# Formal Proofs as Structured Explanations:
Proposing Several Tasks on Explainable Natural Language Inference

Lasha Abzianidze
Institute for Language Sciences
Utrecht University, Utrecht, the Netherlands
<EMAIL_ADDRESS>
###### Abstract

In this position paper, we propose a way of exploiting formal proofs to put
forward several explainable natural language inference (NLI) tasks. The formal
proofs will be produced by a reliable and high-performing logic-based NLI
system. Taking advantage of the in-depth information available in the
generated formal proofs, we show how it can be used to define NLI tasks with
structured explanations. The proposed tasks can be ordered according to
difficulty defined in terms of the granularity of explanations. We argue that
the tasks will suffer with substantially fewer shortcomings than the existing
explainable NLI tasks (or datasets).

##  1 Introduction

Natural Language Inference (NLI) is a popular task that evaluates NLP models
on the capacity of reasoning with natural language: given a premise text and a
hypothesis sentence, a model is expected to predict an inference relation
(entailment, neutral, and contradiction) between the premise and the
hypothesis.111 Originally the task was called Recognizing Textual Entailment
(RTE, Dagan et al. 2005), but later the community started to refer to it as
NLI mainly due to the influential works by MacCartney (2009) and Bowman et al.
(2015).  In this way, NLI is a three-way classification task (sometimes even
two-way) that is supposed to evaluate a model’s capacity for drawing
inferences. To draw correct inferences, we expect a model, where possible, to
parse the relevant part of the premise and the hypothesis, employ a decision
procedure that mostly follows reasonable inference rules, and use necessary
linguistic semantics or knowledge (if applicable). In contrast to the
expectation, multiple works have shown that the high-performing NLI models
have learned how to exploit biases in the NLI datasets rather than how to
approximate sound reasoning. The most notable example of such biases is the
hypothesis-only bias (Poliak et al., 2018; Gururangan et al., 2018; Tsuchiya,
2018), where an NLI dataset allows models to achieve a substantially high
score by observing only the hypothesis sentence.222 There is nothing special
with the hypothesis sentence in the NLI task as such. The bias is due to the
commonly used crowdsourcing method of creating a large NLI dataset where
hypothesis sentences are elicited from crowd workers.

The attempt to evaluate an entire reasoning process between a premise and a
hypothesis with one of the three inference relations can be seen as a highly
simplified way of modeling reasoning. The three-way classification format of
the task and unintended biases and regularities in NLI datasets leave
substantial room for deep learning-based NLI models to optimize their decision
procedure in many ways, which can be meaningfully different from the sound
one. Moreover, such a setup of the task doesn’t teach models to provide any
information on top of the inference label, which makes it unclear why the
model makes a certain prediction. Teaching a model to additionally generate an
explanation supporting the inference label is a way to solve these two issues
simultaneously: (a) to further constraint a model’s decision procedure and
prevent it from learning irrelevant relations, (b) to provide additional
insight supporting a predicted relation.

In the subsequent sections, we will briefly survey several notable explainable
NLI tasks and explain their shortcomings in terms of:

  1. (A)

A lack of reliable automatic evaluation;

  2. (B)

Coarse granularity of the explanations,

  3. (C)

Incomplete set of inference relations.

As a proposal, we will describe a list of tasks that are gradually increasing
in terms of difficulty and the final versions overcome the highlighted
shortcomings.

##  2 Current explainable NLI tasks

##### e-SNLI

We believe that creating the e-SNLI dataset (Camburu et al., 2018) was the
first significant step towards explainable NLI. It is obtained by augmenting
the SNLI dataset (Bowman et al., 2015) with human-elicited explanations in a
natural language form using crowdsourcing: one explanation was collected for
training samples and three for development and evaluation samples.

