltzmann transport model combined with hydrodynamics
for the simulation of jet-medium interaction. The consistency with the
experimental measurement on nuclear modification factor $R_{\text{AA}}$ by CMS
collaboration reveals that the gluon jet quenching is the driving force for
the suppression of $J/\psi$ production in jet. Furthermore, we make
predictions for the dependence of $R_{\text{AA}}$ on the jet transverse
momentum and jet radius $R$, which can be tested in future measurements to
further constrain the flavor dependence of jet quenching.

##  I Introduction

The underlying mechanism for producing $J/\psi$ mesons as bound states of
charm quark pairs ($c\bar{c}$) has been under intensive experimental and
theoretical studies in both hadronic and heavy-ion collisions Andronic  _et
al._ (2016); Chapon  _et al._ (2022); Lansberg (2020). The production of heavy
quarkonium in high-energy particle collisions not only provide valuable
information on non-perturbative QCD physics of hadronization, but also linked
to longstanding challenges in our understanding of the properties of QGP, a
color-deconfined QCD medium, in relativistic heavy-ion collisions Matsui and
Satz (1986). Furthermore, it is proposed that the production yields of
quarkonium at electron ion collisions are critical for studying the production
mechanism and also can be used as an ideal tool to probe the gluon structure
of nucleon and nucleus Flore  _et al._ (2020); D’Alesio  _et al._ (2022); Boer
_et al._ (2021); Chen  _et al._ (2023); Qiu  _et al._ (2021).

The non-relativistic QCD (NRQCD) factorization formalism Bodwin  _et al._
(1995) is by far the most phenomenologically successful framework to describe
heavy quarkonium production in hadronic collisions, although there are still
challenges to simultaneously describe their cross sections and polarisations
Dai and Shrivastava (2017). In recent years, quarkonium production within jets
has been proposed as a probe of heavy-quark hadronization mechanisms Bain  _et
al._ (2016); Baumgart  _et al._ (2014), and was expected to give a new insight
into this puzzle Kang  _et al._ (2017). Recently, the Large Hadron Collider
(LHC) and Relativistic Heavy Ion Collider (RHIC) measured $J/\psi$ mesons
produced in jet in proton-proton (pp) collisions. In particular, the
transverse momentum fraction of jet,
$z_{J/\psi}=p_{\text{T}}^{J/\psi}/p_{\text{T}}^{\text{jet}}$, carried by the
$J/\psi$ have been measured in pp collisions by LHCb Aaij  _et al._ (2017),
CMS Tumasyan  _et al._ (2022); Sirunyan  _et al._ (2020) and STAR Yang
(2021)(preliminary). These measurements indicate significant deviations from
default Monte Carlo PYTHIA simulations Aaij  _et al._ (2017); Tumasyan  _et
al._ (2022); Yang (2021), which use leading order NRQCD complemented with
subsequent parton shower Sjöstrand  _et al._ (2015). Theoretical predictions,
provided in Bain  _et al._ (2016) by using the fragmenting jet functions (FJF)
and the Gluon Fragmentation Improved PYTHIA (GFIP)Bain  _et al._ (2017, 2016),
which correspond to a modified parton shower approach where the quarkonium
fragmentation is only implemented at the end of the shower, provide well
description of the LHCb and CMS data of $J/\psi$ production in jetsSirunyan
_et al._ (2020); Aaij  _et al._ (2017).

In heavy-ion collision, heavy quarkonia production at low transverse momentum
has long been regarded as a signature of color deconfinement and a thermometer
of QGP Matsui and Satz (1986). The interpretation of the experimental data for
$J/\psi$ suppression observed both at RHIC Adare  _et al._ (2007, 2011) and
the LHC Alessandro  _et al._ (2005); Abelev  _et al._ (2012); Adam  _et al._
(2017); Khachatryan  _et al._ (2017); Aaboud  _et al._ (2018), is complicated
due to effects like: sizeable regeneration or recombination of quarkonium from
the uncorrelated charm and anti-charm quark pairs in the QGP Braun-Munzinger
and Stachel (2000); Thews  _et al._ (2001); Zhou  _et al._ (2014); He  _et
al._ (2022), cold nuclear matter effects Vogt (2015) and feed-down
contributions from the decay of excited states. However, in the region of
$p_{\text{T}}^{J/\psi}\gg m_{J/\psi}$ with $m_{J/\psi}$ the $J/\psi$ mass, the
aforementioned dissociation and regeneration effects are negligible while jet
quenching plays a dominant role for the $J/\psi$ suppression Zhang  _et al._
(2022). Recently, CMS Collaboration reported the measurement of the
fragmentation of jets containing a prompt J$/\psi$ meson in PbPb and pp
collisions at $\sqrt{s_{\mathrm{NN}}}=$ 5.02 TeV Tumasyan  _et al._ (2022).
The measurement shows that the $J/\psi$ yield in jets is significantly
suppressed in PbPb collisions comparing to that in pp, and the ratio of yield
in PbPb to pp shows a rising trend with increasing $z_{J/\psi}$, different
with the predicted pattern of longitudinal momentum fraction of $D^{0}$ in
jets due to charm quark energy loss Li  _et al._ (2023). This contrast
questions the intuitive expectation that the nuclear modification for $J/\psi$
production is driven by its constituent charm quark. To the best of our
knowledge, this phenomenon oberved by CMS collaboration has never been
described and explained by any theoretical model calculations.

The above observations motivate us to conduct a systematic study of the
$J/\psi$ production in jet in both pp and AA collisions. Such a systematic
study will not only be critical in our understanding of $J/\psi$ production
mechanism, but also help to constrain the flavor dependent jet quenching
mechanism Zhang  _et al._ (2022). In pp collisions, we utilize a theoretical
framework analogous to the gluon fragmentation improved PYTHIA method Bain
_et al._ (2017, 2016), where we convolve hard QCD partonic cross sections with
leading order NRQCD $J/\psi$ fragmentation functions. The parton propagation
in QGP medium is simulated by the Linear Boltzmann Transport (LBT) model He
_et al._ (2015); Cao  _et al._ (2016) with bulk medium evolution provided by
the CLVisc 3+1D hydrodynamic Pang  _et al._ (2012). The results are compared
with available data in pp and PbPb collisions Tumasyan  _et al._ (2022); Aaij
_et al._ (2017); Yang (2021). In pp collisions, we demonstrate that the
$z_{J/\psi}$ distribution is dominated by the gluon fragmentation Zhang  _et
al._ (2022). In PbPb collisions, we find that the nuclear modification factor
$R_{\text{AA}}(z_{J/\psi})$ is mainly driven by the gluon energy loss effect.
Furthermore, we provide predictions for the jet transverse momentum and jet
radius dependence of $z_{J/\psi}$ distribution and nuclear modification factor
$R_{\text{AA}}(z_{J/\psi})$, which can be tested in future experimental
measurements.

The remainder of the paper is organized as follows. We first introduce the
theoretical framework and present numerical results for $J/\psi$ production in
jet in pp collisions in Sec. III. Based on the pp baseline, we present results
for nuclear modification on $z_{J/\psi}$ distributions in PbPb collisions in
Sec. III. Finally, A summary is given in Sec.IV.

##  II $J/\psi$ production in jet in pp collisions

###  II.1 Theoretical framework

We start this section by introducing the theoretical framework suitable for
describing high transverse momentum $J/\psi$ production in pp collisions.
Within the framework of collinear factorization formalism Collins and Soper
(1982), the distribution of $J/\psi$ meson within a fully reconstructed jet in
pp collisions can be written into the following schematic form Kaufmann  _et
al._ (2015); Kang  _et al._ (2017)

$d\sigma[pp\to({\rm jet}~{}J/\psi)+X]=\sum_{i}d\hat{\sigma}_{pp\to({\rm
jet}~{}i)+X}\otimes D_{i\rightarrow J/\psi},$ (1)

where $\otimes$ denotes convolution product over the partonic momentum
fraction, $\sum_{i}$ represents the sum over all relevant partonic channels.
$d\hat{\sigma}_{pp\to({\rm jet}~{}i)+X}$ is the differential cross section for
inclusive parton $i$ distribution inside a jet in pp collisions, and
$D_{i\rightarrow J/\psi}$ are the fragmentation functions (FFs) describing the
transition of parton $i$ into the final observed $J/\psi$. In our calculation,
we simulate the differential cross section $d\hat{\sigma}_{pp\to({\rm
jet}~{}i)+X}$ with parton $i$ generated by MadGraph Alwall  _et al._ (2014)
and showered using PYTHIA Sjöstrand  _et al._ (2015) down to $2m_{\text{c}}$
to obtain the transverse momentum $p_{\text{T}}^{\text{jet}}$ distribution for
the measured jet. We use the parton distribution function (PDF) set “CT14NLO”
Dulat  _et al._ (2016) in LHAPDF Buckley  _et al._ (2015) to provide the
information for partons that participate in the hard interaction in pp
collisions. The $J/\psi$ FF can be further factorized at an initial scale
$\mu_{0}\sim m_{J/\psi}$ within the framework of NRQCD Ma  _et al._ (2014)

$D_{i\rightarrow
J/\psi}(z,\mu_{0})=\sum_{n}\hat{d}_{i\to[c\bar{c}(n)]}(z,\mu_{0})\langle\mathcal{O}_{[c\bar{c}(n)]}^{J/\psi}\rangle,$
(2)

where $\sum_{n}$ represents the sum over all possible intermediate
nonrelativistic $c\bar{c}$ states, and the quantum number $n$ takes the
standard notation as $n=~{}^{2S+1}L^{[i]}_{J}$ with the superscripts $i=1$ and
$8$ denote color-singlet and color-octet states, respectively.
$\hat{d}_{i\to[c\bar{c}(n)]}$ are the short distance coefficients for the
conversion of parton $i$ to intermediate nonrelativistic $c\bar{c}(n)$ quantum
state, and have been derived, see, e.g., in Refs. Ma  _et al._ (2014); Bodwin
and Lee (2004). $z$ is the longitudinal momentum fraction of parton $i$
carried by the $c\bar{c}$ pair.
$\langle\mathcal{O}_{[c\bar{c}(n)]}^{J/\psi}\rangle$ are the nonperturbative
NRQCD long distance matrix elements (LDMEs), describing the transition of a
$c\bar{c}$ pair in the quantum state $n$ into a final observed $J/\psi$. In
our simulation, the NRQCD fragmentation functions are evaluated at leading
order in QCD.

For $J/\psi$ production, we consider the most important four relevant states:
${}^{3}S^{[1]}_{1}$, ${}^{1}S^{[8]}_{0}$, ${}^{3}S^{[8]}_{1}$,
${}^{3}P^{[8]}_{J}$. The corresponding LDMEs have been extracted from
experimental data by different groups, see, e.g., Bodwin  _et al._ (2014,
2016); Gong  _et al._ (2013); Chao  _et al._ (2012); Ma  _et al._ (2011);
Brambilla  _et al._ (2023). Specifically we take the results from fits to
prompt $J/\psi$ production at large transverse momentum Bodwin  _et al._
(2016), which are more relevant to our case with high $p_{\text{T}}^{\rm jet}$
jet production. Notice that the fragmentation to $J/\psi$ from light quarks
are suppressed by two orders of magnitude comparing to the others Baumgart
_et al._ (2014). Therefore, only the contributions from charm and gluon
fragmentation are considered in our paper.

###  II.2 Phenomenology at RHIC and the LHC energies

In order to compare with the experimental data, we select $J/\psi$ and the
associated jet according to the kinematic cuts adopted by RHIC and the LHC
measurements Tumasyan  _et al._ (2022); Aaij  _et al._ (2017); Yang (2021), as
detailed in Table. 1. We implement the muon cuts by assuming the $J/\psi$ are
unpolarized and, therefore, decays to dilepton isotropically in its rest
frame. The kinematic cut on dilepton clearly suppresses contributions from
partons with low $z$ and, hence, reduce the contribution from soft quark/gluon
initiated $J/\psi$.

