# Pandemic Culture Wars: Partisan Asymmetries in the Moral Language of
COVID-19 Discussions

Ashwin Rao 1, 2, Siyi Guo 1, 2, Sze-Yuh Nina Wang 3, Fred Morstatter 2,
Kristina Lerman 2

###### Abstract

Effective response to the COVID-19 pandemic required coordinated adoption of
mitigation measures, like masking and quarantines, to curb virus’s spread.
However, political divisions that emerged early in the pandemic hindered
consensus on the appropriate response. To better understand these divisions,
our study examines a vast collection of COVID-19-related tweets. We focus on
five contentious issues: coronavirus origins, lockdowns, masking, education,
and vaccines. We describe a weakly supervised method to identify issue-
relevant tweets and employ state-of-the-art computational methods to analyze
moral language and infer political ideology. We explore how ideological
divisions and moral language shape conversations about these issues. Our
findings reveal ideological differences in issue salience and the dynamics of
moral language. We find that conservatives use more negatively-valenced moral
language than liberals, but moral language use by conservatives is less
persistent and appears to be driven by dynamics of the news cycle.
Furthermore, we find that political elites use moral rhetoric to a greater
extent than non-elites across most issues. Examining the evolution and
moralization on divisive issues can provide valuable insights into the
dynamics of COVID-19 discussions and assist policymakers in better
understanding the emergence of ideological divisions.

## Introduction

The COVID-19 pandemic presented a major challenge to US society, testing its
resilience and capacity to respond to a crisis. Slowing the rapidly spreading
illness required coordinated adoption of non-pharmaceutical interventions
recommended by public health experts, such as wearing a mask, staying home,
social distancing, and getting vaccines once they became available. However,
the response to the pandemic became politically polarized, with Republicans
and Democrats disagreeing both on the gravity of the crisis and the
appropriate measures to address it (Funk and Tyson 2020). Some Republican
leaders downplayed the severity of the virus and expressed skepticism about
the effectiveness of measures like mask-wearing and social distancing, while
some Democratic leaders advocated for more stringent measures like closing
schools, businesses, and even parks and beaches. Policy differences in
pandemic mitigation emerged at the state level as well. Republican-led states
lifted restrictions on businesses and public gatherings earlier than
Democratic-led states. When the COVID-19 vaccine was approved, many
Democratic-led states introduced vaccine and mask mandates for returning to
work and school, while some Republican-led states passed bills to ban such
mandates. These political divisions hampered effective pandemic response,
leading to more than a million deaths in US, one of the highest mortality
rates among advanced economies (Mordecai and Connaughton 2020).

To understand how the pandemic response became so polarized, we study a large
corpus of pandemic-related tweets (Chen et al. 2020), comprising over 270M
tweets from 2.1M users around United States. We focus on five highly
politicized and contentious wedge issues: coronavirus origins, lockdowns and
business closures, masking, online education, and vaccines. These issues have
been previously found to be salient during the pandemic (Schaeffer 2020;
Connaughton 2021; Rojas 2020; Luttrell and Trentadue 2023; Chan 2021; Pierri
et al. 2022; Rathje et al. 2022). We classify user ideology based on the
partisanship of information sources users share and also describe a method for
identifying issues raised in a tweet. These models enable us to label millions
of tweets and users, thereby allowing us to study polarized discussions at
scale.

We study online conversations about the pandemic from the lens of Moral
Foundations Theory (MFT) (Haidt and Graham 2007). This theory provides a
framework for understanding how moral values shape people’s political
attitudes and behaviors. MFT proposes that individuals’ values and judgments
can be described by five moral foundations: care/harm, fairness/cheating,
loyalty/betrayal, authority/subversion, and sanctity/degradation. Individuals
and groups vary in how much emphasis they place on each foundation. While both
liberals and conservatives consider the care and fairness foundations to be
important, conservatives are thought to endorse the latter three foundations
to a greater extent than liberals (Graham, Haidt, and Nosek 2009). Studying
online conversations from the perspective of MFT can provide insights into the
ways in which partisans use moral appeals to articulate and defend their
political positions. It can also help us understand how to frame messaging so
as to better appeal to the values of each group.

We organize our investigations around the following research questions:

  1. 1.

What COVID-19 issues did liberals and conservatives discuss online?

  2. 2.

How do liberals and conservatives differ in their use of moral language on
different issues?

  3. 3.

Does moral language persist over time or quickly die out? What moral
foundations resonate with partisans on different issues?

  4. 4.

How does moral language use differ between political elites and non-elites?

We propose and evaluate a weakly-supervised method to identify issue-relevant
tweets. This novel method extracts relevant phrases from publicly available
Wikipedia pages and then uses them to detect issues in the text of tweets. In
addition, we use state-of-the-art computational methods to detect moral
foundations in tweets (Guo, Mokhberian, and Lerman 2023) and to infer the
political ideology of users (Rao et al. 2021). These models enable us to track
partisan attention to polarized issues at scale from the start of the COVID-19
pandemic to November 2021.

We uncover important differences in the issues that liberals and conservatives
discuss over time, as well as their use of moral language over time. We also
unveil differences in the moral appeals of political elites and non-elites
across the ideological spectrum. Understanding the moralization on divisive
issues can provide valuable insights into the emergence of polarization and
help public health experts better tailor messaging to the values and concerns
of each group.

## Related Work

Table 1: Issue Detection: Wikipedia Articles Issue |  Wikipedia Articles
---|---
Origins |  Plandemic, Investigations into the origin of COVID-19, COVID-19 lab leak theory
Lockdowns |  COVID-19 lockdowns, COVID-19 protests in the United States, U.S. federal government response to the COVID-19 pandemic, Protests against responses to the COVID-19 pandemic, Social impact of the COVID-19 pandemic in the United States, Stay-at-home order, Social distancing measures related to the COVID-19 pandemic, Quarantine
Masking |  Face masks during the COVID-19 pandemic, Face masks during the COVID-19 pandemic in the United States, Maskne, Anti-mask sentiment
Education |  Impact of the COVID-19 pandemic on education, Homeschooling during the COVID-19 pandemic, Impact of the COVID-19 pandemic on education in the United States
Vaccines |  COVID-19 vaccination in the United States, COVID-19 vaccine, COVID-19 vaccine hesitancy in the United States, COVID-19 vaccine misinformation and hesitancy, Deaths of anti-vaccine advocates from COVID-19, Herman Cain Award
Baseline |  COVID-19 pandemic, COVID-19 pandemic in the United States, Politics of the United States

#### Polarization

American society has been divided by “culture wars” (Hunter 1992), with
liberals and conservatives disagreeing on controversial issues (also known as
wedge issues) like abortion, women’s and LGBTQ+ rights (DiMaggio, Evans, and
Bryson 1996; Evans 2003; Abramowitz and Saunders 2008). This polarization was
further exacerbated by the COVID-19 pandemic. Partisans disagreed on the
virus’s existence and severity, its origins, mitigation strategies like social
distancing and masking, and vaccine mandates. Political ideology was found to
explain differences in compliance with health guidelines and mandates
(Gollwitzer et al. 2020; Grossman et al. 2020). A study of elite messaging on
Twitter at the onset of the pandemic (Green et al. 2020) found that Democratic
elites discussed the pandemic more frequently and emphasized its threats to
public health, while Republican elites focused on China and the pandemic’s
impact on businesses.

These divisions extend beyond political elites: nearly 3 out of 10 Americans
believed that COVID-19 was created in a lab, with 4 in 10 Republicans
endorsing this claim (Schaeffer 2020). Government efforts to restrict the
spread of COVID-19 through lockdowns were met with resistance in the form of
protests and demonstrations, which were largely driven by conservatives:
nearly $52\%$ of conservatives in the United States preferred to have fewer
COVID-19 restrictions as opposed to only $7\%$ of liberals (Connaughton 2021).
These divisions can have important behavioural consequences: following and
interacting with right-leaning political elites, influencers and hyper-
partisan media sources was found to be associated with vaccine hesitancy in
the US (Pierri et al. 2022; Rathje et al. 2022).

#### Moral Foundations

Differences in moral values may underlie these partisan divisions. We analyze
these through the framework of moral foundations theory (MFT), which proposes
five moral foundations of care, fairness, in-group loyalty, respect for
authority, and purity/sanctity (Haidt and Graham 2007). These foundations can
be further categorized into virtues (care, authority, fairness, loyalty &
purity) and vices (harm, subversion, cheating, betrayal & degradation). A
number of language analysis techniques have been used to quantify the moral
foundations in text data (e.g. (Graham and Haidt 2012; Garten et al. 2016;
Guo, Mokhberian, and Lerman 2023)), with past work suggesting that moral
appeals may increase message diffusion (Brady et al. 2017; Wang and Inbar
2022).

To comprehend and mitigate issue-based political divisions, it is crucial to
understand how individuals attach moral significance to their viewpoints
(Koleva et al. 2012). Some past work has done this in the context of COVID-19.
For example, (Díaz and Cova 2022) found that valuing the care foundation was
correlated with compliance with COVID-19 public health recommendations.
Similarly, (Chan 2021) found that endorsing the fairness and care moral
foundations predicted compliance with calls for staying-at-home, wearing
masks, and social distancing, while purity predicted compliance with face
masks and social distancing. Others have found that framing masking-related
health messaging using ideology-matched moral arguments was effective for
liberals, but not conservatives (Luttrell and Trentadue 2023). Moral values
can also be used to predict county-level vaccination rates: counties with
residents who prioritize moral concerns about purity had lower vaccination
rates while counties whose residents prioritized fairness and loyalty had
higher vaccination rates (Reimer et al. 2022).

