# Transverse momentum and multiplicity dependence of $\Lambda_{c}^{+}/D^{0}$
ratio in $pp$ collisions at $\sqrt{s}=13$ TeV

Jun Song  School of Physical Science and Intelligent Engineering, Jining
University, Shandong 273155, China  Hai-hong Li  School of Physical Science
and Intelligent Engineering, Jining University, Shandong 273155, China  Feng-
lan Shao<EMAIL_ADDRESS>School of Physics and Physical Engineering,
Qufu Normal University, Shandong 273165, China

###### Abstract

We apply an equal-velocity quark combination model to study the
$\Lambda_{c}^{+}/D^{0}$ ratio in the range $p_{T}\lesssim 10$ GeV/c in $pp$
collisions at $\sqrt{s}=13$ TeV. We decompose the ratio into four parts which
are related to quark numbers, light-flavor quark $p_{T}$ spectrum, charm quark
$p_{T}$ spectrum, momentum correlation between light and charm quarks,
respectively. Their influence on $\Lambda_{c}^{+}/D^{0}$ ratio are
individually studied. The curvature property of light-flavor quark $p_{T}$
spectrum is found to be the main reason of the non-monotonic $p_{T}$
dependence of $\Lambda_{c}^{+}/D^{0}$ ratio exhibited in high multiplicity
events. Moreover, the multiplicity dependence of $\Lambda_{c}^{+}/D^{0}$ ratio
as the function of $p_{T}$ is mainly because of the multiplicity dependence of
light-flavor quark $p_{T}$ spectrum. Using the light-flavor quark $p_{T}$
spectrum obtained from experimental data of light-flavor hadrons and charm
quark $p_{T}$ spectrum obtained from FONLL and/or PYTHIA calculations, the
$p_{T}$ dependence of experimental data of $\Lambda_{c}^{+}/D^{0}$ ratio in
high multiplicity events and that in low multiplicity events in $pp$
collisions at $\sqrt{s}=13$ TeV are reasonably understood.

##  I Introduction

In recent years, LHC experiments provided rich experimental data of hadron
production in $pp$ collisions at LHC energies, from which many new features of
hadron production are found. For example, in production of light-flavor
hadrons, experiments found the enhancement of baryon to meson ratio (such as
$p/\pi$, $\Lambda/K_{s}^{0}$, $\Omega/\phi$) at intermediate $p_{T}$ (Acharya
_et al._ , 2021a; Khachatryan  _et al._ , 2011) and the enhancement of multi-
strange hadrons in high multiplicity events (Adam  _et al._ , 2017). We also
find a quark number scaling property for $p_{T}$ spectra of hadrons at mid-
rapidity by further analyzing experimental data of ALICE collaboration (Song
_et al._ , 2017; Zhang  _et al._ , 2020; Li  _et al._ , 2021). In production
of open heavy-flavor hadrons, LHC experiments found the enhancement of
$\Lambda_{c}^{+}/D^{0}$ in the low $p_{T}$ range ($p_{T}\lesssim 10$ GeV/c) in
$pp$ collisions at LHC energies in comparison with measurements in
$e^{+}e^{-}$ and $ep$ collisions at early years (Acharya  _et al._ , 2018;
Sirunyan  _et al._ , 2020; Acharya  _et al._ , 2021b, b, 2022a).

The production enhancement of light-flavor baryons and, in particular, heavy-
flavor baryons such as $\Lambda_{c}^{+}$ attracts lots of theoretical studies.
Many new phenomenological mechanism and/or details at the hadronization of
final-parton system created in $pp$ collisions at LHC energies, either in
fragmentation framework (Ortiz Velasquez  _et al._ , 2013; Bierlich  _et al._
, 2015; Bierlich and Christiansen, 2015; Gieseke  _et al._ , 2018;
Christiansen and Skands, 2015; Fischer and Sjöstrand, 2017; Sjöstrand, 2018)
or in quark (re-)combination framework, are proposed to explain these new
experimental data, which greatly enrich people’s understandings for the
property of hadron production in high energy collisions. Recently, ALICE
collaboration report their precise measurement for the multiplicity dependence
of $\Lambda_{c}^{+}/D^{0}$ ratio in the low $p_{T}$ range (Acharya  _et al._ ,
2022b) and the preliminary data $\Lambda_{c}^{+}/D^{0}$ ratio at small $p_{T}$
($p_{T}<1$ GeV/c) (Wilkinson, 2022; ALICE, 2022). These newest data will
further test the existing hadronization models (He and Rapp, 2019; Minissale
_et al._ , 2021; Chen and He, 2021; Song  _et al._ , 2018).

In this paper, we apply a quark combination model (Song  _et al._ , 2018; Li
_et al._ , 2018; Song  _et al._ , 2017) to study the multiplicity and $p_{T}$
dependence of $\Lambda_{c}^{+}/D^{0}$ ratio in $pp$ collisions at
$\sqrt{s}=13$ TeV. The model applies an equal-velocity combination (EVC)
mechanism to approximately describe the combination of constituent
(anti-)quarks at hadronization. Taking advantage of the analytic feature of
EVC mechanism, we derive the analytical formula of the ratio
$\Lambda_{c}^{+}/D^{0}$ and study several physical ingredients that influence
the multiplicity and $p_{T}$ dependence of the ratio $\Lambda_{c}^{+}/D^{0}$.
We demonstrate how the property of light-flavor quark $p_{T}$ spectra and
charm quark $p_{T}$ spectra lead to, in an intuitive way, the non-monotonic
$p_{T}$ dependence the ratio $\Lambda_{c}^{+}/D^{0}$. Finally, we compare our
theoretical calculations with the latest experimental data of ALICE
collaboration (Acharya  _et al._ , 2022b).

The paper is organized as follows. Sec. II gives a brief introduction to our
quark combination model with EVC mechanism. In Sec. III, we derive the ratio
$\Lambda_{c}^{+}/D^{0}$ and decompose it into four parts. In Secs. IV-VII, we
discuss the influence of these four parts, i.e., quark numbers, light-flavor
quark $p_{T}$ spectra, charm quark $p_{T}$ spectrum, and $p_{T}$ correlation
between charm and light-flavor (anti-)quarks, on the ratio
$\Lambda_{c}^{+}/D^{0}$. In Sec. VIII, we compare our theoretical results of
ratio $\Lambda_{c}^{+}/D^{0}$ with experimental data. In Sec. IX. we give the
summary.

##  II A brief introduction to EVC model

In this paper, we apply a quark combination model proposed in previous works
(Song  _et al._ , 2018; Li  _et al._ , 2018; Song  _et al._ , 2017) to study
ratio $\Lambda_{c}^{+}/D^{0}$. This model is inspired by the quark number
scaling property found from experimental data for $p_{T}$ spectra of light-
flavor hadrons at mid-rapidity in $pp$ and $p$-Pb collisions at LHC energies
(Song  _et al._ , 2017; Gou  _et al._ , 2017). We have applied the model to
describe the yield and $p_{T}$ spectra of light-flavor hadrons and those of
single-charm hadrons in $pp$, $p$-Pb and AA collision at RHIC and LHC
energies, and found generally good agreement with experimental data (Song  _et
al._ , 2021a, 2022, b; Li  _et al._ , 2021; Wang  _et al._ , 2020). In this
section, we briefly introduce the model and, in particular, the relevant
physical approximations and parameters in the model which may influence the
ratio $\Lambda_{c}^{+}/D^{0}$.

###  II.1 general framework

We start from the inclusive momentum distribution of single-charm hadron in
general framework of quark combination mechanism

$\displaystyle f_{M_{c\bar{l}}}(p)$ $\displaystyle=\int
dp_{1}dp_{2}f_{c\bar{l}}(p_{1},p_{2})\,{\cal
R}_{M_{c\bar{l}}}(p_{1},p_{2};p),$ (1) $\displaystyle f_{B_{cll^{\prime}}}(p)$
$\displaystyle=\int
dp_{1}dp_{2}dp_{3}f_{cll^{\prime}}(p_{1},p_{2},p_{3})\,{\cal
R}_{B_{cll^{\prime}}}(p_{1},p_{2},p_{3};p).$ (2)

Here, $f_{c\bar{l}}(p_{1},p_{2})$ is the joint momentum distribution of charm
($c$) quark and light anti-quark ($\bar{l}$). ${\cal
R}_{M_{c\bar{l}}}(p_{1},p_{2};p)$ is the combination function denoting the
probability density for the given $c\bar{l}$ with momenta $p_{1}$, $p_{2}$
combining into a meson $M_{c\bar{l}}$ with momentum $p$. It is similar for the
baryon formula.

In our model, we assume that the charm hadron is formed mainly by the
combination of charm quark with light-flavor (anti-)quarks with equal
velocity. In this approximation, the combination function is simply the
product of Dirac delta functions

$\displaystyle{\cal R}_{M_{c\bar{l}}}(p_{1},p_{2};p)$
$\displaystyle=\kappa_{M_{c\bar{l}}}\prod_{i=1}^{2}\delta(p_{i}-x_{i}p),$ (3)
$\displaystyle{\cal R}_{B_{cll^{\prime}}}(p_{1},p_{2},p_{3};p)$
$\displaystyle=\kappa_{B_{cll^{\prime}}}\prod_{i=1}^{3}\delta(p_{i}-x_{i}p).$
(4)

Here $\kappa_{M_{c\bar{l}}}$ and $\kappa_{B_{cll^{\prime}}}$ are independent
of momentum but are dependent on quarks numbers due to the unitarity
constraint of hadronization. Momentum fraction reads as
$x_{i}=m_{i}/(m_{1}+m_{2})$ in meson formula with momentum conservation
constraint $x_{1}+x_{2}=1$ and $x_{i}=m_{i}/(m_{1}+m_{2}+m_{3})$ in baryon
formula with momentum conservation constraint $x_{1}+x_{2}+x_{3}=1$.

