#  Hadronization effects on the baryon-strangeness correlation in quark
combination models

Feng-lan Shao  School of Physics and Engineering, Qufu Normal University,
Shandong 273165, China  Jun Song  Department of Physics, Jining University,
Shandong 273155, China  Rui-qin Wang  School of Physics and Engineering, Qufu
Normal University, Shandong 273165, China

###### Abstract

The baryon-strangeness correlation in the hadronization of the quark matter is
studied within the quark combination mechanism. We calculate the correlation
coefficient $C_{BS}=-3\big{(}\langle BS\rangle-\langle B\rangle\langle
S\rangle\big{)}/\big{(}\langle S^{2}\rangle-\langle S\rangle^{2}\big{)}$ of
initial hadrons produced from the deconfined free quark system with
$C^{(q)}_{BS}=1$. The competition of the production of baryons against that of
mesons is the key dynamics that is most relevant to the change of baryon-
strangeness correlation during system hadronization. Results of quark
combination under the Poisson statistics agree with the statistical model
predictions for a hadron resonance gas at vanishing chemical potential but
differ from those at relatively large chemical potentials. Results beyond
Poisson statistics are also obtained and are compared with calculations of
Lattice QCD in the phase boundary, giving the best agreement at temperature
$T=163$ MeV. We predict the dependence of the $C_{BS}$ of hadron system on the
baryon chemical potential and strangeness. These predictions are expected to
be tested by the future Lattice QCD calculations at non-zero chemical
potentials and/or by the Beam Energy Scan experiment of STAR Collaboration at
RHIC.

###### pacs:

25.75.Gz, 25.75.Nq

##  I introduction

Baryon-strangeness correlation is an effective diagnostic tool for the
relevant degrees of freedom of the hot nuclear matter produced in relativistic
heavy ion collisions koch05cbs . It has been extensively investigated by
various phenomenological models of high energy collisions cbsByURQMD ;
BMuller06bs ; qMD08cbs ; mayg08cbs ; Mayg10cbs ; PQM12cbs  and first-principle
calculations in Lattice QCD lqcd06cbs ; lqcdCBS . The correlation is usually
quantified by koch05cbs

$C_{BS}\equiv-3\frac{\langle BS\rangle-\langle B\rangle\langle
S\rangle}{\langle S^{2}\rangle-\langle S\rangle^{2}}=-3\frac{\langle
BS\rangle}{\langle S^{2}\rangle}.$ (1)

Here, angle brackets denote the event/ensemble average. In the second step,
strangeness neutrality $\langle S\rangle=0$ is applied to relativistic heavy
ion collisions. For a deconfined system consisting of free quarks and
antiquarks, $C^{(q)}_{BS}=1$ because strangeness is carried only by the
strange (anti-)quarks which carry the baryon number in strict proportion to
their strangeness with the coefficient -1/3, i.e., $B_{s}=-\frac{1}{3}S_{s}$.
In contrast, the relation between baryon number and strangeness in a hadron
system is multiple, e.g., baryon number 1 for a strange baryon but 0 for a
strange meson. $C_{BS}$ of hadron system is usually smaller than one (at low
baryon number density) due to the fact that most strange quarks, at
hadronization, will come into mesons instead of baryons which unlock the
intimate baryon-strangeness correlation existed previously in quarks.
Statistical model estimation of $C_{BS}$ for a hadron resonance gas at zero
baryon number density is about 0.66 koch05cbs . Calculations of Lattice QCD
also show that the $C_{BS}$ of the strong-interacting system at temperature
above the phase transition temperature $T_{c}$ tends to one while near and
below $T_{c}$ the system $C_{BS}$ decreases rapidly lqcd06cbs ; lqcdCBS .

Hadronization refers to the process of the formation of hadrons out of quarks
and/or gluons, accompanying with the change of the correlation between baryon
number and strangeness. Phenomenological models describing the hadronization
should reproduce this change of baryon-strangeness correlation due to the
transformation of the degrees of freedom in the system. Quark combination is
one of the effective mechanisms for the hadronization of the hot quark matter
produced in relativistic heavy ion collisions and has explained lots of
experimental phenomena in heavy ion collisions at SPS, RHIC and recent LHC,
see, e.g. RefsFries22003prl ; Greco2003prl ; hwa04 ; sdqcm ; Shuryak10qcm ;
wqr12 . The change of baryon-strangeness correlation during quark combination
hadronization is intuitive. A strange quark(antiquark) combines with a light
antiquark(quark) to form a strange meson, which completely destroys the
original baryon-strange correlation carried by strange (anti)quark, i.e.,
$B_{(s\bar{q})}=0\times S_{(s\bar{q})}$, but inherits the strangeness
lossless. Occasionally, a strange quark combines with a strange antiquark to
form a hidden-strange meson, which inherits nothing from original quarks. On
the contrary, in the baryon formation a strange quark combines with two light
quarks to form a baryon, which alters the baryon-strangeness correlation
coefficient $B_{(sqq)}=-S_{(sqq)}$. The combination of two strange quarks with
one light quark also alters the correlation coefficient
$B_{(ssq)}=-\frac{1}{2}S_{(ssq)}$. Obviously, the $C_{BS}$ of the system
depends on the relative proportion of the produced baryons to mesons.

In this paper, we study the change of system $C_{BS}$ caused by hadronization
in quark combination mechanism (QCM). We discuss in details how the dynamics
of baryon-meson production competition at hadronization to dominate the
$C_{BS}$ of hadron system. In addition, we study the dependence of $C_{BS}$ on
the strangeness content and the baryon number density of the system, which is
related to the hot quark system produced in relativistic heavy ion collisions
at different collisional energies. We compare our results with the
calculations of Lattice QCD lqcd06cbs ; lqcdCBS  and the prediction of
statistical model for a hadron resonance gas koch05cbs .

The paper is organized as follows. In Sec. II, we present a working model in
QCM for the yields of hadrons. In Sec. III, using the model we explain the
experimental data of the mid-rapidity yields of strange hadrons that are most
relevant to $C_{BS}$ calculations in central Pb+Pb collisions at
$\sqrt{s_{NN}}=2.76$ TeV. In Sec. IV, we calculate the $C_{BS}$ of the hadron
system in which two assumptions are made for initially produced hadrons. One
assumption is that different kinds of hadrons are uncorrelated, which can be
expected in the case that the quark system existed previously is made up of
free quarks and antiquarks ($C^{(q)}_{BS}=1$). In this paper, we only consider
this kind of quark system since the effects of hadronization on system
$C_{BS}$ are already addressed properly. The other assumption is Poisson
statistics for yield fluctuations of hadrons. In Sec. V, we present a
different approach of calculating $C_{BS}$ which is beyond Poisson statistics
and can also reflect intuitively the essence of baryon-strangeness
correlation. Summary and discussion are finally given in Sec. VI.

##  II  A working model of hadron yields

In this section, we present a working model of the yields of hadrons that are
produced from the deconfined quark phase after hadronization in the framework
of quark combination mechanism, for the facility of the subsequent correlation
studies. As discussed in introduction, baryon-strangeness correlation
$C^{(h)}_{BS}$ is strongly dependent on the relative production of baryons to
mesons, which requires our model to well address this point. Therefore, we
introduce the working model according to the following strategy: (1) give the
global properties for the global production of all mesons, all baryons and all
antibaryons, in which the production competition of baryons against mesons is
properly addressed. (2) on the basis of (1), give the yield formulas of
various identified hadrons.

In previous work sjbbar13 , we have studied the properties of the global
production of all mesons, baryons and antibaryons, and obtained their yield
formulas utilizing only the basic ideas of QCM. We start from a deconfined
system consisting of $N_{q}$ constituent quarks and $N_{\bar{q}}$ antiquarks.
Here, gluons contribution to the system on the threshold of hadronization is
replaced by pairs of quark and antiquark. After hadronization, the system
changes the basic degrees of the freedom to become the hadronic system and
produces _in average_ $B(N_{q},N_{\bar{q}})$ baryons,
$\bar{B}(N_{q},N_{\bar{q}})$ antibaryons and $M(N_{q},N_{\bar{q}})$ mesons.
Four properties of hadron production from general principles are used to
constrain the behavior of their yield formulas.

(1) charge conjugation symmetry of the hadron yields,

$\begin{split}B(N_{q},N_{\bar{q}})=\bar{B}(N_{\bar{q}},N_{q}),\\\
M(N_{q},N_{\bar{q}})=M(N_{\bar{q}},N_{q}).\end{split}$ (2)

(2) unitary of the hadronization, i.e. production of mesons, baryons and
antibaryons should exhaust all quarks and antiquarks of the system existed
previously

$\begin{split}M(N_{q},N_{\bar{q}})+3B(N_{q},N_{\bar{q}})=N_{q},\\\
M(N_{q},N_{\bar{q}})+3\bar{B}(N_{q},N_{\bar{q}})=N_{\bar{q}}.\end{split}$ (3)

(3) boundary condition, i.e.,

$\begin{cases}\bar{B}=0,\ \ B=\frac{N_{q}}{3},M=0&\text{if }N_{\bar{q}}=0\\\
\bar{B}=\frac{N_{\bar{q}}}{3},B=0,\ \ M=0&\text{if }N_{q}=0\end{cases}$ (4)

(4) linear response of meson and baryon yields to quark antiquark numbers

$\begin{split}M(\lambda N_{q},\lambda N_{\bar{q}})=\lambda
M(N_{q},N_{\bar{q}}),\\\ B(\lambda N_{q},\lambda N_{\bar{q}})=\lambda
B(N_{q},N_{\bar{q}}),\\\ \bar{B}(\lambda N_{q},\lambda
N_{\bar{q}})=\lambda\bar{B}(N_{q},N_{\bar{q}}).\end{split}$ (5)

Using these properties, we obtained in Ref.sjbbar13  the yield formulas of
baryons, antibaryons and mesons,

$\displaystyle M(x,z)$ $\displaystyle=$
$\displaystyle\frac{x}{2}\\{1-z\frac{(1+z)^{a}+(1-z)^{a}}{(1+z)^{a}-(1-z)^{a}}\\},$
$\displaystyle B(x,z)$ $\displaystyle=$
$\displaystyle\frac{x\,z}{3}\frac{(1+z)^{a}}{(1+z)^{a}-(1-z)^{a}},$ (6)
$\displaystyle\bar{B}(x,z)$ $\displaystyle=$
$\displaystyle\frac{x\,z}{3}\frac{(1-z)^{a}}{(1+z)^{a}-(1-z)^{a}}.$

Here, we have rewritten $x=N_{q}+N_{\bar{q}}$ which characterizes the bulk
property of the system related to the system size or energy and rewritten
$z=(N_{q}-N_{\bar{q}})/x$ which depicts the asymmetry between quarks and
antiquarks in the system ($|z|\leq 1$) and is a measurement of the baryon
number density of the system.

