#  proScript: Partially Ordered Scripts Generation
via Pre-trained Language Models

Keisuke Sakaguchi†, Chandra Bhagavatula†, Ronan Le Bras†,
Niket Tandon†, Peter Clark†, Yejin Choi†‡
†Allen Institute for AI, ‡University of Washington
{keisukes, chandrab, rolanlb, nikett, peterc<EMAIL_ADDRESS>

###### Abstract

Scripts \- standardized event sequences describing typical everyday activities
- have been shown to help understand narratives by providing expectations,
resolving ambiguity, and filling in unstated information. However, to date
they have proved hard to author or extract from text. In this work, we
demonstrate for the first time that pre-trained neural language models (LMs)
can be be finetuned to generate high-quality scripts, at varying levels of
granularity, for a wide range of everyday scenarios (e.g., bake a cake). To do
this, we collected a large (6.4k), crowdsourced partially ordered scripts
(named proScript), which is substantially larger than prior datasets, and
developed models that generate scripts with combining language generation and
structure prediction. We define two complementary tasks: (i) edge prediction:
given a scenario and unordered events, organize the events into a valid
(possibly partial-order) script, and (ii) script generation: given only a
scenario, generate events and organize them into a (possibly partial-order)
script. Our experiments show that our models perform well (e.g., F1=75.7 on
task (i)), illustrating a new approach to overcoming previous barriers to
script collection. We also show that there is still significant room for
improvement toward human level performance. Together, our tasks, dataset, and
models offer a new research direction for learning script knowledge.

##  1 Introduction

Figure 1: We collected 6.4k of partially ordered scripts (proScript) and
developed models that take a scenario (e.g., bake a cake) as the input and
generate a (possibly partial-order) script.

Scripts, originally introduced by Schank and Abelson (1975), represent
structured commonsense knowledge about prototypical events in everyday
situations/scenarios such as bake a cake and fuel a car (Figure 1). However,
while scripts have been shown to help understand narratives by providing
expectations, resolving ambiguity, and filling in unstated information
(Chambers and Jurafsky, 2008; Modi et al., 2017, inter alia), they have proved
hard to author or extract from text, with only small script databases
available Regneri et al. (2010); Chambers (2017); Ostermann (2020).

In this work, we show for the first time that pre-trained neural language
models (LMs) can be adapted to generate high-quality scripts, including
appropriately partial ordering events where a specific temporal ordering is
required only when it is necessary. LMs have previously been shown to
successfully generate stories Rashkin et al. (2020), summaries Lewis et al.
(2020), and commonsense facts Bosselut et al. (2019); Hwang et al. (2020).
Here we investigate their application to script generation. First, we collect
large amount (6.4k) of partially ordered script from crowdsourcing with a
similar but simplified collection method Ciosici et al. (2021). We call the
dataset as proScript (PaRtial Order SCRIPt for generaTion), and this is
substantially larger than prior (crowdsourced) dataset such as DeScript
Regneri et al. (2010) that has 40 scripts. Since the granularity of scripts
(and the events) are inherently vague and subjective Modi et al. (2016), we
collected wider variety of micro and macroscopic scripts than previous
datasets. Additionally, temporal duration of each event is also annotated
(e.g., take the cake out of the oven typically takes one minute in the bake a
cake script), which will potentially link script knowledge with temporal
reasoning in future work.

Second, with the collected data, we introduce two complementary tasks: script
edge prediction and entire script generation. In the edge prediction task,
given a scenario and unordered intermediate events, models must organize the
events as a valid partial-order script. On the other hand, the script
generation task is to generate intermediate events and the partial-order of
those events for a given scenario. This task requires both natural language
generation (for nodes) and graph structure prediction (for edges).

Finally, based on our proposed dataset, we develop models for both edge
prediction and entire script generation tasks. As Chambers (2017) has revealed
that models trained and evaluated on missing events prediction (i.e.,
narrative cloze) are insufficient to assess script knowledge, our evaluation
scheme evaluate the entire script. We compare the models against baselines,
and show that our models outperform the baselines for both the edge prediction
and the script generation tasks. Nonetheless, there is a significant room for
improvement toward human-level performance – e.g., for edge prediction, the
best model achieves 75.71 of F1 score while human achieves 89.28, and for
script generation, the best model obtains a graph edit distance of 4.97 (i.e.,
number of human edits), while human-created scripts achieve 2.98 on average.

Our contributions are thus:

  * •

A new dataset (proScript) of crowdsourced scripts that is substantially larger
than prior (manually crafted) datasets

  * •

Two complementary task definitions against proScript

  * •

Two new models for these task, providing the first demonstration that
generative models can be successfully applied, although it is still
significantly below human levels

##  2 Related Work

#### Script as narrative chain

Mooney and DeJong (1985) and Chambers and Jurafsky (2008, inter alia) have
investigated automatically inducing scripts from (unstructured) corpus. In
particular, Chambers and Jurafsky (2008) introduced scripts as narrative
chain, where verbs with the participants information (e.g., (claimed, subj),
and (accused, obj) ) named narrative events are partially ordered according to
causal and temporal relations. They also introduced narrative cloze task,
where a model is expected to predict one removed narrative event, given all
the other narrative events, while our proposed task requires to generate
scripts as a partial-order graph for a given scenario. The “script as
narrative chain” approach has been actively studied Jans et al. (2012); Modi
and Titov (2014); Pichotta and Mooney (2014); Rudinger et al. (2015);
Granroth-Wilding and Clark (2016); Weber et al. (2018); Belyy and Van Durme
(2020), but it has its drawbacks. First, the source corpora is mainly from a
news domain rather than everyday scenarios, and a number of verbs in news
texts are not script-relevant events such as reporting verbs Mostafazadeh et
al. (2016); Chambers (2017). Second, events are highly abstracted as tuples of
verb and the dependency (subj or obj) Ostermann (2020). Third, the evaluation
scheme for the narrative cloze task is insufficient to evaluate script
knowledge Chambers (2017).

#### Script as paraphrase sets

Script as paraphrase sets Regneri et al. (2010); Modi et al. (2016); Wanzare
et al. (2016) is more recent approach to gather script knowledge, where crowd
workers are asked to write down a sequence of events for a given everyday
scenario (e.g., bake a cake) and the collected sequences (called event
sequence description) are aligned with paraphrased events being clustered. The
collected (partially ordered) scripts cover wide variety of everyday
situations compared to narrative chains (news domain), but one shortcoming of
this approach is the scalability; it is not easy to scale because of the cost
for manual data collection Chambers (2017); Ostermann (2020). In fact, Modi et
al. (2016) crowdsourced 1000 stories that cover only 10 scripts, and similarly
Regneri et al. (2010) end up with collecting 40 scripts. The limited amount of
data hinders learning script knowledge by models. Furthermore, they provide no
evaluation metric on the dataset for assessing model’s script knowledge.

#### Story generation

Neural models have been demonstrated to successfully generate stories Kiddon
et al. (2016); Peng et al. (2018); Zhai et al. (2019); Rashkin et al. (2020).
Our work is related to story generation in terms of generating higher-level
agenda (or plot) of a story. However, a main difference between stories and
scripts is that stories often require surprising and incidental sequence of
events as well as description about character’s mental states and landscape
depiction that make the story attractive for readers, whereas our script
generation expects generating essential core events Chambers (2017) in partial
order.

##  3 Definitions

####  proScript

We define proScript as a directed acyclic graph (DAG), $G(V,E)$ with a given
scenario ($s$), where $V$ is a set of essential events
$\\{v_{1},...v_{i},...v_{|V|}\\}$ and $E$ is a set of temporal ordering
constraints between events $\\{e_{ij}\\}$ which means that the events $v_{i}$
must precede the event $v_{j}$ ($v_{i}\prec v_{j}$).111Technically, proScript
is a transitive reduction of a DAG. In short, transitive reduction of $G$ does
not have any short cut edges between nodes. In proScript, we add a single root
node ($v_{r}$) and scenario ($s$) as a unique leaf node. DAGs effectively
encode the partial-ordering of core events–crucial for representing events
which can be performed in any order. For example, in a bake a cake scenario,
one can “gather the ingredients” and “turn on the oven” in any order (Figure
1). We emphasize that scripts should not include non-core events such as
discourse related events (e.g., reporting verbs) as Chambers (2017) proposed.
In proScript, we also exclude alternative events in a proScript DAG. For
example, in a bake a cake scenario, “get ingredients” and “buy ingredients”
are alternative events with each other because either one is only necessary in
the scenario. By excluding alternative events, we can resolve ambiguity of the
edges in partial order structure as temporal relations or alternative paths.
Regneri et al. (2010) and Modi et al. (2016) do not discriminate this
ambiguity.222In proScript, we focus on events and the (partial-) ordering, and
we leave participants/arguments identification for future work.

With the definition, we introduce proScript task in two complementary
settings: script edge prediction and entire script generation.

#### Edge Prediction

The script edge prediction task is to predict a set of partial-ordered edges
($E$) of the script $G(V,E)$, given a scenario and a set of unordered
intermediate events $v\in V$.

#### Script Generation

The script generation task is to predict a partial order script $G(V,E)$, but
only the scenario is given. Models are additionally expected to generate
events ($V$) in natural language.

