#  $\Omega$-dibaryon production with hadron interaction potential from the
lattice QCD in relativistic heavy-ion collisions

S. Zhang(张松) and Yu-Gang Ma(马余刚)111Email<EMAIL_ADDRESS>Key Laboratory
of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern
Physics, Fudan University, Shanghai $200433$, China

###### Abstract

Recently HAL QCD Collaboration reported the $\Omega-\Omega$ and $N-\Omega$
interaction potentials by Lattice QCD algorithm. Based on these results,
$N\Omega$ (${}^{5}S_{2}$) and $\Omega\Omega$ (${}^{1}S_{0}$) bound state were
proposed with binding energy about a few MeV and $N-\Omega$ HBT correlation
were also calculated and measured by the STAR and the ALICE Collaboration.
These results provided dynamical information if $\Omega$-dibaryons exist from
the interaction aspects. Another necessary point is the experimental
environment where the bound states can produce and survive in the system or
not, such as in relativistic heavy-ion collisions. So there are at least two
necessary conditions to constrain the production probability of
$\Omega$-dibaryons, i.e. the short-range attractive interaction to form bound
states and experimental environment to provide abundant enough strangeness and
multiplicity of nucleons. In this work the $\Omega-\Omega$ and
$\Omega-$nucleon interaction potentials by the lattice QCD were employed to
obtain $\Omega\Omega$ (${}^{1}S_{0}$) and $N\Omega$ (${}^{5}S_{2}$) wave
functions, and the production of $\Omega$-dibaryons was estimated by use of a
dynamical coalescence mechanism with considering the environment in
relativistic heavy-ion collisions at $\sqrt{s_{NN}}=$ 200 GeV and 2.76 TeV.

###### pacs:

25.75.Gz, 12.38.Mh, 24.85.+p

††preprint: 1

##  I Introduction

Strangeness dibaryon has been investigated in theory and experiments for a
long period since $H-$dibaryon was predicted by Jaffe Jaffe (1977). However to
date there is only one stable dibaryon measured as deuteron ($d$).
$\Omega-$dibaryons, namely $N\Omega$ (${}^{5}S_{2}$) and $\Omega\Omega$
(${}^{1}S_{0}$), were proposed in several theoretical works and considered as
the most promising candidates of strangeness dibaryons Clement (2017); Cho et
al. (2017).

Goldman et al. Goldman et al. (1987) predicted the strangeness-3 dibaryons by
use of two different quark models of hadrons. In the framework of the quark
delocalization color screening model and the chiral quark model, the $N\Omega$
dibaryon was further studied and the binding energy was estimated from a few
MeV to hundred MeV within different configurations Wang et al. (1995); Huang
et al. (2015). A baryon-baryon interaction model with meson exchanges also
calculated the $N-\Omega$ two-body system and suggested a quasibound state
with binding energy 0.1 MeV Sekihara et al. (2018). Lattice QCD reported the
results of the spin-2 $N\Omega$ dibaryon with large binding energy 18.9 MeV by
the HAL QCD collaboration by using pion mass $m_{\pi}$ = 875 MeV and kaon mass
$m_{K}$ = 916 MeV Etminan et al. (2014). A further work from the lattice QCD
near the physical point suggested that the binding energy of $N\Omega$
dibaryon is 2.46 MeV and 1.54 MeV, respectively, with and without Coulomb
attraction Iritani et al. (2019).

$\Omega\Omega$ dibaryon with strong $\Omega-\Omega$ attraction was predicted
by a chiral quark model Zhang et al. (2000); Li et al. (2001); Dai et al.
(2006), while a weak repulsion $\Omega-\Omega$ interaction was suggested by
other models Wang et al. (1995); Buchoff et al. (2012). Based on the possible
production channel Yu et al. (2001, 2002), an extended version of a multi-
phase transport (AMPT) model Pal et al. (2005) estimated the production
probability of $\Omega\Omega$ dibaryon in Au+Au collisions at $\sqrt{s_{NN}}$
= 130 GeV. The HAL QCD collaboration’s previous work Yamada et al. (2015)
showed a moderate attraction with $m_{\pi}$ = 700 MeV and recently the HAL QCD
method with $m_{\pi}$ close to the physical point presented $\Omega\Omega$
binding energy 1.6 MeV or 0.7 MeV with the Coulomb repulsion Gongyo et al.
(2018).

The momentum correlation functions of hadron pairs can reflect the hadron-
hadron interaction Adamczyk et al. (2015a, b) which can provide the
information if the pairs can form a bound state. Based on the lattice QCD
simulations of $N-\Omega$ interaction Etminan et al. (2014), the momentum
correlation functions of $N-\Omega$ was calculated in Ref. Morita et al.
(2016) for probing $N\Omega$ dibaryons. The STAR collaboration conducted this
measurement Adam et al. (2019a) in Au+Au collisions at $\sqrt{s_{NN}}$ = 200
GeV and the results favored the p-$\Omega$ bound state with binding energy 27
MeV. By use of the interaction potential from recent lattice QCD calculations
at nearly physical quark masses Gongyo et al. (2018); Iritani et al. (2019),
the $p-\Omega$ and $\Omega-\Omega$ momentum correlation functions were updated
Morita et al. (2020). Recently the ALICE collaboration reported the
measurement of $p-\Omega^{-}$ and $p-\Xi^{-}$ correlations ALICE Collaboration
(2020) in $pp$ collision at $\sqrt{s}$ = 13 TeV and the results of $p-\Xi^{-}$
was in agreement with the predicted functions by the HAL QCD results Morita et
al. (2020), and $p-\Omega^{-}$ correlation should be investigated further in
nucleus-nucleus collisions in theory and experiment aspects.