The production competition of baryons against mesons is often quantified by
the yield ratios $R_{B/M}(z)=B(x,z)/M(x,z)$ and
$R_{\bar{B}/M}(z)=\bar{B}(x,z)/M(x,z)$, which are the function of only $z$.
The factor $a$ in Eq. (6) represents the degree of the baryon-meson
competition by the relation $a=\frac{1}{3R_{B/M}(0)}+1$. Studies in Ref. wqr12
show the $R_{B/M}(0)$ of value about $1/12$, i.e., $a\approx 5$, can well
describe yield ratios of various baryons to mesons in heavy ion collisions at
LHC energy. We note that such a competition as well as yield formulas Eq. (6)
can be properly addressed by a phenomenological combination rule in the quark
combination model developed by Shandong group (SDQCM) sdqcm . In addition,
using vested $R_{B/M}(0)$, we have successfully explained the yield ratios of
various anti-hadrons to hadrons at non-zero $z$ region in relativistic heavy
ion collisions sjbbar13 ; wqr12 ; wqr14 , i.e., the data of these yield ratios
at different collision energies and at different rapidities. We emphasis that
through the dependence of $C^{(h)}_{BS}$ on the factor $a$ or $R_{B/M}(0)$ we
can address the effects of hadronization on the baryon-strangeness correlation
of the system.

To study $C^{(h)}_{BS}$, we have to obtain the yield formulas of identified
hadron. Given the total yield of baryon, that of antibaryon and that of meson,
the inclusive/averaged yields of identified hadrons $M_{i}(q_{1}\bar{q}_{2})$,
$B_{j}(q_{1}q_{2}q_{3})$ are calculated by their individual production
weights,

$\displaystyle N_{M_{i}}$ $\displaystyle=$ $\displaystyle
P_{M_{i}}M(x,z)=C_{M_{i}}P_{q_{1}\bar{q}_{2},M}M(x,z),$ (7) $\displaystyle
N_{B_{j}}$ $\displaystyle=$ $\displaystyle
P_{B_{j}}B(x,z)=C_{B_{j}}P_{q_{1}q_{2}q_{3},B}B(x,z).$ (8)

Here, $P_{q_{1}\bar{q}_{2},M}$ denotes the probability that, as a meson is
known to be produced, the flavor content of this meson is $q_{1}\bar{q}_{2}$.
$C_{M_{i}}$ further denotes the branch ratio of this meson with given flavor
composition $q_{1}\bar{q}_{2}$ to a specific meson state $M_{i}$.
$C_{M_{i}}P_{q_{1}\bar{q}_{2},M}$ thus gives the emerging probability of a
specific meson, $P_{M_{i}}$, when a meson is known to be formed. Similarly,
$P_{q_{1}q_{2}q_{3},B}$ denotes the probability that, as a baryon is known to
produced, the flavor content of this baryon is $q_{1}q_{2}q_{3}$, and
$C_{B_{j}}$ denotes the branch ratio of this baryon with given flavor
composition $q_{1}q_{2}q_{3}$ to a specific baryon state $B_{j}$, and
$C_{B_{j}}P_{q_{1}q_{2}q_{3},B}$ thus gives the emerging probability of a
specific baryon, $P_{B_{j}}$, when a baryon is known to be formed. A similar
formula holds for antibaryons.

The probability $P_{q_{1}\bar{q}_{2},M}$ is posteriorly evaluated by the
proportion of $q_{1}\bar{q}_{2}$ pairs in all quark-antiquark pairs in the
system, $P_{q_{1}\bar{q}_{2},M}=N_{q_{1}\bar{q}_{2}}/N_{q\bar{q}}$, by
considering the fact that every quark/antiquark has both the probability of
entering into a meson and the probability of entering a baryon/antibaryon.
Here, $N_{q}=\sum_{i}N_{q_{i}}$ and $N_{\bar{q}}=\sum_{i}N_{\bar{q}_{i}}$ are
total number of quarks and antiquarks in the system, respectively. We have
$N_{q_{1}\bar{q_{2}}}=N_{q_{1}}N_{\bar{q_{2}}}$ and
$N_{q\bar{q}}=N_{q}N_{\bar{q}}$. Similarly, the probability
$P_{q_{1}q_{2}q_{3},B}$ is evaluated by the proportion of $q_{1}q_{2}q_{3}$
combinations to all three-quark combinations in the system,
$P_{q_{1}q_{2}q_{3},B}=N_{iter}N_{q_{1}q_{2}q_{3}}/N_{qqq}$.
$N_{q_{1}q_{2}q_{3}}$ is the number of $q_{1}q_{2}q_{3}$ combination which
satisfies $N_{q_{1}q_{2}q_{3}}=N_{q_{1}}N_{q_{2}}N_{q_{3}}$ for $q_{1}\neq
q_{2}\neq q_{3}$, $N_{q_{1}q_{2}q_{3}}=N_{q_{1}}(N_{q_{1}}-1)N_{q_{3}}$ for
$q_{1}=q_{2}\neq q_{3}$ and
$N_{q_{1}q_{2}q_{3}}=N_{q_{1}}(N_{q_{1}}-1)(N_{q_{1}}-2)$ for
$q_{1}=q_{2}=q_{3}$. $N_{qqq}=N_{q}(N_{q}-1)(N_{q}-2)$ is the number of all
$qqq$ combination in the system. In large quark number limit we can apply
$N_{qqq}\approx N_{q}^{3}$ and $N_{q_{1}q_{2}q_{3}}\approx
N_{q_{1}}N_{q_{2}}N_{q_{3}}$ always. $N_{iter}$ stands for the number of
possible iterations of $q_{1}q_{2}q_{3}$, which is 1, 3, and 6 for three
identical flavor, two different flavor, and three different flavor cases,
respectively.

$C_{M_{j}}$ and/or $C_{B_{j}}$ denote the branch ratio of a given flavor
composition (e.g., $u\bar{s}$) to a specific hadron state (e.g., $K^{+}$)
under the condition that they are known form a hadron. In the case that when
the ground state $J^{P}=0^{-}$ and $1^{-}$ mesons and $J^{P}=\frac{1}{2}^{+}$
and $\frac{3}{2}^{+}$ baryons are considered only, we have, for mesons,

$C_{M_{j}}=\left\\{\begin{array}[]{ll}{1}/{(1+R_{V/P})}~{}~{}~{}~{}~{}~{}~{}~{}\textrm{for
}J^{P}=0^{-}\textrm{ mesons},\\\
{R_{V/P}}/{(1+R_{V/P})}~{}~{}~{}~{}\textrm{for }J^{P}=1^{-}\textrm{
mesons},\end{array}\right.$

where $R_{V/P}$ represents the ratio of the $J^{P}=1^{-}$ vector mesons to the
$J^{P}=0^{-}$ pseudoscalar mesons of the same flavor composition; and for
baryons,

$C_{B_{j}}=\left\\{\begin{array}[]{ll}{R_{O/D}}/{(1+R_{O/D})}~{}~{}~{}~{}\textrm{for
}J^{P}=({1}/{2})^{+}\textrm{ baryons},\\\
{1}/{(1+R_{O/D})}~{}~{}~{}~{}~{}~{}~{}~{}\textrm{for
}J^{P}=({3}/{2})^{+}\textrm{ baryons},\end{array}\right.$

except that $C_{\Lambda}=C_{\Sigma^{0}}={2R_{O/D}}/{(1+2R_{O/D})}$,
$C_{\Sigma^{*0}}={1}/{(1+2R_{O/D})}$,
$C_{\Delta^{++}}=C_{\Delta^{-}}=C_{\Omega^{-}}=1$ . Here, $R_{O/D}$ stands for
the ratio of the $J^{P}=(1/2)^{+}$ octet to the $J^{P}=(3/2)^{+}$ decuplet
baryons of the same flavor composition. The two parameters $R_{V/P}$ and
$R_{O/D}$ can be determined using the data from different high energy
reactions sdqcm ; wq95 , and they are taken to be 3 and $2$ in this paper,
respectively.

##  III yields of strange hadrons at LHC

With above working model, we can conveniently predict yields of various
identified hadrons. As an illustration, we now explain the experimental data
of mid-rapidity yields of strange hadrons $K$, $\Lambda$, $\Xi$ and
$\Omega^{-}$ in Pb+Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV. These hadrons
are most relevant to $C_{BS}$ calculation.

As applying yield formulas in Sec. II to the finite rapidity window in heavy
ion collisions, we note that Eqs. (2) and (4-5) still hold generally and Eq.
(3) also approximately holds with good numerical accuracy due to the locality
of the hadronization, and therefore no conceptual issues exist for our mid-
rapidity predictions. In addition, our formulas in Sec. II give the averaged
yields of hadrons at the given quark system with the fixed quark numbers while
the experimental data of hadronic yields are event-averaged quantities. The
produced quark system in heavy ion collisions at a given collision energy is
varied in size event-by-event and thus the number of quarks and that of
antiquarks should follow a certain distribution around the averaged quark
number $\langle N_{q_{i}}\rangle$ and antiquark numbers $\langle
N_{\bar{q}_{i}}\rangle$ (where $i=u,d,s$ considered in this paper). Our final
predictions of hadron yields should be the average over this distribution. In
general such averages depend on the precise form of the distribution. Here, we
approximate these averages by taking the corresponding values of the
quantities at the event averages $\langle N_{q_{i}}\rangle$ and $\langle
N_{\bar{q}_{i}}\rangle$, i.e.,

$\displaystyle\langle N_{M_{i}}\rangle$ $\displaystyle=$ $\displaystyle
C_{M_{i}}\overline{P}_{q_{1}\bar{q}_{2},M}M(\langle x\rangle,\langle
z\rangle),$ (9) $\displaystyle\langle N_{B_{j}}\rangle$ $\displaystyle=$
$\displaystyle C_{B_{j}}\overline{P}_{q_{1}q_{2}q_{3},B}B(\langle
x\rangle,\langle z\rangle),$ (10)

where $\langle x\rangle=\langle N_{q}\rangle+\langle N_{\bar{q}}\rangle$,
$\langle x\rangle\langle z\rangle=\langle N_{q}\rangle-\langle
N_{\bar{q}}\rangle$ and $\overline{P}$ is also calculated with the event
averaged quark numbers. Such approximation can be expected in considering that
properties Eqs. (2 \- 5) in our derivation of hadron yields are also
apparently satisfied even for the event-averaged quantities.