Some initial work has assessed COVID-19 vaccine discourse on Twitter.
(Borghouts et al. 2023) found that liberal tweets about the COVID-19 vaccine
expressed care, fairness, liberty111Note that some researchers include a sixth
foundation, liberty/oppression. and authority moral foundations more than
conservative tweets, while oppression and harm were referenced more by
conservatives. Similarly, (Pacheco et al. 2022) found that care/harm was
associated with pro-vaccine sentiment whereas, liberty/oppression was
correlated with anti-vaccine attitudes. While vaccinations are a critical
polarizing issue in the discussion of COVID-19, no work as of yet has explored
differences in moral appeals across a broader range of contentious COVID-19
issues.

## Methods

### Data

We use a publicly available dataset (Chen et al. 2020) consisting of 1.4B
tweets about COVID-19 posted between January 21, 2020 and November 4, 2021.
These tweets contained one or more COVID-19-related keywords, such as
coronavirus, pandemic, and Wuhan, among others. We utilized Carmen (Dredze et
al. 2013), a geo-location identification technique for Twitter data to assign
tweets to locations within US. This method leverages tweet metadata, including
“place” and “coordinates” objects that encode location information, such as
country, city, and geographic coordinates. Additionally, the technique uses
mentions of locations in a user’s bio on Twitter to infer their location. A
manual review confirmed that this approach was effective in identifying a
user’s home state. As a result, we were left with 270 million tweets generated
by 2.1 million geo-located users in the United States.

(a) Origins (b) Lockdowns (c) Masking (d) Education (e) Vaccines

Figure 1: Wordclouds showing phrases identified from Wikipedia articles using
SAGE for each issue. We show the top-50 most frequently occurring phrases in
tweets. Bigrams and trigrams are connected by __ for better visualization.

### Issue Detection

Table 2: Sample tweets highlighting wedge issues in COVID-19 online discussions. Issue |  Tweets
---|---
Origins |  No matter what the Chinese Communist Party says, given the mounting evidence, the most likely origins for the China virus are the Wuhan labs studying bats and coronavirus.
Lockdowns |  This is a GREAT idea. We’re all in this together. Take care of each other. #StayHome #TakeItSeriously #FlattenTheCurve #COVID19
Masking |  We’re in the middle of a pandemic and y’all are still coughing and sneezing without covering your mouths? Come on now.
Education |  More glimmers of hope as we “safely” move forward and open up Texas A&M University while containing #COVID19.
Vaccines |  You are joking right? Zero sympathy for anti-vaxxers who quit their jobs rather than get vaccinated. They put us all at risk and make the pandemic prolonged for the world.

As the pandemic progressed, a set of polarizing issues such as origins of the
coronavirus, lockdowns and masking mandates, school closures and online
schooling and vaccines, emerged and grew to dominate conversations about
COVID-19 (Schaeffer 2020; Connaughton 2021; Rojas 2020; Pierri et al. 2022;
Rathje et al. 2022). Previous studies (Garimella et al. 2018; Brady et al.
2017) relied on a small set of manually-selected keywords and phrases to
harvest issue-relevant tweets. However, there is a lack of data on the
reliability of these methods, and a systematic framework to identify issue-
relevant conversations is largely missing. While issue detection has analogues
to topic-detection (Blei, Ng, and Jordan 2003; Grootendorst 2022), one cannot
control for the stochasticity of topics identified in traditional topic-
detection approaches. Although (Card et al. 2015) presented a corpus of
several thousand news articles annotated with frames on traditional policy
issues such as smoking, same-sex marriages, and immigration, there has not
been an attempt to propose a framework for identifying COVID-19 issues in
social media discourse. The lack of expertly annotated data on the issues we
explore in this work prevents us from employing supervised learning
techniques.

We define the origins issue to encompass discussions surrounding the origins
of the pandemic, including topics such as pangolins, gain of function
research, wet-markets, and bats. The lockdown issue comprises content
pertaining to early state and federal mitigation efforts, such as quarantines,
stay-at-home orders, business closures, reopening, and calls for social
distancing. The masking issue is defined by discussions on the use of face
coverings, mask mandates, mask shortages, and anti-mask sentiment. Education-
related content involves tweets about school closures, reopening of
educational institutions, homeschooling, and online learning during the
pandemic. The vaccines issue pertains to discussions about COVID-19 vaccines,
vaccine mandates, anti-vaccine sentiment, and vaccine hesitancy in the US.

We describe a weakly-supervised method to harvest relevant keywords from
Wikipedia pages discussing these issues (see Table 1). We use SAGE
(Eisenstein, Ahmed, and Xing 2011) to identify distinctive keywords and
phrases that are relevant to a specific issue. SAGE calculates the deviation
in log-frequencies of words from a baseline lexical distribution. As a
baseline, we use Wikipedia articles discussing general aspects of the pandemic
and politics in the US (see 1). We concatenate these pages for each issue.
Depending on how we tokenize the corpus, we can use SAGE to identify issue-
relevant n-grams. In this study, we restrict analysis to unigrams, bigrams and
trigrams. Using SAGE we then compare tokens from issue-relevant articles to
baseline tokens to identify keywords and phrases that uniquely define each
issue. We manually verify the keywords identified by SAGE to ensure precision
and relevance. Word clouds in Fig. 1 show the top 50 keywords/phrases
extracted from Wikipedia articles for each issue. We assess the relevance of a
tweet to a issue by the presence of these keywords and phrases. For example,
terms such as ‘Wuhan labs’ or ‘wet markets’ would indicate a tweet is about
the origins of the pandemic, while ‘cover your mouth’, ‘N-95 masks’ would
indicate a masking-related tweet. Examples of tweets discussing these issues
are shown in Table 2.

Table 3: Inter-rater Agreement and Evaluation of Issue Detection. Issue |  Pairwise |  Multi- |  F1- |  Support
---|---|---|---|---
|  Cohen’s $\kappa$ |  annotator |  Score |
|  |  Fleiss’ $\kappa$ |  |
Origins |  $0.73\pm 0.06$ |  0.45 |  0.51 |  25
Lockdowns |  $0.79\pm 0.04$ |  0.47 |  0.90 |  108
Masking |  $0.92\pm 0.04$ |  0.55 |  0.83 |  101
Education |  $0.71\pm 0.07$ |  0.50 |  0.74 |  54
Vaccines |  $0.87\pm 0.05$ |  0.53 |  0.92 |  84

##### Validation

We evaluate issue detection on a subset of tweets, which we draw at random
with the constraint that there are at least five tweets for each combination
of issues and moral foundation. This gave us a test set of 784 tweets, which
were then labeled by five trained annotators. Table 3 shows that the mean
Cohen’s Kappa between each pair of annotators for different issues were all
above 0.7. The Multi-annotator Fleiss’ Kappa values are lower because it
relies on agreement among all five annotators rather than pairs of annotators.
Finally, we evaluate performance of issue detection method on this test set
and show that F1 scores for most issues are above 0.7 (Table 3), indicating
good model performance.

### Morality Detection

We capture the moral sentiments in tweets based on the Moral Foundations
Theory. We train our morality detection model on top of the transformer-based
pretrained language model BERT (Devlin et al. 2018). We train BERT with three
Twitter datasets, including a manually annotated COVID dataset (Rojecki,
Zheleva, and Levine 2021), the Moral Foundation Twitter Corpus dataset with
six different topics (Hoover et al. 2020), and a dataset of political tweets
published by US congress members (Johnson and Goldwasser 2018). We test our
model on a test subset from the COVID dataset and show the F1 scores from 10
random runs in Table 4. The method has good performance on the major
categories such as care and harm, although the performance inevitably varies
with the amount of support for different moral categories, which is also
observed in previous studies (Hoover et al. 2020; Trager et al. 2022).
However, by fusing an in-domain training set that is also about COVID-19,
along with other datasets consisting of various topics, we are improving the
model generalizability when applied to the target data (Guo, Mokhberian, and
Lerman 2023). We compare our method with a widely used dictionary-based method
Distributed Dictionary Representations (DDR) (Garten et al. 2018) and find our
method outperforms DDR for most moral categories (Refer Table 4).

Table 4: Evaluation of Morality Detection Method on Test Data. We show the mean and standard deviation of F1 scores over 10 random runs, and the median of support of 10 runs. Moral Foundation | DDR | Ours | Support
---|---|---|---
Care | $0.71\pm 0.02$ | $\mathbf{0.82\pm 0.02}$ | 239
Harm | $0.21\pm 0.04$ | $\mathbf{0.75\pm 0.03}$ | 111
Fairness | $0.09\pm 0.08$ | $\mathbf{0.24\pm 0.17}$ | 5
Cheating | $0.10\pm 0.12$ | $\mathbf{0.24\pm 0.18}$ | 7
Loyalty | $0.33\pm 0.08$ | $\mathbf{0.59\pm 0.06}$ | 56
Betrayal | $0.03\pm 0.02$ | $\mathbf{0.16\pm 0.07}$ | 11
Authority | $0.00\pm 0.00$ | $\mathbf{0.68\pm 0.09}$ | 42
Subversion | $0.20\pm 0.15$ | $\mathbf{0.35\pm 0.15}$ | 11
Purity | $\mathbf{0.47\pm 0.26}$ | $0.33\pm 0.11$ | 8
Degradation | $0.05\pm 0.04$ | $\mathbf{0.20\pm 0.35}$ | 2

### Ideology Detection

A number of methods identify the political ideology of social media users,
with recent works (Le et al. 2019; Nikolov, Flammini, and Menczer 2020;
Cinelli et al. 2021; Rao et al. 2021) leveraging URLs that users share in
tweets. We use the method discussed in (Rao et al. 2021) to estimate the
ideology of individual users. This method relies on Media Bias-Fact Check
(MBFC) (Check 2023) to compute ideology scores for individuals. MBFC provides
ideological leaning for over 6K Pay-Level Domains (PLDs) with leaning
categorized as Left/Hardline Liberal ($0$), Left-Center ($0.25$), Least-
Biased/Center ($0.5$), Right-Center ($0.75$), Right/Hardline Conservative
($1$). Each individual’s ideological leaning is estimated to be the weighted
average of the leanings of the URLs they share.