In this work, we reported the production of $N\Omega$ (${}^{5}S_{2}$) and
$\Omega\Omega$ (${}^{1}S_{0}$) dibaryons calculated by a dynamical coalescence
model with considering $N-\Omega$ and $\Omega-\Omega$ interaction potential
from the HAL QCD results Iritani et al. (2019); Gongyo et al. (2018) in Au+Au
collisions at $\sqrt{s_{NN}}$ = 200 GeV and Pb+Pb collisions at
$\sqrt{s_{NN}}$ = 2.76 TeV. It was found that the production probabilities of
$N\Omega$ (${}^{5}S_{2}$) $\sim 10^{-3}$ and $\Omega\Omega$ (${}^{1}S_{0}$)
$\sim 10^{-6}$, respectively.

##  II A brief introduction to algorithm

Dynamical coalescence model is able to describe hadron and light nuclei
production in heavy-ion collisions Schaffner-Bielich et al. (2000); Chen et
al. (2003); Zhang et al. (2010); Sun and Chen (2015, 2017), and the component
interaction was reflected in the overlap wave function. For two-body clustered
object, the multiplicity of the object can be obtained by Schaffner-Bielich et
al. (2000); Chen et al. (2003); Zhang et al. (2010); Sun and Chen (2015,
2017),

$\displaystyle\begin{split}N_{2b}&=g_{2}\int\left(d^{4}x_{1}S_{1}(x_{1},p_{1})\frac{d^{3}p_{1}}{E_{1}}\right)\times\\\
&\left(d^{4}x_{2}S_{2}(x_{2},p_{2})\frac{d^{3}p_{2}}{E_{2}}\right)\times\rho^{W}_{2}(x_{1},x_{2};p_{1},p_{2}),\end{split}$
(1)

where $\rho^{W}_{2}(x_{1},x_{2};p_{1},p_{2})$ is the Wigner density function
which gives the coalescence probability, $g_{2}$ is the coalescence factor
Polleri et al. (1999), 3/8 for deuteron, 5/8 for $N\Omega$ and 1/4 for
$\Omega\Omega$, $S(x,p)$ is the phase space distribution of the components at
coordinate-momentum space $(x,p)$ with energy $E$. Note that the phase space
distribution of neutron ($n$) was assumed the same as that of proton ($p$) if
a certain cluster contains neutron in this work.

The phase-space distribution can be expressed by a blast-wave model Retière
and Lisa (2004); Sun and Chen (2015, 2017); Zhang et al. (2014),

$\displaystyle\begin{split}S(x,p)d^{4}x&=m_{T}\cosh(\eta_{s}-y_{p})f(x,p)J(\tau)\times\\\
&\tau d\tau d\eta_{s}rdrd\varphi_{s},\end{split}$ (2)

where $y_{p}$ and $m_{T}$ are respectively the rapidity and transverse mass of
the hadron, and $r$, $\tau$, $\eta_{s}$ $\varphi_{s}$ are the polar
coordinates, proper time, pseudorapidity and azimuthal angle in coordinate
space. A Gaussian distribution for the freeze-out proper time is given by,
$J(\tau)=\frac{1}{\Delta\tau\sqrt{2\pi}}\exp(-\frac{(\tau-\tau_{0})^{2}}{2(\Delta\tau)^{2}})$,
where $\tau_{0}$ and $\Delta\tau$ are the mean value and the dispersion of the
$\tau$ distribution. The statistical distribution function Cooper and Frye
(1974) $f(x,p)$ is defined by $f(x,p)$ =
$\frac{2s+1}{(2\pi)^{3}}\left[\exp\left(p^{\mu}u_{\mu}/T_{kin}\right)\pm
1\right]^{-1}$, where $s$ is the spin of the particle, $u_{\mu}$ is the four-
velocity of a fluid element in the fireball of the emission source, and
$T_{kin}$ is the kinetic freeze-out temperature. The energy in the local rest
frame of the fluid can be written as, $p^{\mu}u_{\mu}$ =
$m_{T}\cosh\rho\cos(\eta_{s}-y_{p})-p_{T}\sinh\rho\cos(\varphi_{p}-\varphi_{s})$,
where $\varphi_{p}$ is azimuthal angle in momentum space, $\rho$ is the
transverse flow rapidity distribution of the fluid element in the fireball
with a transverse radius $R_{0}$, defined as $\rho$ =
$\rho_{0}\frac{r}{R_{0}}$ without considering the anisotropic part Retière and
Lisa (2004); Sun and Chen (2015); Zhang et al. (2014). Fixing the parameters
of ($\tau_{0}$, $\Delta\tau$, $\rho_{0}$, $R_{0}$, $T_{kin}$), one can obtain
the transverse momentum distribution of the hadrons by,

$\displaystyle\frac{dN}{2\pi p_{T}dp_{T}dy_{p}}$ $\displaystyle=$
$\displaystyle\int S(x,p)d^{4}x.$ (3)

By using transport model to obtain the component phase-space, the dynamical
coalescence model for two-body clustered object can be written as Chen et al.
(2003); Zhang et al. (2010),

$\displaystyle N_{2b}=g_{2}\int
d\vec{r}_{1}d\vec{r}_{2}d\vec{p}_{1}d\vec{p}_{2}\times\left<\rho^{W}_{2}(x_{1},x_{2};p_{1},p_{2})\right>,$
(4)

where $\left<...\right>$ denotes event averaging. In this work a multi-phase
transport (AMPT) model Lin et al. (2005) with version $2.26t7b$ are employed
to provide the phase-space of neutrons, protons and $\Omega$’s. AMPT simulates
the relativistic heavy-ion collisions dynamically in a framework of multi
phases, in which the initial state is given by the Heavy Ion Jet Interaction
Generator (HIJING) model Wang and Gyulassy (1991); Gyulassy and Wang (1994),
and the melted partons from the HIJING interact with each other by the Zhang’s
Parton Cascade (ZPC) model Zhang (1998), and finally the interacting-ceased
partons converts to hadrons by a simple quark coalescence model or the Lund
string fragmentation followed by a relativistic transport model performing the
hadron rescattering Li and Ko (1995). AMPT can well describe physics in
relativistic heavy-ion collisions at the RHIC Lin et al. (2005) and LHC Ma and
Lin (2016) energies, including pion-HBT correlations Lin et al. (2002), di-
hadron azimuthal correlations Ma et al. (2006); Wang et al. (2019), collective
flow Abelev et al. (2008); Bzdak and Ma (2014) and strangeness production Jin
et al. (2018, 2019) and so on.