For convenience, we use factors $\lambda_{s}=\langle
N_{\bar{s}}\rangle/\langle N_{\bar{u}}\rangle=\langle
N_{\bar{s}}\rangle/\langle N_{\bar{d}}\rangle$ and associated
$\lambda^{\prime}_{s}=\langle N_{s}\rangle/\langle N_{u}\rangle=\langle
N_{s}\rangle/\langle N_{d}\rangle$ to denote the suppression of strange anti-
quarks relative to light antiquarks and that of strange quarks to light
quarks, respectively, and simply the yield formulas of hadrons. Here, isospin
symmetry between (anti-)up quark and (anti-)down quark is applied.

Considering the decay contribution from the short-lived resonances, we obtain
the yields of final state hadrons

$\langle N_{h_{j}}^{(f)}\rangle=\langle N_{h_{j}}\rangle+\sum_{k}Br(h_{k}\to
h_{j})\ \langle N_{h_{k}}\rangle,$ (11)

where we use the superscript $(f)$ to denote the results for the final hadrons
to differentiate them from those for the directly produced hadrons. The data
of the decay branch ratios are taken from the PDG PDG .

With $\lambda_{s}=\lambda^{\prime}_{s}$ at LHC ($\langle z\rangle=0$
approximation), we obtain yields of these strange hadrons in final state

$\displaystyle\langle N^{(f)}_{K^{+}}\rangle$ $\displaystyle=$
$\displaystyle\langle N_{K^{+}}\rangle+\langle N_{K^{*+}}\rangle Br(K^{*+}\to
K^{+})+\langle N_{K^{*0}}\rangle Br(K^{*0}\to K^{+})$ (12) $\displaystyle=$
$\displaystyle\Big{(}1+0.493\frac{R_{V/P}}{1+R_{V/P}}\lambda_{s}\Big{)}\frac{\lambda_{s}}{(2+\lambda_{s})^{2}}M(\langle
x\rangle,0),$ $\displaystyle\langle N^{(f)}_{\Lambda}\rangle$ $\displaystyle=$
$\displaystyle\langle N_{\Lambda}\rangle+\langle N_{\Sigma^{0}}\rangle
Br(\Sigma^{0}\to\Lambda)+\langle N_{\Sigma^{*+}}\rangle
Br(\Sigma^{*+}\to\Lambda)$ (13) $\displaystyle+\ \langle
N_{\Sigma^{*0}}\rangle Br(\Sigma^{*0}\to\Lambda)+\langle
N_{\Sigma^{*-}}\rangle Br(\Sigma^{*-}\to\Lambda)$ $\displaystyle=$
$\displaystyle\frac{7.736\lambda_{s}}{(2+\lambda_{s})^{3}}B(\langle
x\rangle,0),$ $\displaystyle N^{(f)}_{\Xi^{-}}$ $\displaystyle=$
$\displaystyle\langle N_{\Xi^{-}}\rangle+\langle N_{\Xi^{*-}}\rangle
Br(\Xi^{*-}\to\Xi^{-})+\langle N_{\Xi^{*0}}\rangle Br(\Xi^{*0}\to\Xi^{-})$
(14) $\displaystyle=$ $\displaystyle
3\frac{\lambda_{s}^{2}}{(2+\lambda_{s})^{3}}B(\langle x\rangle,0),$
$\displaystyle\langle N^{(f)}_{\Omega^{-}}\rangle$ $\displaystyle=$
$\displaystyle\langle
N_{\Omega^{-}}\rangle=\frac{\lambda_{s}^{3}}{(2+\lambda_{s})^{3}}B(\langle
x\rangle,0).$ (15)

Here, we have only taken into account the strong and electric-magnetic (S&EM)
decays of short-lived resonances.

Absolute yields of these hadrons are dependent on the total quark number
$\langle x\rangle$ of the system by $M(\langle x\rangle,0)=2\langle
x\rangle/5$ and $B(\langle x\rangle,0)=\langle x\rangle/30$ with fixed
$R_{B/M}(0)=1/12$. To eliminate this $\langle x\rangle$ dependence, we
consider the relative production of these hadrons to pions, i.e., the yield
ratios of these hadrons to pions, which finally rely only on the strangeness
of the system, beside of the baryon-meson competition in hadronization. Since
the decay contributions to pion are complex, we here give directly the
numerical result of the yield of $\pi^{+}$, i.e., $\langle
N^{(f)}_{\pi^{+}}\rangle=0.213\langle x\rangle$ under S&EM decays, instead of
the detailed compositions like Eqs. (12-15). In Fig. 1, we use Eqs. (12-15) to
explain the experimental data of the mid-rapidity yield ratios
$K^{+}/\pi^{+}$, $\Lambda/\pi^{+}$, $\Xi^{-}/\pi^{+}$ and $\Omega^{-}/\pi^{+}$
in central Pb+Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV aliceKpion13 ;
aliceKLam13 ; aliceXiOmg13 . To incorporate the change of $\lambda_{s}$ at
different collision centralities, a varied strangeness
$\lambda_{s}(N_{part})=(0.43\pm 0.02)/(1+10.5N_{part}^{-1.3})$ is used. We can
see that the hierarchy properties in yields of these strange hadrons and their
$N_{part}$ dependence can be systematically described by our formulas Eqs.
(12-15). Here, we would like to emphasis that the yield difference between $K$
and those hyperons is mainly due to the baryon-meson competition at
hadronization while the hierarchy structures among $\Lambda$, $\Xi^{-}$ and
$\Omega^{-}$ are mainly strangeness relevant.

Figure 1: (Color online) Yield ratios $K^{+}/\pi^{+}$, $\Lambda/\pi^{+}$,
$\Xi^{-}/\pi^{+}$ and $\Omega^{-}/\pi^{+}$ at mid-rapidity as the function of
nuclear participants $N_{part}$ in central Pb+Pb collisions at
$\sqrt{s_{NN}}=2.76$ TeV. Symbols are experimental data from Refs.aliceKpion13
; aliceKLam13 ; aliceXiOmg13 . Shadow regions are our results with a
$N_{part}$-dependent strangeness $\lambda_{s}(N_{part})=(0.43\pm
0.02)/(1+10.5N_{part}^{-1.3})$.

Applying our yield formulas to other hadrons at LHC and those at RHIC energies
with nonzero baryon number densities, we also find a good agreement with
available experimental data. This is not surprise. In fact, QCM has already
shown its effectiveness in explaining the data of hadronic yields and
longitudinal rapidity distributions in relativistic heavy ion collisions at
different collisional energies biro1999 ; alcor2000 ; alcorC2000 ; sdqcm ;
sj09ijmpa ; wqr12 ; lsSLX12 . The related low-$p_{T}$ issues of QCM such as
entropy conservation and pion production have been properly addressed biro1999
; ckm03 ; nonaka05 ; hwa04 ; biro2007entropy ; fries08review ; SJentrop10 .
There are also many successful applications of QCM on correlation studies,
e.g., multi-hadron yield correlations wqr12 ; sjbbar13 ; wqr14 ; wqr15charm ,
baryon-meson correlated emission fries:2004hd ; fries:2007ix  as well as the
charge balance function Bialas04 ; SJ12FB .

##  IV $C_{BS}$ of hadrons under Poisson fluctuations

For initial hadrons produced by the hadronization of deconfined quark system,
the baryon number of the system is $B=\sum_{\alpha}Q_{\alpha,B}N_{\alpha}$ and
the strangeness $S=\sum_{\alpha}Q_{\alpha,S}N_{\alpha}$, where the species
$\alpha$ has baryon number $Q_{\alpha,B}$ and strangeness $Q_{\alpha,S}$. By
definition Eq. (1), the $C_{BS}$ of hadrons is

$C_{BS}^{(h)}=-3\frac{\sum_{\alpha,\beta}Q_{\alpha,B}Q_{\beta,S}C_{\alpha\beta}}{\sum_{\alpha,\beta}Q_{\alpha,S}Q_{\beta,S}C_{\alpha\beta}},$
(16)

where the covariance $C_{\alpha\beta}=\langle
N_{\alpha}N_{\beta}\rangle-\langle N_{\alpha}\rangle\langle N_{\beta}\rangle$
describes the correlation between hadron $\alpha$ and hadron $\beta$. A
general calculation of $C_{\alpha\beta}$ in QCM is still unavailable in the
current progress of hadronization phenomenology since there still has lots of
unsolved dynamics in hadronization due to its non-perturbative feature. On all
“on market” combination models, there are a few ones that can give the
calculation of $C_{\alpha\beta}$ with their own specific model
details/assumptions. Such specific calculations are not the purpose of this
paper since we intend to analyze the hadronization effects in a general and
transparent way. Here, we consider a simple case that after hadronization
different kinds of produced hadrons are uncorrelated, i.e.
$C_{\alpha\beta}=\delta_{\alpha\beta}\sigma^{2}_{\alpha}$. This can be
expected if the quark system existed previously is made up of free quarks and
antiquarks (i.e., $C^{(q)}_{BS}=1$). We note that, above $T_{c}$, the $C_{BS}$
of strong-interacting system calculated by Lattice QCD lqcdCBS  indeed tends
to one and the off-diagonal flavor susceptibilities also tend to be relatively
small. We only consider this kind of quark system in this paper because the
effects of hadronization on system $C_{BS}$ are already addressed clearly .