Table 1: The kinematic cuts applied to $J/\psi$ in jet measurements at RHIC and the LHC.  |  LHCb Aaij  _et al._ (2017) |  CMS Tumasyan  _et al._ (2022) |  STAR Yang (2021)
---|---|---|---
$\sqrt{s}$ |  13 TeV |  5.02 TeV |  500 GeV
Jet |  $R=0.5$, $p_{\text{T}}^{\text{jet}}>20$ GeV/c, $2.5<\eta^{\text{jet}}<4.0$ |  $R=0.3$, $30<p_{\text{T}}^{\text{jet}}<40$ GeV/c, $|\eta^{\text{jet}}|<2.0$ |  $R=0.4$, $p_{\text{T}}^{\text{jet}}>10$ GeV/c, $|\eta^{\text{jet}}|<0.6$
$J/\psi$ |  $p(\mu)>5$ GeV/c, $2.5<\eta^{\mu}<4.0$ |  $p_{\text{T}}^{J/\psi}>6.5$ GeV/c |  $p_{\text{T}}^{J/\psi}>5$ GeV/c
Figure 1: (Color online) Distributions of $J/\psi$ in jet (green line) as well
as from gluon (red dashed line) and charm quark (solid blue line)
fragmentation calculated as function of
$z_{J/\psi}=p_{\text{T}}^{J/\psi}/p_{\text{T}}^{\text{jet}}$ in pp collision
at $\sqrt{s}$: (a) 13 TeV, (b) 5.02 TeV, and (c) 500 GeV. The experimental
data are taken from Refs. Tumasyan  _et al._ (2022); Aaij  _et al._ (2017);
Yang (2021).

In Fig. 1, we show the results for jet fragmentation function for $J/\psi$
production in jet as a function of $z_{J/\psi}$ in pp collision at three
different collision energies: (a) 13 TeV, (b) 5.02 TeV, and (c) 500 GeV, as
well as the comparison with experimental data from LHCb Aaij  _et al._ (2017),
CMS Tumasyan  _et al._ (2022), and STAR Yang (2021), respectively. In order to
satisfy the kinematic cuts and ensure the stability of the evaluated cross
section at $z_{J/\psi}\rightarrow 1$, we limit our comparison with data in the
range $0.2<z_{J/\psi}<0.9$ and normalize the $z_{J/\psi}$ distributions to the
sum of the data in these bins as in Ref. Bain  _et al._ (2017). Our numerical
result, as shown by the green band in Fig. 1, can quantitatively describe the
experimental data within uncertainties from LDMEs (mainly comes from the
counterbalance of $c\bar{c}(^{3}S^{[8]}_{1})$ and $c\bar{c}(^{3}P^{[8]}_{J})$
channel). The contributions from gluon (red dashed lines) and charm (blue
dotted lines) fragmentation are also shown to demonstrate the $J/\psi$
production mechanism. One can see clearly that $J/\psi$ in jet is driven by
gluon fragmentation ($>85\%$) in a wide range of $z_{J/\psi}$, while charm
fragmentation is rather limited, in agreements with the statements in Refs
.Bain  _et al._ (2017); Sirunyan  _et al._ (2020); Zhang  _et al._ (2022) for
inclusive $J/\psi$ production. It is worth to point out that the experimental
errors are generally smaller than the theoretical uncertainties, such a fact
indicates the power of using jet fragmentation function to further constrain
the LDMEs, similar to that for light hadron Gao  _et al._ (2024) and D-meson
FFs Anderle  _et al._ (2017).

It is interesting to note that the $z_{J/\psi}$ distributions are different in
three collision energies, with the peak of $z_{J/\psi}$ shifts to smaller
values from RHIC to LHC energies. This phenomenon can be understood by the
fact that the parton shower at RHIC energy is weaker than that at LHC
energies, therefore less partons are produced within jet, which eventually
lead to harder momentum distributions of charm and gluon in jet at RHIC
energy. Combined with the fact that the gluon fragmentation function peaks at
$z\sim 1$, we expect harder $z_{J/\psi}$ distribution at lower collision
energy, which explains the energy dependence of peak position for $z_{J/\psi}$
distribution shown in Fig. 1.

Figure 2: (Color online) The normalized distributions of $J/\psi$ in jet as a
function of $z_{J/\psi}$ in: (a) three $p_{\text{T}}^{\text{jet}}$ intervals
with jet cone $R=0.3$, (b) $30<p_{\text{T}}^{\text{jet}}<40$ GeV/c interval
with three jet cone R =0.2, 0.4, and 0.6, in p+p collisions at $\sqrt{s}=5.02$
TeV.

The mechanism mentioned above can be tested by a more differential study in
terms of jet transverse momentum $p_{\text{T}}^{\text{jet}}$ and jet radius
$R$ dependencies of $z_{J/\psi}$ distribution. In Fig. 2(a), we present
predictions for $J/\psi$ production within jet in pp collisions as a function
of $z_{J/\psi}$ at $\sqrt{s}=5.02$ TeV in three different
$p_{\text{T}}^{\text{jet}}$ and $R$ intervals. The distributions are
normalized to unity in $0.3<z_{J/\psi}<0.8$. As can be seen from the top panel
that the $z_{J/\psi}$ is shifted to softer regions with the increase of
$p_{\text{T}}^{\text{jet}}$, which is qualitatively consistent with the CMS
measurements Sirunyan  _et al._ (2020). This can be understood in a similar
fashion as in Fig. 1, i.e., more gluons is generated in larger
$p_{\text{T}}^{\text{jet}}$ region, therefore softer momentum distributions
with the increase of $p_{\text{T}}^{\text{jet}}$. We show in Fig. 2(b) the $R$
dependence in $30<p_{\text{T}}^{\text{jet}}<40$ GeV/c with three jet radius
$R=0.3,0.4,0.6$. As we expect that larger size of jet contains more particles
because of more radiated gluons are restored, the peak value of $z_{J/\psi}$
distribution shifts to smaller value with larger jet cone $R$. Such predicted
$p_{\text{T}}^{\text{jet}}$ and $R$ dependent distributions can be tested in
future measurements to confirm the production mechanism of $J/\psi$ in jet.

##  III $J/\psi$ production in jet in AA collisions

###  III.1 Framework for jet propagation in medium

In relativistic heavy ion collisions, the energetic partons from hard
interactions encounter multiple scatterings inside the QGP medium before they
fragment into high-$p_{\text{T}}$ heavy quarkonium. We use the well developed
Linear Boltzmann Transport (LBT) model to incorporate both elastic and
inelastic processes for the charm quarks and gluons scattering with medium
constituents He  _et al._ (2015); Cao  _et al._ (2016). For elastic
scatterings, the evolution of hard partons are simulated by the following
linear Boltzmann transport equation,

$\displaystyle p_{1}\cdot\partial
f_{a}(p_{1})=-\int\frac{d^{3}p_{2}}{(2\pi)^{3}2E_{2}}\int\frac{d^{3}p_{3}}{(2\pi)^{3}2E_{3}}\int\frac{d^{3}p_{4}}{(2\pi)^{3}2E_{4}}$
$\displaystyle\frac{1}{2}\sum_{b(c,d)}[f_{a}(p_{1})f_{b}(p_{2})-f_{c}(p_{3})f_{d}(p_{4})]|M_{ab\rightarrow
cd}|^{2}$ $\displaystyle\times
S_{2}(s,t,u)(2\pi)^{4}\delta^{4}(p_{1}+p_{2}-p_{3}-p_{4})+{\rm inel.}$ (3)

where $f_{i=a,b,c,d}$ are the phase-space distributions of jet shower partons
($a,c$) and medium partons ($b,d$), $|M_{ab\rightarrow cd}|$ are the
corresponding elastic scattering matrix elements which are regulated by a
Lorentz-invariant regulation condition
$S_{2}(s,t,u)=\theta(s>2\mu^{2}_{D})\theta(-s+\mu^{2}_{D}\leq
t\leq-\mu^{2}_{D})$, with $\mu_{D}^{2}=g^{2}T^{2}(N_{c}+N_{f}/2)/3$ the Debye
screening mass. The effect of inelastic scatterings is described by the
higher-twist formalism for induced gluon radiation as follows Guo and Wang
(2000); Zhang  _et al._ (2004),

$\displaystyle\frac{dN_{g}}{dxdk_{\perp}^{2}dt}=\frac{6\alpha_{s}P(x)\hat{q}k_{\perp}^{4}}{\pi(k_{\perp}^{2}+x^{2}M^{2})^{4}}\sin^{2}\left(\frac{t-t_{i}}{2\tau_{f}}\right),$
(4)

Here $x$ denotes the energy fraction of the radiated gluon relative to a
parent parton with mass $M$, $k_{\perp}$ is the transverse momentum. A lower
energy cut-off $x_{min}=\mu_{D}/E$ is applied for the emitted gluon in the
calculation. $P(x)$ is the splitting function in vacuum, and
$\tau_{f}=2Ex(1-x)/(k^{2}_{\perp}+x^{2}M^{2})$ is the formation time of the
radiated gluons in QGP. The dynamic evolution of bulk medium is given by 3+1D
CLVisc hydrodynamical model Pang  _et al._ (2012) with parameters fixed by
reproducing hadron spectra from experimental measurements. The details of LBT
can be found in Ref. Luo  _et al._ (2023), which has been implemented to
successfully describe the suppression of experimental data on hadron,
inclusive jets, $\gamma+$hadron/jets correlations and Z+jet production, as
well as the prompt $J/\psi$ production Zhang  _et al._ (2022). In our
calculation, we follow previous study Zhang  _et al._ (2018) to fix the free
parameter $\alpha_{s}=0.18$ to control the strength of parton-medium
interactions.

###  III.2 Numerical results

Figure 3: (Color online) The nuclear modification factor $R_{\text{AA}}$, as a
function of $z_{J/\psi}$ at $\sqrt{s}=5.02$ TeV and the comparison with CMS
measurement Tumasyan  _et al._ (2022). The medium modifications on the
$J/\psi$ cross section from gluon (red) and charm (blue) fragmentation are
also shown for illustration.

In Fig. 3, we present the nuclear modification factor $R_{\text{AA}}$, defined
as the ratio of $z_{J/\psi}$ distribution between PbPb and pp collisions at
$\sqrt{s}=5.02$ TeV, and compare with the CMS measurement Tumasyan  _et al._
(2022). The $z_{J/\psi}$ distribution is overall suppressed in PbPb collisions
relative to that in pp collisions, and is quantitatively consistent with the
experimental measurements within the uncertainties. The overall suppression
has a natural interpretation in terms of the jet quenching phenomenon. The
yields of jet and $J/\psi$ that pass all the selection cuts in PbPb collisions
are reduced due to parton-medium interactions. The $z_{J/\psi}$ dependent
$R_{\text{AA}}$ show a slight rising trend with increasing $z_{J/\psi}$, this
is because of the reason that lower values of $z_{J/\psi}$ should be populated
with jets with a $J/\psi$ produced late in the parton shower. Such a parton
cascade is expected to have a large degree of interaction with the QGP in the
form of subsequent medium-induced emissions Caucal  _et al._ (2018), as
compared to a jet with a small partonic multiplicity.

To illustrate the underlying mechanism for the suppression of $z_{J/\psi}$, we
plot in Fig. 3 the medium modification on the $z_{J/\psi}$ from gluon and
charm fragmentation as well. We see that $J/\psi$ from gluon fragmentation
(red line) is much more suppressed compared to charm quark fragmentation (blue
line) due to the parton mass and color-charge effect, where gluon lose much
larger fraction of its energy in the medium compared to light-quark and
$c$-quark jet as analyzed in Zhang  _et al._ (2022); Xing  _et al._ (2024).
Considering the fact that gluon fragmentation dominates the $z_{J/\psi}$
distribution in pp collisions, an interesting observation is that the strong
suppression of high-$p_{\text{T}}$ $J/\psi$ production in AA collisions is
mainly driven by the gluon jet quenching. Besides, as shown by the red line in
Fig. 3, the nuclear modification for gluon fragmentation also show a rising
trend in $z_{J/\psi}$, while the one for charm quark represented by blue line
show much weaker dependence on $z_{J/\psi}$. Those modification patterns will
be explained in detail in the following.