The Wigner phase-space densities of the objects ($\rho^{W}_{2}$) can be
obtained from,

$\displaystyle\begin{split}\rho^{W}_{2}(\vec{r},\vec{q})=\int\phi\left(\vec{r}+\frac{\vec{R}}{2}\right)\phi^{*}\left(\vec{r}-\frac{\vec{R}}{2}\right)\times\\\
\exp\left(-i\vec{q}\cdot\vec{R}\right)d\vec{R},\end{split}$ (5)

where $\vec{q}$ = $(m_{2}\vec{p}_{1}-m_{1}\vec{p}_{2})/(m_{1}+m_{2})$ and
$\vec{r}$ = $(\vec{r}_{1}-\vec{r}_{2})$ are the relative momentum and relative
coordinate, respectively, $\phi(\vec{r})$ is the overlap wave function of the
two components. In our previous work for three components and other
calculations for two-body, the overlap wave function always were taken to be a
spherical harmonic oscillator. In this work, the wave function and binding
energy $E_{B}$ for $d$ (${}^{3}S_{1}$), $N\Omega$ and $\Omega\Omega$ at
assumed bound state ${}^{5}S_{2}$ and ${}^{1}S_{0}$ will be obtained by
solving the radial Schrödinger equation with the potential for $n-p$ Friar et
al. (1990), $N-\Omega$ Iritani et al. (2019), $\Omega-\Omega$ Gongyo et al.
(2018), respectively,

$\displaystyle\begin{split}V_{np}(r)&=\sum_{i=1}^{2}C_{i}\frac{e^{-\mu_{i}r}}{r},\\\
V_{N\Omega}(r)&=b_{1}e^{-b_{2}r^{2}}+b_{3}\left(1-e^{-b_{4}r^{2}}\right)\left(\frac{e^{-m_{\pi}r}}{r}\right)^{2},\\\
V_{\Omega\Omega}(r)&=\sum_{i=1}^{3}C_{i}e^{-(r/d_{i})^{2}},\end{split}$ (6)

here the parameters for the potential are listed in Table 1.

Table 1:  The parameters for the potential, $V_{np}(r)$ Friar et al. (1990), $V_{N\Omega}(r)$ Iritani et al. (2019) and $V_{\Omega\Omega}(r)$ Gongyo et al. (2018). $V_{np}(r)$ | $C_{1}~{}(\text{MeV})$ | $C_{2}~{}(\text{MeV})$ | $\mu_{1}~{}(fm^{-1})$ | $\mu_{2}~{}(fm^{-1})$
---|---|---|---|---
| 626.885 | 1438.72 | 1.55 | 3.11
$V_{N\Omega}(r)$ |  $b_{1}$ (MeV) | $b_{2}~{}(fm^{-2})$ | $b_{3}~{}(\text{MeV}\cdot fm^{2})$ | $b_{4}~{}(fm^{-2})$
| -313 | 81.7 | -252 | 0.85
$V_{\Omega\Omega}(r)$ |  $C_{1}$ (MeV) |  $C_{2}$ (MeV) |  $C_{3}$ (MeV) |
| 914 | 305 | -112 |
| $d_{1}~{}(fm)$ | $d_{2}~{}(fm)$ | $d_{3}~{}(fm)$ |
| 0.143 | 0.305 | 0.949 |
Figure 1: Calculated wave function of $d$ ($n$-$p$), $N\Omega$ and
$\Omega\Omega$, and the Hulthén wave function Chen et al. (2003); Hodgson
(1971) for $d$. (color online)

In the calculation the Coulomb interaction was also taken into account for the
charged pairs by adding $\pm\alpha/r$ with $\alpha$ = $e^{2}/4\pi$ to the
potential in equation (6). Figure 1 showed the numerical results of the wave
function $\phi(r)$ for $d$ (${}^{3}S_{1}$), $N\Omega$ (${}^{5}S_{2}$) and
$\Omega\Omega$ (${}^{1}S_{0}$). The Hulthén wave function Chen et al. (2003);
Hodgson (1971) for $d$ which is also presented in Fig. 1 is higher than the
calculated wave function by using potential $V_{np}(r)$ in short relative
distance region, which is because of the repulsion core of the potential for
$n-p$ interaction and this pattern was also found in $\Omega\Omega$ wave
function. The $N-\Omega$ attractive potential resulted in that the wave
function was similar to $d$’s Hulthén wave function. Table 2 showed the
calculated binding energy and they were consistent with the collected
published results Friar et al. (1990); Iritani et al. (2019); Gongyo et al.
(2018).