These scores are then binarized with a threshold of $\leq 0.4$ for liberals
($0$) and $\geq 0.6$ for conservatives ($+1$). The model described in (Rao et
al. 2021) then leverages a fastText embedding model pre-trained on Twitter
data (Pagliardini, Gupta, and Jaggi 2018), to generate embeddings for user’s
tweets over time. These embeddings serve as features to train a Logistic
Regression classifier to predict the binarized ideological leanings for all
users, including users who have not shared URLs from domains in the MBFC. We
were able to identify ideology for 2.1M US users comprising of 1.6M liberal
users and 500K conservatives.

##### Validation

We assess the level of agreement between the ideology scores estimated by the
methods proposed in (Rao et al. 2021) and (Barberá 2015), which uses the
follower graph to estimate user ideology. We discretize the continuous ideal
point estimates in (Barberá 2015) using a threshold of $<0$ for liberals and
$>0$ for conservatives, which we respectively label as $0$ and $+1$. We
identify approximately 35K users who appear in both datasets and calculate the
F1-score between the two models. The resulting F1-score of $0.75$ and Jaccard
overlap of $0.87$ indicates a high level of agreement between the two methods.

Figure 2: Dynamics of attention to issues. Daily fraction of (a) original
tweets, (b) retweets and (c) replies related to each issue. We use 7-day
rolling average to reduce noise. Major events are marked with vertical lines
of the same color as the issue and placed along the timeline. Figure 3:
Difference in the daily share of original tweets by liberals and conservatives
on issues. Positive (negative) values indicate that the issue is $x\%$ more
significant for liberals (resp. conservatives). Seven-day rolling average is
leveraged to reduce noise. Vertical lines mark significant events that brought
shifts in issue-related discussions.

(a) Origins (b) Lockdowns (c) Masking (d) Education (e) Vaccines

Figure 4: Partisan differences in framing each issue. Values show the log odds
ratio of phrases used by liberals (blue) and conservatives (red) in issue-
relevant tweets across the time period.

## Results

Figure 2 shows the daily share of tweets, retweets and replies discussing each
issue, smoothed using a 7-day rolling window, with vertical lines marking
major issue-related events. The five issues we monitored accounted for small
share of all messages at the beginning of the pandemic, but grew to dominate
all discussions. Lockdowns were among the first mitigation efforts carried out
by state governments, with California the first state in the mainland US to
issue a stay-at-home order on March 19, 2020. This date marks the start of an
increase in lockdown related discussions. Although the peak in the masking
discussions did not occur until the Black Lives Matter protests in June 2020,
Fig. 2 indicates an early upward trend on April 3rd, 2020, when the CDC
recommended face coverings to prevent infections. On July 8, 2020 President
Trump promised to cut funding to schools that did not reopen for in-person
instruction, which is associated with a spike in education related
discussions. Similarly, Pfizer-BioNTech’s announcement (Pfizer 2020) on
November 9, 2020 that their COVID-19 vaccine was highly effective in Phase 3
clinical trials created a spike in vaccine related tweets. A spike in the
discussions of COVID-19 origins is seen on May 23, 2021 when the Wall Street
Journal published an article citing an US Intelligence Report that three
researchers from the Wuhan Institute of Virology had sought hospital care in
November 2019 shortly before the outbreak (Gordon, Strobel, and Hinshaw 2021).
The story marked the first major report on the origins of the pandemic by a
major news organization. These results provide a qualitative check of
robustness of our issue detection approach, as well as insights into
importance of issues at different times. Fig. 2 also shows differences in the
engagement on different issues. For example, tweets about masking seemed to
garner more replies, suggesting that they may have generated more discussion
or controversy.

(a) Time series: Education (b) Auto-correlation: Education

Figure 5: Partisan asymmetries in the persistence of moral language. (a) Time
series of moral language used by liberal and conservative accounts in the
discussions of education, separated by moral foundation. We show only the
three more widely present foundations, and for benefits of visualization,
reflect the time series of conservative tweets about the x-axis. White line
shows the difference between the two groups, indicating which side mentions
the issue more. (b) Auto-correlation function (ACF), showing decay of the
correlation of the time series with its lagged version for different lags (in
days).

(a) ACF: Pre-Vaccine (b) ACF: Post-Vaccine

Figure 6: Persistence of moral language. Each cell of the heatmap shows the
minimum lag time in day required for the auto-correlation function to drop
below the upper confidence level when time series are split into the pre-
vaccine period (before 12/11/2020) and post-vaccine period.

(a) Origins (b) Lockdowns (c) Masking (d) Education (e) Vaccines

Figure 7: Partisan differences in moral language use along each issue. Values
indicate percentage of tweets shared by liberals (blue) and conservatives
(red) containing the specific moral foundation. Values for Care and Harm are
downscaled by a factor of two for better visualization.  Figure 8: Moral
language differences between elites and non-elites. Boxplot shows the
differences in the share of daily tweets made by each group on the issues: (a)
origins, (b) lockdowns,(c) masking, (d) education and (e) vaccines.

### Issue Salience by Ideology

Next, we examine the relative prevalence of issues in tweets by conservatives
and liberals. Since the Twitter user population in our sample is ideologically
unbalanced, with three times as many liberals and conservatives, we cannot
directly compare the volume of messages posted by each side. Instead, we
compare the share of all messages posted by liberals and conservatives that
are devoted to each issue. Specifically, we calculate the difference $\Delta$
along an issue $i$ at time (day) $d$ as follows:
$\Delta(i,d)=(\frac{\tau_{l}(i,d)}{\tau_{l}(d)}-\frac{\tau_{c}(i,d)}{\tau_{c}(d)})$
where $\tau_{l}$ and $\tau_{c}$ represent the number of all original tweets
made by liberals and conservatives on that day respectively, and
$\tau_{l}(i,d)$ and $\tau_{c}(i,d)$ represent the number of original tweets
about the issue posted by each group.

Fig. 3 illustrates the difference in the share of issue-relevant original
tweets posted by liberals and conservatives. Positive (negative) values on the
y-axis indicate higher prevalence for liberals (conservatives). Vertical lines
mark events where significant shifts in conversations occurred. While liberals
discussed education more than conservatives, conservatives discussed lockdowns
and COVID-19 origins more than liberals. Early in the pandemic, masking was an
issue marginally more important to liberals, but after George Floyd’s killing,
it was more discussed among conservatives. Ideological divisions about masking
remained small, however, until the summer of 2021. We also see interesting
changes in vaccine-related discussions. While there were minimal differences
between liberals and conservatives prior to the vaccine release, we see that
liberals were discussing the vaccine more than conservatives after it was made
available in the US on December 11, 2020. However, following vaccine mandates
for federal workers on September 9, 2021 (Biden 2021), the issue became a
greater focus for conservatives, likely due to their outrage about the
mandates. After the Wall Street Journal’s report (Gordon, Strobel, and Hinshaw
2021) in May 2021 claiming that COVID-19 originated at the Wuhan Institute of
Virology, there was an increase in conservative’s share of COVID-19 origins
tweets. While we limit our discussion here to original tweets due to space
constraints, we see similar trends in our analysis of retweets and replies.

To characterize differences in liberal and conservative tweets about issues,
we identify adjectives that were used to modify issue-relevant phrases
(anchors) extracted earlier. We leverage SpaCy’s Dependency Parser to extract
dependency relations of the form: XX-$amod\rightarrow ANC$ and
XX–$amod\rightarrow YY\leftarrow amod$–$ANC$, where $XX$ refers to the
extracted adjective and $ANC$ refers to an issue-relevant phrase. We then pair
the extracted adjective with its corresponding anchor as a phrase of the form
$XX$_$ANC$ and compute the log-odds ratio of these phrases being expressed in
liberal and conservative tweets. Wordclouds in Fig. 4 depict the top 10 most-
likely used phrases in liberal (blue) and conservative (red) issue-discourse.

Conservatives employ conspiratorial phrases such as “batshit conspiracy”,
“fake plandemic”, and “military bioweapon” to discuss the origins issue. In
contrast, liberals use phrases like “likely bats” and “medical virology”,
suggesting uncertainty. When it comes to lockdowns, conservatives express
negative sentiment, framing them as “deadly”, “unconstitutional”, and
“stringent”. Liberals, however, approach the topic cautiously, using phrases
such as “premature reopening”, “safe reopening”, and “soft lockdowns”. Similar
patterns emerge in the discourse on masking, with conservatives deeming mask
mandates as “unconstitutional” and “useless”, while liberals focus on
“reusable masks” and “mandatory masking”. Concerning education, conservatives
express distrust in mitigation strategies employed by educational
institutions, using terms like “fascist education” and “unmasked students”,
while liberals highlight “hybrid learning”, “vulnerable students”, and “early
education.” On the issue of vaccines, conservatives exhibit skepticism,
referring to “experimental mRNA” and “dangerous vaccines”. In contrast,
liberals emphasize the benefits, discussing “free vaccinations” and “mobile
vaccines”.