In this section, we first calculate the $C_{BS}$ of the initial hadron system,
assuming the Poisson statistics $\sigma^{2}_{\alpha}\approx\langle
N_{\alpha}\rangle$ for the yield distribution of identified hadrons. Now we
have

$C^{(h)}_{BS}=-3\frac{\langle BS\rangle}{\langle
S^{2}\rangle}\approx-3\frac{\sum_{\alpha}Q_{\alpha,B}Q_{\alpha,S}\langle
N_{\alpha}\rangle}{\sum_{\alpha}Q^{2}_{\alpha,S}\langle N_{\alpha}\rangle}.$
(17)

With Eqs. (9) and (10), we are ready to calculate $C_{BS}$ of hadrons via Eq.
(17),

$\displaystyle\langle BS\rangle$ $\displaystyle=$ $\displaystyle-\\{\langle
N_{\Lambda}\rangle+\langle N_{\Sigma^{\pm,0}}\rangle+\langle
N_{\Sigma^{*\pm,0}}\rangle\\}-2\\{\langle N_{\Xi^{0,-}}\rangle+\langle
N_{\Xi^{*0,-}}\rangle\\}$ $\displaystyle-3\langle N_{\Omega^{-}}\rangle-anti-
hyperons$ $\displaystyle=$ $\displaystyle{-}$
$\displaystyle\frac{12\lambda^{\prime}_{s}+12\lambda^{\prime
2}_{s}+3\lambda^{\prime 3}_{s}}{(2+\lambda^{\prime}_{s})^{3}}B(\langle
x\rangle,\langle
z\rangle)-\frac{12\lambda_{s}+12\lambda^{2}_{s}+3\lambda^{3}_{s}}{(2+\lambda_{s})^{3}}$
(18) $\displaystyle{\times}$ $\displaystyle\bar{B}(\langle x\rangle,\langle
z\rangle),$

and

$\displaystyle\langle S^{2}\rangle$ $\displaystyle=$ $\displaystyle\\{\langle
N_{\Lambda}\rangle+\langle N_{\Sigma^{\pm,0}}\rangle+\langle
N_{\Sigma^{*\pm,0}}\rangle\\}+4\\{\langle N_{\Xi^{0,-}}\rangle+\langle
N_{\Xi^{*0,-}}\rangle\\}$ (19) $\displaystyle+9\langle
N_{\Omega^{-}}\rangle+anti-hyperons$ $\displaystyle+\\{\langle
K^{\pm}\rangle+\langle K^{*\pm}\rangle+\langle K^{0}\rangle+\langle
K^{*0}\rangle+\langle\bar{K^{0}}\rangle+\langle\bar{K}^{*0}\rangle\\}$
$\displaystyle=$ $\displaystyle\frac{12\lambda^{\prime}_{s}+24\lambda^{\prime
2}_{s}+9\lambda^{\prime 3}_{s}}{(2+\lambda^{\prime}_{s})^{3}}B(\langle
x\rangle,\langle
z\rangle)+\frac{12\lambda_{s}+24\lambda^{2}_{s}+9\lambda^{3}_{s}}{(2+\lambda_{s})^{3}}$
$\displaystyle\times\bar{B}(\langle x\rangle,\langle
z\rangle)+\frac{2\lambda_{s}+2\lambda^{\prime}_{s}}{(2+\lambda_{s})(2+\lambda^{\prime}_{s})}M(\langle
x\rangle,\langle z\rangle).$

We note that above correlations are independent of the parameters $R_{O/D}$
and $R_{V/P}$ and thus they are unaffected by S&EM decays. Substituting them
into Eq. (17), we obtain

$C^{(h)}_{BS}=3\frac{\frac{3\lambda^{\prime}_{s}}{2+\lambda^{\prime}_{s}}R_{B/M}(\langle
z\rangle)+\frac{3\lambda_{s}}{2+\lambda_{s}}R_{\bar{B}/M}(\langle
z\rangle)}{\frac{3\lambda^{\prime}_{s}(3\lambda^{\prime}_{s}+2)}{(2+\lambda^{\prime}_{s})^{2}}R_{B/M}(\langle
z\rangle)+\frac{3\lambda_{s}(3\lambda_{s}+2)}{(2+\lambda_{s})^{2}}R_{\bar{B}/M}(\langle
z\rangle)+\frac{2\lambda^{\prime}_{s}+2\lambda_{s}}{(2+\lambda^{\prime}_{s})(2+\lambda_{s})}},$
(20)

which gives the dependence of $C^{(h)}_{BS}$ on baryon-meson competition
factor $R_{B/M}(0)$, strangeness $\lambda_{s}$ and the baryon number density
of the system.

We firstly consider the situation of zero baryon number density $\langle
z\rangle=0$ to study the dependence of $C^{(h)}_{BS}$ on $R_{B/M}(0)$ and
$\lambda_{s}$. With $\lambda^{\prime}_{s}=\lambda_{s}$ and
$R_{B/M}(0)=R_{\bar{B}/M}(0)$, we get

$C^{(h)}_{BS}=3\frac{(2+\lambda_{s})R_{B/M}(0)}{(3\lambda_{s}+2)R_{B/M}(0)+2/3}.$
(21)

Fig. 2 (a) shows the dependence of $C^{(h)}_{BS}$ on the strangeness
$\lambda_{s}$, as the $R_{B/M}(0)$ is taken to 1/12. One can see that
$C^{(h)}_{BS}$ is insensitive to the change of the strangeness, i.e.
$C^{(h)}_{BS}$ only increases about 5% as $\lambda_{s}$ increases from 0.3
(the rough value in $pp$ reactions) to 0.7 (almost maximum value occurred in
heavy ion collisions). In addition, we see that $C^{(h)}_{BS}$ of initial
hadron system is obviously small than one (the value of ideal quark system)
because the produced strange mesons significantly outnumber the strange
baryons in the current baryon-meson competition $R_{B/M}(0)=1/12$. Fig. 2 (b)
shows the dependence of $C^{(h)}_{BS}$ on baryon-meson competition factor
$R_{B/M}(0)$, as the strangeness $\lambda_{s}$ is taken to the saturated value
0.43 in relativistic heavy ion collisions. Here, the saturated $\lambda_{s}$
is extracted from the fit of yields of strange hadrons in Sec. III and in our
previous work wqr12 , and we note that this value is consistent with
calculations of Lattice QCD at $T_{c}$ lqcd06cbs ; lsLQCD06 . Clearly, we see
that the increase of $R_{B/M}(0)$ will enhance the yields of baryons against
mesons and the $C^{(h)}_{BS}$ increases rapidly. With preferred values
$R_{B/M}(0)=1/12$ and $\lambda_{s}=0.43$, $C^{(h)}_{BS}$ is about 0.65, much
smaller than $C^{(q)}_{BS}=1$. To reach the unit correlation for hadrons, an
extremely high baryon-meson competition factor $R_{B/M}(0)\approx 1/5$ is
needed, which is completely unable to reproduce yields of strange mesons and
baryons.

Figure 2: (Color online) (a) The dependence of $C^{(h)}_{BS}$ on strangeness
suppression factor $\lambda_{s}$; (b) on baryon-meson competition factor
$R_{B/M}(0)$.

Subsequently, we study the dependence of $C^{(h)}_{BS}$ on the baryon number
density of the system. In previous discussions, we use the quark-antiquark
asymmetry $\langle z\rangle$ to characterize the baryon number density of the
system. To compare our results with existed predictions of statistical models,
we alternatively use the chemical potential $\mu_{B}$ which relates $\langle
z\rangle$ via

$\langle
z\rangle=\frac{2\sinh(\frac{\mu_{B}}{3T_{c}})}{2\cosh(\frac{\mu_{B}}{3T_{c}})+\lambda_{s}\exp(-\frac{\mu_{B}}{3T_{c}})},$
(22)

under the assumption of Boltzmann distribution for thermalized quarks and
antiquarks. Here, strangeness neutrality $N_{s}=N_{\bar{s}}$ is applied.
$T_{c}$ is the temperature of the quark system at hadronization.

Fig. 3 shows our predictions of the $C^{(h)}_{BS}$ of initial hadrons as the
function of $\mu_{B}$. Here, we have taken into account the fact that both
$T_{c}$ and $\lambda_{s}$ are varied with the $\mu_{B}$ for the hot quark
matter produced in relativistic heavy ion collision. For the $\mu_{B}$
dependence of $T_{c}$, we apply the calculation of Lattice QCD by G. Endrődi
_et al_ tclqcd , i.e., $T_{c}(\mu_{B})=T_{c0}(1-0.0089\mu_{B}^{2}/T_{c0}^{2})$
with $T_{c0}$ being 0.169 GeV, the transition temperature at $\mu_{B}=0$. For
the $\mu_{B}$ dependence of $\lambda_{s}$, we use our previous extractions at
mid-rapidity in heavy ion collisions at different collision energies sdqcm ;
lsSLX12 , i.e., $\lambda_{s}=$ (0.43, 0.43, 0.44, 0.44, 0.48, 0.5, 0.57, 0.8,
0.7) as $\sqrt{s_{NN}}=$ (2760, 200, 130, 62.4, 17.3, 12.3, 8.7, 7.6, 6.3)
GeV, and convert the energy dependence into $\mu_{B}$ dependence by the
formula $\sqrt{s_{NN}}=(1.308\text{GeV}/\mu_{B}-1)/0.273$ GeV in
Refs.statMub11 ; statMub06 . The solid line with filled square symbols are our
result. We also calculate $C^{(h)}_{BS}$ at fixed $\lambda_{s}=0.43$ (dotted
line with down-triangles), at fixed $T_{c}=0.169$ GeV (dashed line with up-
triangles), and at both fixed $T_{c}=0.169$ GeV and fixed $\lambda_{s}=0.43$
(dashed line with open circles), respectively.

Figure 3: (Color online) The dependence of $C_{BS}$ on the chemical potential
$\mu_{B}$ of the system. The lines with filled squares, down-triangles, up-
triangles and open circles are our results at both varied $\lambda_{s}$ and
$T_{c}$, at constant $\lambda_{s}=0.43$, at constant $T_{c}=0.169$ GeV, and at
both constant $\lambda_{s}=0.43$ and $T_{c}=0.169$ GeV, respectively. The
solid line with open cross symbols is our results using the chemical freeze-
out temperature. They are compared with the prediction of statistical model
for hadron resonance gas koch05cbs , the dashed line with stars.