Table 2:  The calculated binding energy together with the collected values from other published results Friar et al. (1990); Iritani et al. (2019); Gongyo et al. (2018). | this work | value/Reference
---|---|---
$E_{d}$ (MeV) | 2.23 | 2.2307 Friar et al. (1990)
$E_{p\Omega}$ (MeV) | 3.09 | 3.00 Iritani et al. (2019)
$E_{n\Omega}$ (MeV) | 1.38 | 1.54 Iritani et al. (2019)
$E_{\Omega\Omega}$ (MeV) | 0.6 | 0.7 Gongyo et al. (2018)

##  III Results and discussion

In blast-wave model Retière and Lisa (2004); Sun and Chen (2015, 2017); Zhang
et al. (2014), the parameters of $T_{kin}$, $\rho_{0}$, $\tau_{0}$,
$\Delta\tau$ and $R_{0}$ can be obtained by fitting experimental transverse
momentum $p_{T}$ spectra of $p$ and $\Omega$ by using equation (3) and the
parameters of $\tau_{0}$, $\Delta\tau$ and $R_{0}$ can be further fixed by
fitting deuteron’s $p_{T}$ spectra from experiments by using equation (1). In
Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV, the data were collected for $p$
from the PHENIX experiment Adler et al. (2004), for $\Omega$ and $d$ from the
STAR experiments Adam et al. (2019b); Adams et al. (2007) (centrality, 5% for
$p$, 0-10% for $\Omega$ and $d$). In Pb+Pb collisions at $\sqrt{s_{NN}}$ =
2.76 TeV, the data of $p$, $\Omega$ and $d$ were taken from the ALICE
experiments Abelev et al. (2013); Adam et al. (2016); Abelev et al. (2014)
(centrality, 5% for $p$, 0-10% for $\Omega$ and $d$). The fitting to the
experimental data was shown in Fig. 2 and the extracted parameters were
$R_{0}$ = 12 $fm$, $\tau_{0}$ = 9 $fm/c$, $\Delta\tau$ = 3.5 $fm/c$, $T_{kin}$
= 111.6 MeV, and $\rho_{0}$ = 0.98 for proton and 0.9 for $\Omega$ in Au+Au
collisions at $\sqrt{s_{NN}}$ = 200 GeV and $R_{0}$ = 19.7 $fm$, $\tau_{0}$ =
15.5 $fm/c$, $\Delta\tau$ = 1 $fm/c$, $T_{kin}$ = 122 MeV, and $\rho_{0}$ =
1.2 for proton and 1.07 for $\Omega$ in Pb+Pb collisions at $\sqrt{s_{NN}}$ =
2.76 TeV.

By using the above configured blast-wave model and coalescence model (BLWC) as
in equation ( 1), the transverse momentum $p_{T}$ spectra of $n\Omega$,
$p\Omega$ and $\Omega\Omega$ dibaryons were calculated and shown in Fig. 2, in
which panel (a) is for Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV and panel
(b) for Pb+Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV. The $p_{T}$ integrated
yields $dN/dy$ of objects at midrapidity were given in Table 3 and the
calculated $dN/dy$ of $p$, $\Omega$ and $d$ were comparable with those from
experimental results from the RHIC data Adler et al. (2004); Adam et al.
(2019b); Adams et al. (2007) as well as from the ALICE data Abelev et al.
(2013); Adam et al. (2016); Abelev et al. (2014). The predicted $dN/dy$ of
$p\Omega$, $n\Omega$ and $\Omega\Omega$ were $7.51\times 10^{-4}$, $7.39\times
10^{-4}$, $1.24\times 10^{-6}$ for the RHIC top energy and $1.31\times
10^{-3}$, $1.27\times 10^{-3}$, $3.15\times 10^{-6}$ for the ALICE at 2.76
TeV, respectively, in nucleus-nucleus collisions. It is seen that the
production of $N\Omega$ and $\Omega\Omega$ at the ALICE energy was about 2
times of those at the RHIC top energy. And these calculated results were
similar to the previous work by using naive coalescence model Shah et al.
(2016) or analytical coalescence model Sun and Chen (2017) as well as using
the AMPT model with $\Omega\Omega$ production channel Pal et al. (2005).

Figure 2: Transverse momentum $p_{T}$ spectra of $p$, $d$, $\Omega$, $n\Omega$, $p\Omega$ and $\Omega\Omega$ in Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV (a), and in Pb+Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV (b), respectively. Lines: blast-wave model coupled with dynamical coalescence model (BLWC); Markers: data from the RHIC Adler et al. (2004); Adam et al. (2019b); Adams et al. (2007) and the ALICE Abelev et al. (2013); Adam et al. (2016); Abelev et al. (2014). (color online)  Table 3:  $dN/dy$ of $p\Omega$, $n\Omega$, $\Omega\Omega$ at mid-rapidity. | $p\Omega$ | $n\Omega$ | $\Omega\Omega$
---|---|---|---
200 GeV |  |  |
BLWC | 7.51$\times 10^{-4}$ | 7.39$\times 10^{-4}$ | 1.24$\times 10^{-6}$
AMPTC | 9.5$\times 10^{-4}$ | 9.5$\times 10^{-4}$ | 3.23$\times 10^{-6}$
| $\pm 7.9\times 10^{-5}$ | $\pm 7.9\times 10^{-5}$ | $\pm 4.6\times 10^{-6}$
2.76 TeV |  |  |
BLWC | 1.31$\times 10^{-3}$ | 1.27$\times 10^{-3}$ | 3.15$\times 10^{-6}$
AMPTC | 1.11$\times 10^{-3}$ | 1.10$\times 10^{-3}$ | 4.30$\times 10^{-6}$
| $\pm 8.57\times 10^{-5}$ | $\pm 8.57\times 10^{-5}$ | $\pm 5.3\times 10^{-6}$
Figure 3: Transverse momentum $p_{T}$ spectra of $p$, $d$, $\Omega$,
$n\Omega$, $p\Omega$ and $\Omega\Omega$ in Au+Au collisions at $\sqrt{s_{NN}}$
= 200 GeV (a), and in Pb+Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV (b),
respectively. Open Marker: AMPT coupled with dynamical coalescence model
(AMPTC); Solid Markers: data from the RHIC Adler et al. (2004); Adam et al.
(2019b); Adams et al. (2007) and the ALICE Abelev et al. (2013); Adam et al.
(2016); Abelev et al. (2014). (color online)