##  4 Datasets

#### Source of Scenarios

We collected scenarios from ROCStories Mostafazadeh et al. (2016), DeScript
Wanzare et al. (2016), and VirtualHome Puig et al. (2018). As ROCStories
consists of sentences instead of scenarios, we extract phrases that match the
manually curated patterns “want(ed) to …”, “need(ed) to …”, “look(ing) to …”
and that do not include personal pronouns or person’s name. The 2,565
scenarios we collected include both high-level long-term ones (e.g., open a
small business) and fine-grained short-term ones (e.g., sign into an email
account). DeScript consists of 40 daily scenarios (e.g., making coffee) and we
use all of them. VirtualHome is constructed to learn activities interactively
in a household in a 3D simulated world. It has 233 indoor tasks (e.g., turn on
light) and we include them as scenarios.

#### Crowdsourcing proScript

For the collected scenarios, we crowdsource the corresponding proScript on the
Amazon Mechanical Turk. Our crowdsourcing procedure is similar but simplified
method to Ciosici et al. (2021). First, crowdworkers are required to describe
five to seven core events that they are essential for the given scenario
Chambers (2017) with the estimated time it takes to complete each event. In
the second question, they are asked to sort them in possibly partial order
(=DAG), which represents the proScript for the scenario.

Due to the complex nature of this crowdsourcing procedure, it is crucial to
maintain the quality. To identify and filter out noisy instances, two
different workers are asked to sort the same set of events in partial order
(i.e., the same as the second question described above). According to our
manual analysis, we decided to retain scripts that have at least 65.0 F1 score
between the workers.333In our crowdsourcing tasks, we maintained a pay rate of
12$/hr or higher. For example, crowd workers were paid $0.8 for the script
creation and $0.4 for the validation. To collect proScript with both micro and
macroscopic scenarios, we iteratively picked two adjacent events in the DAGs
and use them as a source of finer-grained scenarios.

Figure 2: Normalized histogram of time duration in proScript dataset. We see
the dataset contains scripts with various time granularity.

#### Dataset Statistics

In total, we collected 6,414 valid scripts that include 311,502 pairs of
events, and we split the proScript into training (3,252 scenarios),
development (1,085), and test set (2,077). The training and development sets
consist of scenarios collected from ROCStories, and the test set consists of
those from ROCStories, DeScript, and VirtualHome. This helps us evaluate in-
and out-of-domain performance.

The average number of events in proScript scenarios is 5.45 and the maximum
degrees of DAGs in the training set are distributed as follows: 2,198 scripts
(67.6%) for degree 1, 915 scripts (28.1%) for degree 2, 108 scripts (3.3%) for
degree 3, 31 scripts (0.9%) for degree 4 and above.

Figure 2 shows the normalized histogram of the typical time to take for each
script in proScript dataset. Most of the scripts take between a minute and an
hour (e.g., “go to bathroom”, “buy some new clothes”), while there are a
reasonable amount of high-level long-term scripts (e.g., “find a new job”,
“open a small business”).

##  5 proScript Edge Prediction

###  5.1 Models

For the proScript edge prediction task (§3), we implement a two-step approach
baseline (pairwise model) and compare it with our proposed end-to-end neural
method (proScript${}_{\text{edge-pred}}$).

#### Pairwise Model

We implement a two-step baseline where we train a binary classifier to predict
the precedence between pairs of events, followed by building a partial order
script $\hat{G}$ by aggregating the predicted relations across all pairs of
events.

Formally, the classifier takes a pair of events ($v_{i}$, $v_{j}$) and
predicts the precedence ${\hat{e}_{ij}}$ – i.e. the event $v_{i}$ precedes
($\prec$) $v_{j}$ .

$\displaystyle\hat{e}_{ij}=p(v_{i}\prec v_{j}|v_{i},v_{j})$ (1)

Scores by the classifier are used as weights to create an adjacency matrix of
$G$ which is then automatically converted into a partial-order script with
heuristics – when $G$ contains a cycle, we iteratively remove edges by
choosing the one with minimum weight until we get a valid DAG.

####  proScript${}_{\text{edge-pred}}$

Figure 3: Example of input and output for the proScript${}_{\text{edge-pred}}$
model. The input is a flattened sequence of events, and the output is a
flattened sequence of edges of the (predicted) partial-order script.

We propose an an end-to-end neural model, which takes all the (unordered)
events ($v$) and the scenario ($s$) as the input ($x$) and predicts the edges
($\hat{E}$) in a partial-order script ($\hat{G}$) at one time. To represent
$\hat{E}$ in a linear format ($y$), we use DOT, a graph description language
as shown in Figure 3.444Madaan and Yang (2020) have previously shown that
finetuned LMs can generate valid DOT language. By flattening the nodes and
edges of $G$ (and $\hat{G}$), we apply neural encoder-decoder models.
Formally, flattened unordered events and scenario as $x$ are embedded as
continuous representation (emb($x$)) by the encoder, then the decoder will
generate tokens ($y$) as follows:

$\displaystyle p(y_{1},\dots,y_{N}|x_{1},\dots,x_{M})$ $\displaystyle=$ (2)
$\displaystyle\prod_{n=1}^{N}p(y_{n}|\text{emb}(x_{1},\dots,x_{M})$
$\displaystyle,y_{1},\dots,y_{n-1}).$

Compared to the pairwise model, the proScript${}_{\text{edge-pred}}$ model
uses information from all the events jointly to build partial-order script
with a broader context.

| dev | test (all) | test (in domain) | test (out domain)
---|---|---|---|---
Models | F1 | P | R | F1 | P | R | F1 | P | R | F1 | P | R
random | 21.30 | 21.08 | 21.72 | 21.03 | 21.00 | 21.26 | 20.57 | 20.52 | 20.84 | 21.32 | 21.27 | 21.58
Pairwise (RoB) | 65.75 | 67.05 | 64.71 | 61.29 | 62.85 | 60.06 | 63.25 | 64.97 | 61.89 | 59.06 | 60.44 | 57.98
Pairwise (T5) | 70.96 | 71.93 | 69.76 | 67.64 | 69.44 | 66.18 | 69.50 | 71.41 | 67.96 | 65.51 | 67.20 | 64.16
proScr(11B-100) | 56.05 | 56.58 | 55.75 | 52.26 | 52.91 | 51.89 | 54.98 | 55.67 | 54.59 | 49.16 | 49.76 | 48.83
proScr(11B-1k) | 65.98 | 66.49 | 65.71 | 60.55 | 61.24 | 60.15 | 64.64 | 65.40 | 64.20 | 55.89 | 56.51 | 55.54
proScr(Large) | 66.25 | 66.89 | 65.83 | 63.64 | 64.22 | 63.27 | 65.76 | 66.38 | 65.35 | 61.23 | 61.76 | 60.91
proScr(11B-all) | 78.20 | 78.48 | 78.14 | 75.71 | 75.93 | 75.72 | 77.75 | 78.03 | 77.71 | 73.37 | 73.54 | 73.46
Human | 89.32 | 89.60 | 89.21 | 89.28 | 89.91 | 88.86 | 90.04 | 90.54 | 89.74 | 88.71 | 89.44 | 88.18
Table 1: Results for proScript edge prediction task. In this table, proScript
refers to proScript${}_{\text{edge-pred}}$.

###  5.2 Evaluation Metrics

Given $G(V,E)$ as a predicted (partial order) script and $\hat{G}(V,\hat{E})$
as the correct (oracle) script, the F1 score is defined as follows:

Precision $\displaystyle=\frac{|E\&\hat{E}|}{|\hat{E}|},\
\text{Recall}=\frac{|E\&\hat{E}|}{|E|}$ F1 score
$\displaystyle=\frac{2*\text{Precision}*\text{Recall}}{\text{Precision}+\text{Recall}}.$

###  5.3 Experiments

#### Setup

For the binary classifier (pairwise model), we use two variants of the
Transformer Vaswani et al. (2017): RoBERTa-large Liu et al. (2019) and T5-11B
Raffel et al. (2020).

When training (i.e., fine-tuning) RoBERTa,555We used the implementation from
Huggingface Transformers Wolf et al. (2019). we use a grid-search for choosing
the best hyper-parameters from the best performed model on the development
set: epochs {1, 2, 3}, learning rate {1e-5, 1e-6, 1e-7}, batch size {16, 24,
32}. For training the T5 model as the pairwise model, we followed a default
set of hyper-parameters that are recommended in Raffel et al. (2020).

For the proScript${}_{\text{edge-pred}}$ model, we use the T5 with different
model sizes (Large and 11B) and training sizes (100, 1k, and all 3.2k) to see
how these factors affect the performance.666We also used BART Lewis et al.
(2020), but we found that BART did not perform well on this task. We followed
a default set of hyper-parameters for the T5 models.

Figure 4: Model performance by pairwise (T5-11B), proScript${}_{\text{edge-
pred}}$ (T5-11B) and human according to the maximum degree of the script
(DAG).

#### Results

The results are shown in Table 1. We find that the pairwise and
proScript${}_{\text{edge-pred}}$ models significantly outperform the random
baseline where the edges are randomly assigned. The proScript${}_{\text{edge-
pred}}$ T5-11B model outperforms the pairwise T5-11B model. This indicates
that the proScript${}_{\text{edge-pred}}$ model benefits from a larger context
from the input to predict edges more accurately, although there is still a
significant room for improvement toward human-level performance.777We find
that 99% of the outputs from proScript${}_{\text{edge-pred}}$ are valid DOT
language. Regarding the difference between in and out of domain, we find that
the in-domain performance is higher than the out-of-domain performance,
whereas human performance is robust regardless of the domain difference. We
also see that the training set (100, 1k, all) and model sizes (Large, 11B)
significantly affect the performance of proScript${}_{\text{edge-pred}}$.