Substituting Eqs. (3) and (4) into (1) and (2), we obtain

$\displaystyle f_{M_{c\bar{l}}}(p)$
$\displaystyle=\kappa_{M_{c\bar{l}}}f_{c\bar{l}}(x_{1}p,x_{2}p),$ (5)
$\displaystyle f_{B_{cll^{\prime}}}(p)$
$\displaystyle=\kappa_{B_{cll^{\prime}}}f_{cll^{\prime}}(x_{1}p,x_{2}p,x_{3}p).$
(6)

For the integral of joint distribution of quarks, we have

$\displaystyle\int dp_{1}dp_{2}f_{c\bar{l}}(p_{1},p_{2})$
$\displaystyle=N_{c}N_{\bar{l}}\equiv N_{c\bar{l}}$ (7) $\displaystyle\int
dp_{1}dp_{2}dp_{3}f_{cll^{\prime}}(p_{1},p_{2},p_{3})$
$\displaystyle=N_{c}N_{ll^{\prime}}\equiv N_{cll^{\prime}}$ (8)

where $N_{ll^{\prime}}$ equals to $N_{l}N_{l^{\prime}}$ as $l\neq l^{\prime}$
and $N_{l}(N_{l}-1)$ as $l\neq l^{\prime}$. Obviously, $N_{c\bar{l}}$ is the
number of all $c\bar{l}$ pair and $N_{cll^{\prime}}$ is the number of all
possible $cll^{\prime}$ combinations. Under EVC constraint, the integrals
become

$\displaystyle\int dpf_{c\bar{l}}(x_{1}p,x_{2}p)$
$\displaystyle\equiv\frac{N_{c\bar{l}}}{A_{c\bar{l}}},$ (9) $\displaystyle\int
dpf_{cll^{\prime}}(x_{1}p,x_{2}p,x_{3}p)$
$\displaystyle\equiv\frac{N_{cll^{\prime}}}{A_{cll^{\prime}}},$ (10)

where we use coefficient $A_{c\bar{l}}$ to denote the effect of equal-velocity
constraint to the effective number of $c\bar{l}$ pairs and $A_{cll^{\prime}}$
to denote that to effective number of $cll^{\prime}$ combinations.

Integrating Eqs. (5) and (6) over $p$, we obtain the number of charm hadrons

$\displaystyle N_{M_{c\bar{l}}}$
$\displaystyle=\kappa_{M_{c\bar{l}}}\frac{N_{c\bar{l}}}{A_{c\bar{l}}}=N_{c\bar{l}}\frac{\kappa_{M_{c\bar{l}}}}{A_{c\bar{l}}},$
(11) $\displaystyle N_{B_{cll^{\prime}}}$
$\displaystyle=\kappa_{B_{cll^{\prime}}}\frac{N_{cll^{\prime}}}{A_{cll^{\prime}}}=N_{cll^{\prime}}\frac{\kappa_{B_{cll^{\prime}}}}{A_{cll^{\prime}}}.$
(12)

We see that $\frac{\kappa_{M_{c\bar{l}}}}{A_{c\bar{l}}}$ has the meaning of
the average probability of a $c\bar{l}$ pair forming a meson, and it is
similar for baryon term.

If we neglect the contribution of multi-charm hadrons, the unitarity of charm
quark hadronization gives

$\sum_{M_{c\bar{l}}}N_{M_{c\bar{l}}}+\sum_{B_{cll^{\prime}}}N_{B_{cll^{\prime}}}=N_{c},$
(13)

which means

$\sum_{M_{c\bar{l}}}N_{\bar{l}}\frac{\kappa_{M_{c\bar{l}}}}{A_{c\bar{l}}}+\sum_{B_{cll^{\prime}}}N_{ll^{\prime}}\frac{\kappa_{B_{cll^{\prime}}}}{A_{cll^{\prime}}}=1.$
(14)

At charm quark hadronization, light-flavor quarks serve as the background and
their property, i.e., numbers $N_{l}$ and momentum distributions $f_{l}(p)$
relating to $A$, can be freely changed. With this consideration, we expect
$\frac{\kappa_{M_{c\bar{l}}}}{A_{c\bar{l}}}\sim\frac{1}{N_{\bar{q}}}$ and
$\frac{\kappa_{B_{cll^{\prime}}}}{A_{cll^{\prime}}}\sim\frac{1}{N_{q}^{2}}$
with $N_{\bar{q}}=\sum_{\bar{l}}N_{\bar{l}}$ and $N_{q}=\sum_{l}N_{l}$ so that
the unitarity constraint can be satisfied in an easy manner. In this
philosophy, we firstly introduce a dynamic parameter $R_{B/M}^{(c)}$ to denote
the competition between the formation of charm baryon and the formation of
charm meson as a charm quark hadronizes. Then we can write

$\displaystyle\sum_{M_{c\bar{l}}}N_{\bar{l}}\frac{\kappa_{M_{c\bar{l}}}}{A_{c\bar{l}}}$
$\displaystyle=\frac{1}{1+R_{B/M}^{(c)}},$ (15)
$\displaystyle\sum_{B_{cll^{\prime}}}N_{ll^{\prime}}\frac{\kappa_{B_{cll^{\prime}}}}{A_{cll^{\prime}}}$
$\displaystyle=\frac{R_{B/M}^{(c)}}{1+R_{B/M}^{(c)}}.$ (16)

Following the spirit in our previous works (Li  _et al._ , 2018; Song  _et
al._ , 2018), we can take the following parameterizations

$\displaystyle\frac{\kappa_{M_{c\bar{l}}}}{A_{c\bar{l}}}$
$\displaystyle=C_{M_{c\bar{l}}}\frac{1}{N_{\bar{q}}}\frac{1}{1+R_{B/M}^{(c)}}=C_{M_{c\bar{l}}}\frac{1}{N_{c\bar{q}}}\frac{1}{1+R_{B/M}^{(c)}}N_{c},$
(17) $\displaystyle\frac{\kappa_{B_{cll^{\prime}}}}{A_{cll^{\prime}}}$
$\displaystyle=C_{B_{cll^{\prime}}}N_{iter,ll^{\prime}}\frac{1}{N_{qq}}\frac{R_{B/M}^{(c)}}{1+R_{B/M}^{(c)}}$
$\displaystyle=C_{B_{cll^{\prime}}}N_{iter,ll^{\prime}}\frac{1}{N_{cqq}}\frac{R_{B/M}^{(c)}}{1+R_{B/M}^{(c)}}N_{c},$
(18)

which can be understood as follows. In the second equality,
$N_{c}/(1+R_{B/M}^{(c)})$ denotes the number of charm quarks that will form
charm mesons. $N_{c\bar{q}}$ equals to $N_{c}N_{\bar{q}}$ and denotes the
number of all possible $c\bar{q}$ pair. Then
$N_{c}/(1+R_{B/M}^{(c)})/N_{c\bar{q}}$ denotes the average probability of a
$c\bar{q}$ pair forming a charm meson. Because $c\bar{q}$ pair can form the
charm meson with different total angular moments, here we introduce a
parameter $C_{M_{c\bar{l}}}$ to denote the probability of a $c\bar{l}$ pair
forming a given state $M_{c\bar{l}}$. Obviously, unitarity requires
$\sum_{M}C_{M_{c\bar{l}}}=1$ where summation runs over all meson states with
the same $c\bar{l}$ composition. It is similar for Eq. (18) of the baryon.
Here, $N_{iter,ll^{\prime}}$ is the permutation factor of $ll^{\prime}$ and is
take 2 for $l\neq l^{\prime}$ and 1 for $l=l^{\prime}$, respectively. We
expect $C_{M_{c\bar{l}}}$ and $C_{B_{cll^{\prime}}}$ are stable. Because
$\sum_{M_{c\bar{l}}}N_{\bar{l}}C_{M_{c\bar{l}}}=\sum_{\bar{l}}N_{\bar{l}}=N_{\bar{q}}$
and
$\sum_{B_{cll^{\prime}}}C_{B_{cll^{\prime}}}N_{iter,ll^{\prime}}N_{ll^{\prime}}=\sum_{ll^{\prime}}N_{iter,ll^{\prime}}N_{ll^{\prime}}=N_{qq}$,
the unitarity of charm hadronization is naturally satisfied via
$\sum_{M_{c\bar{l}}}N_{M_{c\bar{l}}}=\sum_{M_{c\bar{l}}}N_{c\bar{l}}\kappa_{M_{c\bar{l}}}/A_{c\bar{l}}=N_{c}/(1+R_{B/M}^{(c)})$
and
$\sum_{B_{cll^{\prime}}}N_{B_{cll^{\prime}}}=\sum_{B_{cll^{\prime}}}N_{cll^{\prime}}\kappa_{B_{cll^{\prime}}}/A_{cll^{\prime}}=N_{c}R_{B/M}^{(c)}(1+R_{B/M}^{(c)})$.

Finally, numbers of charm hadrons are

$\displaystyle N_{M_{c\bar{l}}}$
$\displaystyle=C_{M_{c\bar{l}}}\frac{1}{1+R_{B/M}^{(c)}}\frac{N_{c\bar{l}}}{N_{\bar{q}}},$
(19) $\displaystyle N_{B_{cll^{\prime}}}$
$\displaystyle=C_{B_{cll^{\prime}}}\frac{R_{B/M}^{(c)}}{1+R_{B/M}^{(c)}}N_{iter,ll^{\prime}}\frac{N_{cll^{\prime}}}{N_{qq}},$
(20)

and momentum distribution of charm hadrons are

$\displaystyle f_{M_{c\bar{l}}}(p)$
$\displaystyle=N_{M_{c\bar{l}}}f_{c\bar{l}}^{(n)}(x_{1}p,x_{2}p),$ (21)
$\displaystyle f_{B_{cll^{\prime}}}(p)$
$\displaystyle=N_{B_{cll^{\prime}}}f_{cll^{\prime}}^{(n)}(x_{1}p,x_{2}p,x_{3}p),$
(22)

where $f_{c\bar{l}}^{(n)}(x_{1}p,x_{2}p)$ and
$f_{cll^{\prime}}^{(n)}f_{cll^{\prime}}^{(n)}(x_{1}p,x_{2}p,x_{3}p)$ are the
normalized distributions under integral over $p$.