The productions of $p\Omega$, $n\Omega$ and $\Omega\Omega$ bound states were
also calculated by using phase-space data from the AMPT model Lin et al.
(2005) via dynamical coalescence mechanism equation (4) (AMPTC). To fit proton
spectra, some parameters defined in the AMPT model Lin et al. (2005); Ma and
Lin (2016) were adjusted as, $(a,b)$ = $(0.55,0.1)$ for the RHIC energy and
$(0.21,0.075)$ for the LHC energy, (here $a$ and $b$ are the Lund string
fragmentation parameters defined in reference Lin et al. (2005)). And the
coalescence mechanism for $\Omega$ was also developed as in Ref. Wang and Xu
(2019) to fit $\Omega$ spectra. Figure 3 presented the fitted $p_{T}$ spectra
for proton and $\Omega$ as well as the coalesced $p_{T}$ spectra of $p\Omega$,
$n\Omega$ and $\Omega\Omega$ bound states in Au+Au central collisions at
$\sqrt{s_{NN}}$ = 200 GeV (a) and in Pb+Pb central collisions at
$\sqrt{s_{NN}}$ = 2.76 TeV (b), respectively. Based on the adjusted AMPT
parameters and the developed coalescence mechanism for $\Omega$, the $p_{T}$
spectra of $p$ and $\Omega$ could be described well and the $p_{T}$ spectra of
$p\Omega$, $n\Omega$ and $\Omega\Omega$ were similar to those via BLWC shown
in Fig. 2. Yields of $dN/dy$ of $p\Omega$, $n\Omega$ and $\Omega\Omega$ were
also listed in Table 3 and agreed with those from the BLWC.

In this work the particle interaction potential from the lattice QCD was taken
into account and the overlap wave function was assumed to be the $S-$wave
function calculated by solving Schrödinger equation. By using the phase-space
information from the blast-wave model as well as the transport model (AMPT),
the coalesced $p\Omega$, $n\Omega$, $\Omega\Omega$ gave the similar results
and agreed with the previous predictions Shah et al. (2016); Sun and Chen
(2017); Pal et al. (2005). These consistent results implied that $N\Omega$ and
$\Omega\Omega$ could be bounded in $S-$wave state and produced via coalescence
mechanism at final stage in relativistic heavy ion collisions. And it also
illustrated that the hyperon-hyperon ($YY$) and hyperon-nucleon ($YN$)
interactions from first principle calculation, such as the lattice QCD, could
be examined by investigated the production of $\Omega$-dibaryons.

##  IV Summary

The overlap $S-$wave function and Wigner density function are calculated
through solving Schrödinger equation with the $N-\Omega$ and $\Omega-\Omega$
potential from the lattice QCD which was recently published by the HAL QCD
collaboration and the calculated binding energy is consistent with the
published results. In the coalescence mechanism frame, the blast-wave model
and AMPT model are employed to provide the phase-space information to coalesce
$N\Omega$ and $\Omega\Omega$ bound state by using the calculated Wigner
density function, and the production rate of the bound states agree with other
model predicted results. This dynamical coalescence calculation of the
$\Omega$-dibaryons sheds light on the experiment searching for the
(most)-strangeness dibaryon bound states at the STAR and the ALICE experiments
which can help us to understand the $YY$ and $YN$ interactions.

Acknowledgments: This work was supported in part by the National Natural
Science Foundation of China under contract Nos. 11875066, 11421505, 11890714,
11925502, 11961141003, National Key R&D Program of China under Grant No.
2016YFE0100900 and 2018YFE0104600, the Key Research Program of Frontier
Sciences of the CAS under Grant No. QYZDJ-SSW-SLH002, and the Key Research
Program of the CAS under Grant NO. XDPB09.

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# A Note on Bias to Complete

Jia Xu1
Stevens Institute of Technology
<EMAIL_ADDRESS>
Mona Diab 2
Carnegie Mellon University
<EMAIL_ADDRESS>

###### Abstract

Minimizing social bias strengthens societal bonds, promoting shared
understanding and better decision-making. We revisit the definition of bias by
discovering new bias types (e.g., societal status) in dynamic environments and
describe them relative to context, such as culture, region, time, and personal
background. Our framework includes eight hypotheses about bias and a
minimizing bias strategy for each assumption as well as five methods as
proposed solutions in LLM. The realization of the framework is yet to be
completed.

##  1 Introduction

###  1.1 Main Objective

The bias that refers to an inclination or prejudice for or against one person
or group is a major reason for societal division and injustice, as it affects
our perceptions and decisions, often leading to unequal or unfair treatment of
different groups and individuals. We aim to advance LLM technologies to
alleviate communication barriers among different social opinions towards a
connected community with two ultimate goals: (1) minimizing the disparity of
recognizing social bias, contrasting diverse perspectives on dynamically
emerging bias types (often called sensitive or protected attributes); (2)
eliminating the mis(dis)informed content verified from reliable sources.

The notion of bias highly depends on the dynamic societal environments we live
in, our “context”, both geographically and temporally affected and impacted by
events. For example, rights afforded to people of different races change over
time, and ethics standards and norms vary from place to place. In addition to
considering relatively well defined stationary value judgment of bias (e.g.,
gender and race) with stable and predetermined definitions (such as balanced
ratios), we propose discovering new bias types ((e.g., poverty, educational
level, societal status) and redefine them as conditioned on their context such
as culture, region, time, and personal background.