To further clarify the medium modification of $z_{J/\psi}$ distribution, the
relationship between the fraction variable $z$ in AA and pp collisions can be
expressed as

$z^{\text{AA}}=\frac{p_{\text{T,AA}}^{J/\psi}}{p_{\text{T,AA}}^{\text{jet}}}=\frac{(1-f^{J/\psi})p_{\text{T,pp}}^{J/\psi}}{(1-f^{\text{jet}})p_{\text{T,pp}}^{\text{jet}}}=\frac{(1-f^{J/\psi})}{(1-f^{\text{jet}})}z^{\text{pp}},$
(5)

where $z^{\text{AA}}$ and $z^{\text{pp}}$ are the fraction variables after and
before jet quenching, respectively. $f^{J/\psi}$ and $f^{\text{jet}}$ are
energy loss fractions of $J/\psi$ and jet when propagating through the medium.
In the large $z$ region, where the $c$ quark is produced with fewer associated
partons, the radiated gluon from charm quark may still fall in the jet cone
and lead to smaller energy loss comparing to charm quark itself. As a
consequence, we will have $f^{J/\psi}\geq f^{\text{jet}}$, thus find
$z^{\text{AA}}<z^{\text{pp}}$ for $c$ quark fragmentation, similar with the
modification pattern of $D^{0}$ in jet Li  _et al._ (2023). Whereas for gluon
fragmentation, jet has a larger number of sources of energy loss and tend to
lose more energy Pablos (2020), based on which, $f^{J/\psi}$ will be smaller
than $f^{\text{jet}}$, then gives $z^{\text{AA}}>z^{\text{pp}}$. This analysis
is qualitatively in consistent with our numerical results, which demonstrate
the role of gluon jet quenching in driving the nuclear suppression of $J/\psi$
in jet.

Figure 4: (Color online) The nuclear modification factor $R_{\text{AA}}$
evaluated as a function of $z_{J/\psi}$ (a) in three
$p_{\text{T}}^{\text{jet}}$ intervals with jet cone $R=0.3$, (b) in
$30<p_{\text{T}}^{\text{jet}}<40$ GeV/c interval with three jet cone $R=$ 0.2,
0.4, and 0.6 at $\sqrt{s}=5.02$ TeV.

In order to further test the mechanisms mentioned above. We present
predictions in Fig. 4 for the dependence of nuclear modification factor
$R_{\text{AA}}$ on $p_{\text{T}}^{\text{jet}}$ and $R$ . As shown in the top
panel, we predict mild dependence of $R_{\text{AA}}$ on
$p_{\text{T}}^{\text{jet}}$. This can be explained by the fact that
$f^{\text{jet}}$ and $f^{J/\psi}$ have similar downward tendency as shown in
Refs. Zhang  _et al._ (2024, 2022). Therefore, the ratio between $f^{J/\psi}$
and $f^{\text{jet}}$ is only moderately modified, leading to the predicted
mild dependence on $p_{\text{T}}^{\text{jet}}$. Shown in the bottom panel is
the prediction for nuclear modification factor $R_{\text{AA}}$ with three jet
radius over the $p_{\text{T}}$ interval of [30, 40] GeV/c. It turns out that
the modification pattern of $R_{\text{AA}}(z_{J/\psi})$ is significantly
modified with increasing jet radius. For example, the $z_{J/\psi}$ dependent
nuclear modification factor shows a downward (rising) trend at large
$z_{J/\psi}$ with $R=0.6~{}(0.3)$. This is driven by fact that larger fraction
of radiations induced by the medium will be restored with larger jet radius
$R$, thus leads to smaller energy loss fraction, i.e. $f^{\text{jet}}$
decrease with increasing jet radius $R$. Therefore, one would expect smaller
$z^{\text{AA}}$ with $R=0.6$ comparing to $R=0.3$, leading to less suppression
in small $z_{J/\psi}$ and more suppression in large $z_{J/\psi}$. However, we
realize that the shape of $R_{\text{AA}}(z_{J/\psi})$ from charm fragmentation
is hardly changed, simply because that charm jet lose much less fraction of
its energy, leading to weaker jet radius dependence, while that from gluon
fragmentation is changed more significant. As the $J/\psi$ production is
mainly driven by gluon fragmentation, we conclude that
$R_{\text{AA}}(z_{J/\psi})$ should be dominated by the medium modification on
gluon fragmentation processes. The upcoming high precision measurements of jet
radius dependent $R_{\text{AA}}(z_{J/\psi})$ at LHC will put new insight into
our understanding of $J/\psi$ production mechanism, and the charm and gluon
jet quenching mechanism.

##  IV Summary

In the paper, we have studied $J/\psi$ production within a jet in pp and their
modification in heavy-ion collisions. Within the framework of leading power
NRQCD factorization formalism, we convolve the differential cross section of
charm quarks and gluons in jet with leading order NRQCD $J/\psi$ fragmentation
functions. Then a linear Boltzmann transport model combined with hydrodynamics
are implemented for the simulation of jet-medium interaction in nucleus-
nucleus collisions. We found that $J/\psi$ production in jet is mainly
dominated by gluon dynamics, similar to that for inclusive $J/\psi$
suppression in heavy-ion collisions. Furthermore, we present predictions for
the dependence of $J/\psi$ production in jet on jet transverse momentum
$p_{\text{T}}^{\text{jet}}$ and jet radius $R$ in both pp and AA collisions.
We expect mild $p_{\text{T}}^{\text{jet}}$ dependence and strong jet radius
$R$ dependence of the nuclear modification $R_{\text{AA}}$. These predictions
can be tested in future high precision measurements on $J/\psi$ in jet at the
LHC, which can help us to further pin down the $J/\psi$ production mechanism,
as well as the flavor dependent jet quenching mechanism, ultimately, the
properties of the QGP itself.

Acknowledgments: We thank Yiannis Makris, Xiaozhi Bai and Qian Yang for
helpful discussions. This research is supported by the National Natural
Science Foundation of China with Project No. 12147131, 12035007, 12022512, and
Guangdong Major Project of Basic and Applied Basic Research with No.
22020B0301030008, 2022A1515010683.