While e-SNLI provides ample training data (570K problems) for deep neural
networks (DNNs), unfortunately, its automated evaluation method for
explanations is not reliable. In particular, Camburu et al. (2018) concluded
this by showing that mediocre system-produced explanations and the third
reference explanations obtained a BLEU score similar to the first two
reference explanations.333 A somewhat similar finding was concluded by Wang et
al. (2020): a BLEU score with respect to the crowdsourced explanations did not
correlate well with human evaluation, where explanations provided a reason why
nonsensical sentences don’t make sense.  Due to this finding, when using
e-SNLI for evaluation, the papers randomly draw 50-100 gold explanations and
manually compare them to the corresponding system-produced explanations.
Obviously, such a way of evaluation is not representative and replicable.
While e-SNLI is a large information-rich dataset, unfortunately, it fails to
serve as a reliable benchmark for explainable NLI.

##### Fact-rule proofs

Building upon the RuleTaker’s datasets (Clark et al., 2021), Saha et al.
(2020a) and Tafjord et al. (2021) prepared corresponding datasets for
explainable NLI.444 While there are several datasets differing in details such
as open- and close-world assumptions, we focus on the general description of
the task.  In a nutshell, a sample of the dataset represents a pair of a
premise text and a hypothesis sentence, where the premise text is a collection
of sentences representing facts (e.g., Erin is young) and rules (e.g., If
someone is blue then they are kind). Based on the fact and rule premises, a
system is expected to predict whether a hypothesis sentence is true or false
and provide a corresponding proof tree. The proof tree is rooted in the
hypothesis sentence ID, other nodes represent fact and rule IDs, and the
child-parent relation stands for a rule application to the facts. The proofs
serve as explanations, and they are suitable for automatic evaluation due to
their unambiguous structure. During the evaluation, a system-produced proof is
correct if it fully matches the reference proof.

Formatting explanations in terms of proof trees on facts and rules is a step
forward from the evaluation perspective. On the downside, the facts and rules
in the datasets are synthetically created, which results in unnatural
sentences with little diversity in syntactic structure and semantic phenomena.
Note that the diversity in syntactic structures was alleviated in
ParaRules–the version of the dataset, where facts and rules were paraphrased
with crowdsourcing. Some versions of the dataset (with the closed-world
assumption) include only entailment and contradiction relations. The version
with the open-world assumption allows the neutral relation but no proof is
defined for it.555 Providing explanations for neutral relations is challenging
as there is no proof similar to entailment and contradiction relations.
Somewhat related, the fact verification task (Thorne et al., 2018) also omits
an explanation for NotEnoughInfo label in contrast to the Supported and
Refuted labels.  While facts and rules are mostly of a synthetic structure,
strictly speaking, it is not explained how, for example, a rule is applied to
a fact, e.g., does a model somehow match the antecedent of the rule to the
fact, or is the rule application accidental?

##### Entailment Bank

Yet another version of explainable NLI with structured explanations was
introduced by Dalvi et al. (2021). They created the Entailment Bank which
consists of up to 2K entailment trees. The entailment tree is structurally
similar to the fact-rule proofs but differs from them in the following: (i)
The sentences are more naturalistic and diverse, collected from the grade-
school science questions; (ii) trees are manually created by trained
annotators; (iii) some non-terminal nodes in the tree are not among the input
facts and need to be generated on the fly.

Although the entailment trees improve upon the fact-rule proofs in terms of
the naturalness of the sentences and inferences, they have several
shortcomings: (i) The Entailment Bank focuses entirely on the entailment
relation while the contradiction and neutral relations are skipped; (ii)
Creating entailment trees is a time-consuming process ($\approx$20 minutes per
tree as reported by Dalvi et al. (2021)); (iii) Similarly to the fact-rule
proofs, entailment trees don’t explain how sentences entail another sentence.
In other words, each step in the entailment trees can be seen as an instance
of a simple NLI problem labeled as entailment while no explanation is provided
for such sub-problems; (iv) The evaluation based on the tree matching is too
strict as it penalizes any paraphrases of the intermediate nodes generated on
the fly.

It seems the variability in free-text explanations is not sufficiently
constrained by an NLI problem, unlike how reference translations are
constrained by the source text in machine translation. That’s why the
comparison metrics commonly used in machine translation haven’t been
successful in detecting the semantic equivalence of explanations in NLI. To
support an accurate automatic evaluation of explanations, one needs to abandon
free-text explanations and opt for structured explanations. However, existing
approaches to generating structured explanations come with shortcomings. If
the generation is automatized, it results in synthetic data, while the manual
construction of explanations is hard and time-consuming. In the next section,
we propose a method of collecting structured explanations that represents a
substantial improvement over the existing methods.