Figure 4 shows the performance of the pairwise (T5-11B) model,
proScript${}_{\text{edge-pred}}$ (T5-11B) and human according to the (maximum)
degree of the script DAGs. We find that scripts with higher degree are more
difficult to predict for both proScript${}_{\text{edge-pred}}$ and pairwise
models, whereas human shows smaller decrease for predicting higher-degree
scripts.

##  6 proScript Generation

###  6.1 Models

Figure 5: Example of input and output for proScript${}_{\text{gen}}$. The
input is a scenario and number of events to generate in natural text format,
and the output is a sequence of events and edges of the script.

####  proScript${}_{\text{gen}}$

The proScript generation task combines natural language generation (i.e.
generating events in natural language) with structure prediction over the
generated events (i.e. organizing the events into a DAG). Our approach
(proScript${}_{\text{gen}}$) is to formulate it as an end-to-end problem,
similar to the proScript${}_{\text{edge-pred}}$ for the proScript edge
prediction task (§5.1). Given a scenario ($s$) and the number of events to
generate in the script, proScript${}_{\text{gen}}$ generates events and edges
for the partial-order script ($\hat{G}$) in DOT language (Figure 5). Formally,
we use the same encoder-decoder framework (eq.2) except that a scenario and
number of steps to generate are described in natural text as $x$ and the
decoder is expected to generate events as well as the edges (as $y$) in the
script.

#### Transfer learning from WikiHow data

Transfer learning often helps improve the performance when it is (pre-)trained
on a similar task Peters et al. (2018); Devlin et al. (2019). As additional
resource for pre-training proScript${}_{\text{gen}}$, we use procedural texts
extracted from WikiHow,888https://www.wikihow.com/ which contains 130k
instances of a sequence of essential steps for a given topic in various
categories (e.g., health, finance, hobbies, etc.). It is important to note
that all the procedures in WikiHow are formatted as sequences rather than a
partial-order. We refer to this approach as proScript${}_{\text{gen-
transfer}}$.

#### Pipeline approach

An alternative approach is to use proScript${}_{\text{gen}}$ followed by the
proScript${}_{\text{edge-pred}}$ model. The approach relies on
proScript${}_{\text{gen}}$ to generate a set of events but allows to fix the
predicted edges via the proScript${}_{\text{edge-pred}}$ model. We refer to
this approach as proScript${}_{\text{gen-pipe}}$, and study whether it can
improve the performance over proScript${}_{\text{gen}}$.

###  6.2 Evaluation Metrics

Chambers (2017) emphasizes the importance of human annotation for evaluating
script knowledge. However, human evaluation for the proScript generation task
is challenging because it involves natural text generation and structured
prediction. As in the text generation tasks such as machine translation and
text summarization, there are several possible correct answers. Therefore, we
use two complementary evaluation metrics for the proScript generation task:
(i) graph edit distance, and (ii) pairwise comparison. These are the absolute
and relative measures of performance, respectively. Graph edit distance Abu-
Aisheh et al. (2015) computes the distance between two graphs. Formally, given
two graphs $G_{1}$ and $G_{2}$,

$\displaystyle\text{GED}(G_{1},G_{2})=\min_{G_{1}\xrightarrow[]{d_{1},...,d_{k}}G_{2}}\sum_{i=1}^{k}\text{cost}(d_{i})$

where $d_{1},\ldots,d_{k}$ is a list of graph edit operations from $G_{1}$ to
$G_{2}$. The operations include deletion, insertion, and replacement for
vertex and edge. Each operation has its cost and we set the cost to be 1 for
all the operations in our evaluation for simplicity. We use the graph edit
distance between a model-generated script and the revised script by human
annotators. The graph edit distance is indicative of the quality of the
generated scripts; higher-quality scripts must have smaller graph edit
distances to the gold-standard (i.e. they require a smaller number of human
revisions). In addition, we also employ pairwise human judgments where we ask
human annotators to compare the scripts generated by
proScript${}_{\text{gen}}$ with those from the other approaches.

Split | Models | Edit Dist | V-Del | V-Ins | V-Rep | E-Del | E-Ins | E-Rep
---|---|---|---|---|---|---|---|---
| proScript${}_{\text{gen}}$ | 4.73 | 0.426 | 0.192 | 0.581 | 1.558 | 1.308 | 0.671
dev | proScript${}_{\text{gen-transfer}}$ | 4.79 | 0.337 | 0.195 | 0.679 | 1.491 | 1.281 | 0.775
| proScript${}_{\text{gen-pipe}}$ | 4.88 | 0.397 | 0.159 | 0.560 | 1.705 | 1.407 | 0.661
| Human | 2.78 | 0.155 | 0.161 | 0.144 | 1.123 | 1.011 | 0.199
| proScript${}_{\text{gen}}$ | 4.97 | 0.581 | 0.142 | 0.656 | 1.668 | 1.184 | 0.709
test | proScript${}_{\text{gen-transfer}}$ | 5.38 | 0.438 | 0.213 | 0.775 | 1.713 | 1.402 | 0.835
| proScript${}_{\text{gen-pipe}}$ | 5.41 | 0.594 | 0.143 | 0.671 | 1.880 | 1.292 | 0.787
| Human | 2.98 | 0.168 | 0.149 | 0.130 | 1.276 | 1.074 | 0.189
| proScript${}_{\text{gen}}$ | 4.57 | 0.513 | 0.158 | 0.633 | 1.471 | 1.108 | 0.687
test (in domain) | proScript${}_{\text{gen-transfer}}$ | 5.03 | 0.339 | 0.299 | 0.649 | 1.575 | 1.496 | 0.677
| proScript${}_{\text{gen-pipe}}$ | 5.10 | 0.561 | 0.147 | 0.630 | 1.765 | 1.217 | 0.744
| Human | 3.03 | 0.168 | 0.211 | 0.154 | 1.223 | 1.091 | 0.206
| proScript${}_{\text{gen}}$ | 5.43 | 0.659 | 0.124 | 0.681 | 1.894 | 1.270 | 0.735
test (out domain) | proScript${}_{\text{gen-transfer}}$ | 5.76 | 0.549 | 0.115 | 0.916 | 1.867 | 1.296 | 1.013
| proScript${}_{\text{gen-pipe}}$ | 5.81 | 0.659 | 0.116 | 0.795 | 1.961 | 1.267 | 0.941
| Human | 2.91 | 0.170 | 0.074 | 0.102 | 1.340 | 1.054 | 0.170
Table 2: Results for proScript generation task (dev, test, in-domain test and
out-of-domain test set). We measure the average graph edit distance between
generated script and the two human revisions (lower the better). We also show
the average number of each graph edit operation ({Delete, Insert, Replace}
$\times$ {Vertex, Edge}). Random (edge) baseline shows 11.06 edit distance for
the dev set and 10.95 for the test set.

###  6.3 Experiments

#### Setup

For our proScript${}_{\text{gen}}$, we use T5-11B. Similarly to the
proScript${}_{\text{edge-pred}}$, we follow the default set of hyper-
parameters recommended in Raffel et al. (2020). For proScript${}_{\text{gen-
transfer}}$, we pre-train the proScript${}_{\text{gen}}$ with the 130k
procedures, and finetune it on the proScript dataset. For the
proScript${}_{\text{gen-pipe}}$, we first obtain the actions generated by
proScript${}_{\text{gen}}$ (ignoring the edges), and use the set of events as
input for proScript${}_{\text{edge-pred}}$, which is trained (see §5.3) to
predict the edges.

As defined in §3, we use graph edit distance and pairwise judgments to
evaluate the quality of the generated scripts. For computing graph edit
distances, we select 500 scripts (250 for dev and test sets) and ask
crowdworkers to revise the generated scripts as necessary (e.g.,
add/delete/replace the events and the edges).We use the revised scripts as
gold-standard. Each script is revised by two annotators, and we compute the
average of the graph edit distances.

In pairwise judgments, we compare the scripts generated by
proScript${}_{\text{gen}}$ with those from the other approaches. We randomly
select 150 pairs, and ask three crowdworkers to judge whether the script
generated by proScript${}_{\text{gen}}$ is better, worse, or equal to the
other (i.e. transfer, pipeline, or human). We use majority vote to decide the
final pairwise human judgment between the two scripts.

Figure 6: Pairwise judgments (%) between proScript${}_{\text{gen}}$ and the
other approaches. Figure 7: Histograms of graph edit distance (in dev set).
The number of scripts (y-axis) according to the (binned) graph edit distance
(x-axis).

#### Results

The pairwise judgment result is shown in Figure 6. We see that the pipeline
and transfer models show slight preference over the proScript${}_{\text{gen}}$
(except pipeline-dev), although the difference is not large. We also see that
the transfer model constantly have more preference over the
proScript${}_{\text{gen}}$ than the pipeline model in both dev and test sets.
Regarding the pairwise comparison with human-created plans,
proScript${}_{\text{gen}}$ still has a significant room for improvement toward
human level.