Around the central theme of relative bias in dynamic environments, we will
provide a family of 8 hypotheses forming our framework of communication and
information processing for debiasing LLMs. To faciliate these hypotheses we
will design novel NLP techniques for each of the hypotheses and deliver
transformative methods and tools reflected in 5 methods including bias
definition, quantification, data augmentation, mitigation, reasoning, and
analysis using mis(dis)information. The conceptual novelty lies in embracing
the bias from 8 bias classes: context, language, connotation, position, data,
algorithmic, topic, and intention. In the following, we will first review the
existing work limitation then highlight our innovations.

###  1.2 Current Limitations

####  1.2.1 Key Technical Challenges

We will focus on 5 aspects of limitations (L) of previous social bias work.

  * •

(L1) Social bias has been manually pre-defined top-down and measured with a
singular, fixed, and superficial social scale, almost devoid of context.
However, in realistic settings, bias depends on, for example, cultural
assumptions and inferences.

  * •

(L2) Measuring bias is challenging: first, quantifying bias lacks considering
the social norm of the dynamic context; second, bias expressions can sometimes
go through implicit language such as connotation; third, when the speaker, the
listener, and the propagator are transferring information, their different
interpretations of words can cause biased viewpoints and misunderstanding.

  * •

(L3) Existing work in bias mitigation met three bottlenecks: first, the cause
of social bias contains complex factors and has been understudied so far;
second, artificially changing bias such as manipulating ratios across biased
groups may cause injustice; third, sometimes bias is needed to adapt to a
certain cultural background according to their social norm.

  * •

(L4) While biased datasets are in high demand, obtaining labeled bias datasets
can be complex due to privacy, cost, or legal considerations. Furthermore,
existing data augmentation does not adapt to new bias types and definitions,
social norms, as well as the dynamic content of the data, since the original
frameworks are inherently static.

  * •

(L5) The influence of mis/disinformation on bias and vice versa is
understudied. Bias is often due to partial information, such as local news or
minority viewpoints. Partial information or wrong information can lead to
decision bias and discrimination. On the other hand, bias can sometimes cause
misinformation or disinformation in communication, whereas, the misinformation
will likely travel around and even be included in a new LLM training worsening
bias.

####  1.2.2 Proposed Solutions to Challenges

Each of our 5 methods focuses on responding one limitation (L) as follows:

  * •

(M1) Contextualized bias definition with contrastive perspectives. We veer
away from the assumption that bias is absolute, instead developing a notion of
relative bias against sets of norms of cultural standards and time. We will
model a multi-faceted dynamic environment with relevant contexts (e.g.,
cultural, temporal, and regional) as a tensor as a means for grounding
contrastive perspectives with a contextual backdrop. For example, an English
statement is considered biased in a British Christian setting while the same
statement might be considered appropriate within a British Muslim setting.
Accordingly, We propose lifting the blindfold associated with LLMs by
alleviating the potential blindspots inherently present in understanding
generated text due to the biases in the minds of the reader or producer.

  * •

(M2) Bias quantification with social norm and connotation understanding.
First, we will quantify the bias using multilingual consensus, implication
understanding, geometric analysis on semantic and sentiment social norms.
Second, we will take into consideration the speaker and the listener inherent
perspectives as devices of interpretation and inference and factorize the
relative bias to the derived connotations per utterance, namely devising a
novel connotation-based tensor factorization model. Third, we will posit that
a perceived bias in one context might be a norm in another based on the
perspective such as personal background, and we will detect and fill in these
blind spots to bridge the interpretations from the content reader and
producer. Theses methods will be realized for both individual and group as
spheres of bias origin/impact and influence.

  * •

(M3) Data generation using machine translation. We will provide a novel
approach to create labeled synthetic bias data by viewing debiasing as a
multilingual machine translation task, considering the correspondence between
the intersectionality of contexts and language representation as the source
while implications and connotations rendered explicit are the target. We will
also infer the bias implicature by (machine) translating language into
connotations and implications to explore the subtle types of implied language,
incorporating secret languages or cryptolect.

  * •

(M4) Bias mitigation and cause analysis. First, we will classify the cause of
bias using instructive reinforcement learning and provide explanations how
bias gradually yields. Second, we will create a counter-intuitive method of
removing bias-type input features on downstream tasks to minimize algorithmic
bias. Third, we will implement a bias controller to tune on the perspective
and adapt LLM output.

  * •

(M5) Mis(dis)information influence on bias. Our model will control and guide
information content along communication trajectories by understanding how
incomplete knowledge in users or LLMs could lead to unintended or misconstrued
meanings of information, resulting in bias, and how bias can conversely
aggravate mis/disinformation.

###  1.3 Novelty and Key Innovations

Summarized, our project will lead to the following key innovations:

  * •

Novel concepts of contrastive bias definition and dynamic social context.

  * •

A theory of 8 hypotheses on LLMs and communication.

  * •

A paradigm by embracing 8 bias types for the issues arised in the hypotheses.

  * •

New methods to realize our paradigm using tensor networks, machine
translation, social norms, feature interpretation, connotation understanding,
mixture of experts, and so on.

  * •

Products for LLM debiasing either in 1 framework or separately used for 5
methods on bias definition, quantification, data augmentation, mitigation,
explanation, and mis(dis)information.

##  2 Framework

Our solutions are based on our framework composed of lifting barrier framework
and embracing bias strategy.

###  2.1 Bias Hypotheses

Our framework comprises following hypotheses:

  1. 1.

Temporal Agnostic Context-Independence Assumption:  LLMs often exhibit bias
stemming from a lack of temporal context in processing prompts. Unlike humans,
LLMs do not account for temporal gaps between inputs, potentially leading to
erroneous context-dependent interpretations. The LLM’s processing reflects a
temporal context-independent bias, leading to potential misunderstandings or
skewed context comprehension.

  2. 2.