##  3 Designing new explainable NLI tasks

In this section, we describe how necessary data can be semi-automatically
obtained and how several explainable NLI tasks with structured explanations
can be defined based on the collected data.

###  3.1 Natural language theorem prover

The state-of-the-art performance on mainstream NLI datasets is undoubtedly
held by DNN-based NLI models. Despite that, there have been active development
of a couple of NLI systems based on formal logic and theorem proving. The
merits of logic-based NLI systems are the following: their precision is
considerably high due to the underlying methodology, and their decision
procedure is inherently interpretable. To the best of our knowledge, only two
logic-based systems have managed to reach the human-performance level (84%) on
SICK (Marelli et al., 2014): ccg2lambda (Mineshima et al., 2015; Yanaka et
al., 2018) and LangPro (Abzianidze, 2015, 2017). From these two systems, we
find LangPro to be a better fit for collecting data for the new explainable
NLI tasks for three reasons: (i) It has a higher precision on SICK (94% vs
84%); (ii) its underlying natural logic uses logical forms that are not very
specific to a certain meaning representation (in contrast to first-order
logic); (iii) the structure of its proofs is faithful to the underlying proof
calculus, the semantic tableau method (Beth, 1955; Hintikka, 1955).

0-1=dj0-1= 1 $\textbf{{many}}_{{\tt n},{\tt vp},{\tt
s}}~{}\textbf{{bird}}_{\tt n}(\textbf{{high}}_{{\tt vp},{\tt
vp}}~{}\textbf{{hover}}_{\tt vp}):\scalebox{0.85}{$[\kern 3.0pt]$}:\mathbb{T}$
---
many birds hover high
2 $\textbf{{few}}_{{\tt n},{\tt vp},{\tt s}}~{}\textbf{{bird}}_{\tt
n}~{}\textbf{{fly}}_{\tt vp}:\scalebox{0.85}{$[\kern 3.0pt]$}:\mathbb{T}$
few birds fly
0-1= 3 $\textbf{{high}}~{}\textbf{{hover}}:[c]:\mathbb{T}$
---
$c$ hover high
4 $\textbf{{fly}}:[c]:\mathbb{F}$
$c$ fly
0-1= 7 $\textbf{{hover}}:[c]:\mathbb{T}$
---
$c$ hover
0-1= 8 $\times$
---
0-1= 5 $\textbf{{many}}~{}\textbf{{bird}}:[\textbf{{fly}}]:\mathbb{T}$
---
many birds fly
6 $\textbf{{few}}~{}\textbf{{bird}}:[\textbf{{fly}}]:\mathbb{T}$
few birds fly
0-1= 9 $\textbf{{many}}:[\textbf{{bird}},\textbf{{fly}}]:\mathbb{T}$
---
many birds fly
0-1= 10 $\textbf{{few}}:[\textbf{{bird}},\textbf{{fly}}]:\mathbb{T}$
---
few birds fly
0-1= 11 $\times$
---
adj${}^{\subset}_{\mathbb{T}}$ [3]
${\scalebox{1.15}{$\times$}\kern-1.5pt}\\!\sqsubseteq$

[4,7]

a  $>$ [5] a  $>$ [6] ${\scalebox{1.15}{$\times$}\kern-1.5pt}\mid$ [9,10]
$\mathord{\vphantom{\uparrow}\text{\ooalign{$\uparrow$\kern-0.50003pt\raisebox{0.3014pt}{\scalebox{0.88}{$|$}}\cr\kern
0.20004pt\raisebox{-1.99997pt}{$\smallsmile$}\cr}}}$ [1,2]

Figure 1: The tableau proves that “many birds hover high” contradicts “few
birds fly” and vice versa.