Revision types |  generated script (subgraph) |  revised script (subgraph)
---|---|---
missing event |  wait for the plane $\rightarrow$ exit the plane |  wait for the plane $\rightarrow$ get on the plane $\rightarrow$ exit the plane
incorrect order |  get off the car $\rightarrow$ drive to the zoo |  drive to the zoo $\rightarrow$ get off the car
irrelevant or redundant event |  put clothes in dryer $\rightarrow$ place clothes into dryer $\rightarrow$ dry clothes |  put clothes in dryer $\rightarrow$ dry clothes
order by context |  get a visa $\rightarrow$ … $\rightarrow$ get off the plane $\rightarrow$ trip to a foreign country |  get off the plane $\rightarrow$ get a visa (on arrival) $\rightarrow$ trip to a foreign country
granularity |  get out of the bed $\rightarrow$ go to the kitchen |  get out of the bed $\rightarrow$ open the bedroom door $\rightarrow$ go to the kitchen
paraphrased |  move into new apartment |  move to a new apartment
Table 3: Examples for each revision type.

Revision types | proScript | Transfer | Pipeline | Human
---|---|---|---|---
| (edge) incorrect order | 15.79 | 21.62 | 24.32 | 10.00
crucial errors | (node) missing event | 5.26 | 2.70 | 2.70 | 0.00
| (node) irrelevant/redundant event | 10.53 | 13.51 | 2.70 | 0.00
| (edge) order by context | 31.58 | 32.43 | 40.54 | 33.33
minor revisions | (node) granularity | 31.58 | 24.32 | 21.62 | 26.67
| (node) paraphrased event | 0.00 | 0.00 | 5.41 | 6.67
wrong revisions |  | 5.26 | 5.41 | 2.70 | 23.33

Table 4: Revision type distribution (%) by each model.

Table 2 shows the average graph edit distance between the generated script and
the human revisions. We find that neither transfer nor pipeline help to
improve the graph edit distance over proScript${}_{\text{gen}}$, indicating
that proScript${}_{\text{gen}}$ is already a strong baseline (see examples in
Appendix). The reason of no improvement by the transfer approach may be
because WikiHow consists of sequences rather than partially ordered steps. No
improvement by the pipeline approach indicates that the
proScript${}_{\text{gen}}$ can directly generate valid script in both events
and edges. Further studies for improvements are needed for future work.

In terms of the edit types, many of the edits are edge-related, suggesting
that proScript${}_{\text{gen}}$ and the variants are all good at generating
events but struggles with ordering them. Regarding in- and out-of domains in
the test sets, we observe that proScript${}_{\text{gen}}$ and the variants
have slightly better performance for in-domain scripts than out-of domain,
while human created scripts are not affected by domains. These findings are
consistent with the result in the edge prediction task (§5.3).

Figure 7 shows a histogram of the graph edit distance. It is evident that
human created scripts are corrected less often than scripts generate by
proScript${}_{\text{gen}}$, whereas the scripts from
proScript${}_{\text{gen}}$ and the variants often have a large number of edits
(e.g., 4 or more). It is interesting to see that fewer number of scripts have
1 to 3 edits (except scripts created by human). The reason is because one
simple revision tends to yield multiple graph edits (e.g., one node insertion
yields multiple edge insertions).

#### Error Analysis

We performed manual error analysis for the scripts generated by each model. We
selected 40 random scripts that have non-zero graph edit distance and
classified the human revisions into 7 types: (1) incorrect order, (2) missing
event, (3) irrelevant/redundant event, (4) order by context, (5) granularity,
(6) paraphrased event, and (7) wrong correction (examples are shown in Table
3). Approximately, the first three types indicate that the script has crucial
errors, the next three types are trivial revisions where both generated and
revised scripts are plausible. The last type of revision is the one where the
revised script is wrong (or worse).

Table 6.3 shows the statistics of each error type. We see that edge-related
revisions are more frequent than node-related revisions. This is consistent
with the results in graph edit distance. Overall, we find that minor revisions
are more frequent than crucial errors, indicating that
proScript${}_{\text{gen}}$ and the variants generates reasonably good scripts.
In contrast, crucial errors are quite rare in human created scripts,
indicating a significant room for future innovation.

##  7 Conclusions

We show for the first time that pre-trained neural language models can be
adapted to generate partial order scripts. We collect 6,400 partially ordered
script from crowdsourcing (proScript), which is substantially larger than
prior manually crafted datasets. With the proScript dataset, we introduced two
complementary task and models, providing the first demonstration that
generative models can be successfully applied to script generation, although
it is still below human performance. We believe that proScript dataset and
models would advance future work on various NLP tasks such as story
generation, machine comprehension, temporal reasoning, and high-level
planning.

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##  Appendix A Appendices

###  A.1 Plans generated by proScript${}_{\text{gen}}$

Figure 8: Example scripts generated proScript${}_{\text{gen}}$.

We show some example scripts generated by proScript${}_{\text{gen}}$ in Figure
8.

###  A.2 Reproducibility

For the POP model. we use a single CPU with 4GB memory. This model does not
require any training procedure. For training RoBERTa-large as a pairwise
model, we use Quadro RTX 8000 (48GB memory), which takes around 4.5 hours to
train a model. RoBERTa-large consists of 355M parameters with 24 layers, 1,024
of hidden embedding size, and 16 of the attention heads. T5-large model has
770M parameters with 24-layers, 1024-hidden-state, 4096 feed-forward hidden-
state, and 16 attention heads. T5-11B models has 11B parameters with
24-layers, 1024-hidden-state, 65,536 feed-forward hidden-state, 128 attention
heads. We use TPU (v3-8) on google cloud platform. It takes 3 hours in average
to train a edge prediction model, and 5 hours for plan generation models.


# Suppression of thermal vorticity as an indicator of QCD critical point

Sushant K. Singh<EMAIL_ADDRESS>Variable Energy Cyclotron Centre,
1/AF, Bidhan Nagar , Kolkata, India  HBNI, Training School Complex, Anushakti
Nagar, Mumbai 400085, India  Jan-e Alam  Variable Energy Cyclotron Centre,
1/AF, Bidhan Nagar , Kolkata, India  HBNI, Training School Complex, Anushakti
Nagar, Mumbai 400085, India

###### Abstract

We study the impact of the QCD critical point (CP) on the spin polarization of
$\Lambda$-hyperon generated by the thermal vorticity in viscous quark gluon
plasma (QGP). The equations of the relativistic causal viscous hydrodynamics
have been solved numerically in (3+1) dimensions to evaluate the thermal
vorticity. The effects of the CP have been incorporated through the equation
of state (EoS) and the scaling behavior of the transport coefficients. A
significant reduction in the global polarization has been found as the CP is
approached. A drastic change induced by the CP in the rapidity dependence of
the spin polarization is observed which can be used as a signature of the CP.

QGP, Vorticity, Critical Point, Hydrodynamics.

The results from lattice quantum chromodynamics (QCD) and effective field
theoretical models at non-zero temperature ($T$) and baryon chemical potential
($\mu$) reveal a rich and complex phase diagram bazavov . While at high $T$
and low $\mu$ $(\rightarrow 0)$ the quark-hadron transition is a crossover, at
low $T$ and high $\mu$ the transition is of first order in nature. Therefore,
it is expected that between the crossover and the first order transition there
exists a point in the $\mu-T$ plane called the Critical End Point or simply
the Critical Point (CP) where the first order transition ends and crossover
begins Fodor:2004nz . The location of CP is not yet known from first
principles lqcd , however, phenomenological studies indicate its existence
stephanov . The search for the CP in the system formed in collisions of nuclei
at relativistic energies is one of the outstanding problem. At present the
general consensus is that the collisions of nuclei at the top Relativistic
Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) energies produce QGP
with small $\mu$ and high $T$ which reverts to hadronic phase via a crossover.
The ongoing Beam Energy Scan - II program at RHIC and the upcoming Compressed
Baryonic Matter experiment at Facility for Anti-proton and Ion Research (FAIR)
and Nuclotron based Ion Collider fAcility (NICA) are planned to create system
of quarks and gluons with different $\mu$ and $T$ by varying the collision
energies to explore the region close to the CP. The non-monotonic variation of
the fluctuations in multiplicity with collision energy in the center of mass
frame ($\sqrt{s_{NN}}$) nxu , the change of sign of the fourth cumulant of
order parameter with the variations of rapidity (rapidity scan) yyin1  and
beam energy stephanovprl1 , the appearance of negative sign in the kurtosis of
the order parameter fluctuation near the CP stephanovprl2  are some of the
proposed signals of the CP (see yyin  for a review and references therein).

Relativistic hydrodynamics has been used to describe the space-time evolution
of QGP and explain various experimental data quite successfully. One such
crucial observable is the polarization of $\Lambda$-hyperon LambdaStar
generated by the thermal vorticity during the hydrodynamic evolution of the
QGP. Invigorating theoretical activities have been witnessed (see
becattini2020  for a review) to understand various aspects of the polarization
within the scope of hydrodynamics and transport models after the experimental
measurement of the global polarization of the $\Lambda$ hyperon LambdaStar .