A striking behavior of $C^{(h)}_{BS}$ of initial hadrons is that it increases
with the increasing $\mu_{B}$ and in the large $\mu_{B}$ region
($\mu_{B}\gtrsim 0.3$ GeV) it surpasses the unity and become higher at larger
$\mu_{B}$. This is in sharp contrary to the free quark system existed
previously where the correlation coefficient remains strictly unity at all
temperatures and chemical potentials. This is because that, as the $\mu_{B}$
increases, the relative production of baryons to mesons, i.e., $R_{B/M}(z)$ in
our approach, becomes large, and then weight of this item in $C_{BS}$ formula
Eq. (20) increases and thus $C_{BS}$ increases correspondingly. Comparing the
result of $C_{BS}$ at fixed $T_{c}$ with those with varied $T_{c}$, we see
that the change(decrease) of $T_{c}$ at large chemical potential increases the
$C_{BS}$ of the system about several percentages. Comparing the result of
$C_{BS}$ at fixed $\lambda_{s}$ with those with varied $\lambda_{s}$, we find
that the change of the strangeness also slightly influences the $C_{BS}$ of
the system.

We also compare our results with the early prediction of V. Koch _et al_
koch05cbs  (the dotted line with star symbols) for a hadron resonance gas in
the statistical treatment. A $\mu_{B}$ dependent chemical freeze-out
temperature, $T_{chemical}(\mu_{B})\approx
0.17-0.13\mu^{2}_{B}-0.05\mu^{4}_{B}$ in Ref.shm03 , is used in their
prediction, which falls obviously below the transition temperature we used
above at intermediate and large $\mu_{B}$. For comparison, we also calculate
the $C_{BS}$ of initial hadrons using their chemical freeze-out temperature
and the result is presented in Fig. 3 as the solid line with open cross
symbols. We find that our result is close to the prediction of statistical
model at small $\mu_{B}$ but at intermediate and large $\mu_{B}$ region our
result is about 15-20% larger than that of statistical model. We note that
recently F. Becattini _et al_ becattini13tc  have reconstructed the original
hadro-chemical equilibrium temperature after considering the effects of final
hadron expansion phase and the results have closely followed the phase
transition boundary line predicted by Lattice QCD tclqcd .

##  V $C_{BS}$ of hadrons beyond Poisson statistics

In above calculations of $C^{(h)}_{BS}$, Poisson statistics for hadronic
yields is applied as an open approximation. Compared the obtained
$C^{(h)}_{BS}$ at zero $\mu_{B}$ with calculations of Lattice QCD in the phase
boundary lqcdCBS , we find our result conform to Lattice calculations at
$T=150\pm 3$ MeV. This value is lower than the usually expected deconfined
temperature (around 165 MeV) lqcdCBS , and is also lower than the chemical
freeze-out temperature extracted from hadronic yields in statistical models
which is about 166 MeV statMub06 ; statMub11 . Such an underestimation may be
relevant to the applied Poisson statistics approximation since this
application implies that a further sufficient evolution of hadron phase
dominated by inelastic scatterings is probably needed. Therefore, in this
section, we calculate the $C^{(h)}_{BS}$ of initial hadron system by another
approach which is independent of the Poisson statistics assumption and also
reflects intuitively the essence of the baryon-strangeness correlation.

To probe the intrinsic correlation between the baryon number and strangeness
for a hadron system, we suppose that the system has a change by stochastically
emitting a small amount of strange hadrons and then the system strangeness
changes $\delta S^{(h)}$ and the baryon number also changes $\delta B^{(h)}$.
Regarding the baryon number of the system $B^{(h)}$ as the function of the
strangeness $S^{(h)}$, the change of baryon number due to the perturbation of
strangeness can be calculated by

$\delta B^{(h)}=\Big{(}\frac{\partial B^{(h)}_{s}}{\partial
S^{(h)}}\Big{)}\,\delta S^{(h)}+\mathcal{O}\big{(}(\delta
S^{(h)})^{2}\big{)},$ (23)

where $B^{(h)}_{s}$ is the baryon number carried by strange hadrons. To second
order fluctuation of strangeness we get

$C^{(h)}_{BS}=-3\frac{\langle\delta B^{(h)}\delta
S^{(h)}\rangle}{\langle{\delta S^{(h)}}^{2}\rangle}\approx-3\
\left.\frac{\partial B^{(h)}_{s}}{\partial
S^{(h)}}\right|_{\begin{subarray}{c}\scriptscriptstyle\langle
N_{h}\rangle\end{subarray}}.$ (24)

The partial derivative is evaluated at the event average values of hadron
yields $\langle N_{h}\rangle$. Note that here we don’t consider the
correlations between light and strange hadrons induced by the interactions
among hadrons, i.e., we consider an ideal hadron gas system after
hadronization.

The same philosophy can be applied to the quark system before hadronization.
Considering that the quark system is made up of free quarks and antiquarks
with three flavors, the strangeness of the system
$S^{(q)}=-(N_{s}-N_{\bar{s}})$ and the baryon number carried by these strange
quarks and antiquarks $B_{s}^{(q)}=\frac{1}{3}(N_{s}-N_{\bar{s}})$ and we get
$C^{(q)}_{BS}=1$ exactly. The difference of $C_{BS}$ between quarks and
initial hadrons can be illustrated by the different slopes of two phases in
$B$-$S$ plane, as shown in schematic Fig. 4. The cross point between two
phases stands for the hadronization that changes the basic degrees of freedom
of the system and also stands for the global charge conservations during
hadronization.

Figure 4: (Color online) Schematic picture for the transition of ideal quark
phase to hadron phase by hadronization in $B-S$ plane.

With yield formulas Eqs. (7) and (8), we have

$\displaystyle
B^{(h)}_{s}=N_{\Lambda}+N_{\Sigma^{\pm,0}}+N_{\Sigma^{*\pm,0}}+N_{\Xi^{0,-}}+N_{\Xi^{*0,-}}+N_{\Omega^{-}}$
$\displaystyle\hskip 10.0pt-\ anti-hyperons$ $\displaystyle=$
$\displaystyle\frac{6N_{uds}+3N_{uus}+3N_{dds}+3N_{uss}+3N_{dss}+N_{sss}}{N_{qqq}}B(x,z)$
$\displaystyle-$
$\displaystyle\frac{6N_{\bar{u}\bar{d}\bar{s}}+3N_{\bar{u}\bar{u}\bar{s}}+3N_{\bar{d}\bar{d}\bar{s}}+3N_{\bar{u}\bar{s}\bar{s}}+3N_{\bar{d}\bar{s}\bar{s}}+N_{\bar{s}\bar{s}\bar{s}}}{N_{\bar{q}\bar{q}\bar{q}}}\bar{B}(x,z),$

and the strangeness of initial hadrons
$S^{(h)}=\sum_{\alpha}Q_{\alpha,S}N_{\alpha}=-(N_{s}-N_{\bar{s}})$ because all
strange quarks and antiquarks are combined into hadrons after hadronization.
Then, we have

$\begin{split}C^{(h)}_{BS}=3\times\Bigg{\\{}&\frac{24\ B(1,\langle
z\rangle)}{(2+\lambda^{\prime}_{s})^{3}(1+\langle z\rangle)}+\frac{24\
\bar{B}(1,\langle z\rangle)}{(2+\lambda_{s})^{3}(1-\langle z\rangle)}\\\
&+\frac{12\lambda^{\prime}_{s}+6\lambda^{\prime 2}_{s}+6\lambda^{\prime
3}_{s}}{(2+\lambda^{\prime}_{s})^{3}}(1+\langle z\rangle)^{a-1}\\\
&\times\frac{\big{[}(1-\langle z\rangle)^{a-1}(\langle z\rangle^{2}-2a\langle
z\rangle-1)+(1+\langle z\rangle)^{a+1}\big{]}}{3\big{[}(1+\langle
z\rangle)^{a}-(1-\langle z\rangle)^{a}\big{]}^{2}}\\\
&+\frac{12\lambda_{s}+6\lambda^{2}_{s}+6\lambda^{3}_{s}}{(2+\lambda_{s})^{3}}(1-\langle
z\rangle)^{a-1}\\\ &\times\frac{\big{[}(1+\langle z\rangle)^{a-1}(\langle
z\rangle^{2}+2a\langle z\rangle-1)+(1-\langle
z\rangle)^{a+1}\big{]}}{3\big{[}(1+\langle z\rangle)^{a}-(1-\langle
z\rangle)^{a}\big{]}^{2}}\Bigg{\\}}.\end{split}$ (26)

for initial hadron system. This result shows a complex dependence on baryon-
meson competition factor $R_{B/M}(0)$ by the factor
$a=\frac{1}{3R_{B/M}(0)}+1$, strangeness $\lambda_{s}$ and the baryon number
density of the system by $z$.

In the case of zero baryon number density $z=0$, we have
$\lambda^{\prime}_{s}=\lambda_{s}$ and $B(1,0)=\bar{B}(1,0)=1/6a$ and simplify
Eq. (26)

$C^{(h)}_{BS}=1+8\frac{6R_{B/M}(0)-1}{(2+\lambda_{s})^{3}(1+3R_{B/M}(0))}.$
(27)

Fig. 5 (a) shows the dependence of $C^{(h)}_{BS}$ on the strangeness
$\lambda_{s}$, as the $R_{B/M}(0)$ is taken to 1/12. We see that
$C^{(h)}_{BS}$ of initial hadrons in the current approach is also insensitive
to the strangeness of the system, but the result is about 20% larger than the
previous prediction in Fig. 2 (a) in Poisson fluctuation. Fig. 5 (b) shows the
dependence of $C_{BS}$ on the baryon-meson competition factor $R_{B/M}(0)$. We
see that $C^{(h)}_{BS}$ is strongly dependent on the $R_{B/M}(0)$, which is
also qualitatively consistent with the result in Fig. 2 (b).

Figure 5: (Color online) The dependence of $C_{BS}$ on strangeness suppression
factor $\lambda_{s}$, (a), and on baryon-meson competition factor
$R_{B/M}(0)$, (b). Predictions of Lattice QCD are from Ref.lqcdCBS .