## References

  * Andronic  _et al._ (2016) A. Andronic _et al._ , Eur. Phys. J. C 76, 107 (2016), arXiv:1506.03981 [nucl-ex] .
  * Chapon  _et al._ (2022) E. Chapon _et al._ , Prog. Part. Nucl. Phys. 122, 103906 (2022), arXiv:2012.14161 [hep-ph] .
  * Lansberg (2020) J.-P. Lansberg, Phys. Rept. 889, 1 (2020), arXiv:1903.09185 [hep-ph] .
  * Matsui and Satz (1986) T. Matsui and H. Satz, Phys. Lett. B 178, 416 (1986).
  * Flore  _et al._ (2020) C. Flore, J.-P. Lansberg, H.-S. Shao, and Y. Yedelkina, Phys. Lett. B 811, 135926 (2020), arXiv:2009.08264 [hep-ph] .
  * D’Alesio  _et al._ (2022) U. D’Alesio, L. Maxia, F. Murgia, C. Pisano, and S. Rajesh, JHEP 03, 037 (2022), arXiv:2110.07529 [hep-ph] .
  * Boer  _et al._ (2021) D. Boer, C. Pisano, and P. Taels, Phys. Rev. D 103, 074012 (2021), arXiv:2102.00003 [hep-ph] .
  * Chen  _et al._ (2023) L. Chen, H. Xing, and S. Yoshida, Phys. Rev. D 108, 054021 (2023), arXiv:2306.12647 [hep-ph] .
  * Qiu  _et al._ (2021) J.-W. Qiu, X.-P. Wang, and H. Xing, Chin. Phys. Lett. 38, 041201 (2021), arXiv:2005.10832 [hep-ph] .
  * Bodwin  _et al._ (1995) G. T. Bodwin, E. Braaten, and G. P. Lepage, Phys. Rev. D 51, 1125 (1995), [Erratum: Phys.Rev.D 55, 5853 (1997)], arXiv:hep-ph/9407339 .
  * Dai and Shrivastava (2017) L. Dai and P. Shrivastava, Phys. Rev. D 96, 036020 (2017), arXiv:1707.08629 [hep-ph] .
  * Bain  _et al._ (2016) R. Bain, L. Dai, A. Hornig, A. K. Leibovich, Y. Makris, and T. Mehen, JHEP 06, 121 (2016), arXiv:1603.06981 [hep-ph] .
  * Baumgart  _et al._ (2014) M. Baumgart, A. K. Leibovich, T. Mehen, and I. Z. Rothstein, JHEP 11, 003 (2014), arXiv:1406.2295 [hep-ph] .
  * Kang  _et al._ (2017) Z.-B. Kang, J.-W. Qiu, F. Ringer, H. Xing, and H. Zhang, Phys. Rev. Lett. 119, 032001 (2017), arXiv:1702.03287 [hep-ph] .
  * Aaij  _et al._ (2017) R. Aaij _et al._ (LHCb), Phys. Rev. Lett. 118, 192001 (2017), arXiv:1701.05116 [hep-ex] .
  * Tumasyan  _et al._ (2022) A. Tumasyan _et al._ (CMS), Phys. Lett. B 825, 136842 (2022), arXiv:2106.13235 [hep-ex] .
  * Sirunyan  _et al._ (2020) A. M. Sirunyan _et al._ (CMS), Phys. Lett. B 804, 135409 (2020), arXiv:1910.01686 [hep-ex] .
  * Yang (2021) Q. Yang (STAR), PoS HardProbes2020, 072 (2021).
  * Sjöstrand  _et al._ (2015) T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C. O. Rasmussen, and P. Z. Skands, Comput. Phys. Commun. 191, 159 (2015), arXiv:1410.3012 [hep-ph] .
  * Bain  _et al._ (2017) R. Bain, L. Dai, A. Leibovich, Y. Makris, and T. Mehen, Phys. Rev. Lett. 119, 032002 (2017), arXiv:1702.05525 [hep-ph] .
  * Adare  _et al._ (2007) A. Adare _et al._ (PHENIX), Phys. Rev. Lett. 98, 232301 (2007), arXiv:nucl-ex/0611020 .
  * Adare  _et al._ (2011) A. Adare _et al._ (PHENIX), Phys. Rev. C 84, 054912 (2011), arXiv:1103.6269 [nucl-ex] .
  * Alessandro  _et al._ (2005) B. Alessandro _et al._ (NA50), Eur. Phys. J. C 39, 335 (2005), arXiv:hep-ex/0412036 .
  * Abelev  _et al._ (2012) B. Abelev _et al._ (ALICE), Phys. Rev. Lett. 109, 072301 (2012), arXiv:1202.1383 [hep-ex] .
  * Adam  _et al._ (2017) J. Adam _et al._ (ALICE), Phys. Lett. B 766, 212 (2017), arXiv:1606.08197 [nucl-ex] .
  * Khachatryan  _et al._ (2017) V. Khachatryan _et al._ (CMS), Eur. Phys. J. C 77, 252 (2017), arXiv:1610.00613 [nucl-ex] .
  * Aaboud  _et al._ (2018) M. Aaboud _et al._ (ATLAS), Eur. Phys. J. C 78, 784 (2018), arXiv:1807.05198 [nucl-ex] .
  * Braun-Munzinger and Stachel (2000) P. Braun-Munzinger and J. Stachel, Phys. Lett. B 490, 196 (2000), arXiv:nucl-th/0007059 .
  * Thews  _et al._ (2001) R. L. Thews, M. Schroedter, and J. Rafelski, Phys. Rev. C 63, 054905 (2001), arXiv:hep-ph/0007323 .
  * Zhou  _et al._ (2014) K. Zhou, N. Xu, Z. Xu, and P. Zhuang, Phys. Rev. C 89, 054911 (2014), arXiv:1401.5845 [nucl-th] .
  * He  _et al._ (2022) M. He, B. Wu, and R. Rapp, Phys. Rev. Lett. 128, 162301 (2022), arXiv:2111.13528 [nucl-th] .
  * Vogt (2015) R. Vogt, Phys. Rev. C 92, 034909 (2015), arXiv:1507.04418 [hep-ph] .
  * Zhang  _et al._ (2022) S.-L. Zhang, J. Liao, G.-Y. Qin, E. Wang, and H. Xing, (2022), arXiv:2208.08323 [hep-ph] .
  * Li  _et al._ (2023) Y. Li, S. Wang, and B.-W. Zhang, Phys. Rev. C 108, 024905 (2023), arXiv:2209.00548 [hep-ph] .
  * He  _et al._ (2015) Y. He, T. Luo, X.-N. Wang, and Y. Zhu, Phys. Rev. C 91, 054908 (2015), [Erratum: Phys.Rev.C 97, 019902 (2018)], arXiv:1503.03313 [nucl-th] .
  * Cao  _et al._ (2016) S. Cao, T. Luo, G.-Y. Qin, and X.-N. Wang, Phys. Rev. C 94, 014909 (2016), arXiv:1605.06447 [nucl-th] .
  * Pang  _et al._ (2012) L. Pang, Q. Wang, and X.-N. Wang, Phys. Rev. C 86, 024911 (2012), arXiv:1205.5019 [nucl-th] .
  * Collins and Soper (1982) J. C. Collins and D. E. Soper, Nucl. Phys. B 194, 445 (1982).
  * Kaufmann  _et al._ (2015) T. Kaufmann, A. Mukherjee, and W. Vogelsang, Phys. Rev. D 92, 054015 (2015), [Erratum: Phys.Rev.D 101, 079901 (2020)], arXiv:1506.01415 [hep-ph] .
  * Alwall  _et al._ (2014) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, JHEP 07, 079 (2014), arXiv:1405.0301 [hep-ph] .
  * Dulat  _et al._ (2016) S. Dulat, T.-J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, and C. P. Yuan, Phys. Rev. D 93, 033006 (2016), arXiv:1506.07443 [hep-ph] .
  * Buckley  _et al._ (2015) A. Buckley, J. Ferrando, S. Lloyd, K. Nordström, B. Page, M. Rüfenacht, M. Schönherr, and G. Watt, Eur. Phys. J. C 75, 132 (2015), arXiv:1412.7420 [hep-ph] .
  * Ma  _et al._ (2014) Y.-Q. Ma, J.-W. Qiu, and H. Zhang, Phys. Rev. D 89, 094029 (2014), arXiv:1311.7078 [hep-ph] .
  * Bodwin and Lee (2004) G. T. Bodwin and J. Lee, Phys. Rev. D 69, 054003 (2004), arXiv:hep-ph/0308016 .
  * Bodwin  _et al._ (2014) G. T. Bodwin, H. S. Chung, U.-R. Kim, and J. Lee, Phys. Rev. Lett. 113, 022001 (2014), arXiv:1403.3612 [hep-ph] .
  * Bodwin  _et al._ (2016) G. T. Bodwin, K.-T. Chao, H. S. Chung, U.-R. Kim, J. Lee, and Y.-Q. Ma, Phys. Rev. D 93, 034041 (2016), arXiv:1509.07904 [hep-ph] .
  * Gong  _et al._ (2013) B. Gong, L.-P. Wan, J.-X. Wang, and H.-F. Zhang, Phys. Rev. Lett. 110, 042002 (2013), arXiv:1205.6682 [hep-ph] .
  * Chao  _et al._ (2012) K.-T. Chao, Y.-Q. Ma, H.-S. Shao, K. Wang, and Y.-J. Zhang, Phys. Rev. Lett. 108, 242004 (2012), arXiv:1201.2675 [hep-ph] .
  * Ma  _et al._ (2011) Y.-Q. Ma, K. Wang, and K.-T. Chao, Phys. Rev. D 84, 114001 (2011), arXiv:1012.1030 [hep-ph] .
  * Brambilla  _et al._ (2023) N. Brambilla, H. S. Chung, A. Vairo, and X.-P. Wang, JHEP 03, 242 (2023), arXiv:2210.17345 [hep-ph] .
  * Gao  _et al._ (2024) J. Gao, C. Liu, X. Shen, H. Xing, and Y. Zhao, (2024), arXiv:2401.02781 [hep-ph] .
  * Anderle  _et al._ (2017) D. P. Anderle, T. Kaufmann, M. Stratmann, F. Ringer, and I. Vitev, Phys. Rev. D 96, 034028 (2017), arXiv:1706.09857 [hep-ph] .
  * Guo and Wang (2000) X.-f. Guo and X.-N. Wang, Phys. Rev. Lett. 85, 3591 (2000), arXiv:hep-ph/0005044 .
  * Zhang  _et al._ (2004) B.-W. Zhang, E. Wang, and X.-N. Wang, Phys. Rev. Lett. 93, 072301 (2004), arXiv:nucl-th/0309040 .
  * Luo  _et al._ (2023) T. Luo, Y. He, S. Cao, and X.-N. Wang, (2023), arXiv:2306.13742 [nucl-th] .
  * Zhang  _et al._ (2018) S.-L. Zhang, T. Luo, X.-N. Wang, and B.-W. Zhang, Phys. Rev. C 98, 021901 (2018), arXiv:1804.11041 [nucl-th] .
  * Caucal  _et al._ (2018) P. Caucal, E. Iancu, A. H. Mueller, and G. Soyez, Phys. Rev. Lett. 120, 232001 (2018), arXiv:1801.09703 [hep-ph] .
  * Xing  _et al._ (2024) W.-J. Xing, S. Cao, and G.-Y. Qin, Phys. Lett. B 850, 138523 (2024), arXiv:2303.12485 [hep-ph] .
  * Pablos (2020) D. Pablos, Phys. Rev. Lett. 124, 052301 (2020), arXiv:1907.12301 [hep-ph] .
  * Zhang  _et al._ (2024) S.-L. Zhang, E. Wang, H. Xing, and B.-W. Zhang, Phys. Lett. B 850, 138549 (2024), arXiv:2303.14881 [hep-ph] .


# Hypergraph characterization of split matroids

Kristóf Bérczi MTA-ELTE Momentum Matroid Optimization Research Group,
Budapest, Hungary.MTA-ELTE Egerváry Research Group, Budapest,
Hungary.Department of Operations Research, Eötvös Loránd University, Budapest,
Hungary. Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Tamás Király11footnotemark: 1 22footnotemark: 2
33footnotemark: 3 Tamás Schwarcz11footnotemark: 1 33footnotemark: 3 Yutaro
Yamaguchi Department of Information and Physical Sciences, Graduate School of
Information Science and Technology, Osaka University, Osaka, Japan. Email:
<EMAIL_ADDRESS>Yu Yokoi Principles of Informatics
Research Division, National Institute of Informatics, Tokyo, Japan. Email:
<EMAIL_ADDRESS>
###### Abstract

We provide a combinatorial study of split matroids, a class that was motivated
by the study of matroid polytopes from a tropical geometry point of view. A
nice feature of split matroids is that they generalize paving matroids, while
being closed under duality and taking minors. Furthermore, these matroids
proved to be useful in giving exact asymptotic bounds for the dimension of the
Dressian, and also implied new results on the rays of the tropical
Grassmannians.

In the present paper, we introduce the notion of elementary split matroids, a
subclass of split matroids that contains all connected split matroids. We give
a hypergraph characterization of elementary split matroids in terms of
independent sets, and show that the proposed class is closed not only under
duality and taking minors but also truncation. We further show that, in
contrast to split matroids, the proposed class can be characterized by a
single forbidden minor. As an application, we provide a complete list of
binary split matroids.

Keywords: Binary matroids, Excluded minors, Split matroids, Paving matroids

##  1 Introduction

The class of split matroids was introduced by Joswig and Schröter [10] as an
efficient tool in tropical geometry. Their definition was based on a
polyhedral approach, by imposing conditions on the split hyperplanes of the
matroid base polytope. In order to recall the definition, we first overview
the polyhedral background.

Given a polytope $P$, its intersection with a supporting hyperplane is called
a face of $P$. The polytope itself is also considered to be a face. A face is
a facet if it is properly contained in exactly one face, namely $P$. A split
of $P$ is a subdivision without new vertices which has exactly two maximal
cells. The affine span of the intersection of the two cells is then called a
split hyperplane. Two splits are compatible if the corresponding split
hyperplanes do not meet in a relative interior point of $P$. Let $M=(S,r_{M})$
be a matroid on ground set $S$ with rank function $r_{M}$. We denote the rank
of the matroid by $r$, that is, $r_{M}(S)=r$. The convex hull of the
characteristic vectors of the bases of $M$ is called the matroid base polytope
of $M$ and is denoted by $P(M)$. We denote by $\Delta(r,S)$ the
$|S|-1$-dimensional hypersimplex representing the matroid base polytope of the
rank-$r$ uniform matroid on $S$, that is, the convex hull of all zero-one
vectors over $S$ with exactly $r$ ones. For a flat $F$ of $M$, the
$F$-hyperplane is defined as $H(F)=\\{x\in\mathbb{R}^{S}\mid
x(F)=r_{M}(F)\\}$, while the intersection $P(M)\cap H(F)$ is the face of
$P(M)$ defined by $F$. Note that two flats might define the same face. A flat
$F$ is called a flacet if it defines a facet and is inclusionwise minimal
among flats defining $H(F)$. A flacet $F$ is a split flacet if the
corresponding $F$-hyperplane defines a split of $\Delta(r,S)$. Roughly, the
split flacets are the hyperplanes that are used to cut off parts of
$\Delta(r,S)$ to obtain $P(M)$. Using this terminology, a matroid $M$ is a
split matroid if its split flacets form a compatible system of splits of the
affine hull of $P(M)$ intersected with the unit cube $[0,1]^{S}$.

The goal of the present paper is to give a combinatorial understanding of
split matroids.

#### Previous work.

Joswig and Schröter [10] gave a thorough analysis of split matroids in terms
of polyhedral geometry. They observed that it suffices to concentrate on the
connected case, as a matroid is a split matroid if and only if at most one
connected component is a non-uniform split matroid and all other components
are uniform [10, Proposition 15]. For the connected case, they also gave a
characterization that does not rely on polyhedral combinatorics, stating that
a connected matroid is a split matroid if and only if for each split flacet
$F$ the restriction $M|F$ and the contraction $M/F$ both are uniform [10,
Theorem 11].

Besides their applicability in tropical geometry, split matroids are also of
combinatorial interest. In particular, the class of split matroids contains
all paving matroids, a well-studied class with distinguished structural
properties. A conjecture of Crapo and Rota [4], that was made precise by
Mayhew, Newman, Welsh and Whittle [11], suggests that the asymptotic fraction
of matroids on $n$ elements that are paving tends to $1$ as $n$ tends to
infinity. Therefore, an affirmative answer to the conjecture would imply that
almost all matroids are split. A weakness of paving matroids is that their
class is not closed under duality, a property that is desired in many cases.
However, split matroids are closed both under duality and taking minors [10,
Proposition 44], hence they form a large class with strong combinatorial
properties.