### Persistence of Moral Language

The fluctuations in the daily volume of tweets, even when disaggregated by
issue (Fig. 2), obscure the dynamics and complexity of moral expression. To
better understand how partisans use moral appeals, we split original tweets on
a particular issue by moral foundation label and user ideology. Fig. 5 shows
the share of tweets posted by liberals and conservatives that express each
moral foundation in the discussions of education. To ease visualization, the
time series for conservatives tweets is reflected around the x-axis. In this
analysis, both virtues and vices of a moral foundation contribute to its
presence in a tweet. As with other issues, the care/harm foundation is the
most widely expressed dimension of morality, followed by authority/subversion
and fairness/cheating. The two remaining moral foundations are far less
common. The time series of moral expression displays complex dynamics, with
strong symmetry between liberals and conservatives suggesting that both groups
are driven by external events. However, there are also notable differences.
The care/harm language used by liberals shows characteristic weekly patterns
of activity, with dips on weekends, that are less visible in conservative
tweets. In contrast, conservative language shows more short-lived spikes that
dissipate as quickly as they appear.

To quantify differences in the dynamics of moral language, we compute the
auto-correlation function (ACF) of each time series. This function measures
the correlation between a time series and its own past values at different
lags. Fig. 5 shows this function at different lags from 0 to 60 days. By
design, ACF starts at one and decays at larger lags, becoming insignificant
when it drops below the confidence bound (horizontal line). For liberals, the
ACF values for the care/harm and fairness/cheating foundations have peaks at
lag 7, 14, 21, etc. days, a signature of weekly pattern in the time series.
The ACF for fairness/cheating decays quickly for conservatives, becoming
insignificant for lags larger than 10 days. The fast decaying ACF is
associated with short-lived spikes in the time series of moral language that
occur at irregular time intervals. This pattern suggests that conservatives
react to external events but their attention quickly fades. In contrast, a
slowly-decaying ACF for liberals suggests that the moral language resonates
within a community and continues to be used long after the precipitating
event.

We measure how long it takes for ACF to decay to insignificant values. Fig. 6
shows these values for different issues discussed by liberals and
conservatives. We split the time series into two periods: before the first
COVID-19 vaccine became available in the US (December 11, 2020) and after. The
interesting differences in the two heatmaps suggest changes in the moral
language. In general, the care/harm and authority/subversion foundations
resonate with both communities, taking a long time to decay. Moral language
tends to persist longer in the discussions of liberals compared to
conservatives. The exceptions to this observations are seen in the discussions
about vaccines, lockdowns and COVID-19 origins in the post-vaccine period.
Given that pandemic origins is a partisan issue of greater interest to
conservatives, we expect their discussions on this topic to persist longer,
consistently with the trends we observe in the heatmap. In the discussions of
lockdowns, care and fairness resonate more for conservatives, while loyalty
resonates more for liberals. Purity language is no longer resonant for either
community; it is largely driven by the 3–4 day news cycles. Another surprising
reversal is for conservative discussions of vaccines, where purity language
becomes far more resonant in the post-vaccine period. This likely reflects the
worries of conservatives to inject themselves with an “experimental vaccine”.
This is even more surprising as purity language becomes less resonant among
both liberals and conservatives in the post-vaccine period, making its
persistence among conservative discussions of vaccines that much more
remarkable. The findings suggest that informational campaigns aimed at
different groups should appeal to moral foundations that resonate within each
group.

Similar analysis of the moral language of retweets shows that ACF quickly
decays, with correlations losing significance after three to five days. This
suggests that moral language in retweets is driven largely by the news cycle.

### Partisan Differences in Moral Language

To study which moral values are important to liberals and conservatives, we
measure the share of issue-related tweets posted by each group that expresses
a certain moral foundation. We examine moral valence by separately calculating
the share for vices and virtues. The proportion $\rho(m,i)$ of original tweets
related to issue $i$ that express moral foundation vice or virtue $m$ is:
$\rho(m,i)=\frac{\tau(m,i)}{\tau(i)},$ where, $\tau(i)$ returns the number of
original tweets on issue $i$.

Figure 7 shows partisan differences in the proportions $\rho$. Across issues,
we find that care and harm the most prevalent moral virtues and vices,
followed by authority and subversion, with very few tweets expressing purity
and degradation. Conservatives express more vices than virtues. There are
other partisan differences in the use of moral language. For example, while
both appealed to subversion, betrayal, and cheating when discussing the
origins of COVID-19, liberals focus more on harm (see Fig. 7(a)). This could
reflect greater concern among liberals about the potential harms of these
speculations, as they led to a rise in anti-Asian hate crimes across the
nation (Gover, Harper, and Langton 2020).

Across lockdowns, masking, education, and vaccine issues (Figs. 7(b)-(e)),
conservatives expressed greater levels of harm and subversion than liberals,
while liberals expressed more care. This may indicate greater support for
mandates (e.g., school shutdowns and vaccine mandates) among liberals, and
opposition by conservatives, consistent with partisan divides on shutdowns,
social distancing, and masks (Deane, Parker, and Gramlich 2021).

Overall, we find similarities and interesting differences in the moral values
that liberals and conservatives appeal to when discussing COVID-19.
Conservatives appear to make more negatively valenced moral appeals when it
comes to lockdowns, masking, education, and vaccines.

### Differences in Moral Language Use by Elites and Non-Elites

Past research has shown that moral language used by political elites (Clifford
and Jerit 2013; Wang and Inbar 2021) in the US can influence the opinions and
moral reasoning of non-elites (Clayton et al. 2021). In this work, we compare
the moral language used by elites and non-elites. Political elites in US
include prominent members of Congress, politicians, journalists and media
pundits. Previous studies (Wojcieszak et al. 2022; Shugars et al. 2021) have
curated extensive lists of Twitter handles for over 4K US political elites. To
ensure a fair comparison with non-elites, we randomly sample non-elite
original tweets such that the number matches the number of tweets from elites
on a given day. The random sampling also ensures that tweets from non-elites
with higher overall activity are selected. We then calculate the daily
proportion of issue-related tweets that express a certain moral foundation.
Specifically, we calculate the proportion $\rho$ of original tweets relevant
to issue $i$ that express moral foundation $m$ on day $d$:
$\rho(m,i,d)=\frac{\tau(m,i,d)}{\tau(i,d)}$ where, $\tau$ returns the number
of original tweets. We calculate this separately for elites and non-elites. To
account for variation, we bootstrap non-elite tweets 100 times.

Figure 8((a)-(e)) compares moral language use by elites and non-elites across
issues. Boxplots show the distribution of the daily share of tweets in each
category. We do not show loyalty/betrayal and purity/degradation foundations,
as they were not substantially expressed by elites. The lower variance for
non-elites is an attribute of bootstrapping. We rely on the non-parametric
Mann-Whitney U Test to test the significance of our comparisons. All
differences are assessed for significance at $p<0.001$ unless otherwise
specified.

On the issue of origins (Fig. 8(a)), non-elites express more care, subversion
and cheating than elites. Conservative non-elites use more harm language than
conservative elites, a difference that is not significant for liberals.
Lockdowns, masking, and education tweets all show a pattern of elites using
more moral language than non-elites across most moral foundations, with
liberal elites and non-elites tending to refer more to care (Fig. 8(b-d)).

Discussions of vaccines deviate somewhat from these trends (Fig. 8(e)), with
conservative non-elites expressing more harm than both liberals and
conservative elites. Conservatives (both elites and non-elites) also express
more subversion than liberals.

## Conclusions

The COVID-19 pandemic presented a significant challenge to society. The
ensuing response to the pandemic quickly became polarized, with liberals and
conservatives disagreeing on the severity of the pandemic and appropriate
measures to address it. Sharp differences in issue positions were observed in
discussions relating to the origins of the pandemic, lockdowns and business
closures, masking mandates, disruptions to education, and vaccines. Previous
studies (Schaeffer 2020; Connaughton 2021; Rojas 2020; Luttrell and Trentadue
2023; Chan 2021; Pierri et al. 2022; Rathje et al. 2022) attest to the
salience of these issues during the pandemic. We use a massive corpus of over
200M tweets (Chen et al. 2020) to assess these ideological divisions.

We find that importance of these issues varies by political ideology. While
liberals dominated the discussion on education, tweets about COVID-19 origins
and lockdowns were much more prevalent among conservative. Tweets about the
vaccine were predominantly made by liberals once the vaccine became available,
but switched to being more important to conservatives after federal vaccine
mandate were announced. We also found strong differences in moral language
among liberals and conservatives. We found that tweets expressing the
care/harm and authority/subversion foundations were most persistent,
suggesting that these values resonated with both groups. Our time series
analysis of moral expression suggests that conservatives were more reactive,
driven by the dynamics of the news cycle. Our analysis revealed further
differences in the moral language of partisans. While conservatives relied on
negatively-valenced moral language, like harm and subversion, liberals
primarily leveraged care. We also found that political elites in the US tended
to use more moral language compared to non-elites, regardless of ideology. In
sum, our findings suggest that differences in the expressions of moral values
drove ideological divisions on different issues, fueling polarization.

However, it is worth noting that our methods have several limitations. We
infer geo-location of tweets based on tweet metadata and user bios’ which may
not be accurate. Expression of ideological preferences and moral appeals can
be both subtle and highly subjective. Methods leveraged to identify ideology
and moral foundations, while state-of-the-art, may suffer from inconsistencies
and biases given the task’s inherent complexity. While our method to identify
issue relevant tweets is highly scalable, less explicit references to issues
in tweets may be missed by this method. Future work could test a supervised
classifier that may more accurately identify all relevant tweets. While we
made an effort to combine available lists of US political elites on Twitter,
this list is by no means exhaustive.