Social Norm Alignment Assumption: There exists a norm of social opinion on
bias-related statements in a restricted environment. The median of embedding a
statement over various biased statements with opinions can approximate this
norm. The bias angle is the distance between the embedded biased statement and
the norm.

  3. 3.

Individual Interpretation Assumption: In connotation, the interpretation of a
term by different people can vary, leading to a gap of bias and
miscommunication.

  4. 4.

Information Over-writable Assumption: A LLM can be fine-tuned and the new data
will partially or even fully cover the data in the pre-trained model. Thus,
information such as biased opinion can be revised or overwritten.

  5. 5.

Blindspot fill-in Assumption: The blind spots on knowledge is one of the
source of a biased view. Filling in the blind spots will shift the bias
towards the norm.

  6. 6.

Bias No Effect on Performance Assumption: In downstream applications,
attributes like gender and race have minimal impact on the predictive
performance of neural networks.

  7. 7.

LowGram Phenomenon Assumption: This term refers to a prevalent pattern in
information dissemination, particularly within the domain of ’infotainment,’
where content characterized by lower intellectual or cultural value (‘light
weight’ and ’low taste’) exhibits a higher propensity for rapid and widespread
circulation in the public sphere. This phenomenon is particularly notable when
such content requires minimal contextual support for its comprehension and
propagation. The “LowGram Effect" encapsulates the observation that
simplistic, sensational, or superficial topics often achieve greater and
faster distribution compared to more substantive, educational, or complex
information, emphasizing the ease of dissemination of lower-quality,
illusionary content over more intellectually rigorous material.

  8. 8.

Consistent Fact Narrative Validity Assumption: The validity of a narrative
hinges on its construction from verified facts that demonstrate logical
coherence. For a narrative to be considered valid, these facts must not only
be consistent with each other but must also synergize to form a comprehensive
and complete story. A narrative constructed from coherent facts but lacking in
completeness may still hold validity, yet it remains unverified until it forms
a fully realized and comprehensive story.

We will verify these hypotheses on our dataset using deep learning and
geometric methods on the embedding space.

###  2.2 Embracing Bias Strategy

Our central theme is to embrace bias to mitigate bias. The rationale is that
bias is mainly due to limited knowledge and narrowed view, and the inclusive
of a broad spectrum of biased inputs will enrich the understanding of the
“world map" comprehensive to multi-angle perspectives, leading to a more
objective conclusion on sensitive manners. We categorize the bias in the
following types:

  1. 1.

Context bias Bias based on context refers to the understanding and judgment
influenced by the dynamic environment relative to the cultural, temporal, and
regional information. We will collect data containing different dynamic
environments to fill in our tensor factorization on the dependency modeling.
The evaluation will also be carried out on multi-environments in multiple
dimensions.

  2. 2.

Language bias Bias from using different languages and perspectives arises
because each language shapes thought differently, reflecting unique cultural
contexts and values; for example, certain nuances or concepts may be inherent
in one language but not easily translatable or understood in another. We will
make use of the existing machine translation systems of co-PI Xu for
multilingual LLM bias analysis and mitigation. The evaluation will be
performed on English, Arabic, German, and Chinese on WMT datasets.

  3. 3.

Connotation bias Bias from connotation and implication occurs when the
emotional or cultural associations of words, beyond their literal meanings,
influence perception and judgment, leading to interpretations that reflect
personal or societal prejudices rather than objective assessments. We will
view the connotation as a machine translation task, where the source language
is the connotation, and the target language is its debiased version in plain
text. We will craft manual datasets on connotation examples for training and
testing. The evaluation will measure the F-score on the label and prediction
matches.

  4. 4.

Position bias Bias from different personal experiences and viewpoints,
combined with blindspots in knowledge, can significantly shape how individuals
perceive, understand, and interpret information, where what is seen or
understood is deeply influenced by what is known, believed, and overlooked. We
will measure the position of an individual or group view using the context-
dependent bias definition to quantify the relative distance of a specific view
to the social norm and evaluate manually with counterexamples.

  5. 5.

Data bias Bias due to data and domain occurs when the information used to
train algorithms or make decisions reflects limitations, inaccuracies, or
partial perspectives inherent in the data sources or the specific field of
study, leading to skewed or prejudiced outcomes. We will take data from
multiple domains, such as news, TED talks, and scientific articles, to form
multiple perspectives. Different experts and their backgrounds will agree upon
manual evaluation.

  6. 6.

Algorithmic bias Algorithmic bias refers to systematic and repeatable errors
in a computer system that create unfair outcomes, such as privileging one
arbitrary group of users over others. Here, we will consider the bias that
comes from features and models other than data. We will eliminate algorithmic
bias by excluding protected attributes such as gender and race when conducting
predictions on downstream tasks of LLMs. Evaluation will be performed on
criminal charge prediction and loan credibility.

  7. 7.

Topic bias The "LowGram Phenomenon Assumption" affects the selection of topics
in information dissemination by favoring unhealthy and unethical content that
can spread quickly and widely, influencing public discourse and knowledge with
less intellectually demanding and, figurative and vigorous content. We will
classify the ethics of the topics and give preference to those that are more
ethical and serious. We will construct sample tests on various topics and
evaluate them manually.

  8. 8.

Blind spot bias When given an utterance, we will decompose it into several
statements with ideally small semantic and logic overlap. Then, for each
statement, we verify its fact. We will form statement verification using a
matrix, where the x-coordinate indicates each statement, and the y-coordinate
indicates the viewpoints, such as different languages. The missing slot in the
matrix is the blind spot, which will be filled by query knowledge using
information retrieval, and the content of each of the slots will be checked
against that of other slots with logical inference. Consensus slots will stay
otherwise notified.