The magnetization of uncharged objects induced by mechanical rotation, called
Barnett effect barnett  and its inverse, that is, the rotation generated by
varying magnetization, called the Einstein-de Haas effect deHaas  originate
due to the conversion between spin ($S$) and orbital angular momentum ($L$)
via spin-orbit coupling constrained by the conservation of total angular
momentum $\vec{J}\ (=\vec{L}+\vec{S})$. Similar kind of coupling between $L$
and $S$ in the system formed in relativistic heavy ion collision results into
the spin polarization of particles. The initial orbital angular momentum (OAM)
imparted by the spectators in non-central heavy ion collisions makes the
fireball of QGP to rotate and polarize the quarks ztliang . This rotation may
then appear as local vorticities in the fireball, the exact mechanism for
which is not yet fully understood. Vorticity is a measure of the local
spinning of fluid elements. The coupling of fluid vorticity and quantum
mechanical spin has been experimentally demonstrated for the first time in
Ref. spinhydro . Such an information is reflected in the spin polarization of
final state hadrons. However, the vorticity can be generated by the viscous
stresses of the system even in the absence of an initial OAM. Hence, spin
polarization of hadrons has two contributions: one coming through OAM and
another generated through viscosities of the system. The first contribution
depends on the details of mechanism of transfer of initial OAM to vorticity
and is sensitive to the initial condition. The second contribution depends on
the transport properties of the system and will be sensitive to the EoS. In
this letter, we focus on the second contribution and discuss about the first
one in the supplemental material supplement . The goal here is to understand
the CP induced change in the vorticity and its consequences on the spin
polarization of $\Lambda$ hyperon. In other words if
$\varpi_{\mu\nu}^{\text{CP}}$ ($\varpi_{\mu\nu}$) is the vorticity in the
presence (absence) of CP then what is the value of
$\Delta\varpi_{\mu\nu}=(\varpi_{\mu\nu}^{\text{CP}}-\varpi_{\mu\nu}$) and the
corresponding change on the spin polarization of $\Lambda$-hyperon. We show
that as the CP is approached the local vorticity and hence the polarization
effect is suppressed.

The presence of CP in the EoS affects the expansion of the system due to
suppression of the sound wave and divergence of some of the transport
coefficients hasan . This will affect the evolution of local vorticity and
hence the $\Lambda$-polarization through vorticity-spin coupling. Apart from
polarization, the effect of CP on the separation of baryon and anti-baryon due
to chiral vortical effect is another interesting facet cve .

Here we use natural unit $c=\hbar=k_{B}=1$ where $c$ is the speed of light in
vacuum, $h=2\pi\hbar$ is the Planck’s constant and $k_{B}$ is the Boltzmann’s
constant. The signature metric for flat space time is taken as
$g_{\mu\nu}=\text{diag}(1,-1,-1,-1)$.

We numerically solve (3+1)-dimensional relativistic viscous causal
hydrodynamics using the algorithm detailed in Ref. karpenko2014 . The code
contains the effect of CP through the EoS and the scaling behavior of the
transport coefficients. The initial condition and the EoS models that we use
to solve hydrodynamic equations have been extensively tested by reproducing
the results available in Refs. chunshen2020  and parotto2020  respectively.
The CORNELIUS code Cornelius  has been used to find the constant energy-
density hyper surface. Our numerical results in the absence of CP have been
contrasted with the known analytical results of Ref. gubser2010  and with
numerical results from other publicly available codes: AZHYDRO azhydro , MUSIC
music  and vHLLE karpenko2014 . The reliability of our code can be further
appreciated by contrasting its output with the transverse momentum, rapidity
and azimuthal angle dependence of various experimental observables (see the
supplemental material supplement  for details).

The relativistic hydrodynamic equations that we solve are:

$\displaystyle\partial_{\mu}T^{\mu\nu}=0,$
$\displaystyle\partial_{\mu}N^{\mu}=0,$ (1)

where $T^{\mu\nu}$ is the energy-momentum tensor and $N^{\mu}$ is the net-
baryon number current. Here we work in the Landau frame of reference where the
$T^{\mu\nu}$ and $N^{\mu}$ are given by

$\displaystyle T^{\mu\nu}$
$\displaystyle=\varepsilon\,u^{\mu}\,u^{\nu}-(p+\Pi)\Delta^{\mu\nu}+\pi^{\mu\nu},$
(2) $\displaystyle N^{\mu}$ $\displaystyle=n_{B}\,u^{\mu}+V^{\mu},$ (3)

where $\Pi$ is the bulk pressure, $\pi^{\mu\nu}$ is the shear-stress tensor
which is symmetric, traceless and orthogonal to $u^{\mu}$,
$V^{\mu}(=0,\text{here})$ is the baryon diffusion 4-current and
$\Delta^{\mu\nu}=g^{\mu\nu}-u^{\mu}\,u^{\nu}$. The viscous terms obey the
following evolution equations,

$\displaystyle u^{\alpha}\partial_{\alpha}\Pi$
$\displaystyle=-\frac{\Pi-\Pi_{NS}}{\tau_{\Pi}}-\frac{4}{3}\Pi\partial_{\alpha}u^{\alpha},$
(4) $\displaystyle\langle u^{\alpha}\partial_{\alpha}\pi^{\mu\nu}\rangle$
$\displaystyle=-\frac{\pi^{\mu\nu}-\pi^{\mu\nu}_{NS}}{\tau_{\pi}}-\frac{4}{3}\pi^{\mu\nu}\partial_{\alpha}u^{\alpha},$
(5)

where $\langle\cdot\rangle$ is defined as,

$\langle
A^{\mu\nu}\rangle=\left(\frac{1}{2}\Delta^{\mu}_{\alpha}\Delta^{\nu}_{\beta}+\frac{1}{2}\Delta^{\nu}_{\alpha}\Delta^{\mu}_{\beta}-\frac{1}{3}\Delta^{\mu\nu}\Delta_{\alpha\beta}\right)A^{\alpha\beta},$

$\Pi_{NS}$ and $\pi_{NS}^{\mu\nu}$ are the Navier-Stokes limit of $\Pi$ and
$\pi^{\mu\nu}$ respectively, given by

$\Pi_{NS}=-\zeta\theta\quad,\quad\pi_{NS}^{\mu\nu}=2\eta\left\langle\partial^{\alpha}u^{\beta}\right\rangle.$
(6)

The coefficients of shear ($\eta$) and bulk ($\zeta$) viscosities are
positive, _i.e._ $\eta,\zeta>0$.

The hydrodynamical equations are solved in $(\tau,x,y,\eta_{s})$ coordinates
where, $\tau=\sqrt{t^{2}-z^{2}}$ and $\eta_{s}=\text{tanh}^{-1}(z/t)$. The
space-time evolution begins at time $\tau_{0}$. For lower energies the initial
time, $\tau_{0}$ is taken as the time required by the nuclei to pass through
one other ($\sim\frac{2R}{\gamma_{z}v_{z}}$) and for higher energies
($\sqrt{s_{NN}}\geq 62.4$ GeV), $\tau_{0}=1$ fm as shown in Table 1. The
initial energy density profile at $\tau_{0}$ is taken as:

$\varepsilon(x,y,\eta_{s};\tau_{0})=e(x,y)\,f(\eta_{s}).$ (7)

A symmetric rapidity profile, $f(\eta_{s})$, with the local energy-momentum
conservation puts a constraint on $e(x,y)$ as shown in Ref. chunshen2020 . The
energy deposited in the transverse plane, $e(x,y)$ depends on the number of
wounded nucleons per unit area which has been calculated by using the optical
Glauber model for given impact parameter ($b$) at different $\sqrt{s_{NN}}$.
The quantity, $e(x,y)$ is related to the the number of wounded nucleons per
unit area in the transverse plane, $n_{A}(x,y)$ and $n_{B}(x,y)$ of the
colliding nuclei A and B respectively as:
$e(x,y)\propto\sqrt{n_{A}^{2}+n_{B}^{2}+2n_{A}n_{B}\text{cosh}(2y_{\tt
beam})}$, where $y_{\tt beam}=\text{cosh}^{-1}(\sqrt{s_{NN}}/2)$. Here we
consider Au+Au collisions at $b=5.6$ fm for different $\sqrt{s_{NN}}$ that
corresponds to 15-25% centrality supplement . The thickness function of the Au
nucleus has been calculated by assuming Woods-Saxon profile for nuclear
density with nuclear radius, $R_{0}=6.37$ fm, and surface thickness,
$\delta=0.535$. The p+p inelastic cross-section,
$\sigma^{\texttt{in}}_{NN}(\sqrt{s_{NN}})$, needed for the calculation of the
number of wounded nucleons in the Glauber model has been taken from Refs.
sigmaNN_parametrization1 ; sigmaNN_parametrization2 .

The initial velocity profile is taken as:

$u^{\mu}(x,y,\eta_{s};\tau_{0})=\left(\cosh(\eta_{s}),0,0,\sinh(\eta_{s})\right),$
(8)

The initial density profiles for energy and net baryon number have been
computed with the parameters used in Ref. chunshen2020 . The viscous terms
have been initialized with their corresponding Navier-Stokes limit.