The proofs produced by LangPro will be the source of the data for the proposed
tasks. Figure 1 illustrates a tableau proof from LangPro. The nodes of the
trees represent terms of simply typed $\lambda$-calculus backed up with the
semantics of higher-order logic (Church, 1940). The main idea behind the proof
is to find a counterexample for the target relation, e.g., the contradiction
relation. A counterexample of contradiction is a situation where the premise
and the hypothesis are true, hence the tableau proof starts with 1,2 marked
with $\mathbb{T}$. The tableau grows by applying rules from the predefined set
of rules to the existing nodes and breaking the antecedent terms into smaller
pieces. The annotations of the edges show this process. For instance, 3-6
nodes are obtained by applying
($\mathord{\vphantom{\uparrow}\text{\ooalign{$\uparrow$\kern-0.50003pt\raisebox{0.3014pt}{\scalebox{0.88}{$|$}}\cr\kern
0.20004pt\raisebox{-1.99997pt}{$\smallsmile$}\cr}}}$) rule to 1,2.666 The rule
($\mathord{\vphantom{\uparrow}\text{\ooalign{$\uparrow$\kern-0.50003pt\raisebox{0.3014pt}{\scalebox{0.88}{$|$}}\cr\kern
0.20004pt\raisebox{-1.99997pt}{$\smallsmile$}\cr}}}$) exploits upward
monotonicity of many in its second argument, e.g., _many birds $V_{1}$_
entails _many birds $V_{2}$_ if $V_{1}$ entails $V_{2}$. The idea behind the
rule is as follows. If $f^{\uparrow}(x)=1$ and $g(y)=1$ where $f$ and $g$ are
Boolean functions and $f$ is upward monotone, then there are two
possibilities: either $x\not<y$ (the left branch) or $x<y$, hence
$f^{\uparrow}(y)=1$ (the right branch). Note that $A~{}B~{}C:[]$, $A~{}B:[C]$,
and $A:[B,C]$ are semantically the same nodes only differing in terms of
formatting that distinguishes functions from their argument list.  The
branching represents two possibilities in searching for a counterexample. In
the example, both branches are closed due to inconsistencies spotted in the
corresponding possible situations. The left branch represents an inconsistent
situation due to some entity $c$ hovering but not flying (see 4,7) while the
right branch is closed as few and many cannot be true on the same arguments.
The tableau with all branches closed represents a failure to find a
counterexample. Therefore, the relation for which the refutation failed is
proved and the closed tableau represents its proof.

###  3.2 From proofs to explainable NLI tasks

####  3.2.1 Answering several concerns

Before describing the design of specific tasks, we would like to answer
several questions that naturally arise about the methodology that utilizes
proofs as explanations.

##### Is it feasible to get a substantial amount of proofs?

LangPro achieves $84\%$ accuracy on SICK with $94\%$ of precision (Abzianidze,
2020), however, many false proofs are due to noisy gold labels in SICK. While
SICK is not a challenging dataset, similar NLI problems do occur in other NLI
datasets. Considering the number and the size of the existing human-annotated
NLI datasets (in total more than 1M problems), it is realistic to obtain
proofs for at least 20K of NLI problems.777 We expect most of these problems
to be drawn from SICK, SNLI, MED (Yanaka et al., 2020), Breaking NLI (Glockner
et al., 2018), NegNLI (Hossain et al., 2020), and ConjNLI (Saha et al.,
2020b).  This is already an order of magnitude larger than the Entailment
Bank. The advantage of the method is that the data collection process is
automatized and produces detailed proofs with extremely high consistency and
correctness. For instance, matching such levels of detail, consistency, and
correctness will also be challenging with trained annotators.

##### Will not the proved NLI problems be homogeneous?

The number of inference rules used to build tableau proofs is up to 80. The
rules cover syntactic categories and constructions such as prepositional
phrases, expletives, light verb constructions, open compounds, verb
subcategorization, passives, copula, and coordination. This will allow
sufficient syntactic diversity in the proved problems and will represent a
substantial improvement over the existing inference datasets with structured
explanations (Saha et al., 2020a; Tafjord et al., 2021; Dalvi et al., 2021).
In the collected proofs, there might be relatively many proofs that are
structurally identical to other proofs. However, such proofs will differ from
each other in terms of lexical entities and pose a challenge for systems from
a generalization perspective.