###  2.3 Related Work

There have been extensive literature investigating bias issues in NLP,
including but not limited to the following work:

  * •

Methods for objectively quantifying bias (e.g., relative to a specific
collection of texts and other content): Delobelle et al. (2021); Holtermann et
al. (2022); Weidinger et al. (2021); Fleisig et al. (2023); Jourdan et al.
(2023); Kong et al. (2023); Krishna et al. (2022); Li et al. (2023); Teo et
al. (2023); Zhao et al. (2023); Giguere et al. (2022); Kaneko and Bollegala
(2022); Tao et al. (2022); Feng and Shah (2022); Hu and Zhang (2022);
Spliethöver et al. (2022); Zhou et al. (2022); Han et al. (2021); Nejadgholi
et al. (2022); Stahl et al. (2022); Han et al. (2022); Hessenthaler et al.
(2022); Wang et al. (2021b); Du et al. (2021); Ahn and Oh (2021); Patel and
Pavlick (2021); Spinde et al. (2022); Zhong (2021); Levy et al. (2021); Wang
et al. (2021a); Keidar et al. (2021); Huang et al. (2021); He et al. (2021);
Shin et al. (2020); Chen et al. (2020); Manzoor and Shah (2021); Yuan et al.
(2021); Pryzant et al. (2020); Dev et al. (2020); Yang and Feng (2020); Jiang
et al. (2020); Kim and Johnson (2022); Liu and Hulden (2021); Chowdhury and
Chaturvedi (2022); Hiranandani et al. (2020); Cho et al. (2020); Mandal et al.
(2020); Ji et al. (2020); Ahmadian et al. (2020); Ding et al. (2021); Singh et
al. (2021); Esmaeili et al. (2021); Zhang and Long (2021); van Breugel et al.
(2021); Hossain et al. (2021); Maity et al. (2021); Meyer et al. (2021); Zuo
et al. (2022); Ebadian et al. (2022); Wang et al. (2022c); Ray Chaudhury et
al. (2022); Qi et al. (2022); Gardner et al. (2022); Liu et al. (2022a);
Psomas and Verma (2022); Lohaus et al. (2022); Ko et al. (2022); Gaucher et
al. (2022); Soen et al. (2022); Grazzi et al. (2022); Colombo et al. (2022);
Chai and Wang (2022); Xu et al. (2022); Mehrotra and Vishnoi (2022); Wang et
al. (2022b); An et al. (2022a); Jesus et al. (2022); Wang et al. (2022a);
Dooley et al. (2022); van der Linden et al. (2022); Czarnowska et al. (2021)

  * •

Computational techniques to characterize perspective spaces and measure the
differences between perspectives: Schrouff et al. (2022); Wan (2021); Dullerud
et al. (2022); Costa-jussà et al. (2022); Ding et al. (2022); Xie et al.
(2022); Cheng et al. (2022); Wambsganss et al. (2022); Liang et al. (2020);
Lei et al. (2022); Kaneko et al. (2022b); Chopra et al. (2020); Zhang et al.
(2022b); Liu et al. (2022b); Rungta et al. (2022); Blevins and Zettlemoyer
(2022); Kim and Skiena (2022); Tzeng et al. (2020); Ma et al. (2021); Chen et
al. (2022c); Gupta and Dukkipati (2022); Liang et al. (2022); Mukherjee et al.
(2022)

  * •

Using human-LLMs interactions to identify analysts’ blind spots and induce
perspectives representative of those blind spots: Ma et al. (2023); Hertweck
and Räz (2022); Kaneko et al. (2022a); Das and Balke (2022); Shwartz and Choi
(2020); Wu et al. (2022b); Lauscher et al. (2021); Liu et al. (2021)

  * •

Induction of outputs representing diverse perspectives (e.g., “How would a
particular group or organization interpret this event?”): Omrani et al.
(2023); Feng et al. (2023); Jiang et al. (2022); Elsafoury et al. (2022); Kim
and Johnson (2022); He et al. (2022b); Subramanian et al. (2021); Zhang et al.
(2020); Feldman et al. (2020); Sen and Ganguly (2020); Amplayo et al. (2022);
Bansal et al. (2022); Orgad et al. (2022); Ziems et al. (2022); Li et al.
(2022); Zhao et al. (2022); He et al. (2022c); Pujari et al. (2022); Meissner
et al. (2022); Qian et al. (2022); Gaci et al. (2022); He et al. (2022a); Qiu
et al. (2022); Ye et al. (2022); Feng et al. (2022); Wu et al. (2022c);
Sridhar et al. (2022); Wiher et al. (2022); Ruoss et al. (2020); El Halabi et
al. (2020); Duan et al. (2021); Venkatesh et al. (2021); Lapata et al. (2021);
Haghtalab et al. (2022); Zou et al. (2022); An et al. (2022b); Chen et al.
(2022b); Bommasani et al. (2022); Mao et al. (2022); Chen et al. (2022a);
Gautam et al. (2022); Tan et al. (2022); Espinosa Zarlenga et al. (2022); Wu
et al. (2022a); Chakraborty et al. (2022); Aziz et al. (2022); Zhang et al.
(2022a); Jian et al. (2022); Zhang et al. (2022c); Alghamdi et al. (2022)

  * •

Deriving insights from simulated dialogue between LLMs with different
biases/perspectives: Salewski et al. (2023); Hassan et al. (2021); Weller et
al. (2022); Falk and Lapesa (2022); Rusert et al. (2022); Yeo and Chen (2020);
Holtermann et al. (2022)

The existing work mostly focuses on the typical bias types, yet we are
interested in new bias types and their flexible definition relative to the
context.

##  3 Conclusion

We introduce the notion of relative bias depending on the context, with
thoughts on novel solutions to quantify and reduce LLM bias to join the effort
for a more equal, fair, and connected society.

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