The qualitative consistency between Fig. 5 and Fig. 2 can be naturally
understood via the definition $C_{BS}=-3\langle BS\rangle/\langle
S^{2}\rangle$. Because both the nominator $\langle BS\rangle$ and denominator
$\langle S^{2}\rangle$ are dependent on the strangeness mainly by the single-
strange hadrons, the strangeness dependence are thus partially offset in their
ratio and we observe a weak dependence of $C^{(h)}_{BS}$ on $\lambda_{s}$. On
the other hand, since the nominator $\langle BS\rangle$ receives the larger
contributions from multi-strange baryons than the denominator $\langle
S^{2}\rangle$ which is always dominated by single-strange mesons, we would
observe a slightly increase of $C^{(h)}_{BS}$ with the increasing
$\lambda_{s}$. In addition, the increase of $R_{B/M}(0)$ enhances the
production weights of (strange) baryons against mesons and contributes largely
to the nominator $\langle BS\rangle$ and we observe a rapid increase of
$C^{(h)}_{BS}$ with the increasing $R_{B/M}(0)$.

Since results in current approach hold more generally than those in Sec. IV
under Poisson statistics, we can compare our result (solid line in Fig. 5 (b))
with the calculation of Lattice QCD lqcdCBS  (box area) in the phase
transition boundary. With the critical temperature $T_{c}=0.16-0.17$ GeV,
Lattice QCD lqcdCBS  predicts the $C_{BS}$ of the system is about $0.74-0.85$,
which has a striking overlap with our results as $R_{B/M}(0)$ is within
$[1/9.5,1/13.5]$. In the preferred situation of $R_{B/M}(0)=1/12$ and
$\lambda_{s}=0.43$, our calculated $C_{BS}$ is 0.78, which best conforms to
the Lattice QCD calculations $C_{BS}=0.77-0.80$ at $T_{c}=0.163$ GeV within
its theoretical uncertainty lqcdCBS . Such a agreement with Lattice QCD
calculations suggests that quark combination mechanism reflect some basic
dynamics of the hadronization process of the strong-interacting quark system
and thus it is indispensable for the hadronization phenomenology.

Figure 6: (Color online) The dependence of $C_{BS}$ on the chemical potential
$\mu_{B}$ of the system. The filled squares, up-triangles, open squares and
open circles are our results at both varied $\lambda_{s}$ and $T_{c}$, at
constant $\lambda_{s}=0.43$, at constant $T_{c}=0.166$ GeV, and at both
constant $\lambda_{s}=0.43$ and $T_{c}=0.166$ GeV, respectively. They are
compared with the prediction of statistic method for a hadron resonance gas
koch05cbs .

We also predict the dependence of $C_{BS}$ on the baryon number density of the
system using the chemical potential $\mu_{B}$. The results are shown in Fig.
6. Here, the transformation of variable $z$ to $\mu_{B}$ is by Eq. (22) and
the $\mu_{B}$ dependence of $T_{c}$ and $\lambda_{s}$ are same as those in
Sec. IV. The results of statistical model from Ref.koch05cbs  at different
$\mu_{B}$ are also presented as the comparison. The solid line with filled
squares are our results at varied $\mu_{B}$ and $T_{c}$. It is shown that they
are about $15\%$ higher than the predictions of statistical model, i.e.,
dashed line with stars, in the full $\mu_{B}$ region. Comparing our results at
varied $\mu_{B}$ and $T_{c}$ with those at fixed $\mu_{B}$ and/or $T_{c}$, we
find that decrease of $T_{c}$ at large chemical potential $\mu_{B}$ will
slightly increase the $C_{BS}$ of the system and that the change of system
strangeness at different $\mu_{B}$ slightly influences the $C_{BS}$ of the
system. These effects of $\mu_{B}$ and $\lambda_{s}$ on the $C_{BS}$ of the
system are similar to those calculated in Sec. IV. In addition, we see that
results of small $\mu_{B}$ region calculated here are higher than those in
Fig. 3 in Sec. IV but the results at large $\mu_{B}$ are close to the previous
ones. We also note that another difference between the current calculations of
$C_{BS}$ and the previous ones in Fig. 3 is the curvature of $C_{BS}$ as the
function of $\mu_{B}$. The $C_{BS}$ in Fig. 3 is the concave function at
intermediate and large $\mu_{B}$ while the current $C_{BS}$ in Fig. 6 is
almost linear function at intermediate and large $\mu_{B}$. By noticing that
the predictions of statistical model in this region is a slightly convex
function, these different predictions may be identified in future by
calculating the higher-order susceptibilities for the conservative charges of
the system. The other difference between QCM prediction and statistical model
is the position that the $C_{BS}$ of hadrons reaches one. QCM predicts the
position at $\mu_{B}\approx 0.3$ GeV corresponding to the AA collisions at
collisional energy about 12 GeV while statistical model predicts the position
at $\mu_{B}\approx 0.4$ GeV corresponding to the collisional energy about 9
GeV. The Beam Energy Scan programme of STAR collaboration can distinguish
these two different predictions.

##  VI Summary and discussion

Hadronization describes the process of the formation of hadrons out of quarks
and/or gluons. In the meanwhile, the correlation properties between
conservative charges of the system change also due to the transformation of
basic degrees of freedom of the system. Phenomenological models mechanisms of
the hadronization should reproduce these changes of charge correlation
properties during hadronization process for the system. In this paper, we have
studied the change of the baryon-strangeness correlation caused by
hadronization in the quark combination mechanism. We found that this
correlation is a good quantity of studying the hadronization because of its
sensitivity on the dynamics of production competition between baryons and
mesons. We calculated the correlation coefficient $C_{BS}=-3(\langle
BS\rangle-\langle B\rangle\langle S\rangle)/\langle S^{2}\rangle$ of initial
hadrons produced from the deconfined free quark system with $C^{(q)}_{BS}=1$.
The calculated $C^{(h)}_{BS}$ of initial hadrons under Poisson fluctuation is
about 0.65 at zero baryon chemical potential, which is consistent with the
prediction of statistical model for the hadron resonance gas. Beyond Poisson
statistics, we calculated $C^{(h)}_{BS}$ of initial hadrons by
$C^{(h)}_{BS}=-3(\partial B^{(h)}_{s}/\partial S^{(h)})_{L}$. The resulting
$C^{(h)}_{BS}$ at zero baryon chemical potential is about 0.78, comparable to
the calculations of Lattice QCD in the phase boundary at $T_{c}=163$ MeV. This
suggests that the quark combination, as a phenomenological microscopic
mechanism, is able to describe the change of conservative charge correlations
at hadronization, revealing certain basic dynamics of the realistic
hadronization process. We also predicted the correlation coefficient of
hadrons at different baryon chemical potentials and compared them with the
existed calculations of statistical method. These predictions are expected to
be tested by the Beam Energy Scan experiment of STAR Collaboration at RHIC
and/or by the future Lattice QCD calculations at non-zero chemical potentials.

Finally we would like to comment the measurement of $C_{BS}$ directly by the
definition Eq. (1) either in realistic experiments or in model simulations
using an event generator, which already has lots of discussions in
literatures. To measure the $C_{BS}$ in general we usually have to select a
finite accept window, e.g. mid-rapidity region $|y|<0.5$. Selecting a small
window size will increase the Poisson statistical fluctuations while selecting
a wide window size will inevitably involve the global conservation effect of
baryon number and in particular of the strangeness. Amending such a finite
window effect is usually complex. These considerations just prompt us to
calculate the $C_{BS}$ of the system in Sec. V by the response of baryon
number of the system with respect to the change of the system strangeness.
This method is less relevant to the accept window size and thus maybe more
closely reflect the intrinsic baryon-strangeness correlation of the hadron
system.

## Acknowledgments

The work is supported in part by the National Natural Science Foundation of
China under grant 11175104 and 11305076, and by the Natural Science Foundation
of Shandong Province, China under grant ZR2012AM001.

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Existence, uniqueness, localization and minimization property of positive
solutions for non-local problems

involving discontinuous Kirchhoff functions

BIAGIO RICCERI

Abstract: Let $\Omega\subset{\bf R}^{n}$ be a smooth bounded domain. In this
paper, we prove a result of which the following is a by-product: Let
$q\in]0,1[$, $\alpha\in L^{\infty}(\Omega)$, with $\alpha>0$, and $k\in{\bf
N}$. Then, the problem

$\cases{-\tan\left(\int_{\Omega}|\nabla u(x)|^{2}dx\right)\Delta
u=\alpha(x)u^{q}&in $\Omega$\cr&\cr u>0&in $\Omega$\cr&\cr u=0&on
$\partial\Omega$\cr&\cr(k-1)\pi<\int_{\Omega}|\nabla
u(x)|^{2}dx<(k-1)\pi+{{\pi}\over{2}}\cr}$

has a unique weak solution $\tilde{u}$ which is the unique global minimum in
$H^{1}_{0}(\Omega)$ of the functional

$u\to{{1}\over{2}}\tan\left(\int_{\Omega}|\nabla\tilde{u}(x)|^{2}dx\right)\int_{\Omega}|\nabla
u(x)|^{2}dx-{{1}\over{q+1}}\int_{\Omega}\alpha(x)|u^{+}(x)|^{q+1}dx\ ,$

where $u^{+}=\max\\{0,u\\}$.

Keywords: Discontinuous Kirchhoff function; existence; uniqueness;
localization; minimization property

2020 MSC: 35J15, 35J25, 35J61, 49J35

1\. Introduction

First, we stress that we have chosen the above long title just to summarize
the main features and novelties of our results.

Throughout the sequel, $\Omega\subset{\bf R}^{n}$ is a smooth bounded domain
and $K$ is a real-valued function defined in $[0,+\infty[$.

Given a function $\varphi:\Omega\times{\bf R}\to{\bf R}$, we are interested in
the problem

$\cases{-K\left(\int_{\Omega}|\nabla u(x)|^{2}dx\right)\Delta
u=\varphi(x,u)&in $\Omega$\cr&\cr u>0&in $\Omega$\cr&\cr u=0&on
$\partial\Omega$\ .\cr}$

We define weak solution of this problem any $u\in H^{1}_{0}(\Omega)$, with
$u>0$ in $\Omega$, such that, for every $v\in H^{1}_{0}(\Omega)$, the function
$\varphi(\cdot,u(\cdot))v(\cdot)$ lies in $L^{1}(\Omega)$ and one has

$K\left(\int_{\Omega}|\nabla u(x)|^{2}dx\right)\int_{\Omega}\nabla u(x)\nabla
v(x)dx-\int_{\Omega}\varphi(x,u(x))v(x)dx=0\ .$

This is a Kirchhoff-type problem, $K$ being the Kirchhoff function.
Unquestionably, it is among the most studied nonlinear problems of the last
two dacades. For a lucid introduction to the subject jointly with the relevant
bibliography, we refer to the recent [5].