Table 1: Values of $\tau_{0}$ used at different colliding energies $\sqrt{s_{NN}}$ (GeV) | 14.5 | 19.6 | 27 | 39 | 62.4 | 200
---|---|---|---|---|---|---
$\tau_{0}$ (fm) | 2.2 | 1.8 | 1.4 | 1.3 | 1.0 | 1.0

The EoS parotto2020  employed here to solve the hydrodynamic equations
reproduces the lattice QCD results at zero baryon chemical potential. The
parameters $w$, $\rho$ and $\alpha_{1}$ that appear in the linear mapping from
Ising model to QCD in Ref. parotto2020  have been fixed as $w=1$, $\rho=2$ and
$\alpha_{1}=3.85^{\tt o}$. The other parameters are same as Ref. parotto2020 .
The transport coefficients are expected to diverge near the critical point
following a scaling behavior amonnai :

$\zeta\sim\xi^{3}\quad,\quad\eta\sim\xi^{0.05}.$

where $\xi(\mu,T)$ is the equilibrium correlation length, which is obtained
through mapping QCD to 3D Ising model in the critical region. In the Ising
model, $\xi$ is computed by taking the derivative of equilibrium
magnetization, $M(r,h)$, with respect to the magnetic field, $h$, at fixed
$r=(T-T_{c})/T_{c}$, as amonnai

$\xi^{2}=\frac{1}{H_{0}}\left(\frac{\partial M}{\partial h}\right)_{r},$

where $H_{0}$ is a dimensionful parameter to get the correct dimensions of
$\xi$. We shall take $H_{0}=1$ in our calculations and $\left(\frac{\partial
M}{\partial h}\right)_{r}$ is obtained from the EoS model parotto2020  using
chain rule of differentiation. The extent of the critical domain in the
$\mu-T$ plane is determined by the condition: $\xi(\mu,T)=\xi_{0}$, where
$\xi_{0}$ is taken as 1.75 fm. The possibility of divergent behavior is
incorporated through the following expressions of the transport coefficients
amonnai

$\zeta=\zeta_{0}\left(\frac{\xi}{\xi_{0}}\right)^{3}\ ,\
\eta=\eta_{0}\left(\frac{\xi}{\xi_{0}}\right)^{0.05}$ (9)

Outside the critical region the values of the shear and bulk viscosities
denoted by $\eta_{0},\ \zeta_{0}$ respectively are chosen as denicol2018 ;
denicol2014 :

$\displaystyle\eta_{0}(\mu,T)$
$\displaystyle=C_{\eta}\left(\frac{\varepsilon+p}{T}\right)\quad,\quad\zeta_{0}(\mu,T)=15\
\eta_{0}\left(\frac{1}{3}-c_{s}^{2}\right)^{2}$

The above parametrization is consistent with the estimates of the temperature-
dependent specific shear and bulk viscosity extracted using Bayesian method
bernhard  away from the critical region. The authors of Ref. Martinez  compute
the critical contribution to the bulk viscosity which is an order of magnitude
less than that of the noncritical contribution. The effect of reduced
$\zeta_{0}$ in the critical region has been discussed in supplement .

The dependence of the relaxation times appeared in Eqs.(4) and (5) on $\xi$
are parameterized as:

$\tau_{\pi}=\tau^{0}_{\pi}\left(\frac{\xi}{\xi_{0}}\right)^{0.05}\quad,\quad\tau_{\Pi}=\tau^{0}_{\Pi}\left(\frac{\xi}{\xi_{0}}\right)^{3}.$
(10)

where $\tau^{0}_{\pi}$ and $\tau^{0}_{\Pi}$ are the relaxation times outside
the critical region which are given by denicol2018 ; denicol2014 ,

$\frac{\tau^{0}_{\pi}}{5}=\tau^{0}_{\Pi}=\frac{C_{\eta}}{T}$

with $C_{\eta}=0.08$.

Figure 1: Trajectories traced by the center of the fireball _i.e._
$x=y=\eta_{s}=0$ in the $\mu-T$ plane for different $\sqrt{s_{NN}}$. The
critical point is indicated by solid black dot at $(\mu,T)=(350,143.2)$ MeV.

The critical region and the trajectories traced by the center of the fireball
in the ($\mu-T$) plane at different $\sqrt{s_{NN}}$ are shown in Fig. 1. The
black dot indicates the location of the CP at
$(\mu_{c},T_{c})=(350\text{MeV},143.2\text{MeV})$ parotto2020 . The
trajectories have been calculated by solving the hydrodynamic equations with
(denoted by EoS-CP) and without (noCP) the effects of CP. The CORNELIUS code
is then used to find the freeze-out hyper surface $\Sigma_{\varepsilon}$,
defined by $\varepsilon=0.3\text{GeV}/\text{fm}^{3}$. The spin polarization
has been evaluated on this hyper surface. The trajectories for
$\sqrt{s_{NN}}=$14.5 GeV and 19.6 GeV pass through the critical domain (shown
by the closed contour in Fig. 1) and those for higher $\sqrt{s_{NN}}$ remain
outside the critical domain. We evaluate the thermal vorticity and
subsequently the polarization of $\Lambda$ for system evolving along
trajectories passing through both inside and outside the critical domain. The
effect of the CP on the polarization is expected to be larger for
$\sqrt{s_{NN}}=14.5$ GeV as the trajectory for this case is closer to the CP
compared to other values of $\sqrt{s_{NN}}$ considered here.

The thermal vorticity at any space-time point of the fluid is given by
Becattini1 ; Becattini2 :

$\varpi_{\mu\nu}=\frac{1}{2}\left[\partial_{\nu}\beta_{\mu}-\partial_{\mu}\beta_{\nu}\right]$
(11)

where $\beta_{\mu}=u_{\mu}/T$. The time evolution of the $x\eta$ component of
the thermal vorticity, averaged over the spatial coordinates and weighted by
the energy density, with and without the effects of CP are shown in Fig. 2.
Initially the system has zero vorticity. The vorticity generated by the
viscous effects increase at first to attain some maximum value and then
decreases subsequently. The evolution of the vorticity is affected by several
factors. The hydrodynamic expansion does not create or destroy vortices but
reduces it through redistribution. The shear viscous coefficient is
responsible for its diffusion and the stretching and baroclinic torque enhance
the vorticity. Near the CP, absorption of sound wave affects the expansion
directly and the diverging nature of the transport coefficients reduces the
vorticity as seen in Fig. 2. The observed suppression of the vorticity due to
CP at $\sqrt{s_{NN}}=14.5$ GeV is expected to influence some of the
experimental results. As the evolution trajectory for
$\sqrt{s_{NN}}=62.4\text{GeV}$ (and higher energies) remain outside the
critical domain the results with and without the CP essentially overlap.

---
Figure 2: Time evolution of the mean of the $x\eta$-component of thermal
vorticity for $\sqrt{s_{NN}}=14.5\text{GeV}$ and $62.4\text{GeV}$.

The local thermal vorticity and the $\Lambda$-polarization is calculated by
using the following expression for mean spin vector of a spin-1/2 particle
with four-momentum $p_{\nu}$ Smu  as,

$S^{\mu}(x,p)=-\frac{1}{8m}(1-n_{F})\epsilon^{\mu\nu\rho\sigma}p_{\nu}\varpi_{\rho\sigma}(x)+O(\varpi)^{2}$

where $m$ is the mass of the particle, $\epsilon^{\mu\nu\rho\sigma}$ is the
Levi-Civita tensor and $n_{F}$ is the Fermi-Dirac distribution. Since the mass
of the $\Lambda$ is much larger than the temperature range being considered in
this study, we assume that $1-n_{F}\approx 1$ and $n_{F}\approx n_{B}$, where
$n_{B}$ is the Boltzmann distribution. Consequently the expression for the
mean spin vector becomes

$S^{\mu}(x,p)=\frac{1}{8m}\epsilon^{\mu\nu\rho\sigma}p_{\nu}\partial_{\rho}\beta_{\sigma}$

In the rest frame of the particle, the spin vector is
$S^{*\mu}=(0,{\bf{S}^{*}})$, which is obtained by using the Lorentz
transformation as:

${\bf{S}^{*}}={\bf{S}}-\frac{{\bf{p}}.{\bf{S}}}{E(E+m)}{\bf{p}}$

The mean spin averaged over the surface $\Sigma$ is then given by Becattini1 ,

$S^{\mu}(p)=\frac{\int d\Sigma_{\lambda}p^{\lambda}\
e^{-\beta(p.u-\mu)}S^{\mu}(x,p)}{\int d\Sigma_{\lambda}p^{\lambda}\
e^{-\beta(p.u-\mu)}}$ (12)

The net spin is obtained by integrating over azimuthal angle
($0\leq\phi<2\pi$), rapidity ($|y|<1$) and transverse momentum
($0<p_{T}<3\text{ GeV}$) following the procedure of Ref. wu2019 . Finally the
spin polarization of $\Lambda$ is given by,

${\bf{P}}=2{\bf{S}^{*}}$

In view of an ongoing puzzle on the issue of the variation of the longitudinal
polarization with azimuthal angle ($\phi$) ( signpuzzle1 ; signpuzzle2 ;
STARlambda ), we display the variation of the different components of the
polarization, $P_{x}$, $P_{y}$ and $P_{z}$ with $\phi$ in Fig. 3 for
$\sqrt{s_{NN}}=14.5$ GeV and 62.4 GeV with and without the effects of CP. A
systematic suppression of the polarization is observed ($xz$ is the reaction
plane and $y$-axis is the axis of rotation here) at $\sqrt{s_{NN}}=14.5$ GeV
which originates from several competing factors like enhancement of various
transport coefficients, slower expansion, changes in baroclinic torque and
vortex stretching near the CP. The polarization with and without the effect of
CP overlap at $\sqrt{s_{NN}}=62.4$ GeV which is obvious as the trajectory for
this case remains outside the critical region (Fig. 1).

---
Figure 3: $x$, $y$ and $z$ components of $\Lambda$-polarization are plotted
respectively in the upper, middle and the lower panels as a function of
azimuthal angle in momentum space for
$\sqrt{s_{NN}}=14.5\text{GeV}\,\,\text{and}\,\,62.4\text{GeV}$.