In the light of minor-closedness, it is natural to ask what the excluded
minors are for the class of split matroids. It is an easy exercise to show
that uniform matroids are exactly the $U_{0,1}\oplus U_{1,1}$ minor-free
matroids. The broader class of paving matroids coincides with the family of
$U_{0,1}\oplus U_{2,2}$-minor-free matroids [14]. For split matroids, Joswig
and Schröter [10, Question A] identified five forbidden minors, and Cameron
and Mayhew [3] later verified that the list is complete. It is worth
mentioning that $U_{0,\ell}\oplus U_{k,k}$-minor-free matroids were studied
for positive integers $k$ and $\ell$ in general [5], while [18] gave excluded-
minor characterizations for the class of so-called nearly-uniform and nearly-
paving matroids.

#### Our results.

In [10], split matroids were introduced via polyhedral geometry. The
polyhedral point of view gives an insight into the geometry of the base
polytope of split matroids, which in turn leads to a series of fundamental
structural results. However, the polyhedral approach has two shortcomings when
it comes to optimization. First, the definition is difficult to work with as
it relies on the joint structure of split flacets. For example, it is not
immediate to see what the independent sets are, or how the rank of a set can
be determined. Furthermore, as it was already observed in [10], the notion of
split matroids is a bit subtle in the disconnected case. This is strengthened
by the fact that while uniform or paving matroids can be characterized by a
single excluded minor, the class of split matroids requires five of those.

The above observations suggest that there might be an intermediate matroid
class that captures all the good characteristics of split matroids (i.e.
closed under duality and taking minors) but is more convenient to work with in
terms of optimization. We show that this is indeed true and introduce a class
that we call elementary split matroids. The proposed class is a proper
subclass of split matroids which includes all connected split matroids. The
definition follows a combinatorial approach by setting the independent sets of
the matroid to be the family of sets having bounded intersections with certain
hyperedges. An analogous characterization was previously known for paving
matroids, see [9, 19]: for a non-negative integer $r$, a ground set $S$ of
size at least $r$, and a (possibly empty) family
$\mathcal{H}=\\{H_{1},\dots,H_{q}\\}$ of proper subsets of $S$ such that
$|H_{i}\cap H_{j}|\leq r-2$ for $1\leq i<j\leq q$, the set system
$\mathcal{B}_{\mathcal{H}}=\\{X\subseteq S\mid|X|=r,\ X\not\subseteq H_{i}\
\text{for }i=1,\dots,q\\}$ forms the set of bases of a paving matroid, and in
fact every paving matroid can be obtained in this form. Elementary split
matroids satisfy similar constraints; nevertheless, the underlying hypergraph
might have more complex structure.

We show that the proposed class has various nice properties that partially
follow from connected split matroids being special cases. However, elementary
split matroids are closed not only under duality and taking minors but also
truncation. Furthermore, the class can be characterized by the single
forbidden minor $U_{0,1}\oplus U_{1,2}\oplus U_{1,1}$, therefore fitting in
the list of earlier results. Based on the excluded-minor characterization, we
give a new proof for the result of [3] on forbidden minors of split matroids
and a complete description of binary split matroids.

The rest of the paper is organized as follows. Basic notation and definitions
are introduced in Section 2. The hypergraph representation of elementary split
matroids is presented in Section 3. Section 4 gives an excluded-minor
characterization of the proposed class. Finally, Section 5 gives a new proof
for the list of forbidden minors of split matroids, and further provides a
complete list of binary split matroids.

##  2 Preliminaries

Let $S$ be a ground set of size $n$. A clutter (or Sperner family) is a
collection $\mathcal{F}$ of subsets of $S$ in which none of the sets is a
subset of another. For subsets $X,Y\subseteq S$, the difference of $X$ and $Y$
is denoted by $X-Y$. If $Y$ consists of a single element $y$, then $X-\\{y\\}$
and $X\cup\\{y\\}$ are abbreviated as $X-y$ and $X+y$, respectively.

A matroid is a pair $M=(S,\mathcal{I})$ where $\mathcal{I}\subseteq 2^{S}$ is
the family of independent sets satisfying the so-called independence axioms:

  1. (I1)

$\emptyset\in\mathcal{I}$,

  2. (I2)

if $X\subseteq Y$ and $Y\in\mathcal{I}$, then $X\in\mathcal{I}$,

  3. (I3)

for every subset $X\subseteq S$ the maximal subsets of $X$ which are in
$\mathcal{I}$ have the same cardinality.

For a set $X\subseteq S$, the maximum size of an independent subset of $X$ is
the rank of $X$ and is denoted by $r_{M}(X)$. The subscript $M$ is dismissed
when the matroid is clear from the context. The inclusionwise maximal members
of $\mathcal{I}$ are called bases. An inclusionwise minimal non-independent
set forms a circuit, while a loop is a circuit consisting of a single element.
The dual of $M$ is the matroid $M^{*}=(S,\mathcal{I}^{*})$ where
$\mathcal{I}^{*}=\\{X\subseteq S\mid S-X\ \text{contains a basis of $M$}\\}$.
A cocircuit or coloop of $M$ is a circuit or loop of $M^{*}$, respectively.
The matroid is connected if for any two elements $e,f\in S$ there exists a
circuit containing both. This can be shown to be equivalent to
$r_{M}(X)+r_{M}(S-X)>r_{M}(S)$ for every $\emptyset\neq X\subsetneq S$. A set
$X\subseteq S$ is closed or is a flat if $r_{M}(X+e)>r_{M}(X)$ for every $e\in
S-X$. The closure of a set $X\subseteq S$, that is, the inclusionwise minimal
closed set containing $X$ is denoted by $\operatorname{cl}_{M}(X)$. Two non-
loop elements $e,f\in S$ are parallel if $r_{M}(\\{e,f\\})=1$. A flat of rank
one is called a parallel class. A flat is proper if it has nonzero rank and it
is not the ground set of the matroid. A subset $Z\subseteq S$ is cyclic if it
is the (possibly empty) union of circuits, or equivalently, the matroid
restricted to $Z$ has no coloops. Bonin and de Mier [2] rediscovered the
following axiom scheme for the cyclic flats of a matroid, first proved by Sims
[15].

######  Proposition 1.

Let $\mathcal{Z}$ be a collection of subsets of a ground set $S$ and
$r\colon\mathcal{Z}\to\mathbb{Z}_{\geq 0}$ a function. There is a matroid $M$
on $S$ for which $\mathcal{Z}$ is the set of cyclic flats and $r$ is the rank
function of $M$ restricted to $\mathcal{Z}$ if and only if the following
conditions are satisfied:

  1. (Z0)

$\mathcal{Z}$ is a lattice under inclusion,

  2. (Z1)

$r(0_{\mathcal{Z}})=0$ where $0_{\mathcal{Z}}$ is the zero of this lattice,

  3. (Z2)

$0<r(Y)-r(X)<|Y-X|$ for all $X,Y\in\mathcal{Z}$ with $X\subsetneq Y$,

  4. (Z3)

$r(X)+r(Y)\geq r(X\vee Y)+r(X\wedge Y)+|(X\cap Y)-(X\wedge Y)|$ for all
$X,Y\in\mathcal{Z}$ with join $X\vee Y$ and meet $X\wedge Y$.

In this case, the independent sets of $M$ are $\mathcal{I}=\\{I\subseteq
S\mid|I\cap Z|\leq r(Z)\text{ for each }Z\in\mathcal{Z}\\}$.

For a non-negative integer $r\leq n$, the uniform matroid $U_{r,n}$ is defined
on an $n$-element set by setting every subset of size at most $r$ to be
independent, that is, $\mathcal{I}=\\{X\subseteq S\mid|X|\leq r\\}$. When
$r=n$, the matroid is called a free matroid, while the choice $r=0$ results in
a rank-0 matroid. For technical reasons, we allow the ground set of the
matroid to be the empty set, i.e. $n=0$, in which case the matroid is simply
the empty matroid $M=(\emptyset,\\{\emptyset\\})$. A matroid of rank $r$ is
called paving if every set of size at most $r-1$ is independent, or in other
words, every circuit of the matroid has size at least $r$.

The direct sum $M_{1}\oplus M_{2}$ of matroids $M_{1}=(S_{1},\mathcal{I}_{1})$
and $M_{2}=(S_{2},\mathcal{I}_{2})$ on disjoint ground sets is the matroid
$M=(S_{1}\cup S_{2},\mathcal{I})$ whose independent sets are the disjoint
unions of an independent set of $M_{1}$ and an independent set of $M_{2}$,
that is, $\mathcal{I}=\\{I_{1}\cup I_{2}\mid I\in\mathcal{I}_{1}\text{ and
}I_{2}\in\mathcal{I}_{2}\\}$. Given a non-negative integer $k$, the
$k$-truncation of $M=(S,\mathcal{I})$ is the matroid
$(M)_{k}=(S,\mathcal{I}_{k})$ with
$\mathcal{I}_{k}=\\{X\in\mathcal{I}\mid|X|\leq k\\}$. Given a subset
$S^{\prime}\subseteq S$, the restriction of $M$ to $S^{\prime}$ is again a
matroid $M|S^{\prime}=(S^{\prime},\mathcal{I}^{\prime})$ with independence
family $\mathcal{I}^{\prime}=\\{I\in\mathcal{I}\mid I\subseteq S^{\prime}\\}$.
We also say that $M|S^{\prime}$ is obtained by the deletion of $S-S^{\prime}$,
denoted by $M\backslash(S-S^{\prime})$. The contraction of a subset
$S^{\prime\prime}\subseteq S$ results in a matroid
$M/S^{\prime\prime}=(S-S^{\prime\prime},\mathcal{I}^{\prime\prime})$ where
$\mathcal{I}^{\prime\prime}=\\{I\in\mathcal{I}\mid I\subseteq
S-S^{\prime\prime},\ |I|=r_{M}(S^{\prime\prime}\cup
I)-r_{M}(S^{\prime\prime})\\}$. A matroid $N$ that can be obtained from $M$ by
a sequence of deletions and contractions is called a minor of $M$. For uniform
matroids, it is not difficult to see that $U_{k^{\prime},\ell^{\prime}}$ is a
minor of $U_{k,\ell}$ if and only if $k^{\prime}\leq k$ and
$\ell-\ell^{\prime}\geq k-k^{\prime}$ hold. The following well-known result is
[13, Theorem 4.3.1].

######  Proposition 2.

Let $e$ be an element of a connected matroid $M$. Then $M/e$ or $M\backslash
e$ is connected.

A class $\mathcal{M}$ of matroids is minor-closed if for any member $M$ of
$\mathcal{M}$, each minor of $M$ is also contained in $\mathcal{M}$. For a
minor-closed class $\mathcal{M}$, a nearly-$\mathcal{M}$ matroid is a matroid
$M$ such that $M/e\in\mathcal{M}$ or $M\backslash e\in\mathcal{M}$ for each
element $e$. We will use the following observation of [16, 18].

######  Proposition 3.

The class of nearly-$\mathcal{M}$ matroids is minor-closed for each minor-
closed class $\mathcal{M}$ of matroids.

The rank-2 wheel $M(\mathcal{W}_{2})$ is the matroid obtained from $U_{2,3}$
by adding a parallel copy of one of the elements of the ground set. The
following is a consequence of a result of Gershkoff and Oxley [8].

######  Proposition 4.

Every connected non-uniform matroid contains $M(\mathcal{W}_{2})$ as a minor.

For connected matroids, the following proposition summarizes the relations
between the different notions of flacets and characterizes compatibility, see
[3, 10, 7, 6].

######  Proposition 5.

Let $M$ be a connected matroid on ground set $S$ with rank function $r_{M}$.

  1. (a)

A subset $Z\subseteq S$ is a flacet of $M$ if and only if it is a proper flat
such that both $M|Z$ and $M/Z$ are connected.

  2. (b)

A flacet $Z$ is a split flacet if and only if $|Z|\geq 2$, or equivalently, if
$Z$ is cyclic.

  3. (c)

For distinct split flacets $F$ and $G$, the splits obtained from the $F$\- and
$G$-hyperplanes are compatible if and only if $|F\cap G|+r_{M}(S)\leq
r_{M}(F)+r_{M}(G)$.