### Broader perspective, ethics and competing interests

All data used in this study is publicly available (Chen et al. 2020; Check
2023). The study was deemed exempt from review by the Institutional Review
Board (IRB) at the University of Southern California, as it relied solely on
publicly available data. Our study also adheres to Twitter’s terms of service
(Twitter 2022). Tweet objects contain user information and this brings user
anonymity into question. Users can restrict their tweets by switching to a
private account or by deleting their tweets. Additionally, we preserve user
anonymity during analysis by relying on user IDs instead of screen names.
However, we acknowledge that many Twitter users may not be aware their data is
used for research purposes (Fiesler and Proferes 2018).In this article, we
present aggregated statistics to address further mitigate this concern. The
authors declare that there are no competing interests.

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# Hadron yield correlation and constituent quark degree of freedom in heavy
ion collisions

Rui-qin Wang  Department of Physics, Qufu Normal University, Shandong 273165,
People’s Republic of China  Feng-lan Shao<EMAIL_ADDRESS>Department
of Physics, Qufu Normal University, Shandong 273165, People’s Republic of
China  Jun Song  Department of Physics, Shandong University, Shandong 250100,
People’s Republic of China  Qu-bing Xie  Department of Physics, Shandong
University, Shandong 250100, People’s Republic of China

###### Abstract

Based on the assumption of the production of deconfined quark matter, we use a
quark combination model to systematically investigate hadron yields in heavy
ion collisions from RHIC $\sqrt{s_{NN}}=200,130,62.4$ GeV to SPS
$E_{beam}=158,80,40,30,20$ AGeV. We find that as the collision energy is
greater than or equal to 30 AGeV the yields of various hadrons, their
correlations, in particular, the observables
$A=\frac{\overline{\Lambda}~{}k^{-}~{}p}{\Lambda~{}k^{+}~{}\overline{p}}$ and
$B=\frac{\Lambda~{}k^{-}~{}\overline{\Xi}^{+}}{\overline{\Lambda}~{}k^{+}~{}\Xi^{-}}$
, are all reproduced; however, as the collision energy drops to 20 AGeV quark
combination fails. This indicates that the constituent quark degrees of
freedom represent a decisive factor in thermal hadron production above 30 AGeV
and seem to be invalid at 20 AGeV. In addition, hadron yields as well as
particle ratios at midrapidity in the most central Pb+Pb collisions at
$\sqrt{s_{NN}}=5.5$ TeV are predicted.

###### pacs:

25.75.Dw, 25.75.Gz, 25.75.Nq, 25.75.-q

##  I INTRODUCTION

The question at which collision energy in heavy ion collisions the
deconfinement is first reached has attracted more and more attentions in
recent years Andronic2009PLB ; YGMa2009nuclex ; Hove1982PLB ; Gazdzicki ;
Gazdz ; GAGo99 ; Bugaev2003PLB ; Akkelin . The Beam Energy Scan programme of
NA49 experiment at the CERN-SPS has suggested a preliminary answer — around 30
AGeV20A30A2008PRC . The ongoing Beam Energy Scan programme of STAR
Collaboration at Brookhaven National Lab provides an opportunity to study it
in more detailstarESC ; besRHIC . Once the deconfined hot and dense quark
matter is produced in heavy ion collisions, the observables of various thermal
hadrons after hadronization, e.g. yields and momentum spectra etc, have some
correlations originated from early quark degrees of freedom. One of the most
typical examples is the elliptic flow (v2) of hadrons measured at RHIC
energies. As both v2 and transverse momentum($p_{T}$) are divided by the
constituent quark number of hadron, the rescaled v2 of various baryons and
mesons, which are just that of constituent quarks, almost coincide with each
other in the intermediate $p_{T}$ range AAdare2007PRL . If the hot and dense
quark matter is hadronized by quark (re-)combination/coalescenceFries2003PRL ;
Greco2003PRL ; FLShao2005PRC , as is commonly accepted, these correlations of
hadrons can be beautifully explained. In quark (re-)combination/coalescence
scenario, quarks and antiquarks are available in unbound state before
hadronization and they can coalesce freely into various hadrons, and thereby
these correlations from early quark degrees of freedom among different hadron
species are naturally formed. On the other hand, if the deconfined quark
matter is not produced at all in collisions, there is no free quarks and
antiquarks (much less their subsequent combination) and these so-called
“quark-level” correlations of hadrons maybe disappear or contort. Therefore,
we can study whether the deconfinement is achieved by investigating these
correlations among various hadrons produced in heavy ion collisions.

Hadron yield is one of the most significant observables from which one can
obtain a lot of important information on the hot nuclear matter produced at
the early stage of relativistic heavy ion collisions. At the high collision
energies RHIC and top SPS, the yields of different hadron species have shown
an explicit “quark-level” correlation in quark (re-)combination/coalescence
scenario A.Bialas1998PLB ; J.Zimanyi2000PLB ; FLShao2005PRC . In this paper,
we make an energy scan from RHIC energies $\sqrt{s_{NN}}=200,130,62.4$ GeV to
SPS energies $E_{beam}=158,80,40,30,20$ AGeV to study at which collision
energy this “quark-level” correlation of hadron yields first breaks. In
particular, we define two correlation quantities
$A=\frac{\overline{\Lambda}~{}k^{-}~{}p}{\Lambda~{}k^{+}~{}\overline{p}}$ and
$B=\frac{\Lambda~{}k^{-}~{}\overline{\Xi}^{+}}{\overline{\Lambda}~{}k^{+}~{}\Xi^{-}}$,
sensitive to quark degrees of freedom. The values of A and B are equal to one
in the framework of quark (re-)combination/coalesce, independent of models.
The deviation of A and B from one or not can be regarded as a possible signal
of deconfinement in heavy ion collisions. We apply a quark combination model,
which can exclusively describe hadron production and well reproduce the yields
and momentum spectra of final-state hadrons in relativistic heavy ion
collisionsFLShao2007PRC ; CEShao2009PRC ; JSong2009MPA ; DMWei2008MPA ;
YFWang2008CPC ; WHan2009PRC , to carry out the concrete calculations.

The paper is organized as follows. In Sec.II, we present the relations among
yields of various hadrons in quark (re-)comibination/coalescence scenario. In
Sec.III, we use the quark combination model to calculate the yields of various
hadrons, their yield ratios and correlation quantities
$A=\frac{\overline{\Lambda}~{}k^{-}~{}p}{\Lambda~{}k^{+}~{}\overline{p}}$ and
$B=\frac{\Lambda~{}k^{-}~{}\overline{\Xi}^{+}}{\overline{\Lambda}~{}k^{+}~{}\Xi^{-}}$
at midrapidity in the most central A+A collisions at different energies.
Sec.IV summaries our work.

##  II hadron yields in quark (re-)combination/coalescence scenario

Let us start from the general inclusive formula of hadron production in quark
(re-)combination/coalescence scenario

$\displaystyle N_{M}$ $\displaystyle=$ $\displaystyle\int
dp_{1}dp_{2}F_{q\bar{q}}(p_{1},p_{2})\,\mathcal{R}_{M}(p_{1},p_{2})$ (1)
$\displaystyle N_{B}$ $\displaystyle=$ $\displaystyle\int
dp_{1}dp_{2}dp_{3}F_{qqq}(p_{1},p_{2},p_{3})\,\mathcal{R}_{B}(p_{1},p_{2},p_{3}).$
(2)

Here $F_{q\bar{q}}$ ($F_{qqq}$) is the joint quark-antiquark (three quark)
distribution. $\mathcal{R}_{M}$ ($\mathcal{R}_{B}$) is the combination
function which stands for the formation probability of quark antiquark (three
quarks) into a meson (baryon), dominated by chromodynamics. In sudden
approximation, it is equal to the overlap between two (three) quark wave
functions and the wave function of meson (baryon). Neglecting exotic (multi-
quark) states, mesons and baryons exhaust all fate of quarks and antiquarks.
One reaches the following relations : $\sum N_{M}+3\sum N_{B}=\sum N_{q}$ and
$\sum N_{M}+3\sum N_{\bar{B}}=\sum N_{\bar{q}}$, where $N_{q}$ ($N_{\bar{q}}$)
is the quark (antiquark) number of flavor $q$($\bar{q}$). Extracting $N_{q}$
and $N_{\bar{q}}$ from the joint two (three) quark distribution $F_{q\bar{q}}$
($F_{qqq}$) and putting them out of the integral, one has the following
schematic relations between hadron yields and quark numbers after integrating
over quark momenta

$N_{M(q\bar{q})}\propto C_{M}N_{q}N_{\bar{q}},\hskip
28.45274ptN_{B({qqq})}\propto C_{B}N_{q}N_{q}N_{q},$ (3)

and quark number conservation will fix the proportionality coefficient. The
effects of combination function on hadron yields are characterized with the
factors $C_{M}$ and $C_{B}$, and $C_{M}=C_{\overline{M}}$ and
$C_{B}=C_{\overline{B}}$ are assumed.