##### But human reasoning is different from deductive reasoning

The inferences that humans validate but LangPro fails to prove will be out of
reach when collecting proofs with the help of LangPro. The NLI problems that
LangPro proves but get different inference labels from human annotators will
be manually verified by experts. In many cases, the gold label can be
overridden by LangPro’s prediction given that human annotations often disagree
with each other for various reasons. This means that some of the problems that
were complex for crowd workers to correctly classify will be identified by
LangPro and be eventually correctly classified. We see it unnecessary to dumb
down NLI problems by aligning their labels to crowd workers’ judgments. For
example, it is known that humans are not good at using modus tollens rule
Wason (1968), but this doesn’t mean that we need to eliminate the use of modus
tollens when reasoning with NLI problems.

####  3.2.2 New explainable NLI tasks

We describe a list of explainable NLI tasks that will be defined by the
datasets derived from the tableau proofs of LangPro. We present the list
following the order based on the explanation richness. An illustration sample
of the datasets will be based on the proof in Figure 1.

##### Lexical relations as explanations

One of the simplest structured explanations is a set of lexical relations that
are necessary for the inference. For example, the explanation of the NLI
problem in Figure 1 could be two lexical relations:
$\textbf{{hover}}\sqsubset\textbf{{fly}}$ and
$\textbf{{many}}|\textbf{{few}}$, where the relations stand for the
subsumption and disjoint/alternation relations. All such necessary relations
can be easily extracted from tableau proofs. In some cases, the relations can
be over the short phrases if it is not further decomposable, e.g.,
$\textbf{{mouse}}\sqsubset\textbf{{small animal}}$. The NLI problems that use
no lexical relation, e.g., _Not all birds fly_ entailing _Some bid doesn’t
fly_ , will simply carry an empty explanation. Optionally, such problems can
also be omitted for this type of explanation.

##### Rules and lexical relations as explanations

To provide a relatively simple explanation for NLI problems that require no
lexical relations, we propose to augment the (possibly empty) set of lexical
relations with the multiset of inference rules used in the tableau proof. In
the case of the example proof, the multiset of rules will be
[($\mathord{\vphantom{\uparrow}\text{\ooalign{$\uparrow$\kern-0.50003pt\raisebox{0.3014pt}{\scalebox{0.88}{$|$}}\cr\kern
0.20004pt\raisebox{-1.99997pt}{$\smallsmile$}\cr}}}$),(adj${}^{\subset}_{\mathbb{T}}$)].
Here, we ignore the argument pushing rule (a  $>$ ) as it has no semantic
contribution and also omit (${\scalebox{1.15}{$\times$}\kern-1.5pt}\mid$) and
(${\scalebox{1.15}{$\times$}\kern-1.5pt}\\!\sqsubseteq$) as they are redundant
beside the lexical relations. The multiset of the example entailment requiring
no lexical relation will be [($\neg$), ($\forall_{\mathbb{F}}$),
($\exists_{\mathbb{F}}$), ($\neg$)]. The multiset reflects the complexity of
the entailment: two negation rules correspond to _not_ and _doesn’t_ while the
rules for universal and existential quantifiers correspond to _all_ and _some_
where their subscript makes sure that _all_ is in the negative polarity
context and _some_ in the hypothesis.

In the current and the previous tasks, we recommend using F-score. In the case
of multisets of inference rules, the comparison should be based on exact
matching to avoid inflating the score.

##### Unlabeled proofs as explanations

To add more details and structure to the explanations, we propose the proof
trees without inference rule labels as explanations in addition to the lexical
relations. The example of an unlabeled proof tree is illustrated in Figure 2.
Alternatively, the lexical relations can be encoded in terms of the closure
rules and their references to the antecedent nodes.

It will be too much to ask an NLI model to generate exact $\lambda$-terms of
the nodes as this automatically includes semantic parsing tasks when
considering the initial nodes. Given that we want to prevent NLI models from
committing to a certain meaning representation, the node entries in the gold
proofs will be converted into the original surface forms (as shown with gray
text in Figure 1). Fortunately, this is possible with the help of the
naturalness of the $\lambda$-terms, as each term represents a constituent or a
constituent with a trace. It is important to emphasize the naturalness of
natural logic formulas. With the help of this property, NLI systems are not
forced to learn semantic parsing along with reasoning. Moreover, to decrease
the depth of the trees, it is recommended to omit the structure-related
inference rules, e.g., (a  $>$ ), that have no semantic contribution. This
simplification results in a shorter right-hand side branch in Figure 2.