As we will show, the inequality of (c) motivates a matroid class slightly
different from that of split matroids.

##  3 Hypergraph representation

In this section, we introduce the notion of elementary split matroids.
Similarly to paving matroids, the definition is via hypergraphs, which will
immediately imply that the proposed class is closed under duality, taking
minors, and truncation.

######  Theorem 6.

Let $S$ be a ground set of size at least $r$,
$\mathcal{H}=\\{H_{1},\dots,H_{q}\\}$ be a (possibly empty) collection of
subsets of $S$, and $r,r_{1},\dots,r_{q}$ be non-negative integers satisfying

(H1) $|H_{i}\cap H_{j}|\leq r_{i}+r_{j}-r\ \text{for $1\leq i<j\leq q$}.$

Then $\mathcal{I}=\\{X\subseteq S\mid|X|\leq r,\ |X\cap H_{i}|\leq r_{i}\
\text{for $1\leq i\leq q$}\\}$ forms the independent sets of a matroid with
rank function $r_{M}(Z)=\min\big{\\{}r,|Z|,\displaystyle\min_{1\leq i\leq
q}\\{|Z-H_{i}|+r_{i}\\}\big{\\}}$. If furthermore

(H2) $|S-H_{i}|+r_{i}\geq r\ \text{for $i=1,\dots,q$}$

holds, then the rank of the matroid is $r$.

###### Proof.

The first two independence axioms clearly hold. A nice trick of the proof is
that the third independence axiom (I3) and the rank formula is proved
simultaneously. For any subset $Z\subseteq S$, let $I\subseteq Z$ be a maximal
member of $\mathcal{I}$ in the sense that $I$ cannot be extended by an element
of $Z$ to a member of $\mathcal{I}$. If $|I|=\min\\{r,|Z|\\}$ then we are
done, hence assume that strict inequality holds. Since $I$ is maximal in $Z$,
there is a hyperedge $H_{z}\in\mathcal{H}$ for every $z\in Z-I$ such that
$|I\cap H_{z}|=r_{z}$ and $z\in H_{z}$. Furthermore, if
$z^{\prime},z^{\prime\prime}\in Z-I$ are distinct elements, then the
corresponding hyperedges $H_{z^{\prime}}$ and $H_{z^{\prime\prime}}$ are
identical as otherwise

$\displaystyle|H_{z^{\prime}}\cap H_{z^{\prime\prime}}|{}$
$\displaystyle{}\geq|I\cap H_{z^{\prime}}\cap H_{z^{\prime\prime}}|$
$\displaystyle{}=|I\cap H_{z^{\prime}}|+|I\cap
H_{z^{\prime\prime}}|-|I\cap(H_{z^{\prime}}\cup H_{z^{\prime\prime}})|$
$\displaystyle{}\geq r_{z^{\prime}}+r_{z^{\prime\prime}}-|I|$
$\displaystyle{}>r_{z^{\prime}}+r_{z^{\prime\prime}}-r,$

contradicting (H1). Therefore there exists a hyperedge, say $H_{i}$, such that
$Z-I\subseteq H_{i}$ and $|I\cap H_{i}|=r_{i}$. Thus we get $|I|=|I\cap
H_{i}|+|I-H_{i}|=r_{i}+|Z-H_{i}|$, implying that the cardinality of $I$
depends only on $Z$. Therefore the third independence axiom holds, and the
rank formula is also verified. If (H2) holds, then the rank formula implies
$r_{M}(S)=\min\big{\\{}r,|S|,\min_{1\leq i\leq
q}\\{|S-H_{i}|+r_{i}\\}\big{\\}}=r$, concluding the proof of the theorem. ∎

We call the matroids that can be obtained in the form provided by Theorem 6
elementary split matroids. When (H2) fails for some $1\leq i\leq q$, the rank
of the matroid is less than $r$ by the rank formula. In such a case, replacing
$r$ with $r^{\prime}=\min_{1\leq i\leq q}\\{|S-H_{i}|+r_{i}\\}$ does not
violate (H1) while $\mathcal{I}$ remains the same. Thus a rank-$r$ elementary
split matroid can be represented by a hypergraph
$\mathcal{H}=\\{H_{1},\dots,H_{q}\\}$ and values $r,r_{1},\dots,r_{q}$
satisfying both (H1) and (H2). It is not difficult to see that the underlying
hypergraph can be chosen in such a way that

(H3) $\displaystyle r_{i}\leq r-1\ \text{for $i=1,\dots,q$,}$ (H4)
$\displaystyle|H_{i}|\geq r_{i}+1\ \text{for $i=1,\dots,q$.}$

Indeed, if a pair $(H_{i},r_{i})$ violates (H3) or (H4), then the
corresponding constraint $|X\cap H_{i}|\leq r_{i}$ is redundant. Therefore, we
call the representation non-redundant if all of (H1)–(H4) hold.

Elementary split matroids generalize paving matroids. Indeed, paving matroids
correspond to the special case when $r_{i}=r-1$ for $i=1,\dots,q$. If, in
addition, $|H_{i}|=r$ holds for $i=1,\dots,q$, then we get back the class of
sparse paving matroids.

######  Remark 7.

The definition of elementary split matroids is closely related to the
construction of matroids by cyclic flats, described in Proposition 1. Consider
a non-redundant hypergraph representation
$\mathcal{H}=\\{H_{1},\dots,H_{q}\\}$, $r,r_{1},\dots,r_{q}$ of a rank-$r$
elementary split matroid. In order to exclude extreme cases, assume that
$q\geq 1$, the $r_{i}$ values are strictly positive, and (H2) holds with
strict inequality for $1\leq i\leq q$. We claim that the family
$\mathcal{Z}=\\{\emptyset,H_{1},\dots,H_{q},S\\}$ satisfies the conditions of
Proposition 1 with $r_{M}(\emptyset)=0$, $r_{M}(H_{i})=r_{i}$ and
$r_{M}(S)=r$. Indeed, for different indices $i$ and $j$ we have $|H_{i}\cap
H_{j}|\leq r_{i}+r_{j}-r\leq(|H_{i}|-1)-1=|H_{i}|-2$ by (H1), (H3) and (H4),
hence $H_{i}\not\subseteq H_{j}$. Thus condition (Z0) is satisfied and
$H_{i}\wedge H_{j}=\emptyset$, $H_{i}\vee H_{j}=S$ for each $i\neq j$.
Condition (Z1) holds by $r_{M}(\emptyset)=0$. Condition (Z2) for $X=\emptyset$
and $Y=H_{i}$ translates to $0<r_{i}<|H_{i}|$, for $X=\emptyset$ and $Y=S$ it
translates to $0<r<|S|$, and for $X=H_{i}$ and $Y=S$ it translates to
$0<r-r_{i}<|S-H_{i}|$, all of which are satisfied by our assumptions.
Conditions (Z0)–(Z2) imply that (Z3) is satisfied if either $X$ or $Y$ is
$0_{\mathcal{Z}}$ or $1_{\mathcal{Z}}$, or if $X=Y$. If $X=H_{i}$ and
$Y=H_{j}$ for $i\neq j$, then (Z3) is equivalent to $r_{i}+r_{j}\geq
r+|H_{i}\cap H_{j}|$, that is, to (H1). Therefore, Proposition 1 provides
another proof for $M$ being a rank-$r$ matroid whose system of cyclic flats is
$\mathcal{Z}$. However, the addition of the missing extreme cases ensures that
our class is minor-closed.

A nice feature of the class of split matroids is that it is closed under
duality and taking minors. We show that the same holds for elementary split
matroids. In addition, the class of elementary split matroids is closed for
truncation, a property that split matroids do not satisfy in general. To see
the latter, consider the matroid $M=(U_{1,2}\oplus U_{1,2}\oplus U_{1,2}\oplus
U_{1,2})_{3}$, that is, the $3$-truncation of the direct sum of four rank-$1$
uniform matroids on 2 elements. Then it is not difficult to check that $M$ is
connected and has a $U_{0,1}\oplus U_{1,2}\oplus U_{1,1}$-minor, therefore it
is not a split matroid, see Theorem 11 later.

######  Theorem 8.

The class of elementary split matroids is closed under duality, taking minors,
and truncation.

###### Proof.

Let $M=(S,\mathcal{I})$ be a rank-$r$ elementary split matroid and
$\mathcal{H}=\\{H_{1},\dots,H_{q}\\}$, $r,r_{1},\dots,r_{q}$ be a
representation satisfying (H1) and (H2). For a non-negative integer $k<r$, the
$k$-truncation of $M$ is the matroid $(M)_{k}=(S,\mathcal{I}_{k})$ where
$\mathcal{I}_{k}=\\{X\subseteq S\mid|X|\leq k,\ |X\cap H_{i}|\leq r_{i}\
\text{for $1\leq i\leq q$}\\}$. As $|H_{i}\cap H_{j}|\leq r_{i}+r_{j}-r\leq
r_{i}+r_{j}-k$, $(M)_{k}$ is an elementary split matroid.

Now consider a set $Z\subseteq S$. The deletion of $Z$ results in a matroid
$M\backslash Z=(S-Z,\mathcal{I}_{S-Z})$ where $\mathcal{I}_{S-Z}=\\{X\subseteq
S-Z\mid|X|\leq r,\ |X\cap(H_{i}-Z)|\leq r_{i}\ \text{for $1\leq i\leq q$}\\}$.
As $|(H_{i}\cap H_{j})-Z|\leq|H_{i}\cap H_{j}|\leq r_{i}+r_{j}-r$,
$M\backslash Z$ is an elementary split matroid. Note that (H2) might not hold
for the restriction as the size of the ground set decreased, hence the rank of
$M\backslash Z$ might be smaller than $r$.

Finally, define $\overline{H}_{i}:=S-H_{i}$, $\overline{r}:=|S|-r$, and
$\overline{r}_{i}:=|\overline{H}_{i}|-r+r_{i}$ for $i=1,\dots,q$. Then
$\overline{r}\leq|S|$ and
$\overline{r},\overline{r}_{1},\dots,\overline{r}_{q}$ are non-negative by
$r\leq|S|$ and (H2). By (H1), for $1\leq i<j\leq q$, we obtain

$\displaystyle|\overline{H}_{i}\cap\overline{H}_{j}|{}$
$\displaystyle{}=|S|-|H_{i}|-|H_{j}|+|H_{i}\cap H_{j}|$
$\displaystyle{}\leq|S|-|H_{i}|-|H_{j}|+r_{i}+r_{j}-r$
$\displaystyle{}=\left(|\overline{H}_{i}|-r+r_{i}\right)+\left(|\overline{H}_{j}|-r+r_{j}\right)-\left(|S|-r\right)$
$\displaystyle{}=\overline{r}_{i}+\overline{r}_{j}-\overline{r}.$

By $r_{i}\geq 0$, for $i=1,\dots,q$, we obtain

$|S-\overline{H}_{i}|+\overline{r}_{i}=|H_{i}|+|S|-|H_{i}|-r+r_{i}\geq\overline{r}.$

Therefore
$\overline{\mathcal{H}}=\\{\overline{H}_{1},\dots,\overline{H}_{q}\\}$,
$\overline{r},\overline{r}_{1},\dots,\overline{r}_{q}$ satisfies all the
conditions of Theorem 6, hence $\\{X\subseteq S\mid|X|\leq\overline{r},\
|X\cap\overline{H}_{i}|\leq\overline{r}_{i}\ \text{for $i=1,\dots,q$}\\}$
forms the independent sets of a rank-$\overline{r}$ elementary split matroid
$\overline{M}$. For a set $X\subseteq S$ of size $r$, $|X\cap H_{i}|\leq
r_{i}$ holds if and only if
$|\overline{X}\cap\overline{H}_{i}|\leq\overline{r}_{i}$ holds, where
$\overline{X}=S-X$. That is, the bases of $M$ are exactly the complements of
the bases of $\overline{M}$, thus $\overline{M}$ coincides with the dual
$M^{*}$ of $M$.