One realized method of quark number conservation during combination is adding
a factor $b_{q}$ for each quark flavor in above equations, as did in ALCOR
model alcor95 ,

$\displaystyle N_{M(q\bar{q})}$ $\displaystyle=$ $\displaystyle
C_{M}(b_{q}N_{q})(b_{\bar{q}}N_{\bar{q}}),$ (4) $\displaystyle N_{B({qqq})}$
$\displaystyle=$ $\displaystyle C_{B}(b_{q}N_{q})(b_{q}N_{q})(b_{q}N_{q}).$
(5)

Then $b_{q}$ can be uniquely determined by quark number conservation.
According to Eqs.(4) and (5), we have the following relations between hadrons
and the corresponding antihadrons

$\displaystyle\frac{\overline{p}}{p}$ $\displaystyle=$
$\displaystyle(\frac{b_{\overline{q}}\overline{q}}{b_{q}q})^{3},\hskip
45.52458pt\frac{k^{+}}{k^{-}}=(\frac{b_{q}q}{b_{\overline{q}}\overline{q}})(\frac{b_{\overline{s}}\overline{s}}{b_{s}s}),$
$\displaystyle\frac{\overline{\Lambda}}{\Lambda}$ $\displaystyle=$
$\displaystyle(\frac{b_{\overline{q}}\overline{q}}{b_{q}q})^{2}\frac{b_{\overline{s}}\overline{s}}{b_{s}s},\hskip
25.6073pt\frac{\Xi^{-}}{\overline{\Xi}^{+}}=(\frac{b_{s}s}{b_{\overline{s}}\overline{s}})^{2}\frac{b_{q}q}{b_{\overline{q}}\overline{q}}.$

Here, we use particle symbols stand for their numbers for short, $q$ for light
quark number and $s$ for strange quark number. Hiding the quark content in
hadron yield, we obtain the following interesting relations among different
hadron species

$\frac{\overline{\Lambda}~{}k^{-}}{\Lambda~{}k^{+}}=\frac{\overline{p}}{p},\hskip
28.45274pt\frac{\Lambda~{}k^{-}}{\overline{\Lambda}~{}k^{+}}=\frac{\Xi^{-}}{\overline{\Xi}^{+}}.$

We define correlation quantities $A$ and $B$ as follows

$A=\frac{\overline{\Lambda}~{}k^{-}~{}p}{\Lambda~{}k^{+}~{}\overline{p}}\hskip
28.45274ptB=\frac{\Lambda~{}k^{-}~{}\overline{\Xi}^{+}}{\overline{\Lambda}~{}k^{+}~{}\Xi^{-}}.$
(6)

If the quark matter exists and hadronizes via sudden (re-)combination/coalesce
in heavy ion collisions, A and B should be equal to one for directly produced
hadrons. It is a general result under the constraint of quark number
conservation, which is independent of specific models.

Different from the well-known recombination and coalescence models
Fries2003PRL ; Greco2003PRL , the quark combination model FLShao2005PRC ;
QBXie1988PRD  is unique for its combination rule. The main idea of the
combination rule is to line up the (anti)quarks in a one-dimensional order in
phase space, e.g., in rapidity, and then let them combine into initial hadrons
one by one according to this orderFLShao2005PRC . Three (anti)quarks or a
quark-antiquark pair in the neighborhood form a (anti)baryon or a meson,
respectively. At last all quarks and antiquarks are combined into hadrons. The
relations between hadron yields and the corresponding quark numbers are easily
obtained CEShao2009PRC . With this rule, the model can give the yields and
momentum distributions of all hadrons (included in the model) in an event,
possessing some exclusive nature. The decay of the short-life resonances is
systematically taken into account in the model. The model has been realized in
Monte Carlo program and has described many properties of hadron production in
relativistic heavy ion collisions FLShao2007PRC ; CEShao2009PRC ; JSong2009MPA
; DMWei2008MPA ; YFWang2008CPC ; WHan2009PRC .

##  III RESULTS AND DISCUSSIONS

In this section, we use the quark combination model to calculate the hadron
yields and their ratios as well as correlation quantities
$A=\frac{\overline{\Lambda}~{}k^{-}~{}p}{\Lambda~{}k^{+}~{}\overline{p}}$ and
$B=\frac{\Lambda~{}k^{-}~{}\overline{\Xi}^{+}}{\overline{\Lambda}~{}k^{+}~{}\Xi^{-}}$
at midrapidity in the most central A+A collisions at
$\sqrt{s_{NN}}=200,~{}130,~{}62.4$ GeV and
$E_{beam}=158,~{}80,~{}40,~{}30,~{}20$ AGeV. The predictions at LHC are also
presented. The necessary input of the model, i.e., quark distribution just
before hadronization, can be obtained by applying the hydrodynamics to
describe the evolution of hot and dense quark matter just before
hadronization, see Appendix A for details.

###  III.1 Hadron yields, their ratios and correlation quantities A and B

Table 1: The calculated hadron yields $dN/dy$ at midrapidity in the most
central A+A collisions at different energies. The experimental data are taken
from Refs20A30A2008PRC ; 20A30A40A80A158A2006PRC ; 20A30A40A80A158A2008PRC ;
40A80A158A2004PRL ; 40A158A2005PRL ; 158A2002PRC ; 622009nuclex ; 1302002PRL ;
1302004PRL ; 1302004PRC ; 2002004PRC ; 2002007PRL .

| Au+Au 200 GeV | Au+Au 130 GeV | Au+Au 62.4 GeV | Pb+Pb 158 AGeV
---|---|---|---|---
| data | model | data | model | data | model | data | model
$\pi^{+}$ | $286.4\pm 24.2$ | 281.0 | $276\pm 3\pm 35.9$ | 267.2 | $233\pm 17$ | 227.4 | $170.1\pm 0.7\pm 9$ | 165.1
$\pi^{-}$ | $281.8\pm 22.8$ | 281.8 | $270\pm 3.5\pm 35.1$ | 270.5 | $237\pm 17$ | 233.5 | $175.4\pm 0.7\pm 9$ | 176.5
$k^{+}$ | $48.9\pm 6.3$ | 48.6 | $46.7\pm 1.5\pm 7.0$ | 45.2 | $37.6\pm 2.7$ | 38.3 | $29.6\pm 0.3\pm 1.5$ | 27.2
$k^{-}$ | $45.7\pm 5.2$ | 46.1 | $40.5\pm 2.3\pm 6.1$ | 42.4 | $32.4\pm 2.3$ | 32.2 | $16.8\pm 0.2\pm 0.8$ | 16.2
$p$ | $18.4\pm 2.6$ | 17.0 | $28.7\pm 0.9\pm 4.0$ | 25.7 | $29.0\pm 3.8$ | 29.1 | $29.6\pm 0.9\pm 2.9$ | 30.1
$\overline{p}$ | $13.5\pm 1.8$ | 12.5 | $20.1\pm 1.0\pm 2.8$ | 18.2 | $13.6\pm 1.7$ | 13.5 | $1.66\pm 0.17\pm 0.16$ | 1.91
$\Lambda$ | $16.7\pm 0.2\pm 1.1$ | 15.3 | $17.3\pm 1.8\pm 2.8$ | 14.5 | $14.9\pm 0.2\pm 1.49$ | 13.7 | $10.9\pm 1.0\pm 1.3$ | 13.3
$\overline{\Lambda}$ | $12.7\pm 0.2\pm 0.9$ | 12.1 | $12.7\pm 1.8\pm 2.0$ | 10.9 | $8.02\pm 0.11\pm 0.8$ | 7.28 | $1.62\pm 0.16\pm 0.2$ | 1.66
$\Xi^{-}$ | $2.17\pm 0.06\pm 0.19$ | 2.05 | $2.04\pm 0.14\pm 0.2$ | 1.93 | $1.64\pm 0.03\pm 0.014$ | 1.67 | $1.44\pm 0.10\pm 0.15$ | 1.18
$\overline{\Xi}^{+}$ | $1.83\pm 0.05\pm 0.20$ | 1.69 | $1.74\pm 0.12\pm 0.17$ | 1.53 | $0.989\pm 0.057\pm 0.057$ | 1.02 | $0.31\pm 0.03\pm 0.03$ | 0.23
$\Omega^{-}$ |  |  |  |  |  |  | $0.14\pm 0.03\pm 0.01$ | 0.11
$\overline{\Omega}^{+}$ | $0.53\pm 0.04\pm 0.04$ | 0.56 | $0.56\pm 0.11\pm 0.06$ | 0.52 | $0.356\pm 0.046\pm 0.014$ | 0.379 | $0.07\pm 0.02\pm 0.01$ | 0.04
$\chi^{2}/ndf$ | $2.8/8$ | $2.0/8$ | $2.1/8$ | $8.5/9$