###  II.2 application to high-energy collisions

In this paper, we study the production of charm hadron at mid-rapidity in $pp$
collisions at $\sqrt{s}=13$ TeV and compare our theoretical results with
experimental data at mid-rapidity. The quark momentum distribution $dN/dp$ at
rapidity $y=0$ in our model is reduced to $f(p_{T})\equiv dN/dp_{T}$. The
joint quark momentum distributions are also reduced to
$f_{c\bar{l}}(p_{T,c},p_{T,\bar{l}})$ and
$f_{cll^{\prime}}(p_{T,c},p_{T,l},p_{T,l^{\prime}})$. The rapidity densities
of charm and light-flavor quarks are denoted as $N_{l}$ and $N_{c}$ for
convenience.

In experimental measurement, momentum spectra and yield densities of hadrons
are mainly reported via their averaged values in the selected event class. We
can extend the above formulas to follows experimental statistics by re-
defining quark momentum distributions $f_{c\bar{l}}(p_{T,c},p_{T,\bar{l}})$
and $f_{cll^{\prime}}(p_{T,c},p_{T,l},p_{T,l^{\prime}})$ and quark numbers
$N_{l}$, $N_{c}$ as these in the select event class.

Finally, $p_{T}$ spectra of hadrons in our model are

$\displaystyle f_{M_{c\bar{l}}}(p_{T})$ $\displaystyle=\left\langle
N_{M_{c\bar{l}}}\right\rangle f_{c\bar{l}}^{(n)}(x_{1}p_{T},x_{2}p_{T}),$ (23)
$\displaystyle f_{B_{cll^{\prime}}}(p_{T})$ $\displaystyle=\left\langle
N_{B_{cll^{\prime}}}\right\rangle
f_{cll^{\prime}}^{(n)}(x_{1}p_{T},x_{2}p_{T},x_{3}p_{T}),$ (24)

with

$\displaystyle\left\langle N_{M_{c\bar{l}}}\right\rangle$
$\displaystyle=C_{M_{c\bar{l}}}\frac{1}{1+R_{B/M}^{(c)}}\left\langle\frac{N_{c\bar{l}}}{N_{\bar{q}}}\right\rangle,$
(25) $\displaystyle\left\langle N_{B_{cll^{\prime}}}\right\rangle$
$\displaystyle=C_{B_{cll^{\prime}}}\frac{R_{B/M}^{(c)}}{1+R_{B/M}^{(c)}}N_{iter,ll^{\prime}}\left\langle\frac{N_{cll^{\prime}}}{N_{qq}}\right\rangle.$
(26)

In the following studies, we also assume several symmetry property for numbers
and momentum distributions of quarks produced at mid-rapidity in $pp$
collisions at LHC energy. We consider the iso-spin symmetry between up and
down quarks, i.e., $\left\langle N_{u}\right\rangle=\left\langle
N_{d}\right\rangle$ and $f_{u}(p_{T})=f_{d}(p_{T})$, and also the charge
conjugation symmetry, i.e. , $\left\langle N_{l}\right\rangle=\left\langle
N_{\bar{l}}\right\rangle$ and $f_{l}(p_{T})=f_{\bar{l}}(p_{T})$. These
symmetry assumptions will greatly simplify our theoretical expressions.

##  III $p_{T}$ dependence of ratio $\Lambda_{c}^{+}/D^{0}$

According to Eqs. (5) and (6), $p_{T}$ spectra of $\Lambda_{c}^{+}$ and
$D_{0}$ are

$\displaystyle f_{\Lambda_{c}}(p_{T})$
$\displaystyle=\kappa_{\Lambda_{c}}f_{cud}\left(p_{T,c},p_{T,u},p_{T,d}\right),$
(27) $\displaystyle f_{D^{0}}(p_{T})$
$\displaystyle=\kappa_{D^{0}}f_{c\bar{u}}\left(x_{c}^{{}^{\prime}}p_{T},x_{\bar{u}}^{{}^{\prime}}p_{T}\right),$
(28)

where $x_{u}=x_{d}=m_{u}/(2m_{u}+m_{c})$, $x_{c}=m_{c}/(2m_{u}+m_{c})$ for the
baryon and $x_{\bar{u}}^{{}^{\prime}}=m_{u}/(m_{u}+m_{c})$,
$x_{c}^{{}^{\prime}}=m_{c}/(m_{c}+m_{u})$ for the meson.

We express the joint distribution functions of (anti-)quarks as

$\displaystyle f_{cud}\left(p_{T,c},p_{T,u},p_{T,d}\right)$ (29)
$\displaystyle=f_{c}(p_{T,c})f_{u}(p_{T,u})f_{d}(p_{T,d})\left[1+\mathcal{C}_{cud}\left(p_{T,c},p_{T,u},p_{T,d}\right)\right],$
$\displaystyle
f_{c\bar{u}}\left(p_{T,c},p_{T,\bar{u}}\right)=f_{c}(p_{T,c})f_{\bar{u}}(p_{T,\bar{u}})\left[1+\mathcal{C}_{c\bar{u}}\left(p_{T,c},p_{T,\bar{u}}\right)\right],$
(30)

where $f_{q_{i}}(p_{T})$ is inclusive distribution of $q_{i}$-flavor quarks
and $\mathcal{C}_{cud}\left(p_{T,c},p_{T,u},p_{T,d}\right)$ is the correlation
term. By rewriting $f_{q_{i}}(p_{T})=N_{q_{i}}f_{q_{i}}^{(n)}(p_{T})$ with
$f_{q_{i}}^{(n)}(p_{T})$ being the normalized distribution, we can further
write the joint distribution as

$\displaystyle f_{cud}\left(p_{T,c},p_{T,u},p_{T,d}\right)$
$\displaystyle=N_{cud}f_{c}^{(n)}(p_{T,c})f_{u}^{(n)}(p_{T,u})f_{d}^{(n)}(p_{T,d})\left[1+\mathcal{C}_{cud}\left(p_{T,c},p_{T,u},p_{T,d}\right)\right],$
(31) $\displaystyle f_{c\bar{u}}\left(p_{T,c},p_{T,\bar{u}}\right)$
$\displaystyle=N_{c}N_{\bar{u}}f_{c}^{(n)}(p_{T,c})f_{\bar{u}}^{(n)}(p_{T,\bar{u}})\left[1+\mathcal{C}_{c\bar{u}}\left(p_{T,c},p_{T,\bar{u}}\right)\right].$
(32)

With the above form of $f_{cud}\left(p_{T,c},p_{T,u},p_{T,d}\right)$ and
$f_{\bar{u}c}\left(p_{T,c},p_{T,\bar{u}}\right)$, we can calculate $p_{T}$
spectra of $\Lambda_{c}^{+}$ and $D_{0}$ by formulas in Sec. II.1 and then
consider the event average convention in Sec. II.2, and finally we get the
ratio

$\displaystyle\frac{f_{\Lambda_{c}}(p_{T})}{f_{D_{0}}(p_{T})}$
$\displaystyle=R_{B/M}^{(c)}\frac{C_{\Lambda_{c}}}{C_{D^{0}}}N_{iter,ud}\frac{\left\langle\frac{N_{cud}}{N_{qq}}\right\rangle}{\left\langle\frac{N_{c\bar{u}}}{N_{\bar{q}}}\right\rangle}\frac{A_{\Lambda_{c}}}{A_{D_{0}}}\frac{f_{c}^{(n)}(x_{c}p_{T})}{f_{c}^{(n)}(x_{c}^{{}^{\prime}}p_{T})}\frac{\left[f_{u}^{(n)}(x_{u}p_{T})\right]^{2}}{f_{u}^{(n)}(x_{u}^{{}^{\prime}}p_{T})}\frac{1+\mathcal{C}_{cud}\left(x_{c}p_{T},x_{u}p_{T},x_{d}p_{T}\right)}{1+\mathcal{C}_{\bar{u}c}\left(x_{c}^{{}^{\prime}}p_{T},x_{u}^{{}^{\prime}}p_{T}\right)}$
$\displaystyle=R^{(N_{q_{i}})}\left[\frac{A_{\Lambda_{c}}}{A_{D_{0}}}R_{\Delta
x_{c}}^{(c)}\left(p_{T}\right)R_{\Delta
x_{l}}^{(l)}\left(p_{T}\right)R_{corr}^{(cl)}\left(p_{T}\right)\right].$ (33)

Here, we have used the symmetry property
$f_{u}^{(n)}(p_{T})=f_{d}^{(n)}(p_{T})=f_{\bar{u}}^{(n)}(p_{T})=f_{\bar{d}}^{(n)}(p_{T})$
in the mid-rapidity range at LHC energies. In the second line, we split the
ratio into several parts

$\displaystyle R^{(N_{q_{i}})}$ $\displaystyle\equiv
2R_{B/M}^{(c)}\frac{C_{\Lambda_{c}}\left\langle
N_{cud}/N_{qq}\right\rangle}{C_{D_{0}}\left\langle
N_{c\bar{u}}/N_{\bar{q}}\right\rangle},$ (34) $\displaystyle R_{\Delta
x_{c}}^{(c)}\left(p_{T}\right)$
$\displaystyle\equiv\frac{f_{c}^{(n)}(x_{c}p_{T})}{f_{c}^{(n)}(x_{c}^{{}^{\prime}}p_{T})},$
(35) $\displaystyle R_{\Delta x_{l}}^{(l)}\left(p_{T}\right)$
$\displaystyle\equiv\frac{\left[f_{u}^{(n)}(x_{u}p_{T})\right]^{2}}{f_{u}^{(n)}(x_{u}^{{}^{\prime}}p_{T})},$
(36) $\displaystyle R_{corr}^{(cl)}\left(p_{T}\right)$
$\displaystyle=\frac{1+\mathcal{C}_{cud}\left(x_{c}p_{T},x_{u}p_{T},x_{d}p_{T},\right)}{1+\mathcal{C}_{c\bar{u}}\left(x_{c}^{{}^{\prime}}p_{T},x_{u}^{{}^{\prime}}p_{T}\right)}$
(37)

which reflect the influence of different physical ingredients on the ratio
$\Lambda_{c}^{+}/D^{0}$.