Here is a most remarkable corollary of our main result.

THEOREM 1.1. - Assume that there exists an open interval
$I\subseteq]0,+\infty[$ such that the restriction of $K$ to $I$ is increasing
and $K(I)=]0,+\infty[$.

Then, for each $q\in]0,1[$ and for each $\alpha\in L^{\infty}(\Omega)$, with
$\alpha>0$, the problem

$\cases{-K\left(\int_{\Omega}|\nabla u(x)|^{2}dx\right)\Delta
u=\alpha(x)u^{q}&in $\Omega$\cr&\cr u>0&in $\Omega$\cr&\cr u=0&on
$\partial\Omega$\cr&\cr\int_{\Omega}|\nabla u(x)|^{2}dx\in I\cr}$

has a unique weak solution $\tilde{u}$ which is the unique global minimum in
$H^{1}_{0}(\Omega)$ of the functional

$u\to{{1}\over{2}}K\left(\int_{\Omega}|\nabla\tilde{u}(x)|^{2}dx\right)\int_{\Omega}|\nabla
u(x)|^{2}dx-{{1}\over{q+1}}\int_{\Omega}\alpha(x)|u^{+}(x)|^{q+1}dx\ .$ $None$

Moreover, $\tilde{u}$ satisfies the inequality

$\left(K\left(\int_{\Omega}|\nabla\tilde{u}(x)|^{2}dx\right)\right)^{{2}\over{1-q}}\int_{\Omega}|\nabla\tilde{u}(x)|^{2}dx\leq\left[\left({{2}\over{q+1}}\right)^{2}\left({{\hbox{\rm
ess
sup}_{\Omega}\alpha}\over{\lambda_{1}}}\right)^{q+1}\right]^{{1}\over{1-q}}\int_{\Omega}\alpha(x)dx\
,$

where

$\lambda_{1}=\inf_{u\in
H^{1}_{0}(\Omega)\setminus\\{0\\}}{{\int_{\Omega}|\nabla
u(x)|^{2}dx}\over{\int_{\Omega}|u(x)|^{2}dx}}\ .$

The main novelties of Theorem 1.1 are the lack of continuity of $K$ in
$[0,+\infty[$, the localization of the solution given by $\int_{\Omega}|\nabla
u(x)|^{2}dx\in I$ and the property that $\tilde{u}$ minimizes the functional
$(1.1)$ (which depends on $\tilde{u}$ itself). Actually, as far as we know,
the continuity of $K$ in $[0,+\infty[$ is an assumption present in each paper
devoted to this subject, but [9]. More precisely, in [9], we assumed that, for
some $r>0$, $K$ is continuous and increasing in $[0,r[$, with $\lim_{t\to
r^{-}}\int_{0}^{t}K(s)ds=+\infty$.

So, for instance, given $q\in]0,1[$, $\alpha\in L^{\infty}(\Omega)$, with
$\alpha>0$, $k\in{\bf N}$, $c>0$ and $s\in]0,1[$, the problems

$\cases{-\log\left(\int_{\Omega}|\nabla u(x)|^{2}dx\right)\Delta
u=\alpha(x)u^{q}&in $\Omega$\cr&\cr u>0&in $\Omega$\cr&\cr u=0&on
$\partial\Omega$\cr&\cr\int_{\Omega}|\nabla u(x)|^{2}dx>1\cr}$
$\cases{-\tan\left(\int_{\Omega}|\nabla u(x)|^{2}dx\right)\Delta
u=\alpha(x)u^{q}&in $\Omega$\cr&\cr u>0&in $\Omega$\cr&\cr u=0&on
$\partial\Omega$\cr&\cr(k-1)\pi<\int_{\Omega}|\nabla
u(x)|^{2}dx<(k-1)\pi+{{\pi}\over{2}}\cr}$

and

$\cases{-\left|c-\int_{\Omega}|\nabla u(x)|^{2}dx\right|^{-s}\Delta
u=\alpha(x)u^{q}&in $\Omega$\cr&\cr u>0&in $\Omega$\cr&\cr u=0&on
$\partial\Omega$\cr&\cr\int_{\Omega}|\nabla u(x)|^{2}dx<c\cr}$

cannot be covered by any of the results known up to now. Thanks to Theorem
1.1, each of these problems has a unique weak solution $\tilde{u}$ which
minimizes the functional $(1.1)$. But even when $I=]0,+\infty[$, Theorem 1.1
turns out to be new. Actually, the known results for the problem

$\cases{-K\left(\int_{\Omega}|\nabla u(x)|^{2}dx\right)\Delta u=u^{q}&in
$\Omega$\cr&\cr u>0&in $\Omega$\cr&\cr u=0&on $\partial\Omega$\cr}$

where $q\in]0,1[$, require that $K$, besides being continuous in
$[0,+\infty[$, is non-increasing ([1], Theorem 4; [3], Example 1) or
$K(t)=at+b$, with $a,b>0$ ([10], Theorem 1.2; [11], Corollary 1.1), or with
$a>0$ and $b\geq 0$ ([4], Theorem 5.4).

2\. Results

If $X$ is a topological space, a function $f:X\to{\bf R}$ is said to be inf-
compact (resp. sup-compact) provided that, for each $r\in{\bf R}$, the set
$f^{-1}(]-\infty,r])$ (resp. $f^{-1}([r,+\infty[)$) is compact.

We will obtain our main result via the following abstract theorem.

THEOREM 2.1. - Let $X$ be a topological space and let $\Phi:X\to{\bf R}$, with
$\Phi^{-1}(0)\neq\emptyset$, and $J:X\to{\bf R}$ be two functions such that,
for each $\lambda>0$, the function $\lambda\Phi-J$ is lower semicontinuous,
inf-compact and admits a unique global minimum. Moreover, assume that $J$ has
no global maxima in $X$. Furthermore, let $I\subseteq]0,+\infty[$ be an open
interval and $\Psi:I\to{\bf R}$ be an increasing function such that
$\Psi(I)=]0,+\infty[$.

Then, there exists a unique $\tilde{x}\in X$ such that $\Phi(\tilde{x})\in I$
and

$\Psi(\Phi(\tilde{x}))\Phi(\tilde{x})-J(\tilde{x})=\inf_{x\in
X}(\Psi(\Phi(\tilde{x}))\Phi(x)-J(x))\ .$

PROOF. Clearly, the function $\Psi^{-1}$ is increasing and continuous in
$]0,+\infty[$, with $\lim_{\lambda\to 0^{+}}\Psi^{-1}(\lambda)=\inf I$. So,
setting $\Psi^{-1}(0)=\inf I$, we think of $\Psi^{-1}$ as an increasing and
continuous function in $[0,+\infty[$. Now, consider the function
$\varphi:X\times[0,+\infty[\to{\bf R}$ defined by

$\varphi(x,\lambda)=\lambda\Phi(x)-J(x)-\int_{0}^{\lambda}\Psi^{-1}(t)dt$

for all $(x,\lambda)\in X\times[0,+\infty[$. Of course, for each $x\in X$, the
function $\varphi(x,\cdot)$ is concave, while, for each $\lambda>0$, the
function $\varphi(\cdot,\lambda)$ is lower semicontinuous, inf-compact and
admits a unique global minimum. Consequently, in view of Theorem 1.1 of [8],
we have

$\sup_{\lambda>0}\inf_{x\in X}\varphi(x,\lambda)=\inf_{x\in
X}\sup_{\lambda>0}\varphi(x,\lambda)\ .$

But, by continuity, we have

$\sup_{\lambda\geq 0}\varphi(x,\lambda)=\sup_{\lambda>0}\varphi(x,\lambda)$

and so

$\inf_{x\in X}\sup_{\lambda\geq 0}\varphi(x,\lambda)=\inf_{x\in
X}\sup_{\lambda>0}\varphi(x,\lambda)\leq\sup_{\lambda\geq 0}\inf_{x\in
X}\varphi(x,\lambda)$

from which it follows that

$\sup_{\lambda\geq 0}\inf_{x\in X}\varphi(x,\lambda)=\inf_{x\in
X}\sup_{\lambda\geq 0}\varphi(x,\lambda)\ .$ $None$

Of course, $\lim_{\lambda\to+\infty}\int_{0}^{\lambda}\Psi^{-1}(t)dt=+\infty$.
By assumption, there is some $x_{0}\in X$ such that $\Phi(x_{0})=0$, and so we
have $\lim_{\lambda\to+\infty}\varphi(x_{0},\lambda)=-\infty$. Hence,
$\varphi(x_{0},\cdot)$ is sup-compact. Then, from $(2.1)$ it follows that
there exists $(\tilde{x},\tilde{\lambda})\in X\times[0,+\infty[$ such that

$\sup_{\lambda\geq
0}\varphi(\tilde{x},\lambda)=\varphi(\tilde{x},\tilde{\lambda})=\inf_{x\in
X}\varphi(x,\tilde{\lambda})\ .$

Consequently

$\sup_{\lambda\geq
0}\left(\lambda\Phi(\tilde{x})-\int_{0}^{\lambda}\Psi^{-1}(t)dt\right)=\tilde{\lambda}\Phi(\tilde{x})-\int_{0}^{\tilde{\lambda}}\Psi^{-1}(t)dt$
$None$

and

$\inf_{x\in
X}(\tilde{\lambda}\Phi(x)-J(x))=\tilde{\lambda}\Phi(\tilde{x})-J(\tilde{x})\
.$ $None$

Notice that $\tilde{\lambda}>0$. Indeed, otherwise, by $(2.3)$, $\tilde{x}$
would be a global maximum of $J$, against an assumption. On the other hand, by
$(2.2)$ it follows that $\Psi^{-1}(\tilde{\lambda})=\Phi(\tilde{x})$. Hence,
$\Phi(\tilde{x})\in I$, $\tilde{\lambda}=\Psi(\Phi(\tilde{x}))$, and $(2.3)$
gives the conclusion, for the existence part. Now, let us prove the uniqueness
of $\tilde{x}$. So, let $\tilde{y}\in X$ be such that $\Phi(\tilde{y})\in I$
and

$\Psi(\Phi(\tilde{y}))\Phi(\tilde{y})-J(\tilde{y})=\inf_{x\in
X}(\Psi(\Phi(\tilde{y}))\Phi(x)-J(x))\ .$

Arguing by contradiction, assume that $\tilde{y}\neq\tilde{x}$. We claim that
$\Phi(\tilde{y})=\Phi(\tilde{x})$. Indeed, if, for instance,
$\Phi(\tilde{y})<\Phi(\tilde{x})$, then
$\Psi(\Phi(\tilde{y}))<\Psi(\Phi(\tilde{x}))$, and so, by Proposition 3.1 of
[8], we would have $\Phi(\tilde{y})>\Phi(\tilde{x})$. So, if we set
$\lambda:=\Phi(\tilde{x})$, the points $\tilde{x}$ and $\tilde{y}$ would be
two distinct global minima of the function $\lambda\Phi-J$, against an
assumption. The proof is complete.