One might find some visual resemblance between the unlabeled proof trees and
the entailment trees of the Entailment Bank (Dalvi et al., 2021). However,
this resemblance is superficial. In entailment trees, the tree edges represent
entailment arrows while in tableau proofs the edges are conjunctions; A branch
in a tableau represents a situation modeling a conjunction of all terms
sitting on the branch.

0-1=dj0-1= $\texttt{many birds hover high}:\mathbb{T}$
---
$\texttt{few birds fly}:\mathbb{T}$
0-1= $\texttt{$c$ hover high}:\mathbb{T}$
---
$\texttt{$c$ fly}:\mathbb{F}$
0-1= $\texttt{$c$ hover}:\mathbb{T}$
---
$\times$
0-1= $\texttt{many birds fly}:\mathbb{T}$
---
$\texttt{few birds fly}:\mathbb{T}$
$\times$

Figure 2: The unlabeled proof represents the simplified version of the proof
from Figure 1. Note that all word forms in the nodes are identical to those in
the premise and the hypothesis.

##### Complete proofs as explanations

The final version of the explainable NLI task employs the entire proof as an
explanation. The lexical relations and the inference rules will be all encoded
in the tree. Like in the unlabeled proof trees, the trees here will also be
shortened by pruning the semantically vacuous inference rules.

##  4 Conclusion

We have presented four explainable NLI tasks that exploit tableau proofs to
extract relevant information that serves as structured explanations. The key
components that make this feasible are three characteristics of the LangPro:
the naturalness of terms, an intuitively simple reasoning calculus, and almost
perfect precision of the prover with a decent accuracy.

All the described tasks are compatible with the automatic evaluation as their
structured explanations don’t suffer from linguistic variability: each lexical
item can be represented in terms of character offsets in an NLI problem. The
granularity of the explanations becomes finer when moving from the simplest
task to the most information-rich one. In all four tasks, explanations are
based on fine-grained units such as phrases and lexical units. The tasks cover
both entailment and contradiction relations. Structuring explanations for
neutral cases require special care and we leave this for the feature research.

We have started building a proof bank based on the described approach. The
proof bank consists of NLI problems and their corresponding formal proofs with
structured explanations in various formats. While the term _formal proof_
might suggest a long and entangled sequence of symbols, in fact, the tableau
proofs for NLI problems are user-friendly as illustrated in Figure 2.

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# Impact of nuclear structure on the CME background in ${}^{96}_{44}$Ru +
${}^{96}_{44}$Ru and ${}^{96}_{40}$Zr + ${}^{96}_{40}$Zr collisions at
$\sqrt{s_{NN}}$ = 7.7 $\sim$ 200 GeV from a multiphase transport model

Fei Li(李飞)  Key Laboratory of Nuclear Physics and Ion-beam Application (MOE),
Institute of Modern Physics, Fudan University, Shanghai 200433, China  Yu-Gang
Ma(马余刚) ID<EMAIL_ADDRESS>Key Laboratory of Nuclear Physics and Ion-
beam Application (MOE), Institute of Modern Physics, Fudan University,
Shanghai 200433, China  Shanghai Research Center for Theoretical Nuclear
Physics， NSFC and Fudan University, Shanghai 200438, China  Song Zhang(张松) ID
Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of
Modern Physics, Fudan University, Shanghai 200433, China  Shanghai Research
Center for Theoretical Nuclear Physics， NSFC and Fudan University, Shanghai
200438, China  Guo-Liang Ma(马国亮) ID Key Laboratory of Nuclear Physics and Ion-
beam Application (MOE), Institute of Modern Physics, Fudan University,
Shanghai 200433, China  Shanghai Research Center for Theoretical Nuclear
Physics， NSFC and Fudan University, Shanghai 200438, China  Qiye Shou(寿齐烨) ID
<EMAIL_ADDRESS>Key Laboratory of Nuclear Physics and Ion-beam
Application (MOE), Institute of Modern Physics, Fudan University, Shanghai
200433, China  Shanghai Research Center for Theoretical Nuclear Physics， NSFC
and Fudan University, Shanghai 200438, China