##  IV The property of $R^{(N_{q_{i}})}$

With Eqs. (9), (10) and (31), we can see that

$\displaystyle A_{\Lambda_{c}}\int dp_{T}$
$\displaystyle\left[f_{u}^{(n)}(x_{u}p_{T})\right]^{2}f_{c}^{(n)}(x_{c}p_{T})$
$\displaystyle\times\left[1+\mathcal{C}_{cud}\left(x_{c}p_{T},x_{u}p_{T},x_{d}\text{$p_{T}$}\right)\right]=1,$
(38) $\displaystyle A_{D_{0}}\int dp_{T}$ $\displaystyle
f_{c}^{(n)}(x_{c}^{{}^{\prime}}p_{T})f_{\bar{u}}^{(n)}(x_{u}^{{}^{\prime}}p_{T})$
$\displaystyle\times\left[1+\mathcal{C}_{c\bar{u}}\left(x_{c}^{{}^{\prime}}p_{T},x_{u}^{{}^{\prime}}p_{T}\right)\right]=1,$
(39)

which means that terms in square bracket in Eq. (33), as a global quantity,
mainly influences the shape of the $\Lambda_{c}^{+}/D^{0}$ ratio but not its
global magnitude. Therefore, $R^{(N_{q_{i}})}$ in Eq. (33) plays the role of
controlling the global magnitude of $\Lambda_{c}^{+}/D^{0}$ ratio. In this
section, we discuss the property of $R^{(N_{q_{i}})}$ defined in Eq. (34).

We take

$\displaystyle\left\langle\frac{N_{udc}}{N_{qq}}\right\rangle$
$\displaystyle\approx\frac{\left\langle N_{u}\right\rangle^{2}\left\langle
N_{c}\right\rangle}{\left\langle N_{q}\right\rangle^{2}},$ (40)
$\displaystyle\left\langle\frac{N_{c\bar{u}}}{N_{\bar{q}}}\right\rangle$
$\displaystyle\approx\frac{\left\langle N_{\bar{u}}\right\rangle\left\langle
N_{c}\right\rangle}{\left\langle N_{\bar{q}}\right\rangle}.$ (41)

Here, we use the approximations $\left\langle
N_{\bar{u}}\right\rangle=\left\langle N_{\bar{d}}\right\rangle=\left\langle
N_{u}\right\rangle=\left\langle N_{d}\right\rangle$ and $\left\langle
N_{\bar{q}}\right\rangle=\left\langle N_{q}\right\rangle$ in the mid-rapidity
range at LHC energies. We neglect quark number correlations between different
flavors by considering the small off-diagonal susceptibilities of quark
flavors in Lattice-QCD calculations (Ding  _et al._ , 2015).

In order to obtain intuitive expression, we define a strangeness suppression
factor

$\lambda_{s}=\frac{\left\langle N_{\bar{s}}\right\rangle}{\left\langle
N_{\bar{u}}\right\rangle}.$ (42)

By noticing that $\left\langle N_{q}\right\rangle=\left\langle
N_{u}\right\rangle+\left\langle N_{d}\right\rangle+\left\langle
N_{s}\right\rangle=(2+\lambda_{s})\left\langle N_{u}\right\rangle$, we obtain

$R^{\left(N_{q_{i}}\right)}=2R_{B/M}^{(c)}\frac{C_{\Lambda_{c}}}{C_{D^{0}}}\frac{1}{2+\lambda_{s}}.$
(43)

The branch fraction parameter $C_{\Lambda_{c}}$ and $C_{D^{0}}$ directly
influence $R^{\left(N_{q_{i}}\right)}$. However, experimental data of
$\Lambda_{c}^{+}$ and $D^{0}$ usually contain the contribution of strong and
electromagnetic decays of other single-charm hadrons. Considering the decay
contributions of $\Sigma_{c}^{++,+,0}$ and $\Sigma_{c}^{*++,+,0}$, the yield
of final-state $\Lambda_{c}^{+}$ is

$\displaystyle N_{\Lambda_{c}^{+}}^{(final)}=$
$\displaystyle\left(C_{\Lambda_{c}}+C_{\Sigma_{c}^{+}}\right)N_{iter,ud}\left\langle\frac{N_{udc}}{N_{qq}}\right\rangle$
$\displaystyle+\left(C_{\Sigma^{++}}+C_{\Sigma^{*++}}\right)N_{iter,uu}\left\langle\frac{N_{uuc}}{N_{qq}}\right\rangle$
$\displaystyle+\left(C_{\Sigma^{0}}+C_{\Sigma^{*0}}\right)N_{iter,dd}\left\langle\frac{N_{ddc}}{N_{qq}}\right\rangle$
$\displaystyle=\left\langle\frac{N_{uuc}+2N_{udc}+N_{ddc}}{N_{qq}}\right\rangle$
$\displaystyle\approx\frac{4}{\left(2+\lambda_{s}\right)^{2}}\left\langle
N_{c}\right\rangle.$ (44)

Because the measured $D^{0}$ contains the decay contribution of $D^{*0,+}$,
the yield of final-state $D^{0}$ is

$\displaystyle N_{D^{0}}^{(final)}$
$\displaystyle=C_{D^{0}}\left\langle\frac{N_{c\bar{u}}}{N_{\bar{q}}}\right\rangle+C_{D^{*0}}\left\langle\frac{N_{c\bar{u}}}{N_{\bar{q}}}\right\rangle+\mathcal{B}_{D^{*+}\to
D^{0}}C_{D^{*+}}\left\langle\frac{N_{c\bar{d}}}{N_{\bar{q}}}\right\rangle$
$\displaystyle=\left(1+\mathcal{B}_{D^{*+}\to
D^{0}}C_{D^{*+}}\right)\left\langle\frac{N_{c\bar{u}}}{N_{\bar{q}}}\right\rangle$
$\displaystyle\approx\left(1+\mathcal{B}_{D^{*+}\to
D^{0}}C_{D^{*+}}\right)\frac{1}{2+\lambda_{s}}\left\langle N_{c}\right\rangle$
(45)

with decay branch ratio $\mathcal{B}_{D^{*+}\to D^{0}}=0.677$ from particle
data group (Olive  _et al._ , 2014). The branch fraction parameter
$C_{D^{*+}}$ is about 0.6 since the ratio $D^{*0}/D^{0}$ is about 0.43
(Acharya  _et al._ , 2019). With Eqs. (44) and (45) we obtain
$R^{\left(N_{q_{i}}\right)}$ for final-state $\Lambda_{c}^{+}/D^{0}$ ratio

$\displaystyle R^{\left(N_{q_{i}},final\right)}$
$\displaystyle=R_{B/M}^{(c)}\frac{1}{1+0.677C_{D^{*+}}}\frac{4}{2+\lambda_{s}}$
$\displaystyle=R_{B/M}^{(c)}\frac{2.85}{2+\lambda_{s}}.$ (46)

According to our estimations in previous work (Zhang  _et al._ , 2020; Gou
_et al._ , 2017), strangeness suppression factor $\lambda_{s}$ in low
multiplicity events is about 0.3 and that in high multiplicity events is about
0.36 in $pp$ collisions at LHC energies. This change magnitude of
$\lambda_{s}$ causes little influence on $R^{\left(N_{q_{i}}\right)}$ through
the term $2+\lambda_{s}$ and therefore contribute little multiplicity
dependence of $\Lambda_{c}^{+}/D^{0}$ ratio.

$R_{B/M}^{(c)}$ is a dynamic parameter in our model that denotes the
production competition between baryon formation via combination with two
light-flavor quarks and meson formation via combination with an anti-quark at
a charm quark hadronization. In the environment of abundant light-flavor
quarks and antiquarks, charm quark has sufficient chance to interact with
surrounding light-flavor quarks and antiquarks, and the baryon to meson
production competition is sufficient and therefore we expect $R_{B/M}^{(c)}$
should be saturated.

However, in the low multiplicity events this baryon-to-meson production
competition is not sufficient due to the restricted numbers of light-flavor
quarks and/or anti-quarks. For example, in event multiplicity class X with
mid-rapidity $\left\langle dN_{ch}/d\eta\right\rangle=2.52$ in $pp$ collisions
at $\sqrt{s}=$ 13 TeV (Acharya  _et al._ , 2020a), the up quark number density
is $\left\langle dN_{u}/dy\right\rangle=1.48$ at mid-rapidity according to our
previous study of multiplicity dependence of light-flavor hadron production
(Zhang  _et al._ , 2020). Events with a charm quark and up/down quark with
numbers $(N_{u},N_{d})=(2,0),(0,2)$ can not form $\Lambda_{c}^{+}$. Events
with $(N_{u},N_{d})=(2,1)$ only provide one effective $ud$ pair for
$\Lambda_{c}^{+}$ formation but can provide three light-flavor antiquarks for
charm meson formation. Therefore, the formation of $\Lambda_{c}^{+}$ in these
events is suppressed relative to $D$ mesons. The fraction of these events
should not be too small due to the small values of $\left\langle
dN_{u,d}/dy\right\rangle$ in low multiplicity events. Therefore, we can expect
the suppression of $R_{B/M}^{(c)}$ and corresponding $\Lambda_{c}^{+}/D^{0}$
ratio to a certain extent in low multiplicity events in $pp$ collisions at
$\sqrt{s}=$ 13 TeV.