$\bigtriangleup$

Here is our main result.

THEOREM 2.2. - Assume that there exists an open interval
$I\subseteq]0,+\infty[$ such that the restriction of $K$ to $I$ is increasing
and $K(I)=]0,+\infty[$. Let $f:[0,+\infty[\to[0,+\infty[$ be a continuous
function, with $f(0)=0$, such that the function $\xi\to{{f(\xi)}\over{\xi}}$
is decreasing in $]0,+\infty[$ and
$\lim_{\xi\to+\infty}{{f(\xi)}\over{\xi}}=0$, $\lim_{\xi\to
0^{+}}{{f(\xi)}\over{\xi}}=+\infty\ .$

Then, for each $\alpha\in L^{\infty}(\Omega)$, with $\alpha>0$, the problem

$\cases{-K\left(\int_{\Omega}|\nabla u(x)|^{2}dx\right)\Delta
u=\alpha(x)f(u)&in $\Omega$\cr&\cr u>0&in $\Omega$\cr&\cr u=0&on
$\partial\Omega$\cr&\cr\int_{\Omega}|\nabla u(x)|^{2}dx\in I\cr}$

has a unique weak solution $\tilde{u}$ which is the unique global minimum in
$H^{1}_{0}(\Omega)$ of the functional

$u\to{{1}\over{2}}K\left(\int_{\Omega}|\nabla\tilde{u}(x)|^{2}dx\right)\int_{\Omega}|\nabla
u(x)|^{2}dx-\int_{\Omega}\left(\alpha(x)\int_{0}^{u^{+}(x)}f(t)dt\right)dx\ .$

PROOF. First of all, extend $f$ to ${\bf R}$ putting $f(\xi)=0$ for all
$\xi<0$. We are going to apply Theorem 2.1 taking $X=H^{1}_{0}(\Omega)$
endowed with the weak topology, $\Psi=K$ and defining $\Phi,J$ by

$\Phi(u)=\int_{\Omega}|\nabla u(x)|^{2}dx\ ,$
$J(u)=2\int_{\Omega}\alpha(x)F(u^{+}(x))dx$

for all $u\in H^{1}_{0}(\Omega)$, where $F(\xi)=\int_{0}^{\xi}f(t)dt$. The
functionals $\Phi,J$ are $C^{1}$ with derivatives given by

$\Phi^{\prime}(u)(v)=2\int_{\Omega}\nabla u(x)\nabla v(x)dx$
$J^{\prime}(u)(v)=2\int_{\Omega}\alpha(x)f(u(x))v(x)dx$

for all $u,v\in H^{1}_{0}(\Omega)$. Moreover, due to the sub-critical growth
of $f$, $J$ is sequentially weakly continuous. Fix $\lambda>0$ and choose
$\epsilon\in\left]0,{{\lambda\lambda_{1}}\over{2}}\right[$, where
$\lambda_{1}$ is the constant defined in Theorem 1.1. Since
$\lim_{\xi\to+\infty}{{F(\xi)}\over{\xi^{2}}}=0$, there is $c_{\epsilon}>0$
such that

$F(\xi)\leq{{\epsilon}\over{M}}|\xi|^{2}+c_{\epsilon}$

for all $\xi\in{\bf R}$, where $M=\hbox{\rm ess sup}_{\Omega}\alpha$.
Consequently, we have

$J(u)\leq
2\epsilon\int_{\Omega}|u(x)|^{2}dx+2c_{\epsilon}\int_{\Omega}\alpha(x)dx$

and so

$\lambda\Phi(u)-J(u)\geq\lambda\int_{\Omega}|\nabla
u(x)|^{2}dx-2\epsilon\int_{\Omega}|u(x)|^{2}dx-2c_{\epsilon}\int_{\Omega}\alpha(x)dx$
$\geq\left(\lambda-{{2\epsilon}\over{\lambda_{1}}}\right)\int_{\Omega}|\nabla
u(x)|^{2}dx-c_{\epsilon}\int_{\Omega}\alpha(x)dx$

for all $u\in H^{1}_{0}(\Omega)$. Hence, due to the choice of $\epsilon$, we
have

$\lim_{\|u\|\to+\infty}(\lambda\Phi(u)-J(u))=+\infty\ .$

This fact, jointly with the reflexivity of $H^{1}_{0}(\Omega)$ and the
Eberlein-Smulyan theorem, implies that the sequentially weakly lower
semicontinuous functional $\lambda\Phi-J$ is weakly inf-compact. We now show
that it has a unique global minimum in $H^{1}_{0}(\Omega)$. Indeed, its
critical points are exactly the weak solutions of the problem

$\cases{-\Delta u={{1}\over{\lambda}}\alpha(x)f(u)&in $\Omega$\cr&\cr u=0&on
$\partial\Omega$\ .\cr}$

In turn, since the right-hand side of the equation is non-negative, the non-
zero weak solutions of the problem are positive in $\Omega$. Moreover, since,
for each $x\in\Omega$, the function ${{\alpha(x)f(\xi)}\over{\lambda}\xi}$ is
decreasing in $]0,+\infty[$, Theorem 1 of [2] ensures that the problem has at
most one positive weak solution, and so it has at most one non-zero weak
solution. As a consequence, we infer that the functional $\lambda\Phi-J$ has a
unique global minimum in $H^{1}_{0}(\Omega)$, since otherwise, in view of
Corollary 1 of [6], it would have at least three critical points. Now, fix any
positive function $u\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$, with
$u=0$ on $\partial\Omega$, such that

$\lambda_{1}\int_{\Omega}|u(x)|^{2}dx=\int_{\Omega}|\nabla u(x)|^{2}dx\ .$

Also, fix $\gamma\in]0,\hbox{\rm ess sup}_{\Omega}\alpha[$. Of course, the set

$\Omega_{\gamma}:=\\{x\in\Omega:\alpha(x)\geq\gamma\\}$

has a positive measure. Furthermore, fix $M>0$ so that

$M>{{\lambda\lambda_{1}\int_{\Omega}|u(x)|^{2}dx}\over{\gamma\int_{\Omega_{\gamma}}|u(x)|^{2}dx}}\
.$

Since $\lim_{\xi\to 0^{+}}{{F(\xi)}\over{\xi^{2}}}=+\infty$, there is
$\delta>0$ such that

$F(\xi)\geq M\xi^{2}$

for all $\xi\in[0,\delta]$. Now, set $v=\mu u$, where
$\mu={{\delta}\over{\sup_{\overline{\Omega}}u}}$. Then, we have

$J(v)\geq 2M\int_{\Omega}\alpha(x)|v(x)|^{2}dx\geq
2M\gamma\int_{\Omega_{\gamma}}|v(x)|^{2}>\lambda\lambda_{1}\int_{\Omega}|v(x)|^{2}dx=\lambda\int_{\Omega}|\nabla
v(x)|^{2}dx=\lambda\Phi(v)\ .$

This shows that $0$ is not a global minimum for the functional
$\lambda\Phi-J$. Consequently, the global minimum of this functional agrees
with its only non-zero critical point. Finally, let us show that $J$ has no
global maxima. Arguing by contradiction, suppose that $\hat{u}\in
H^{1}_{0}(\Omega)$ is a global maximum of $J$. Clearly, $J(\hat{u})>0$.
Consequently, the set

$A:=\\{x\in\Omega:\alpha(x)f(\hat{u}(x))>0\\}$

has a positive measure. Fix a closed set $C\subset A$ of positive measure. Let
$v\in H^{1}_{0}(\Omega)$ be such that $v\geq 0$ and $v(x)=1$ for all $x\in C$.
Then, we have

$\int_{\Omega}\alpha(x)f(\hat{u}(x))v(x)dx\geq\int_{C}\alpha(x)f(\hat{u}(x))dx>0$

and so $J^{\prime}(\hat{u})\neq 0$, which is absurd. Therefore, each
assumption of Theorem 2.1 is satisfied. As a consequence, there exists a
unique $\tilde{u}\in H^{1}_{0}(\Omega)$, with
$\int_{\Omega}|\nabla\tilde{u}(x)|^{2}dx\in I$, such that

$K\left(\int_{\Omega}|\nabla\tilde{u}(x)|^{2}dx\right)\int_{\Omega}|\nabla\tilde{u}(x)|^{2}dx-2\int_{\Omega}\alpha(x)F(\tilde{u}(x))dx$
$=\inf_{u\in
H^{1}_{0}(\Omega)}\left(K\left(\int_{\Omega}|\nabla\tilde{u}(x)|^{2}dx\right)\int_{\Omega}|\nabla
u(x)|^{2}dx-2\int_{\Omega}\alpha(x)F(u^{+}(x))dx\right)\ .$

Clearly, from what seen above, the function $\tilde{u}$ satisfies the
conclusion. $\bigtriangleup$

REMARK 2.1. - In Theorem 1.1, the inequality satisfied by the solution follows
directly from Theorem 1.2 of [7].

Acknowledgements. The author has been supported by the Gruppo Nazionale per
l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the
Istituto Nazionale di Alta Matematica (INdAM) and by the Università degli
Studi di Catania, PIACERI 2020-2022, Linea di intervento 2, Progetto ”MAFANE”.

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