| Pb+Pb 80 AGeV | Pb+Pb 40 AGeV | Pb+Pb 30 AGeV | Pb+Pb 20 AGeV
---|---|---|---|---
| data | model | data | model | data | model | data | model
$\pi^{+}$ | $132.0\pm 0.5\pm 7$ | 129.9 | $96.6\pm 0.4\pm 6$ | 97.8 | $83.0\pm 0.4\pm 4.2$ | 84.8 | $72.9\pm 0.3\pm 3.6$ | 73.6
$\pi^{-}$ | $140.4\pm 0.5\pm 7$ | 141.8 | $106.1\pm 0.4\pm 6$ | 110.1 | $96.5\pm 0.5\pm 4.8$ | 99.3 | $84.8\pm 0.4\pm 4.2$ | 85.0
$k^{+}$ | $24.6\pm 0.2\pm 1.2$ | 23.8 | $20.1\pm 0.3\pm 1.0$ | 19.4 | $21.2\pm 0.8^{+1.5}_{-0.9}$ | 21.6 | $16.4\pm 0.6\pm 0.4$ | 16.6
$k^{-}$ | $11.7\pm 0.1\pm 0.6$ | 11.9 | $7.58\pm 0.12\pm 0.4$ | 7.27 | $7.8\pm 0.1\pm 0.2$ | 7.3 | $5.58\pm 0.07\pm 0.11$ | 5.01
$p$ | $30.1\pm 1.0\pm 3.0$ | 31.5 | $41.3\pm 1.1\pm 4.1$ | 38.6 | $42.1\pm 2.0\pm 4.2$ | 37.7 | $46.1\pm 2.1\pm 4.6$ | 37.0
$\overline{p}$ | $0.87\pm 0.07\pm 0.09$ | 0.83 | $0.32\pm 0.03\pm 0.03$ | 0.36 | $0.16\pm 0.02\pm 0.02$ | 0.17 | $0.06\pm 0.01\pm 0.006$ | 0.07
$\Lambda$ | $13.5\pm 0.7\pm 1.0$ | 14.5 | $15.3\pm 0.6\pm 1.0$ | 14.7 | $14.7\pm 0.2\pm 1.2$ | 14.2 | $13.4\pm 0.1\pm 1.1$ | 11.5
$\overline{\Lambda}$ | $1.06\pm 0.08\pm 0.1$ | 0.95 | $0.42\pm 0.04\pm 0.04$ | 0.49 | $0.21\pm 0.02\pm 0.02$ | 0.20 | $0.10\pm 0.02\pm 0.01$ | 0.08
$\Xi^{-}$ | $1.22\pm 0.14\pm 0.13$ | 1.23 | $1.15\pm 0.11\pm 0.13$ | 1.10 | $1.17\pm 0.13\pm 0.13$ | 1.38 | $0.93\pm 0.13\pm 0.10$ | 0.92
$\overline{\Xi}^{+}$ | $0.21\pm 0.03\pm 0.02$ | 0.15 | $0.07\pm 0.01\pm 0.01$ | 0.08 | $0.05\pm 0.01\pm 0.01$ | 0.07 | $------$ | 0.02
$\Omega^{-}+\overline{\Omega}^{+}$ | $------$ | 0.13 | $0.10\pm 0.02\pm 0.02$ | 0.09 | $------$ | 0.14 | $------$ | 0.07
$\chi^{2}/ndf$ | $2.9/7$ | $3.0/8$ | $5.6/7$ | $15.3/6$

Table 1 shows the hadron density dN/dy at midrapidity in the most central A+A
collisions at RHIC energies 200, 130, 62.4 GeV and SPS energies 158, 80, 40,
30, 20 AGeV. The experimental data are taken from Refs20A30A2008PRC ;
20A30A40A80A158A2006PRC ; 20A30A40A80A158A2008PRC ; 40A80A158A2004PRL ;
40A158A2005PRL ; 158A2002PRC ; 622009nuclex ; 1302002PRL ; 1302004PRL ;
1302004PRC ; 2002004PRC ; 2002007PRL . The agreement between the calculation
results and the experimental data is good except at 20 AGeV where
$\chi^{2}/ndf$ is far greater than one. We note that ALCOR model alcor95  can
also describe well the hadron yields at top SPS energy, which gives a cross-
verification of quark combination. On the contrary, HIJING or HIJING/B model
which is in compliance with the hadronic scenario for the early evolution of
heavy ion collisions via a fragmentation hadronization, can not self-
consistently explain the data of multi-strange hadrons topor94 ; Csizmadia99 .

Fig. 1 shows the ratios of antihadrons to hadrons at midrapidity as the
function of collision energy. The filled triangles are the calculated results,
and the experimental data are taken from Refs20A30A2008PRC ;
20A30A40A80A158A2006PRC ; 20A30A40A80A158A2008PRC ; 40A80A158A2004PRL ;
158A2002PRC ; 622009nuclex ; 1302002PRL ; 1302004PRC ; 2002004PRC ; 2002007PRL
. These ratios are mainly influenced by the no-zero baryon number density. As
the collision energy increases, the nuclear transparency power becomes strong
and baryon number density at midrapidity becomes small. This results in a
rapid increase for ratios of $K^{-}/K^{+}$, $\overline{p}/p$,
$\overline{\Lambda}/\Lambda$ with the increasing collision energy.
$\pi^{-}/\pi^{+}$ follows a different pattern. At low collision energies
$\pi^{-}/\pi^{+}$ is slightly higher than one, while it is close to one at
high energies. This is caused by the asymmetry of decay contribution from
hyperons and anti-hyperons (e.g. $\Lambda\rightarrow p\,\pi^{-}$). As the
collision energy increases, the yields of hyperons are close to that of anti-
hyperons and their decay contributions to pion yields are almost the same and
therefore the ratio of $\pi^{-}/\pi^{+}$ is close to one.

Figure 1: (Color online)The yield ratios of antihadrons to hadrons at
midrapidity as the function of the collision energy. The filled symbols are
the calculated results, and the experimental data, open symbols with error
bar, are from Refs20A30A2008PRC ; 20A30A40A80A158A2006PRC ;
20A30A40A80A158A2008PRC ; 40A80A158A2004PRL ; 158A2002PRC ; 622009nuclex ;
1302002PRL ; 1302004PRC ; 2002004PRC ; 2002007PRL .

Fig. 2 shows the ratios of baryons to mesons $p/\pi$ and the strangeness
ratios $K/\pi$ at midrapidity as the function of collision energy. The filled
triangles (up and down) are the computed results, and the experimental data
are taken from Refs20A30A2008PRC ; 20A30A40A80A158A2006PRC ; 158A2002PRC ;
622009nuclex ; 1302002PRL ; 1302004PRC ; 2002004PRC . The big splits between
$p/\pi^{+}$ and $\overline{p}/\pi^{-}$ and between $K^{+}/\pi^{+}$ and
$K^{-}/\pi^{-}$ at low collision energies is due to the high baryon number
density. At 20 AGeV, the result of $p/\pi^{+}$ deviates seriously from the
data. This is probably because the participant nucleons are not broken
completely in collisions. These nucleon fragments deposited in midrapidity
region lead to the extra contribution of proton production besides those from
quark combination and lead to the excessively high ratio of $p/\pi^{+}$.

Figure 2: (Color online)The values of the relative production yields $p/\pi$
and $k/\pi$ at different energies. The triangles down are the numerical
results for $p/\pi^{+}$ and $k^{+}/\pi^{+}$, and the triangles up are the
numerical results for $\overline{p}/\pi^{-}$ and $k^{-}/\pi^{-}$. The
experimental data, open symbols with error bar, are from Refs20A30A2008PRC ;
20A30A40A80A158A2006PRC ; 158A2002PRC ; 622009nuclex ; 1302002PRL ; 1302004PRC
; 2002004PRC .

Following the experimental data in Table 1, A and B at different collision
energies are evaluated and the results are presented with squares in Fig.3. We
find that the data of A and B at RHIC energies are almost equal to one while
at SPS energies they deviate from one; particularly, at 20 AGeV the value of A
amounts to two, seriously deviating from one (the data for B are unavailable).
This is probably because the decay of resonances will blur A and B to a
certain degree. In order to explore the decay effect, we use the quark
combination model to compute the values of A and B for the directly produced
hadrons and the final-state hadrons, respectively. The dashed lines are the
results of directly produced hadrons. Just as analyzed above, the dashed lines
keep an invariant value of one for both A and B, independent of collision
energy. The very small fluctuations of dashed lines are due to the fact that
the rapidity distributions of the formed hadrons are slightly different from
those of quarks, which leads to a small amount of hadrons formed by
midrapidity quarks escape from the midrapidity region FLShao2007PRC . The
filled triangles down are model results of final-state hadrons. We see that as
collision energy is greater than 30 AGeV these triangles agree well with the
experimental data within statistical uncertainties. Removing the part of
resonance decay, the data of A and B for directly produced hadrons should be
equal to one, respectively. This suggests the existence of the “quark level”
correlation of directly produced hadrons at these collision energies. However,
at 20 AGeV the value of A calculated via quark combination for final state
hadrons seriously deviates the data. This in itself indicates the quark
degrees of freedom do not represent a decisive factor in hadron production at
20 AGeV.

Figure 3: (Color online)The correlation quantities A and B as the function of
collision energy. The experimental data, open symbols with error bar, are from
Refs20A30A2008PRC ; 20A30A40A80A158A2006PRC ; 20A30A40A80A158A2008PRC ;
40A80A158A2004PRL ; 158A2002PRC ; 622009nuclex ; 1302002PRL ; 1302004PRL ;
1302004PRC ; 2002004PRC ; 2002007PRL .

From the above analysis of hadron yields, hadron ratios and correlation
quantities A and B, we find that as the collision energy is greater than or
equal to 30 AGeV the quark combination can describe reasonably well all these
observables which indicates the existence of constituent quark degrees of
freedom in the energy region. When the collision energy drops to 20 AGeV,
however, the quark combination mechanism can not self-consistently describe
these quantities, in particular the correlation quantity A. Note that in order
to best fit the data of hadron yields, we have adjusted the strangeness
parameter $\lambda_{s}$ (see Table 4) to the quite high values at 20, 30 AGeV
which are far greater than the saturated values at higher collision energies
and are also greater than those at lower energies thermalreview . But even so
the quark combination can not self-consistently explain the data at 20 AGeV.
In addition, the high strangeness entangled with baryon density leads to the
Sawtooth-like shape for calculated final-state A and B at low SPS energies. In
fact this peak behavior of strangeness (i.e. $K^{+}/\pi^{+}$ ratio or
$\lambda_{s}$ determined mainly by the former ) at low SPS energies has been
interpreted in Ref Gazdzicki ; GAGo99  as a result of onset of deconfinement,
i.e. the result of strangeness carrier changing from strange hadrons to
strange quarks at the onset of deconfinement. The failure of quark combination
energy indicates the (partial) disappearance of constituent quark degrees of
freedom and closely relates to the onset of deconfinement around 30 AGeV
observed by NA49 Collaboration 20A30A2008PRC .