The exact magnitude of such suppression is dependent on the property of quark
number distribution event-by-event, which is little known at present. Here, we
take the Poisson distribution as an example to roughly estimate the magnitude
of this suppression. We assume the joint distribution of quark numbers as

$P(N_{u},N_{d})=\textrm{Poi}(N_{u})\textrm{Poi}(N_{d})$ (47)

with mean number $\left\langle N_{u}\right\rangle=\left\langle
N_{d}\right\rangle$. The number of $\bar{u}$ and that of $\bar{d}$ are taken
to be $N_{\bar{u}}=N_{u}$ and $N_{\bar{d}}=N_{d}$, respectively. That is, we
consider the pair production in each event, for simplicity. For events with
V0M multiplicity classes IX and X with $\left\langle
dN_{ch}/d\eta\right\rangle=2.52,4.64$ (Acharya  _et al._ , 2020a), the $u$
quark number is about $\left\langle N_{u}\right\rangle=1.5-2.5$ according to
our previous study of light-flavor hadrons in $pp$ at $\sqrt{s}=13$ TeV (Zhang
_et al._ , 2020). The fraction of events that can form $D^{0}$ but not
$\Lambda_{c}^{+}$ is

$P^{{}^{\prime}}=\frac{1}{P_{nor}}\sum_{N_{u}=1}^{\infty}P(N_{u},0)=0.08-0.18$
(48)

with $P_{nor}=\sum_{N_{u},N_{d}}P(N_{u},N_{d})-P(0,0)$. In addition, the
fraction of events that can form $D^{0}$ but not $\Sigma_{c}^{0}$ or
$\Sigma_{c}^{*0}$ which finally decay into $\Lambda_{c}^{+}$ is

$P^{{}^{\prime\prime}}=\frac{1}{P_{nor}}\sum_{N_{u}=1}^{\infty}P(N_{u},1)=0.19-0.27.$
(49)

The fraction of events that can not form $\Sigma_{c}^{++}$ or
$\Sigma_{c}^{*++}$ is also $P^{{}^{\prime\prime}}$. These estimation therefore
indicates a magnitude of about 20% suppression for $R_{B/M}^{(c)}$ in low
multiplicity events in comparison with that in high multiplicity events where
the above suppression is absent. In our previous works in study of single-
charm hadrons in $pp$ and $p$Pb collisions at LHC energies (Li  _et al._ ,
2018; Song  _et al._ , 2018), we use $R_{B/M}^{(c)}=0.425\pm 0.025$. If we
take this value as the possibly saturated value, we estimate that
$R_{B/M}^{(c)}$ in low multiplicity events is about 0.34. Therefore, we expect

$\displaystyle R^{\left(N_{q_{i}},final\right)}$
$\displaystyle=R_{B/M}^{(c)}\frac{2.85}{2+\lambda_{s}}$
$\displaystyle\approx\begin{cases}0.51&\textrm{in high $\left\langle
dN_{ch}/d\eta\right\rangle$events}\\\ 0.42&\textrm{in low $\left\langle
dN_{ch}/d\eta\right\rangle$events}\end{cases}.$ (50)

##  V The property of $R_{\Delta x_{l}}^{(l)}\left(p_{T}\right)$

Because $m_{c}\approx 5m_{u}$, the momentum fraction of up/down quark
$x_{u}^{{}^{\prime}}=m_{u}/(m_{c}+m_{u})=0.167$ in $D^{0}$ is close to
$x_{u}=m_{u}/(m_{c}+2m_{u})\approx 0.143$ in $\Lambda_{c}^{+}$. The difference
between them is quite small

$\Delta
x_{u}=x_{u}^{{}^{\prime}}-x_{u}=\frac{m_{u}^{2}}{(m_{c}+m_{u})(m_{c}+2m_{u})}\approx
0.024.$ (51)

In the studied range $p_{T}\lesssim 8$ GeV/c, the momentum difference $\Delta
x_{u}p_{T}\lesssim 0.2$ GeV/c is also small. In order to simplify $R_{\Delta
x_{l}}^{(l)}\left(p_{T}\right)$, we can take Taylor expansion for
$f_{u}(x_{u}p_{T})$

$\displaystyle f_{u}^{(n)}\left(x_{u}p_{T}\right)$
$\displaystyle=f_{u}^{(n)}\left(x_{u}^{{}^{\prime}}p_{T}\right)\Bigg{[}1-\frac{f_{u}^{(n)^{\prime}}\left(x_{u}^{{}^{\prime}}p_{T}\right)}{f_{u}^{(n)}\left(x_{u}^{{}^{\prime}}p_{T}\right)}\Delta
x_{u}p_{T}$
$\displaystyle+\frac{1}{2}\frac{f_{u}^{(n)^{\prime\prime}}\left(x_{u}^{{}^{\prime}}p_{T}\right)}{f_{u}^{(n)}\left(x_{u}^{{}^{\prime}}p_{T}\right)}\left(\Delta
x_{u}p_{T}\right)^{2}+....\Bigg{]},$ (52)

where the contribution of perturbative terms are quite small, i.e., about
$0.01\sim 0.07$ as $p_{T}<8$ GeV/c. Therefore, the light-flavor part
$R_{\Delta x_{l}}^{(l)}\left(p_{T}\right)$ becomes

$\displaystyle R_{\Delta
x_{l}}^{(l)}\left(p_{T}\right)=\frac{\left[f_{u}^{(n)}(x_{u}p_{T})\right]^{2}}{f_{u}^{(n)}(x_{u}^{{}^{\prime}}p_{T})}$
$\displaystyle=f_{u}^{(n)}\left(x_{u}^{{}^{\prime}}p_{T}\right)\Bigg{[}1-\frac{f_{u}^{(n)^{\prime}}\left(x_{u}^{{}^{\prime}}p_{T}\right)}{f_{u}^{(n)}\left(x_{u}^{{}^{\prime}}p_{T}\right)}\Delta
x_{u}p_{T}$ $\displaystyle\hskip
70.0pt+\frac{f_{u}^{(n)^{\prime\prime}}\left(x_{u}^{{}^{\prime}}p_{T}\right)}{f_{u}^{(n)}\left(x_{u}^{{}^{\prime}}p_{T}\right)}\left(\Delta
x_{u}p_{T}\right)^{2}+....\Bigg{]}^{2}$
$\displaystyle=f_{u}^{(n)}\left(x_{u}^{{}^{\prime}}p_{T}\right)\left[1-2\frac{f_{u}^{(n)^{\prime}}\left(x_{u}^{{}^{\prime}}p_{T}\right)}{f_{u}^{(n)}\left(x_{u}^{{}^{\prime}}p_{T}\right)}\Delta
x_{u}p_{T}+....\right]$ $\displaystyle\approx
f_{u}^{(n)}\left((x_{u}^{{}^{\prime}}-2\Delta x_{u})p_{T}\right).$ (53)

We see that $R_{\Delta x_{l}}^{(l)}\left(p_{T}\right)$ directly depends on the
property of $p_{T}$ spectrum of up/down quarks.

In Fig. 1(a), we show the normalized $p_{T}$ spectra of up quarks
$f_{u}^{(n)}(p_{T})$ in the low $p_{T}$ range (i.e., $p_{T,u}\lesssim 2$
GeV/c) in three multiplicity classes in $pp$ collisions at $\sqrt{s}=13$ TeV,
which are obtained in previous work (Zhang  _et al._ , 2020) by fitting
experimental data for $p_{T}$ spectra of proton at mid-rapidity using our EVC
model. We see that $f_{u}^{(n)}(p_{T})$ is different in three multiplicity
classes. In Fig. 1(b), we show the property of $R_{\Delta
x_{l}}^{(l)}\left(p_{T}\right)$ in high multiplicity class I with
$\left\langle dN_{ch}/d\eta\right\rangle=25.75$ at mid-rapidity. Here, the V0M
multiplicity classes are defined in Ref.(Acharya  _et al._ , 2020a). The solid
line denotes the result of direct calculation of $R_{\Delta
x_{l}}^{(l)}\left(p_{T}\right)$ by the definition Eq. (36) and the dashed line
is the approximation Eq. (53). We see that the approximation is very good. We
further see that $R_{\Delta x_{l}}^{(l)}\left(p_{T}\right)$ increases with
$p_{T}$ in the range $0<p_{T}\lesssim 3$ GeV/c and decreases with $p_{T}$ at
larger $p_{T}$. This property will cause the non-monotonic $p_{T}$ dependence
of ratio $\Lambda_{c}^{+}/D^{0}$ which is exhibited in experimental data in
high multiplicity class. We emphasize that the property of
$f_{u}^{(n)}(p_{T})$ actually determine the $p_{T}$ dependence of the ratio
$\Lambda_{c}^{+}/D^{0}$ to a large extent. In Fig. 1(c), we show the property
of $R_{\Delta x_{l}}^{(l)}\left(p_{T}\right)$ in low multiplicity class IX
with $\left\langle dN_{ch}/d\eta\right\rangle=4.64$ at mid-rapidity. We see
that $R_{\Delta x_{l}}^{(l)}\left(p_{T}\right)$ only has a small shoulder
structure in the range $p_{T}\lesssim 1$ GeV/c and then decrease with $p_{T}$.
This is the direct consequence of $f_{u}^{(n)}(p_{T})$ in the low multiplicity
class which exhibits less thermal behavior in the low $p_{T}$ range, see the
dot-dashed line in Fig. 1(a). Comparing Fig. 1 (b) and (c), we emphasize that
the multiplicity dependence of light-flavor quark spectrum
$f_{u}^{(n)}(p_{T})$ will lead to the significant multiplicity dependence of
$\Lambda_{c}^{+}/D^{0}$ ratio as a function of $p_{T}$.

Figure 1: The normalized $p_{T}$ spectra of up quarks in three multiplicity
classes in $pp$ collisions at $\sqrt{s}=13$ TeV (a), the calculated $R_{\Delta
x_{l}}^{(l)}\left(p_{T}\right)$ in multiplicity class I with $\left\langle
dN_{ch}/d\eta\right\rangle=25.75$ at mid-rapidity (b) and that in multiplicity
class IX with $\left\langle dN_{ch}/d\eta\right\rangle=$4.64 (c). The solid
lines in (b) and (c) are directly calculated from the definition by Eq. (36)
and the dashed lines are the approximations by Eq. (53).

##  VI The property of $R_{\Delta x_{c}}^{(c)}\left(p_{T}\right)$

Because the difference between momentum fraction of charm quarks in $D^{0}$
and that in $\Lambda_{c}^{+}$

$\Delta
x_{c}=x_{c}^{{}^{\prime}}-x_{c}=\frac{m_{c}m_{u}}{(m_{c}+m_{u})(m_{c}+2m_{u})}\approx
0.12$ (54)

is not quite small, the momentum difference is $\Delta x_{c}p_{T}\lesssim 1$
GeV/c in the studied range $p_{T}\lesssim 10$ GeV/c. Taylor expansion is not
generally effective to simplify $R_{\Delta
x_{c}}^{(c)}\left(p_{T}\right)=f_{c}^{(n)}(x_{c}p_{T})/f_{c}^{(n)}(x^{\prime}_{c}p_{T})$.
Therefore, we take the specific form of $f_{c}^{(n)}(p_{T})$ and directly
evaluate $R_{\Delta x_{c}}^{(c)}\left(p_{T}\right)$.