As every minor of a matroid can be obtained by a series of deletions and
contractions, and $M/Z=(M^{*}\backslash Z)^{*}$, the minor-closedness of the
class of elementary split matroids follows. ∎

######  Remark 9.

Assume that the representation of $M$ is non-redundant, that is, all of
(H1)–(H4) are satisfied. By (H4), for $i=1,\dots,q$, we obtain

$\overline{r}_{i}=|\overline{H}_{i}|-r+r_{i}=|S|-|H_{i}|-r+r_{i}\leq\overline{r}-1.$

Furthermore, by (H3), for $i=1,\dots,q$, we obtain

$|\overline{H}_{i}|\geq|\overline{H}_{i}|-r+r_{i}+1=\overline{r}_{i}+1.$

That is,
$\overline{\mathcal{H}}=\\{\overline{H}_{1},\dots,\overline{H}_{q}\\}$,
$\overline{r},\overline{r}_{1},\dots,\overline{r}_{q}$ satisfies (H3) and (H4)
as well, hence then the representation of the dual provided by the proof of
Theorem 8 is also non-redundant.

The following observation will be helpful when characterizing binary split
matroids.

######  Theorem 10.

Consider a non-redundant representation
$\mathcal{H}=\\{H_{1},\dots,H_{q}\\},r,r_{1},\dots,r_{q}$ of an elementary
split matroid $M$ on ground set $S$. Then $M|H_{i}\cong U_{r_{i},|H_{i}|}$ and
$M/H_{i}\cong U_{r-r_{i},|S-H_{i}|}$ for $i=1,\dots,q$.

###### Proof.

Let $X\subseteq H_{i}$ be a subset of size $r_{i}$ for some $1\leq i\leq q$.
Then $|X|=r_{i}<r$ by (H3) and $|X\cap H_{j}|\leq|H_{i}\cap H_{j}|\leq
r_{i}+r_{j}-r<r_{j}$ by (H1) and (H3) for each index $j\neq i$, hence $X$ is
independent in $M$. As each independent subset of $H_{i}$ has size at most
$r_{i}$, we get that $M|H_{i}\cong U_{r_{i},|H_{i}|}$. Considering the
hypergraph representation of the dual matroid $M^{*}$ constructed in the proof
of Theorem 8, it follows that $M^{*}|\overline{H}_{i}\cong
U_{\overline{r}_{i},|\overline{H}_{i}|}=U_{|\overline{H}_{i}|-r+r_{i},|\overline{H}_{i}|}$,
hence $M/H_{i}=(M^{*}|\overline{H}_{i})^{*}\cong
U_{r-r_{i},|\overline{H}_{i}|}$. ∎

##  4 Excluded-minor characterization

The aim of this section is to give an excluded-minor characterization of
elementary split matroids. In contrast to split matroids where five forbidden
minors are needed, elementary split matroids can be characterized by a single
one. The next theorem determines the unique forbidden minor, and establishes a
connection between elementary and connected split matroids.

######  Theorem 11.

The following are equivalent for a matroid $M$ on ground set $S$.

  1. (i)

$M$ is an elementary split matroid.

  2. (ii)

$M$ has no $U_{0,1}\oplus U_{1,2}\oplus U_{1,1}$-minor.

  3. (iii)

$M$ is a loopless and coloopless matroid whose proper cyclic flats form a
clutter, or $M$ is the direct sum of a uniform matroid with either a rank-0
matroid or a free matroid.

  4. (iv)

$M$ is a connected split matroid or the direct sum of two uniform matroids.

###### Proof.

(i) $\Rightarrow$ (ii) The class of elementary split matroids is closed under
taking minors by Theorem 8, hence it suffices to show that $M=U_{0,1}\oplus
U_{1,2}\oplus U_{1,1}$ is not an elementary split matroid. Suppose to the
contrary that there exists a hypergraph $\mathcal{H}=\\{H_{1},\dots,H_{q}\\}$
and values $r_{1},\dots,r_{q}$ satisfying (H1)–(H4) with $r=2$ which define
$M$. As $M$ has exactly one loop, there is an index $i$ such that $|H_{i}|=1$
and $r_{i}=0$. We claim that $H_{i}$ is the unique hyperedge in $\mathcal{H}$.
Indeed, for an arbitrary index $j\neq i$, we have $0\leq|H_{i}\cap H_{j}|\leq
r_{j}-r<0$ which is not possible. Hence $i=q=1$ and $M\cong U_{0,1}\oplus
U_{2,3}$, a contradiction.

(ii) $\Rightarrow$ (iii) Suppose that $M$ is $U_{0,1}\oplus U_{1,2}\oplus
U_{1,1}$-minor-free and has proper cyclic flats $X$ and $Y$ such that
$X\subsetneq Y$. As $X$ is a flat and $Y$ is cyclic, $(M|Y)/X$ is loopless and
not free, hence it has a $U_{1,2}$-minor. Let $x\in X$ and $z\in S-Y$ and
consider the matroid $M^{\prime}=(M|(Y+z))/(X-x)$. As $X$ is cyclic, $x$ is a
loop in $M/(X-x)$, and it is also a loop in $M^{\prime}$. As $Y$ is a flat,
$z$ is a coloop in $M|(Y+z)$, hence it is a coloop in $M^{\prime}$ as well. We
get that $M^{\prime}\cong U_{0,1}\oplus(M|Y)/X\oplus U_{1,1}$ where $(M|Y)/X$
has a $U_{1,2}$-minor. Therefore $M^{\prime}$ has a $U_{0,1}\oplus
U_{1,2}\oplus U_{1,1}$-minor, hence so does $M$. This contradiction proves
that the proper cyclic flats of $M$ form a clutter.

It remains to consider the case when $M$ is $U_{0,1}\oplus U_{1,2}\oplus
U_{1,1}$-minor-free and has a loop or a coloop. By duality, we may assume that
$M$ contains a loop. Let $M^{\prime}$ be the matroid obtained by deleting all
the loops and coloops from $M$. If $M^{\prime}$ is empty, then $M$ is the
direct sum of a rank-0 and a (possibly empty) free matroid. Otherwise, as
$M^{\prime}$ is loopless and coloopless, it has a $U_{1,2}$-minor. This
implies that $M$ is coloopless, as otherwise it contains a $U_{0,1}\oplus
U_{1,2}\oplus U_{1,1}$-minor, contradicting the assumption. We also get that
$M^{\prime}$ is connected, as otherwise it has a $U_{1,2}\oplus
U_{1,2}$-minor, meaning that $M$ contains $U_{0,1}\oplus U_{1,2}\oplus
U_{1,2}$ as a minor. By Proposition 4, each connected non-uniform matroid
contains $M(\mathcal{W}_{2})$ as a minor. However, $M(\mathcal{W}_{2})$ has a
$U_{1,2}\oplus U_{1,1}$-minor, hence $M^{\prime}$ is necessarily a uniform
matroid. Therefore $M$ is the direct sum of a rank-0 and a uniform matroid.

(iii) $\Rightarrow$ (iv) The implication is immediate if $M$ is the direct sum
of a uniform matroid with either a rank-$0$ matroid or a free matroid. Hence
we may assume that $M$ is loopless, coloopless and its cyclic flats form a
clutter. If $M$ is connected, we need to show that any two distinct split
flacets $F$ and $G$ are compatible, that is, $|F\cap G|+r_{M}(S)\leq
r_{M}(F)+r_{M}(G)$ by Proposition 5. As proper cyclic flats form a clutter and
each split flacet is a proper cyclic flat, this inequality is equivalent to
the cyclic flat axiom (Z3) for $X=F$ and $Y=G$.

Consider the case when $M$ is disconnected and let $Z_{1},\dots,Z_{t}$ denote
its connected components. As $M$ is loopless and coloopless,
$Z_{1},\dots,Z_{t}$ are proper cyclic flats. This implies $t=2$, as otherwise
$Z_{1}$ and $Z_{1}\cup Z_{2}$ are both proper cyclic flats with
$Z_{1}\subsetneq Z_{1}\cup Z_{2}$, contradicting the assumption that proper
cyclic flats form a clutter. Assume that there is a proper cyclic flat
$Z\not\in\\{Z_{1},Z_{2}\\}$. Then in the lattice of cyclic flats $Z\wedge
Z_{i}=\emptyset$ and $Z\vee Z_{i}=S$, hence (Z3) implies that
$r_{M}(Z)+r_{M}(Z_{i})\geq r_{M}(S)+|Z\cap Z_{i}|$ holds for $i=1,2$. Thus we
get

$\displaystyle r_{M}(Z)+1{}$ $\displaystyle{}\leq|Z|$ $\displaystyle{}=|Z\cap
Z_{1}|+|Z\cap Z_{2}|$
$\displaystyle{}\leq(r_{M}(Z)+r_{M}(Z_{1})-r_{M}(S))+(r_{M}(Z)+r_{M}(Z_{2})-r_{M}(S))$
$\displaystyle{}=2r_{M}(Z)-r_{M}(S)$ $\displaystyle{}\leq r_{M}(Z)-1,$

a contradiction. Therefore, the only proper cyclic flats of $M$ are $Z_{1}$
and $Z_{2}$, hence $M|Z_{1}$ and $M|Z_{2}$ are uniform matroids. We proved
that $M$ is the direct sum of two uniform matroids.

(iv) $\Rightarrow$ (i) Assume first that $M$ is the direct sum of a
rank-$r_{1}$ uniform matroid on ground set $H_{1}$ and a rank-$r_{2}$ uniform
matroid on ground set $H_{2}$. Let $r:=r_{1}+r_{2}$. Then $M$ is the
elementary split matroid on ground set $S:=H_{1}\cup H_{2}$ corresponding to
the hypergraph $\mathcal{H}=\\{H_{1},H_{2}\\}$ and non-negative integers
$r,r_{1},r_{2}$. Indeed, $r\leq|S|$ holds and (H1) is satisfied as
$0=|H_{1}\cap H_{2}|\leq r_{1}+r_{2}-r=0$.

Now consider the case when $M$ is a connected split matroid. Let $r$ denote
the rank of $M$, $\mathcal{H}=\\{H_{1},\dots,H_{q}\\}$ be the collection of
split flacets, and set the value of $r_{i}$ to be the rank of $H_{i}$ for
$i=1,\dots,q$. Then $r\leq|S|$ and the values $r,r_{1},\dots,r_{q}$ are non-
negative. As $M$ is a split matroid, any two split flacets are compatible,
therefore (H1) is satisfied. By Theorem 6,
$\mathcal{H}=\\{H_{1},\dots,H_{q}\\}$, $r,r_{1},\dots,r_{q}$ define an
elementary split matroid $\hat{M}$. We claim that $\hat{M}$ is identical to
$M$. Indeed, if a set $B\subseteq S$ is a basis in $M$ then, by definition, it
is also a basis in $\hat{M}$. If $B$ is not a basis in $M$, then the
charasteristic vector of $B$ is not contained in the base polytope of $M$. As
the base polytope is completely determined by the flacet inequalities (see
e.g. [12, Theorem 2.6]), this means that $|B\cap F|>r_{M}(F)$ for some flacet
$F$ of $M$. Since $M$ is connected, necessarily $|F|\geq 2$. By Proposition
5(b), $F$ is a split flacet, hence $B$ is not a basis of $\hat{M}$ either. ∎

##  5 Applications

As an application of our results, we give a new proof for the result of
Cameron and Mayhew [3]. Furthermore, we further give a complete list of binary
split matroids.

###  5.1 Split matroids

Based on the previous results, we give a different and shorter proof of the
excluded minor characterization of split matroids originally proved by Cameron
and Mayhew [3]. As already observed in [10], the only disconnected excluded
minor for the class of split matroids is $M(\mathcal{W}_{2})\oplus
M(\mathcal{W}_{2})$. This follows from Proposition 4 and the fact that a
disconnected matroid is a split matroid if and only if it is the direct sum of
a connected split matroid and uniform matroids (see [3, Proposition 2.7]).
Joswig and Schröter [10] also identified four connected rank-3 excluded minors
on 6 elements, these matroids $S_{1},S_{2},S_{3},S_{4}$ are given by their
geometric representations on Figure 1, see also [3].