###  III.2 Predictions at LHC

With the increasing collision energy, the strangeness $\lambda_{s}$ of the hot
and dense quark matter in Table 4 tends to be saturated and the squared sound
velocity $c_{s}^{2}$ approaches an ideal value $1/3$ and the baryon number
density $n_{0}$ decreases regularly. So we take $\lambda_{s}$=0.43,
$c_{s}^{2}=1/3$ and $n_{0}=0.0fm^{-3}$ at $\sqrt{s_{NN}}=5.5$ TeV. The initial
entropy density of the central point at the beginning of hydrodynamic
evolution is taken to be 256.5 /fm3 according to Eq. (13). In Tables 2 and 3,
we present our predictions of hadron yields and hadron ratios at LHC energy
($\sqrt{s_{NN}}=5.5$ TeV). Our computed particle ratios are consistent with
the results of the thermal model in RefAndronic2009PLB .

Table 2: The predicted $dN/dy$ of identified hadrons at midrapidity in the most central ($0-5\%$) Pb+Pb collisions at $\sqrt{s_{NN}}=5.5$ TeV. $\pi^{+}$ | $\pi^{-}$ | $k^{+}$ | $k^{-}$ | $p$ | $\overline{p}$ | $\Lambda$ | $\overline{\Lambda}$ | $\Xi^{-}$ | $\overline{\Xi}^{+}$ | $\Omega^{-}$ | $\overline{\Omega}^{+}$
---|---|---|---|---|---|---|---|---|---|---|---
435.5 | 435.5 | 71.6 | 71.6 | 34.6 | 34.6 | 20.5 | 20.5 | 2.9 | 2.9 | 0.4 | 0.4
Table 3: The predicted hadron ratios at midrapidity in the most central Pb+Pb collisions at $\sqrt{s_{NN}}=5.5$ TeV. $p/\pi^{+}$ | $\overline{p}/\pi^{-}$ | $k^{+}/\pi^{+}$ | $k^{-}/\pi^{-}$ | $\Lambda/\pi^{-}$ | $\Xi^{-}/\pi^{-}$ | $\Omega^{-}/\pi^{-}$
---|---|---|---|---|---|---
0.079 | 0.079 | 0.165 | 0.165 | 0.047 | 0.007 | 0.001

##  IV Summary

In this paper, we used the quark combination model to make a systematical
study of the hadron yields in heavy ion collision in a broad collision energy
region. At the collision energies where the deconfined quark matter has been
created, the yields of various hadrons after hadronization have some
correlations inherited from the early quark degrees of freedom. We investigate
at which collision energy this “quark-level” correlation of hadron yields
first breaks. We apply the hydrodynamics to describe the evolution of
deconfined quark matter and to obtain the quark distribution just before
hadronization, then utilize the quark combination model to describe
hadronization. We find that as the collision energy is greater than or equal
to 30 AGeV, the quark combination well reproduce the yields of various
hadrons, their ratios and correlation quantities A and B; however, as the
collision energy drops to 20 AGeV, the mechanism can not self-consistently
describe these quantities. This indicates that the constituent quark degrees
of freedom represent a decisive factor in thermal hadron production above 30
AGeV and seem to be invalid at 20 AGeV. It is related to the onset of
deconfinement observed at collision energy 20-30 AGeV. Finally, we predict the
yields of various hadrons and their ratios in the most central Pb+Pb
collisions at $\sqrt{s_{NN}}=5.5$ TeV.

### ACKNOWLEDGMENTS

The authors thank Z. T. Liang, Q. Wang and G. Li for helpful discussions. R.
Q. Wang would like to thank L. G. Pang and J. Deng for fruitful discussions.
The work is supported in part by the National Natural Science Foundation of
China under the grant 10775089, 10947007 and 10975092.

##  Appendix A Quark distribution just before hadronization

Table 4: The values of the initial baryon density $n_{0}$, strangeness factor $\lambda_{s}$ and squared sound velocity $c_{s}^{2}$ in central A+A collisions at different energies. Energy | 200GeV | 130GeV | 62.4GeV | 158AGeV | 80AGeV | 40AGeV | 30AGeV | 20AGeV
---|---|---|---|---|---|---|---|---
$n_{0}$ ($fm^{-3}$) | 0.30 | 0.34 | 0.74 | 1.31 | 1.46 | 1.64 | 1.74 | 1.56
$\lambda_{s}$ | 0.43 | 0.43 | 0.43 | 0.44 | 0.50 | 0.57 | 0.80 | 0.70
$c_{s}^{2}$ | $1/3.1$ | $1/3.4$ | $1/4.0$ | $1/6.0$ | $1/6.0$ | $1/6.0$ | $1/6.0$ | $1/6.0$

The quark distribution just before hadronization is needed for the quark
combination to describe the hadron production in relativistic heavy ion
collisions. Here, we use relativistic hydrodynamics Landau53 ; DirkH1995NPA ;
Kolb0305084nuth  to describe the time-space evolution of the hot and dense
quark matter before hadronization. The evolution equation of hydrodynamics
follows from the local conservation laws for energy, momentum, and other
conserved charges, e.g. baryon number,

$\displaystyle\partial_{\mu}T^{\mu\nu}(x)=0,~{}~{}(\nu=0,1,2,3)$ (7)
$\displaystyle\partial_{\mu}j^{\mu}(x)=0,$ (8)

by inserting the ideal fluid decomposition

$\displaystyle
T^{\mu\nu}(x)=\bigg{(}e(x)+p(x)\bigg{)}u^{\mu}(x)u^{\nu}(x)-g^{\mu\nu}p(x),$
(9) $\displaystyle j^{\mu}(x)=n(x)u^{\mu}(x).$ (10)

Here, $u^{\mu}(x)=\gamma(1,v_{x},v_{y},v_{z})$ with
$\gamma=1/\sqrt{1-v_{x}^{2}-v_{y}^{2}-v_{z}^{2}}$ is the local four velocity
of a thermalized fluid cell; $e(x)$ is the energy density, $p(x)$ the
pressure, and $n(x)$ the conserved number density.

As the energy density of the fluid cell drops to 1.0 GeV/fm3 (a common
criteria of phase transition from Lattice QCDKarsch2002NPA ), we stop the
hydrodynamic evolution and let the constituent quarks and antiquarks freeze
out according to Cooper-Frye formalism Coopfrye74

$E\frac{dN_{i}}{d^{3}p}=\frac{dN_{i}}{dyp_{T}dp_{T}d\varphi}=\frac{g_{i}}{(2\pi)^{3}}\int_{\Sigma}f_{i}(p\cdot
u(x),x)p\cdot d^{3}\sigma(x),$ (11)

where $d^{3}\sigma(x)$ is the outward normal vector on the freeze-out surface
$\Sigma(x)$, $g_{i}$ the degeneracy factor of quarks ($g_{i}=6$). The phase-
space distribution $f$ in the formula is taken to be a local equilibrium
distribution,

$f_{i}(E,x)=\frac{1}{\exp[(E-\mu_{i}(x))/T(x)]+1}.$ (12)

We consider only the freeze-out of light and strange quarks and antiquarks.
The three chemical potentials $\mu_{q}$, $\mu_{\bar{q}}$ and
$\mu_{s}=\mu_{\bar{s}}$ at freeze-out can be determined uniquely by the global
conservation of energy and baryon number plus an ancillary constraint of
strangeness.

We study in this paper the hadron production in midrapidity region only. The
code of KolbKolb2000PRC ; Kolb2001NPA ; Kolb2003PRC  for 2+1- dimensional
hydrodynamics with the longitudinal boost invariance is used to simulate the
evolution of quark system before hadronization.

These frozen-out quarks and antiquarks with momentum distributions in Eq. (11)
are hadronized by the quark combination modelFLShao2005PRC ; QBXie1988PRD . In
this method of hydrodynamics + quark combination, the quark combination model
is default setting while some inputs for hydrodynamics need fixing. The first
is initial entropy density $s_{0}$ of the central point at the beginning of
hydrodynamic evolution. The distribution of transverse entropy density is
determined by optical Glauber modelGlauber1959 . We apply the following
empirical formula of $s_{0}$entropy2008nuth  as the function of collision
energy $\sqrt{s_{NN}}$

$s_{0}(\sqrt{s_{NN}})=6.99\times
10^{-3}\Big{(}312.5\log_{10}{\sqrt{s_{NN}}}-64.8\Big{)}^{3/2}.$ (13)

The second is the initial baryon number density $n_{0}$ at central point fixed
by the data of the net-proton rapidity density. The distribution of transverse
baryon number density is also determined by optical Glauber model. The third
is the equation of state $p=p(e)$ for the hot and dense quark matter which is
taken to be the simplest pattern $p=c_{s}^{2}e$. The squared sound velocity
$c_{s}^{2}$ is obtained by fitting the data of transverse momentum spectra of
protons. The fourth is the strangeness of hot and dense quark matter denoted
by the factor $\lambda_{s}=2\langle s\bar{s}\rangle/\langle
u\bar{u}+d\bar{d}\rangle$. In order to make a better description of strange
hadrons, we regard it as a free parameter in the present paper. The values of
$n_{0}$, $\lambda_{s}$ and $c_{s}^{2}$ in central A+A collisions at different
energies are shown in Table 4. We note that the extracted $\lambda_{s}$ around
30 AGeV are quite high which are consistent with the analytic results of
thermal model thermalreview .

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