In practice, we take the following parameterization for $p_{T}$ spectrum of
charm quarks

$f_{c}^{(n)}\left(p_{T}\right)=\mathcal{N}p_{T}^{b}\left(1+\frac{\sqrt{p_{T}^{2}+M^{2}}-M}{nc}\right)^{-n},$
(55)

where $b$, $n$, $M$, $c$ are parameters with positive values and $\mathcal{N}$
is the normalization coefficient. This function can usually well fit the
$p_{T}$ spectrum of charm quarks obtained from theoretical method as well as
these of single-charm hadrons measured by experiments. Substituting Eq. (55)
into (35), we obtain

$\displaystyle R_{\Delta x_{c}}^{(c)}\left(p_{T}\right)$
$\displaystyle=\frac{f_{c}^{(n)}(x_{c}p_{T})}{f_{c}^{(n)}(x_{c}^{{}^{\prime}}p_{T})}$
$\displaystyle=\left(\frac{x_{c}}{x_{c}^{{}^{\prime}}}\right)^{b}\left(\frac{nc-M+\sqrt{\left(x_{c}^{{}^{\prime}}p_{T}\right)^{2}+M^{2}}}{nc-M+\sqrt{\left(x_{c}p_{T}\right)^{2}+M^{2}}}\right)^{n},$
(56)

which is slowly increasing with $p_{T}$ due to the positive exponent $n$ and
$x_{c}^{{}^{\prime}}>x_{c}$.

We apply the online calculator of FONLL theoretical method (Cacciari  _et al._
, 1998, 2001) to calculate $p_{T}$ spectrum of charm quarks in $pp$ collisions
at $\sqrt{s}=$ 13 TeV. The result is shown in Fig. 2 (a), where the shadow
area denotes the scale uncertainties in calculations. The solid line is the
fit to center values of FONLL calculations by Eq. (55) with parameter values
$b=1.3$, $M=6.4$ GeV/c, $n=4.16$, and $c=0.34$ GeV/c. Fig. 2(b) show the
calculated $R_{\Delta x_{c}}^{(c)}\left(p_{T}\right)$. We see that it
monotonically increases with $p_{T}$ and the change magnitude is about two in
the range $p_{T}<10$ GeV/c. We also run PYTHIA 8 (version 8.243) (Sjöstrand
_et al._ , 2015) to obtain the $p_{T}$ spectra of charm quarks in three
multiplicity classes and calculate their $R_{\Delta
x_{c}}^{(c)}\left(p_{T}\right)$. Here, the low, intermediate and high
multiplicity classes are defined as events within multiplicity interval
[1,10], [11,19] and [20,60] with the multiplicity $\left\langle
dN_{ch}/d\eta\right\rangle=$ 4.4, 14.3 and 25.0 within the pseudo-rapidity
interval $\left|\eta\right|<0.5$, respectively. Results of $R_{\Delta
x_{c}}^{(c)}\left(p_{T}\right)$ are shown in Fig. 2(b). We see that $R_{\Delta
x_{c}}^{(c)}\left(p_{T}\right)$ calculated by PYTHIA8 has a certain
multiplicity dependence and its $p_{T}$ dependence is similar with that of
FONLL calculations.

Figure 2: Panel (a): the differential cross-section of charm quarks at mid-
rapidity in $pp$ collisions at $\sqrt{s}=13$ TeV calculated by FONLL method.
The shadow area denotes scale uncertainties. The solid line shows the fit of
central points of theoretical calculations with Eq. (55). Panel (b):
$R_{\Delta x_{c}}^{(c)}\left(p_{T}\right)$ for the result of FONLL and these
of PYTHIA8 in three multiplicity classes.

##  VII The possible property of $R_{corr}^{(cl)}\left(p_{T}\right)$

The term $R_{corr}^{(cl)}\left(p_{T}\right)$ defined in Eq. (37) denotes
effects of momentum correlations between different quark flavors on the
$p_{T}$ dependence of $\Lambda_{c}^{+}/D^{0}$ ratio. It contains two
correlation functions $C_{cud}(x_{c}p_{T},x_{u}p_{T},x_{d}p_{T})$ and
$C_{c\bar{u}}(x_{c}^{{}^{\prime}}p_{T},x_{u}^{{}^{\prime}}p_{T})$. In the case
of vanishing momentum correlations $C_{cud}=0$ and $C_{c\bar{l}}=0$,
$R_{corr}^{(cl)}\left(p_{T}\right)=1$ and contribute trivially to
$\Lambda_{c}^{+}/D^{0}$ ratio.

In high energy $pp$ collisions, charm quarks are dominantly produced by
initial hard parton-parton scattering process and possibly interact with
neighboring light-flavor (anti-)quarks before they hadronize. Part of these
interactions is non-perturbative and is difficult to calculate from first
principles. On the other hand, quark momentum correlations are also difficult
to reversely obtained from experimental data which are mainly of inclusive
distributions of particles at present.

In order to roughly understand the possible property of
$R_{corr}^{(cl)}\left(p_{T}\right)$ and its influence on
$\Lambda_{c}^{+}/D^{0}$ ratio in $pp$ collisions, we apply the event generator
PYTHIA 8 (version 8.243) with default tunes, as an example, to study the
possible property of $R_{corr}^{(cl)}\left(p_{T}\right)$. We use PYTHIA8 to
firstly calculate $f_{c\bar{u}}(p_{T,c},p_{T,\bar{u}},\Delta\phi_{c\bar{u}})$
at $\left|y\right|<0.5$ where $\Delta\phi_{c\bar{u}}$ is the azimuthal angle
difference between $\bar{u}$ and $c$. Then we take the limit $p_{T,\bar{u}}\to
p_{T,c}m_{u}/m_{c}$ and $\Delta\phi_{c\bar{u}}\to 0$ for
$f_{c\bar{u}}(p_{T,c},p_{T,\bar{u}},\Delta\phi_{c\bar{u}})$ to obtain the
$f_{c\bar{u}}\left(x_{c}^{{}^{\prime}}p_{T},x_{u}^{{}^{\prime}}p_{T}\right)$
and corresponding
$\mathcal{C}_{c\bar{u}}\left(x_{c}^{{}^{\prime}}p_{T},x_{u}^{{}^{\prime}}p_{T}\right)$
by Eq. (30).

Because the fraction of the event simultaneously consisting of $u$, $d$, and
$c$ in the mid-rapidity range $\left|y\right|<0.5$ is very small, it is very
hard to directly calculate
$f_{cud}\left(x_{c}p_{T},x_{u}p_{T},x_{d}p_{T}\right)$, in particular, in low
multiplicity events. In the general multi-variable statistics $\left\langle
abc\right\rangle=\left\langle a\right\rangle\left\langle
b\right\rangle\left\langle c\right\rangle+\left\langle a\right\rangle
C_{bc}+\left\langle b\right\rangle C_{ac}+\left\langle c\right\rangle
C_{ab}+C_{abc}$, $C_{ab}$ is standard two-body correlation and $C_{abc}$ is
three-body correlation which is a higher order term compared with two-body
correlations. Here, we neglect the contribution of high-order correlation
$C_{cud}$ and obtain

$\displaystyle\mathcal{C}_{udc}\left(x_{u}p_{T},x_{d}p_{T},x_{c}p_{T}\right)$
(57)
$\displaystyle\approx\mathcal{C}_{ud}\left(x_{u}p_{T},x_{d}p_{T}\right)+\mathcal{C}_{cu}\left(x_{c}p_{T},x_{u}p_{T}\right)+\mathcal{C}_{cd}\left(x_{c}p_{T},x_{d}p_{T}\right),$

and then we obtain

$\displaystyle R_{corr}^{(cl)}\left(p_{T}\right)$
$\displaystyle=\frac{1}{1+\mathcal{C}_{c\bar{u}}\left(x_{c}^{{}^{\prime}}p_{T},x_{u}^{{}^{\prime}}p_{T}\right)}\Big{(}1+\mathcal{C}_{ud}\left(x_{u}p_{T},x_{d}p_{T}\right)$
$\displaystyle\hskip
60.0pt+\mathcal{C}_{cu}\left(x_{c}p_{T},x_{u}p_{T}\right)+\mathcal{C}_{cd}\left(x_{c}p_{T},x_{d}p_{T}\right)\Big{)}$
$\displaystyle\approx
1+\mathcal{C}_{ud}\left(x_{u}p_{T},x_{d}p_{T}\right)+2\mathcal{C}_{cu}\left(x_{c}p_{T},x_{u}p_{T}\right)$
$\displaystyle-\mathcal{C}_{c\bar{u}}\left(x_{c}^{{}^{\prime}}p_{T},x_{u}^{{}^{\prime}}p_{T}\right).$
(58)

In the second line, we have assumed that $\mathcal{C}_{c\bar{u}}$ is small and
$\mathcal{C}_{cu}\approx\mathcal{C}_{cd}$.

In Fig. 3, we show results of two-quark correlation $\mathcal{C}_{q_{1}q_{2}}$
as the function of $p_{T}$ in low multiplicity events with $\left\langle
dN_{ch}/d\eta\right\rangle=4.4$ and those in high multiplicity events with
$\left\langle dN_{ch}/d\eta\right\rangle=25.0$. We see that
$\mathcal{C}_{ud}$, $\mathcal{C}_{cu}$ and $\mathcal{C}_{c\bar{u}}$ are very
small in both low and high multiplicity events. We emphasize that these
statistics only serve as a qualitatively estimation because of the following
reasons. Quarks and antiquarks are only small fractions of final-state parton
system in PYTHIA simulations. Gluons that take large fraction of the parton
system are not involved in current statistics. Non-perturbative interactions
of these quarks and gluons near hadronization are not fully simulated in
PYTHIA8, which may generate some momentum correlations among quarks and
antiquarks of different flavors. In addition, collectivity or collective flow
of partons may be formed in high multiplicity events (Khachatryan  _et al._ ,
2017; Zhao  _et al._ , 2018; Nagle and Zajc, 2018; Bierlich  _et al._ , 2018;
Acharya  _et al._ , 2020b), which may also generate some momentum correlations
among quarks and antiquarks. These possible influence on
$R_{corr}^{(cl)}\left(p_{T}\right)$ are left for future study. In this paper,
we take $R_{corr}^{(cl)}\left(p_{T}\right)\approx 1$ for the moment in the
following studies.