The variation of the $y$-component of the spin-polarization with rapidity
($y$) has been displayed in Fig. 4. A drastic change is induced by the CP in
the rapidity distribution of spin-polarization at $\sqrt{s_{NN}}=14.5$ GeV. At
$\sqrt{s_{NN}}=62.4$ GeV, the $P_{y}$ with and without CP is identical as
expected. It is intriguing to note that the CP not only reduces the
polarization around mid-rapidity but also introduces strong qualitative
changes in the slopes of the curves as shown in Fig. 5.

---
Figure 4: $y$-component of $\Lambda$-polarization plotted as a function of
momentum rapidity for
$\sqrt{s_{NN}}=14.5\,\,\text{GeV}\,\,\text{and}\,\,62.4\text{GeV}$.

Finally, on integration over $p_{T}$, $\phi$ and $y$ the global polarization
is obtained as a function of $\sqrt{s_{NN}}$. The suppression of polarization
is conspicuous for the trajectories passing through the critical domain at
lower $\sqrt{s_{NN}}$ (Fig. 6). The slope of the curve without CP is much
steeper than the one with CP at lower $\sqrt{s_{NN}}$. We have taken the
initial orbital angular momentum (OAM) of the fireball as zero resulting in
smaller polarization compared to experimental value LambdaStar . Inclusion of
OAM through the non-zero initial value of the velocity profile Becattini1
will enhance the magnitude of $-P_{y}$, however, the difference in the
polarization observed here with and without CP will still persist. Results
with the inclusion of OAM have been discussed in the supplemental material
supplement . The sensitivity of this result on other parameters has also been
presented in the supplemental material supplement .

Figure 5: Negative slope of the $y$-component of the polarization of
$\Lambda$-hyperon plotted as a function of rapidity. Figure 6: Global
polarization of $\Lambda$-hyperon plotted as a function of $\sqrt{s_{NN}}$.

_Conclusions_ \- It is well-known that the local vorticity of the fluid
couples with the quantum mechanical spin of the particles and polarize them.
We have evaluated the spin polarization of $\Lambda$-hyperon with and without
the effects of CP and found a strong change in the spin polarization around
mid-rapidity as the system approaches the CP. The thermal vorticity and
consequently the polarization of the $\Lambda$ hyperon for different colliding
energies have been estimated and found to be suppressed as the CP is
approached. There are various physical processes which collectively contribute
to the suppression. Although we have solved the relativistic equation to
estimate the vorticity, we consider below the evolution equation for kinematic
vorticity ($\vec{\omega}=\nabla\times\vec{v}$) for a compressible fluid with
constant $\zeta$ and $\eta$ in the non-relativistic limit because in this form
contributions from various terms appear clearly,

$\displaystyle\frac{\partial\vec{\omega}}{\partial t}$
$\displaystyle=\left(\vec{\omega}\cdot\vec{\nabla}\right)\vec{v}-\left(\vec{v}\cdot\vec{\nabla}\right)\vec{\omega}-\theta\vec{\omega}+\frac{1}{\rho^{2}}\vec{\nabla}\rho\times\vec{\nabla}p$
$\displaystyle-\frac{1}{\rho^{2}}\left(\zeta+\frac{1}{3}\eta\right)\vec{\nabla}\rho\times\vec{\nabla}\theta-\frac{\eta}{\rho^{2}}\vec{\nabla}\rho\times\nabla^{2}\vec{v}+\frac{\eta}{\rho}\nabla^{2}\vec{\omega}.$

Here $\rho$ denotes the density of fluid and $\theta=\nabla\cdot\vec{v}$. As
$\theta$ is a measure of the expansion of the system, a larger expansion
results in smaller vorticity as suggested by the negative sign of the term
$\theta\vec{\omega}$. The terms depending on transport coefficients only, are
written in the second line of the above equation. The term proportional to
$\nabla^{2}\vec{\omega}$ is responsible for diffusion of vorticity in space,
the diffusion coefficient being $\frac{\eta}{\rho}$. The term of particular
interest is proportional to $\vec{\nabla}\rho\times\vec{\nabla}\theta$ which
suggests that the vorticity dissipates if there is a gradient in expansion
rate for fluid cells i.e. the fluid cells having less density and expanding
faster will oppose the vorticity of the denser fluid cells expanding slowly.
The strength of this effect is proportional to
$\left(\zeta+\frac{1}{3}\eta\right)$. It is clear that the suppression of
vorticity and hence polarization, is a combined effect of the absorption of
sound wave and the enhancement of various transport coefficients in presence
of CP. The drastic qualitative and quantitative changes induced by CP in the
rapidity distribution of $P_{y}$ can be used to detect the CP experimentally
as the polarization of $\Lambda$ has already been measured by STAR
collaboration LambdaStar . It is important to mention at this point that the
effects of CP on the $p_{T}$ spectra of the hadrons and on the $p_{T}$ and $y$
dependence of directed and elliptic flow are found to be small singh2022 .

Some comments on the application of hydrodynamics near the CP are in order
here. Near the CP, the fluctuating modes do not relax faster than the
timescale of changes in slow/conserved variables due to which the local
thermal equilibrium is not maintained making hydrodynamics inapplicable. The
validity of the hydrodynamics can, however, be extended by adding a scalar
variable representing the slow non-hydrodynamic modes connected to the
relaxation rate of the critical fluctuation (see yin  and stephanov2018  for
details). It has been explicitly shown that the modes associated with the
scalar variable lags behind the hydrodynamic modes resulting in back reactions
on the hydrodynamic variables rajagopal . Further, it has been demonstrated in
Ref. rajagopal  that the back reaction has negligible effects on the
hydrodynamic variables. In view of this, the results presented in this work
will be useful in detecting the CP. Moreover, we may also recall that if a
system is not too close to CP then hydrodynamics can still be applied in a
domain around the CP Stanley .

We thank Sandeep Chatterjee and Tribhuban Parida for helpful discussions
regarding the UrQMD transport code.

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##  I Supplementary Material

In this supplemental material, we present a few test results from our
hydrodynamic code with the inclusion of the effects of critical point (CP)
through the equation of state (EoS) and scaling behaviour of transport
coefficients, and contrast the results without CP (denoted as noCP in the text
and figures). We shall also discuss the effect of non-zero orbital angular
momentum (OAM) on our results. We have already shown a comparison of our
numerical results with the analytical Gubser solution in (2+1) dimensions in
Ref. singh2022 . There is no analytical result available in (3+1)-dimensions.
Therefore, to test the code, we compare our result on the rapidity
distribution of positively charged pion with the output of publicly available
MUSIC code musiccode  without the resonance decays in Fig. 7. For the next
check, we reproduce the PHOBOS data on $p_{T}$ dependence of elliptic flow in
0-50% centrality of Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV phobos2005  in
Fig. 8.

Figure 7: The comparison of the rapidity distribution of $\pi^{+}$ from our
code and the publicly available MUSIC code.

We next reproduce the charged particle pseudorapidity distribution for two
colliding energies ($\sqrt{s_{NN}}$) and different centralities in Fig. 9. To
generate the plots in Fig. 9, we use the switching energy density
$\varepsilon_{\text{sw}}=0.3$ GeV/fm3. The constant energy density
hypersurface ($\varepsilon_{\text{sw}}$) is obtained using the CORNELIUS code
cornelius  which is then given as input to the UrQMD transport code urqmd
(which does not include spin effects) and generate 1000 events. It should be
mentioned here that the width of the experimental distribution is slightly
underestimated because we have used a single impact parameter and not an
event-by-event simulation that would consist of a mixture of several impact
parameters. The results of Fig. 9 include the effects due to CP. However, on
comparing with the results without CP, the effect is negligible as
demonstrated in Fig. 10 (see also Ref. singh2022 ).

Figure 8: Our numerical result on $p_{T}$ dependence of elliptic flow,
$v_{2}$, compared to the experimental data from Ref. phobos2005 .

Now we implement a non-zero OAM in the initial condition. The initial
condition model that we use from Ref. shen2020  has been generalized to
include a non-zero OAM in Ref. shen2021 . This is done by introducing a
parameter, $f$, that takes value in the interval [0,1] and it controls the
fraction of longitudinal momentum that can be attributed to the flow velocity.
$f=0$ corresponds to the Bjorken flow scenario. The assumption for the initial
energy-momentum current in Ref. shen2021  can be achieved through the
following choice of rest frame quantities:

$\displaystyle p$ $\displaystyle=\varepsilon\ ,\ u^{x}=u^{y}=0,$
$\displaystyle u^{\tau}$ $\displaystyle=\cosh\left(\frac{y_{L}}{2}\right)\ ,\
u^{\eta}=\frac{1}{\tau_{0}}\sinh\left(\frac{y_{L}}{2}\right),$

where $p$, $\varepsilon$, and $u^{\mu}$, respectively, denote the pressure,
the energy density, and the fluid four flow-velocity. Also,
$y_{L}=fy_{\text{CM}}$ denotes the local longitudinal rapidity variable and
$y_{\text{CM}}$ is the local center-of-mass rapidity variable (defined in the
main article). The components of the initial energy-momentum tensor then take
the following form:

$\displaystyle T^{\tau\tau}$
$\displaystyle=(\varepsilon+p)(u^{\tau})^{2}-p=\varepsilon\cosh(y_{L}),$
$\displaystyle T^{\tau\eta}$
$\displaystyle=(\varepsilon+p)u^{\tau}u^{\eta}=\frac{\varepsilon}{\tau_{0}}\sinh(y_{L}),$
$\displaystyle T^{\eta\eta}$
$\displaystyle=(\varepsilon+p)u^{\eta}u^{\eta}=\frac{\varepsilon}{\tau_{0}^{2}}\cosh(y_{L}),$

which is consistent with the assumption of Ref. shen2021 .