######  Theorem 12.

The only connected excluded minors for split matroids are $S_{1},S_{2},S_{3}$
and $S_{4}$.

$S_{1}$

$S_{2}$

$S_{3}$

$S_{4}$

Figure 1: The connected excluded minors for split matroids [3].

###### Proof.

Let $M$ be a connected matroid which is not a split matroid but each of its
proper minors is. By Theorem 11, connected split matroids coincide with
connected elementary split matroids, hence $M$ is not elementary split while
it is nearly-elementary-split by Proposition 2. As $M$ is not elementary
split, $(M/X)\backslash Y\cong U_{0,1}\oplus U_{1,2}\oplus U_{1,1}$ for some
subsets $X,Y\subseteq S$ by Theorem 11. By $M$ being loopless and coloopless,
the sets $X$ and $Y$ are nonempty. Let $X^{\prime}\subseteq X$ be a nonempty
subset and consider the matroid $N=M/X^{\prime}$. Notice that $N$ is
disconnected since it is split but not elementary split. Moreover, $N$ is
coloopless since $M$ is coloopless.

We claim that $N$ contains exactly one loop. If $N$ is loopless, then each of
its connected components has a $U_{1,2}$-minor and each of its non-uniform
components has an $M(\mathcal{W}_{2})$-minor by Proposition 4. As $N$ is not
the direct sum of two uniform matroids by Theorem 11, we get that it either
contains $M(\mathcal{W}_{2})\oplus U_{1,2}$ or $U_{1,2}\oplus U_{1,2}\oplus
U_{1,2}$ as a minor. This contradicts Proposition 3 since neither of these two
matroids is nearly-elementary-split. Similarly, if $N$ has at least two loops,
then it has a non-uniform connected component or at least two loopless
connected components. Thus $N$ contains either $U_{0,2}\oplus
M(\mathcal{W}_{2})$ or $U_{0,2}\oplus U_{1,2}\oplus U_{1,2}$ as a minor, both
of which contradict Proposition 3. This proves that $N$ has exactly one loop.

Suppose that $|X|\geq 2$ and choose distinct elements $x_{1},x_{2}\in X$. By
our previous observation, $M/x_{1}$ contains exactly one loop $l_{1}$ and
$M/x_{2}$ contains exactly one loop $l_{2}$, that is, $\\{x_{1},l_{1}\\}$ and
$\\{x_{2},l_{2}\\}$ are parallel classes of $M$. If $x_{2}=l_{1}$, then
$M/\\{x_{1},x_{2}\\}$ is loopless. Otherwise, parallel classes
$\\{x_{1},l_{1}\\}$ and $\\{x_{2},l_{2}\\}$ are disjoint and
$M/\\{x_{1},x_{2}\\}$ contains $l_{1}$ and $l_{2}$ as loops. Both of these
cases contradict that $M/\\{x_{1},x_{2}\\}$ contains exactly one loop. We
conclude that $|X|=1$. The class of split matroids is closed under duality,
hence the dual $M^{*}$ is also an excluded minor. Applying the previous
argument to $(M^{*}\backslash X)/Y=((M/X)\backslash Y)^{*}=U_{1,1}\oplus
U_{1,2}\oplus U_{0,1}$, we get that $|Y|=1$ holds as well.

We proved that $M$ is a rank-3 matroid on 6 elements. Denote the element of
$X$ by $a$, the loop of $M/a$ by $b$, the element of $Y$ by $e$, the coloop of
$M\backslash e$ by $f$, and the remaining two elements of the ground set by
$c$ and $d$. Then $\\{e,f\\}$ is a cocircuit of $M$, thus $M|\\{a,b,c,d\\}$ is
a loopless rank-2 matroid containing the parallel class $\\{a,b\\}$, hence it
is isomorphic to either $U_{1,2}\oplus U_{1,2}$ or $M(\mathcal{W}_{2})$. The
former case gives the matroid $S_{1}$. In the latter case consider the lines
$\\{a,b,c,d\\}$ and $\operatorname{cl}_{M}(\\{e,f\\})$. If
$\operatorname{cl}_{M}(\\{e,f\\})=\\{e,f\\}$, we get $S_{4}$. Otherwise, the
intersection of $\\{a,b,c,d\\}$ and $\operatorname{cl}_{M}(\\{e,f\\})$ is a
rank-1 flat, thus it is $\\{a,b\\}$, $\\{c\\}$ or $\\{d\\}$. The first case
gives $S_{2}$ and the latter two cases give $S_{3}$. ∎

###  5.2 Binary split matroids

Acketa [1] gave a complete list of binary paving matroids: $U_{r,n}$ for
$r\in\\{0,1,n-1,n\\}$, loopless rank-2 matroids with at most three parallel
classes, $M(K_{4}-e)$, $M(K_{4})$, $M(K_{2,3})$, $F_{7}$, $F_{7}^{*}$ and
$AG(3,2)$ (see also [13] for the definition of the latter three matroids).
Based on this and our previous results, we extend this list to contain all
binary split matroids. As each split matroid has at most one non-uniform
connected component, we only consider the connected case. Recall that the only
forbidden minor for binary matroids is $U_{2,4}$ by Tutte [17].

######  Theorem 13.

The following is a complete list of connected binary split matroids on at
least two elements.

  1. (a)

Matroids obtained by adding (possibly zero) parallel copies to an element of
$U_{r-1,r}$ for any $r\geq 2$.

  2. (b)

Loopless rank-2 matroids with exactly three parallel classes, and their duals.

  3. (c)

Connected binary (sparse) paving matroids of rank and corank at least three:
$M(K_{4})$, $F_{7}$, $F_{7}^{*}$, $AG(3,2)$.

###### Proof.

It is not difficult to check that all the listed matroids are connected binary
split matroids. It remains to prove that each binary split matroid $M$ is
included in the list. Connected matroids of rank or corank one are $U_{1,n}$
and $U_{n-1,n}$, and these are included in (a). Connected binary matroids of
rank or corank two are exactly the matroids listed as (b). Thus we may assume
that $M$ has rank and corank at least three. As (c) contains binary paving
matroids of rank and corank at least three from the list of Acketa [1], it
only remains to consider the non-paving case.

Let $M$ be given by a non-redundant hypergraph representation
$\mathcal{H}=\\{H_{1},\dots,H_{q}\\}$, $r,r_{1},\allowbreak\dots,r_{q}$. As
$M$ is non-paving, $r_{i}\leq r-2$ holds for some index $i$. By Theorem 10,
$M/H_{i}\cong U_{r-r_{i},|S-H_{i}|}$ where $2\leq r-r_{i}$ by our assumption
and $r-r_{i}+1\leq|S-H_{i}|$ as $M$ is coloopless. As $M$ contains no
$U_{2,4}$-minor, nececssarily $|S-H_{i}|=r-r_{i}+1$. Then
$|H_{i}|=|S|-r+r_{i}-1\geq r_{i}+2$ as $M$ has corank at least 3. Furthermore,
$M|H_{i}\cong U_{r_{i},|H_{i}|}$ contains no $U_{2,4}$-minor, implying
$r_{i}=1$ and $|S-H_{i}|=r$. Suppose that $q\geq 2$ and pick an index $j\neq
i$. Applying (H1) and (H3), we get $1+r_{j}-r\geq|H_{i}\cap
H_{j}|\geq|H_{j}|-|S-H_{i}|\geq r_{j}+1-r$, hence $S-H_{i}\subseteq H_{j}$ and
$|H_{j}|=r_{j}+1$. Then $S-H_{i}\subseteq H_{j}$ implies that
$r=|S-H_{i}|\leq|H_{j}|=r_{j}+1\leq r$, hence $S-H_{j}=H_{j}$ and $r_{j}=r-1$.
Therefore, $S$ is the disjoint union of the rank-1 set $H_{i}$ and the
rank-$(r-1)$ set $H_{j}$, contradicting the connectivity of $M$. This proves
that $q=1$, so $M\cong(U_{1,|H_{i}|}\oplus U_{r,r})_{r}$ as described in (a).
∎

## Acknowledgement

The work was supported by the Lendület Programme of the Hungarian Academy of
Sciences – grant number LP2021-1/2021 and by the Hungarian National Research,
Development and Innovation Office – NKFIH, grant numbers FK128673 and
TKP2020-NKA-06. Yutaro Yamaguchi was supported by JSPS KAKENHI Grant Numbers
JP20K19743 and JP20H00605 and by Overseas Research Program in Graduate School
of Information Science and Technology, Osaka University. Yu Yokoi was
supported by JSPS KAKENHI Grant Number JP18K18004 and JST PRESTO Grant Number
JPMJPR212B.

## References

  * [1] D. M. Acketa.  On binary paving matroids.  Discrete Mathematics, 70(1):109–110, 1988.
  * [2] J. E. Bonin and A. de Mier.  The lattice of cyclic flats of a matroid.  Annals of Combinatorics, 12(2):155–170, 2008.
  * [3] A. Cameron and D. Mayhew.  Excluded minors for the class of split matroids.  Australasian Journal of Combinatorics, 79(2):195–204, 2021.
  * [4] H. H. Crapo and G.-C. Rota.  On the Foundations of Combinatorial Theory: Combinatorial Geometries.  MIT press Cambridge, Mass., 1970.
  * [5] G. Drummond.  A generalisation of uniform matroids.  Advances in Applied Mathematics, 131:102248, 2021.
  * [6] E. M. Feichtner and B. Sturmfels.  Matroid polytopes, nested sets and Bergman fans.  Portugaliae Mathematica, Nova Série, 62(4):437–468, 2005.
  * [7] S. Fujishige.  A characterization of faces of the base polyhedron associated with a submodular system.  Journal of the Operations Research Society of Japan, 27(2):112–129, 1984.
  * [8] Z. Gershkoff and J. Oxley.  A notion of minor-based matroid connectivity.  Advances in Applied Mathematics, 100:163–178, 2018.
  * [9] J. Hartmanis.  Lattice theory of generalized partitions.  Canadian Journal of Mathematics, 11:97–106, 1959.
  * [10] M. Joswig and B. Schröter.  Matroids from hypersimplex splits.  Journal of Combinatorial Theory, Series A, 151:254–284, 2017.
  * [11] D. Mayhew, M. Newman, D. Welsh, and G. Whittle.  On the asymptotic proportion of connected matroids.  European Journal of Combinatorics, 32(6):882–890, 2011.
  * [12] S. Oh and D. Xiang.  The facets of the matroid polytope and the independent set polytope of a positroid.  arXiv preprint arXiv:1701.08483, 2021.
  * [13] J. Oxley.  Matroid Theory, volume 21 of Oxford Graduate Texts in Mathematics.  Oxford University Press, Oxford, second edition, 2011.
  * [14] J. G. Oxley.  Ternary paving matroids.  Discrete Mathematics, 91(1):77–86, 1991.
  * [15] J. A. Sims.  Some Problems in Matroid Theory.  PhD thesis, University of Oxford, 1980.
  * [16] J. L. Tejeda.  Exploring matroid minors.  Master’s thesis, California State University, San Bernardino, Electronic Theses, Projects, and Dissertations, 2020.
  * [17] W. T. Tutte.  A homotopy theorem for matroids, II.  Transactions of the American Mathematical Society, 88(1):161–174, 1958.
  * [18] V. N. Vega.  Excluded minors for nearly-paving matroids.  Master’s thesis, California State University, San Bernardino, Electronic Theses, Projects, and Dissertations, 2020.
  * [19] D. J. A. Welsh.  Matroid Theory.  L. M. S. Monographs, No. 8. Academic Press, London-New York, 1976.