Figure 3: Two quark correlations $\mathcal{C}_{q_{1}q_{2}}$ as the function of
$p_{T}$ in low and high multiplicity events calculated by PYTHIA8.

##  VIII Result of $\Lambda_{c}^{+}/D^{0}$ ratio

In this section, we combine effects of above four terms to calculate the
$\Lambda_{c}^{+}/D^{0}$ ratio and compare with the latest data of ALICE
collaboration (Acharya  _et al._ , 2022b). Here, we neglect the effect of the
momentum correlation $R_{corr}^{(cl)}\left(p_{T}\right)$ between charm quarks
and light-flavor (anti-)quarks by taking
$R_{corr}^{(cl)}\left(p_{T}\right)\approx 1$. This is equivalent to the
independence approximation for the joint momentum distribution
$f_{c\bar{l}}\left(p_{T,c},p_{T,\bar{l}}\right)=f_{c}\left(p_{T,c}\right)f_{\bar{l}}\left(p_{T,\bar{l}}\right)$
and
$f_{cll^{\prime}}\left(p_{T,c},p_{T,l},p_{T,l^{\prime}}\right)=f_{c}\left(p_{T,c}\right)f_{l}\left(p_{T,l}\right)f_{l^{\prime}}\left(p_{T,l^{\prime}}\right)$.
In this case,

$\frac{f_{\Lambda_{c}}(p_{T})}{f_{D_{0}}(p_{T})}\approx
R^{(N_{q_{i}})}\left[\frac{A_{\Lambda_{c}}}{A_{D_{0}}}R_{\Delta
x_{c}}^{(c)}\left(p_{T}\right)R_{\Delta x_{l}}^{(l)}\left(p_{T}\right)\right]$
(59)

where the property of $R^{(N_{q_{i}})}$, $R_{\Delta
x_{c}}^{(c)}\left(p_{T}\right)$ and $R_{\Delta x_{l}}^{(l)}\left(p_{T}\right)$
are already shown in Eqs. (46), (53) and (56). Coefficients $A_{D_{0}}$ and
$A_{\Lambda_{c}}$ can be calculated by Eqs. (9) and (10) with the inclusive
distributions of charm and light-flavor quarks shown in Figs. 1 and 2.

In order to study the multiplicity dependence of the $\Lambda_{c}^{+}/D^{0}$
ratio, we take the light-flavor quark spectrum $f_{l}(p_{T})$ in event class
$\left\langle dN_{ch}/d\eta\right\rangle=25.75$ (class I) and $\left\langle
dN_{ch}/d\eta\right\rangle=4.64$ (class IX) as examples of high multiplicity
events and low multiplicity events, respectively. For charm quark $p_{T}$
spectrum $f_{c}(p_{T})$, we have two choices. One is that of FONLL calculation
and we use it in both high multiplicity events and low multiplicity events.
Another is that of PYTHIA8 calculation where in Sec.VI we have obtained
$f_{c}(p_{T})$ in high multiplicity events $\left\langle
dN_{ch}/d\eta\right\rangle=25.0$ and that in low multiplicity events
$\left\langle dN_{ch}/d\eta\right\rangle=4.4$. The term $R^{(N_{q_{i}})}$
contains two multiplicity-dependent factors $\lambda_{s}$ and $R_{B/M}^{(c)}$.
As shown by Eq. (46), strangeness factor $\lambda_{s}$ causes little changes
on $\Lambda_{c}^{+}/D^{0}$ ratio and we can safely neglect its multiplicity
dependence. Baryon-to-meson production competition factor $R_{B/M}^{(c)}$ is a
relatively-free parameter which cannot be fixed in our model at present. In
Sec. IV, we have discussed its possible multiplicity dependence.

In Fig. 4(a), we show results of $\Lambda_{c}^{+}/D^{0}$ ratio as the function
of $p_{T}$ in high multiplicity events and low multiplicity events at given
$R_{B/M}^{(c)}=0.425$. The purpose of taking the same $R_{B/M}^{(c)}$ is to
let us focus on the influence of $p_{T}$ spectra of light-flavor quarks and
charm quarks on the $p_{T}$ dependence of $\Lambda_{c}^{+}/D^{0}$ ratio. Here,
high multiplicity corresponds to $\left\langle
dN_{ch}/d\eta\right\rangle\approx 25$ since we combine effect of
$f_{l}(p_{T})$ in event class $\left\langle dN_{ch}/d\eta\right\rangle=25.75$
(class I) and that of $f_{c}(p_{T})$ in events with $\left\langle
dN_{ch}/d\eta\right\rangle=25$ in PYTHIA8 calculations. Low multiplicity
corresponds to $\left\langle dN_{ch}/d\eta\right\rangle\approx 4.5$ since we
combine effect of $f_{l}(p_{T})$ in event class $\left\langle
dN_{ch}/d\eta\right\rangle=4.64$ (class IX) and that of $f_{c}(p_{T})$ in
events with $\left\langle dN_{ch}/d\eta\right\rangle=4.4$ in PYTHIA8
calculations. The dot-dashed line with shadow band is the result when
$f_{c}(p_{T})$ calculated by FONLL and $f_{l}(p_{T})$ in high multiplicity
events are used. The dotted line with shadow band is the result when
$f_{c}(p_{T})$ calculated by FONLL and $f_{l}(p_{T})$ in low multiplicity
events are used. Comparing two results, we see an obvious multiplicity
dependence of $\Lambda_{c}^{+}/D^{0}$ ratio caused by that of light-flavor
quark spectrum. As $p_{T}\gtrsim 3$ GeV/c, the $\Lambda_{c}^{+}/D^{0}$ ratio
in high multiplicity events is obviously higher than that in low multiplicity
events. This is because $f_{l}(p_{T})$ in high multiplicity events is
obviously flatter than that in low multiplicity events, see Fig. 1(a). As
$p_{T}\lesssim 3$ GeV/c, the situation is reversed. This is because at the
same $R_{B/M}^{(c)}$ the ratio of $p_{T}$-integrated yield of
$\Lambda_{c}^{+}$ to $D^{0}$ is almost the same, the suppress of
$\Lambda_{c}^{+}$ at $p_{T}\gtrsim 3$ GeV/c in low multiplicity events is
offset by the enhancement of $\Lambda_{c}^{+}$ at low $p_{T}$. We also show
the result of $\Lambda_{c}^{+}/D^{0}$ ratio when the $f_{c}(p_{T})$ calculated
by PYTHIA8 in high multiplicity events is applied. The result, the solid line,
is higher than that with FONLL calculated $f_{c}(p_{T})$ in low $p_{T}$ range
and is slightly smaller than the latter as $p_{T}\gtrsim 3$ GeV/c. In the
range $3\lesssim p_{T}\lesssim 8$ GeV/c, the result of $\Lambda_{c}^{+}/D^{0}$
ratio with $f_{c}(p_{T})$ calculated by PYTHIA8 in low multiplicity events,
the dashed line, is also slightly smaller than that with FONLL calculated
$f_{c}(p_{T})$. Comparing results using FONLL calculated $f_{c}(p_{T})$ with
those using PYTHIA8 calculated $f_{c}(p_{T})$, we see a weak influence of
$f_{c}(p_{T})$ uncertainty on $\Lambda_{c}^{+}/D^{0}$ ratio.

In addition, we see that $\Lambda_{c}^{+}/D^{0}$ ratios in our model in both
high and low multiplicity events always exhibit a non-monotonic $p_{T}$
dependence. This is mainly because of the property of $R_{\Delta
x_{l}}^{(l)}\left(p_{T}\right)$ in Eq. (53), which is determined by the
property of $p_{T}$ spectrum of light-flavor quark as shown in Fig. 1(b) and
(c). The term $R_{\Delta x_{c}}^{(c)}\left(p_{T}\right)$, as shown in Fig.
2(b), strengthen the increase of the $\Lambda_{c}^{+}/D^{0}$ ratio in the low
$p_{T}$ range ($p_{T}\lesssim 2-3$ GeV/c) but offsets the decrease of the
$R_{\Delta x_{l}}^{(l)}\left(p_{T}\right)$ as $p_{T}\gtrsim 3$ GeV/c to a
certain extent and leads to the relatively weak decrease of the
$\Lambda_{c}^{+}/D^{0}$ ratio as $p_{T}\gtrsim 3$ GeV/c.

Figure 4: $\Lambda_{c}^{+}/D^{0}$ ratio as the function of $p_{T}$ in low and
high multiplicity events in $pp$ collisions at $\sqrt{s}=13$ TeV (a) and the
comparison with experimental data (Acharya  _et al._ , 2022b) (b). See text
for the detailed discussions.

In Fig. 4(b), we test above theoretical results by experimental data of
$\Lambda_{c}^{+}/D^{0}$ ratio in $pp$ collisions at $\sqrt{s}=13$ TeV (Acharya
_et al._ , 2022b). Here, we show data of $\Lambda_{c}^{+}/D^{0}$ ratio in V0M
multiplicity class with $\left\langle dN_{ch}/d\eta=31.5\right\rangle$ as an
example of high multiplicity events and data in V0M multiplicity class with
$\left\langle dN_{ch}/d\eta=4.4\right\rangle$ as an example of low
multiplicity events. Because the light-flavor quark $p_{T}$ spectrum used in
our calculation is extracted from experimental data of light-flavor hadrons in
V0M event class I with $\left\langle dN_{ch}/d\eta\right\rangle=25.75$ and
class IX with $\left\langle dN_{ch}/d\eta\right\rangle=4.64$, the underlying
events of our theoretical calculations are not exactly same as those for data
of $\Lambda_{c}^{+}/D^{0}$ ratio but the difference should be small due to the
similar multiplicity. In high multiplicity events, we see that theoretical
result with FONLL calculated $f_{c}(p_{T})$, the dot-dashed line, is
consistent with the first two datum points with $p_{T}<3$ GeV/c and is higher
than data at larger $p_{T}$ to a certain extent. Theoretical result with
PYTHIA8 calculated $f_{c}(p_{T})$ in high multiplicity events, the solid line,
is more close to experimental data in comparison with that with FONLL
calculated $f_{c}(p_{T})$. Here, the baryon-to-meson competition factor
$R_{B/M}^{(c)}$ is taken to be 0.425, the possible saturation value in high
multiplicity events.