Figure 9: The numerical result for $dN_{\text{ch}}/d\eta$ is compared with
experimental data by PHOBOS in different centralities phobos2003 ; phobos2006
.

Figure 10: $dN_{\text{ch}}/d\eta$ shown as a function of pseudorapidity with
(CP) and without (noCP) the critical point in EoS for colliding energies
(left) 14.5 GeV at impact parameter $b=4$ fm, and (right) 19.6 GeV at impact
parameter $b=5.6$ fm.

The trace of the energy-momentum tensor is given by

$T^{\mu}_{\mu}=T^{\tau\tau}-\tau^{2}T^{\eta\eta}=0.$

Hence, our choice for the rest frame quantities does not violate the condition
for the trace of the energy-momentum tensor. The viscous stresses are
initialized to their corresponding Navier-Stokes limit, the expressions of
which are:

$\displaystyle\pi^{\tau\tau}$ $\displaystyle=-2\eta\tau
u^{\tau}(u^{\eta})^{2}-\frac{2\eta}{3\tau}u^{\tau}\left[1-(u^{\tau})^{2}\right],\
\pi^{\tau x}=-\eta\partial_{x}u^{\tau},$ $\displaystyle\pi^{\tau y}$
$\displaystyle=-\eta\partial_{y}u^{\tau},\
\pi^{\tau\eta}=-\eta\tau(u^{\eta})^{3}-\frac{\eta
u^{\eta}}{\tau}-\frac{\eta}{3\tau}(u^{\tau})^{2}u^{\eta},$
$\displaystyle\pi^{xx}$ $\displaystyle=\frac{2\eta}{3\tau}u^{\tau},\
\pi^{xy}=0,\ \pi^{x\eta}=-\eta\partial_{x}u^{\eta},\
\pi^{yy}=\frac{2\eta}{3\tau}u^{\tau},$ $\displaystyle\pi^{y\eta}$
$\displaystyle=-\eta\partial_{y}u^{\eta},\
\pi^{\eta\eta}=-\frac{4\eta}{3\tau}u^{\tau}\left[\frac{1}{\tau^{2}}+(u^{\eta})^{2}\right]$

It was stated in Ref. shen2021  that the parameter $f$ has negligible effects
on most of the global observables such as the pseudorapidity distributions,
particle yields, and elliptic flow. We have checked this and conclude the
same. The rest of the analysis that follows is carried out on a constant
energy density hypersurface, $\varepsilon=0.3$ GeV/fm3, we shall denote this
surface as $\Sigma_{\varepsilon}$ below.

By setting $f=0.2$, impact parameter, $b=8.7$ fm, and using the same set of
values for other parameters of the IC model for Au+Au collision at
$\sqrt{s_{NN}}=200$ GeV, described in the main article, we compute the
negative $y$-component of global polarization, $-P_{y}$, of $\Lambda$-hyperon
for 20-60% centrality. Our result is $-P_{y}$=0.254%. The corresponding
experimental value from STAR is 0.277$\pm$ 0.040 (stat) and using the updated
PDG value of $\alpha_{\Lambda}$ is 0.243$\pm$ 0.035% (stat) star2018 . We also
show the pseudorapditiy and transverse momentum dependence in the same
centrality in Figs.11 and 12. Again, we use EoS with CP to generate these
results. Because the difference between CP and noCP equation of states is
negligible at such large colliding energy. The azimuthal angle dependence of
the longitudinal component of the spin-polarization, $P_{z}$, is shown in Fig.
13. The sign of our numerical results is opposite to that of experimental
data. This problem is known as the longitudinal sign puzzle in the literature
(see becattini2020  for a review).

Figure 11: The numerical result for $y$-component of $\Lambda$-polarization as
function of pseudorapidity is compared with the experimental data by STAR from
Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV in 20%-60% centrality star2018 .
Vertical lines are the statistical uncertainties only. Figure 12: The
numerical result for $y$-component of $\Lambda$-polarization as function of
transverse momentum ($p_{T}$) is compared with the experimental data by STAR
from Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV for 20-60% centrality
star2018 . Vertical lines are the statistical uncertainties only.

Having validated our code we now carry out our simulation near the critical
point at colliding energy $\sqrt{s_{NN}}=14.5$ GeV. The negative $y$-component
of the global polarization is suppressed in the presence of CP as compared to
the case when the CP is absent for a given value of $f$. This illustrates that
the values of global polarization with and without CP are different at fixed
$f$ which is clearly observed in the main article (Fig. 5) for zero initial
OAM (corresponding to $f=0$). We carry out an exercise to verify whether the
same global polarization of $\Lambda$ hyperon can be obtained with and without
CP by tuning the parameter $f$ in the presence of OAM. The values $f=0.45$
with CP and $f=0.53$ without CP corresponds to the same value of
$-P_{y}$=0.92% of $\Lambda$-hyperon for $b=5.6$ fm which nearly reproduces the
experimental value measured by the STAR collaboraion for 20%-60% centrality.
This indicates that the data on global polarization can not be used to
exclusively determine the CP effects, because other parameters can be tuned to
the data.

Figure 13: Azimuthal angle dependence of $z$-component of spin polarization of
$\Lambda$-hyperon. STAR measurements are from Ref.star2019 .

However, once the value of $f$ is tuned to the global polarization data then
the use of the same value of $f$ (which fixes the OAM) predicts significant
difference in the rapidity distribution of $P_{y}$ with and without CP,
clearly indicating that the rapidity distribution of $-P_{y}$ is sensitive to
CP. We plot the rapidity distribution of $-P_{y}$ in the top panel of Fig. 14.
We observe a suppression of about $25\%$ in $-P_{y}$ at mid-rapidity. The
change in other observables like $p_{T}$ spectra, elliptic flow,
$dN_{\text{ch}}/dy$ is at most 8% on the surface $\Sigma_{\varepsilon}$
singh2022 . We also compute the derivative of $P_{y}$ with respect to rapidity
and plot as a function of rapidity in the bottom panel of Fig. 14. We observe
a slight negative slope for $-dP_{y}/dy$ at mid-rapidity opposite to the case
when there is no critical point. We cannot confirm the negative sign of the
slope by further approaching the critical point due to the limitations of the
EoS model which is valid for $\mu_{B}<450$ MeV. Beyond 450 MeV, the speed of
sound starts to give unphysical results. So close to CP, many fluid cells
whose trajectories cross $\mu_{B}=450$ MeV become problematic. However, it
should be mentioned that the suppression that we observe is when the center of
the fireball created in $\sqrt{s_{NN}}=14.5$ GeV energy is still 100 MeV away
from the critical point along the $\mu_{B}$ axis. We expect the effect to get
enhanced on further approach toward the critical point.

Figure 14: The top panel shows rapidity dependence of $y$ component of
$\Lambda$-hyperon polarization at $\sqrt{s_{NN}}=14.5$ GeV and $b=5.6$ fm with
(CP) and without (noCP) critical point EoS, and the bottom panel shows the
derivative as a function of rapidity.

The sensitivity of the spin polarization to the EoS can be understood from the
following expression for the spin polarization in the rest frame of
$\Lambda$-hyperon at any point on $\Sigma_{\varepsilon}$ becattini2020 :

$\vec{S}^{*}(x,p)\propto\frac{\gamma}{T^{2}}\vec{v}\times\nabla
T+\frac{1}{T}\left(\vec{\omega}-(\vec{\omega}\cdot\vec{v})\vec{v}\right)+\frac{1}{T}\gamma\vec{A}\times\vec{v},$

where $\vec{\omega}$ is the vorticity ($=\nabla\times\vec{v}$), $\gamma$ is
the Lorentz factor and $A$ denotes the acceleration of the fluid element. In
nutshell, spin polarization depend on the gradients of temperature and curl of
flow-velocity $(=\omega$). These gradients in turn depend on the expansion
dynamics of the system and the expansion is strongly influenced by the speed
of sound which is obtained from EoS. Since the sound wave gets suppressed at
the critical point, the system undergoes a slow expansion that results into
smaller gradients of temperature and flow-velocity, as they will not change
much. This should then result into a suppression of spin-polarization as
confirmed by our simulations. The effects of CP on other observables have been
presented in Ref. singh2022  by solving the same hydrodynamic equations with
and without the critical point.

Figure 15: Suppression in $y$-component of spin polarization for $f=0$,
$\zeta^{\prime}_{0}=\zeta_{0}/10$ and $\xi^{\prime}_{0}=0.8$ fm.

To further demonstrate the robustness of our prediction, we show in Fig. 15
the suppression in polarization when the bulk viscosity away from critical
region, denoted by $\zeta_{0}$ in Eq.(9) of main article, is decreased by a
factor of 10 i.e. $\zeta^{\prime}_{0}=\frac{\zeta_{0}}{10}$ and the length
scale, $\xi_{0}$ in Eq.(9) of main article, that marks the boundary of the
critical region is taken as $\xi^{\prime}_{0}=0.8$ fm. We still see a
suppression of about 30% at mid-rapidity.

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