According to discussions in Sec. IV, we expect a suppression of
$R_{B/M}^{(c)}$ in low multiplicity events and a rough estimation based on
Poisson distribution of quark numbers gives about 20% suppression. Therefore,
here we use $R_{B/M}^{(c)}=0.34$ to calculate the $\Lambda_{c}^{+}/D^{0}$
ratio in low multiplicity events. We see a good agreement with experimental
data in the available $p_{T}$ range.

In above comparison with experimental data, we have considered the decay
contributions of other single-charm hadrons to $\Lambda_{c}^{+}/D^{0}$ ratio
by the yield term $R^{(N_{q_{i}})}$ Eq. (46). The decay influence on the shape
of $\Lambda_{c}^{+}/D^{0}$ ratio (as the function of $p_{T}$) is numerically
studied and is found to be quite small and therefore the comparison with
experimental data in Fig. 4(b) is little changed.

In Fig. 5, we show $p_{T}$ spectra of $D^{0}$, $D_{s}^{+}$ and
$\Lambda_{c}^{+}$ in inelastic $pp$ collisions at $\sqrt{s}=13$ TeV and
compare them with experimental data (Acharya  _et al._ , 2022b). Here, $p_{T}$
spectra of light-flavor quarks in inelastic events have been obtained in Ref.
(Zhang  _et al._ , 2020) and $p_{T}$ spectrum of charm quarks is taken from
FONLL calculations shown in Fig. 2 (a) with a $p_{T}$-integrated yield density
$dN_{c}/dy=0.025$. The model parameter $R_{B/M}^{(c)}$ is set to be 0.38,
which falls in between the value in high multiplicity events and that in low
multiplicity events. The decay contributions from other single-charm hadrons
in ground state are systematically considered. We see that experimental data
of $D^{0}$, $D_{s}^{+}$ and $\Lambda_{c}^{+}$ are self-consistently explained
in our model.

Figure 5: $p_{T}$ spectra of $D^{0}$, $D_{s}^{+}$ (a) and $\Lambda_{c}^{+}$
(b) in inelastic $pp$ collisions at $\sqrt{s}=13$ TeV. Lines are model results
and symbols are experimental data (Acharya  _et al._ , 2022b).

##  IX Summary

We have applied an equal-velocity quark combination model to study the
$\Lambda_{c}^{+}/D^{0}$ ratio as the function of $p_{T}$ in $pp$ collisions at
$\sqrt{s}=13$ TeV. Taking advantage of the analytic feature of the model, we
decomposed the $\Lambda_{c}^{+}/D^{0}$ ratio into four parts and studied their
individual influence on the ratio. Finally, we combined effects of these parts
to calculate $\Lambda_{c}^{+}/D^{0}$ ratio as the function of $p_{T}$ and
compared theoretical results with experimental data of $\Lambda_{c}^{+}/D^{0}$
ratio in high and low multiplicity events in $pp$ collisions at $\sqrt{s}=13$
TeV.

The first part of $\Lambda_{c}^{+}/D^{0}$ decomposition is the term of light-
flavor quark numbers. We summarized this part into a compact form containing a
strangeness factor and a baryon-to-meson competition factor defined in our
model. The multiplicity dependence of strangeness has very weak influence on
the $\Lambda_{c}^{+}/D^{0}$ ratio. The global effect of light-flavor quark
numbers is manifested by the baryon-to-meson competition factor. In low
multiplicity events where quark numbers are relatively small, charm quark has
relatively small chance to interact with two light-flavor quarks and thus the
production of charm baryon should be suppressed to a certain extent. We
adopted a Poisson distribution as an example to roughly estimate this
suppression and found about 20% suppression of the $\Lambda_{c}^{+}/D^{0}$
ratio in low multiplicity events with mid-rapidity $\left\langle
dN_{ch}/d\eta\right\rangle\approx 4.5$ in comparison with that in high
multiplicity events with $\left\langle dN_{ch}/d\eta\right\rangle\approx 25$.

The second part of $\Lambda_{c}^{+}/D^{0}$ decomposition is the term
containing the normalized $p_{T}$ spectrum of up/down quarks. Considering the
small difference between momentum fraction ($x_{u}$) of up/down quark in
$\Lambda_{c}^{+}$ and that ($x_{u}^{{}^{\prime}}$) in $D^{0}$, we adopted the
Taylor expansion method and reduced this part to the inclusive distribution of
up/down quarks $f_{u}^{(n)}\left((2x_{u}-x_{u}^{{}^{\prime}})p_{T}\right)$
with a good numerical accuracy. This suggests that the shape of up quark
$p_{T}$ spectrum directly transmit to the $p_{T}$ dependence of the
$\Lambda_{c}^{+}/D^{0}$ ratio. In particular, $p_{T}$ spectrum of up/down
quarks in high multiplicity events behaves a thermal like distribution in the
low $p_{T}$ range, which will lead to an obviously non-monotonic $p_{T}$
dependence of $\Lambda_{c}^{+}/D^{0}$ ratio in the low and intermediate
$p_{T}$ range.

The third part of $\Lambda_{c}^{+}/D^{0}$ decomposition is the term containing
the normalized $p_{T}$ spectrum of charm quarks. We adopted FONLL method and
PYTHIA8 to calculate charm quark $p_{T}$ spectrum, respectively. This part
increases with $p_{T}$ and therefore weaken the effect of the second part. We
found that results of PYTHIA8 are roughly close to that of FONLL calculations
but have a weak multiplicity dependence. The fourth part of
$\Lambda_{c}^{+}/D^{0}$ decomposition is the term containing momentum
correlation between up/down quarks and charm quarks. PYTHIA8 simulation on
two-quark correlations show a negligible momentum correlation between up/down
quarks and charm quarks. Therefore, we neglect the effect of this part in
calculation of $\Lambda_{c}^{+}/D^{0}$ ratio in this paper.

Finally, we combined these parts and calculate $\Lambda_{c}^{+}/D^{0}$ ratio
as the function of $p_{T}$ in $pp$ collisions at $\sqrt{s}=13$ TeV. In
calculations, we chosen the $p_{T}$ spectrum of up/down quarks in multiplicity
class IX with $\left\langle dN_{ch}/d\eta\right\rangle=4.64$ as an example of
low multiplicity events and that in multiplicity class I with $\left\langle
dN_{ch}/d\eta\right\rangle=25.75$ as an example of high multiplicity events.
$p_{T}$ spectrum of charm quarks from PYTHIA8 calculation with $\left\langle
dN_{ch}/d\eta\right\rangle=4.4$ and that with $\left\langle
dN_{ch}/d\eta\right\rangle=25$ are taken in order to be consistent with the
selected $p_{T}$ spectrum of light-flavor quarks at similar $\left\langle
dN_{ch}/d\eta\right\rangle$. Finally, we obtained $\Lambda_{c}^{+}/D^{0}$
ratio as the function of $p_{T}$ in high multiplicity events with
$\left\langle dN_{ch}/d\eta\right\rangle\approx 25$ and that in low
multiplicity events with $\left\langle dN_{ch}/d\eta\right\rangle\approx 4.5$.
We compared our theoretical results with experimental data of
$\Lambda_{c}^{+}/D^{0}$ ratio in multiplicity class with $\left\langle
dN_{ch}/d\eta\right\rangle=31.5$ and those with $\left\langle
dN_{ch}/d\eta\right\rangle=4.4$. In high multiplicity events with a baryon-to-
meson competition factor $R_{B/M}^{(c)}=0.425$, we found that theoretical
result with PYTHIA8 calculated charm $p_{T}$ spectrum is in better agreement
with experiment data than that with FONLL calculated charm $p_{T}$ spectrum.
In low multiplicity events with a suppressed $R_{B/M}^{(c)}=0.34$, theoretical
result with FONLL calculated charm $p_{T}$ spectrum and that with PYTHIA8
calculated charm $p_{T}$ spectrum are both in good agreement with experimental
data.

$R_{B/M}^{(c)}$ is a dynamical parameter of our model which denotes the
competition between the formation of charm baryon and that of charm meson at a
charm quark hadronization and can not be predicted by the model of present
version. Based on the present work and our previous studies Song  _et al._
(2018); Li  _et al._ (2018, 2021); Song  _et al._ (2022), we found that
$R_{B/M}^{(c)}$ is about $0.385-0.425$ in inelastic and relatively-high
multiplicity events $pp$ and $p$Pb collisions and is suppressed in small
multiplicity events but it may also increase to a certain extent in AA
collisions Wang  _et al._ (2020). This possibly multiplicity-dependent
property of $R_{B/M}^{(c)}$ is an interesting property of charm quark
hadronization, which is not seen in the hadronization of light-flavor quarks
where we found an relatively stable baryon-to-meson production competition
Shao  _et al._ (2017). Its explanation is beyond the model of current version
and should introduce further dynamic considerations. For example, by
considering the possible production of some $cll^{\prime}$ resonances which is
predicted by quark model but is not observed yet in experiments He and Rapp
(2020); Chen and He (2021), the hadronization channel of charm quark to charm
baryons should increase which may prompt the possible increase of
$R_{B/M}^{(c)}$ in high multiplicity events where those resonances will have
more practical chance to produce. Also, by considering the competition of
quark combination mechanism and quark fragmentation mechanism and, in
particular, the multiplicity dependence of such competition Plumari  _et al._
(2018); Minissale  _et al._ (2021), the effective $R_{B/M}^{(c)}$ can also be
changed in our model. These possible dynamics at charm quark hadronization are
deserved to study in the future to improve our model.

##  X Acknowledgments

This work is supported by Shandong Provincial Natural Science Foundation
(Grants No. ZR2019YQ06 and No. ZR2019MA053), and Higher Educational Youth
Innovation Science and Technology Program of Shandong Province (Grant No.
2019KJJ010).

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