###### Abstract

Impacts of nuclear structure on multiplicity ($N_{ch}$) and anisotropic flows
($v_{2}$ and $v_{3}$) in the isobaric collisions of ${}^{96}_{44}$Ru +
${}^{96}_{44}$Ru and ${}^{96}_{40}$Zr + ${}^{96}_{40}$Zr at $\sqrt{s_{NN}}$ =
7.7, 27, 62.4 and 200 GeV are investigated by using the string melting version
of A MultiPhase Transport (AMPT) model. In comparison with the experimental
data released recently by the STAR collaboration, it is found that the impact
of quadrupole deformation $\beta_{2}$ on the $v_{2}$ difference is mainly
manifested in the most central collisions, while the octupole deformation
$\beta_{3}$ is in the near-central collisions, and the neutron skin effect
dominates in the mid-central collisions. Viewing from the energy dependence,
these effects are magnified at lower energies.

##  I Introduction

The present theories predict that a local parity $(\textit{P})$ and charge
parity $(\textit{CP})$ violation region could be formed by strong interaction
in relativistic heavy-ion collisions [1, 2, 3, 4], where a charge number
imbalance of light quarks can be achieved. In the process of a non-central
heavy-ion collision, the CP-violating region is affected by the strong
magnetic field produced by high speed protons passing through [5, 6],
resulting in charge separation along the magnetic field. This phenomenon is
also known as the Chiral Magnetic Effect (CME) [7, 8, 9, 10, 11]. The
confirmation of the existence of the CME will not only lead to a deeper
understanding for QCD vacuum, but also imply the existence of the strong-
interaction CP-violating region as well as the restoration of the chiral
symmetry in Quark Gluon Plasma (QGP). This phenomenon can be attributed by the
influence of strongest known electromagnetic field on the collision region [2,
12, 13] etc. Various anomalous chiral phenomena and possible detectable
methods have been discussed in literature, e.g., [13, 14, 15, 16, 17, 18, 19,
20, 21]. Finding an experimental signature that can conclusively confirm the
CME is one of current major challenges in relativistic heavy-ion physics.

Since the magnetic field is usually perpendicular to the reaction plane
($RP$), defined by the impact parameter and the beam momentum in heavy-ion
collisions, the CME-sensitive charge separation is measured with respect to
the reaction plane, and the most widely used observable at present is the
“$\gamma$ correlator”: $\gamma_{\alpha\beta}=\left\langle
cos(\phi_{\alpha}+\phi_{\beta}-2\Psi_{RP})\right\rangle$ [22], where
$\phi_{\alpha}$ and $\phi_{\beta}$ are the azimuthal angle of charged
particles of interest, and $\Psi_{RP}$ is the angle of the reaction plane.
However, some non-CME signal sources (e.g. local charge conservation and/or
transverse momentum conservation entwined with the elliptic flow, $v_{2}$) can
also contribute to the $\gamma$ value, which makes it difficult to quantify
the CME effect in this way [23, 24, 25].

In order to disentangle the contribution of non-CME background to $\gamma$,
many ideas have been proposed, among which the isobar collisions (e.g.
${}^{96}_{44}$Ru + ${}^{96}_{44}$Ru and ${}^{96}_{40}$Zr + ${}^{96}_{40}$Zr)
are expected to provide the best solution to this problem. Since
${}^{96}_{44}$Ru +${}^{96}_{44}$Ru and ${}^{96}_{40}$Zr + ${}^{96}_{40}$Zr
have the same nucleon numbers but different charges, it is expected that
different CME strengths can be quantitatively extracted from similar flow-
driven backgrounds [26, 27]. The experimental project of ${}^{96}_{44}$Ru +
${}^{96}_{44}$Ru and ${}^{96}_{40}$Zr + ${}^{96}_{40}$Zr isobar collisions at
$\sqrt{s_{NN}}$ = 200 GeV was launched at RHIC in 2018, and the results were
recently released by the STAR collaboration [4]. The isobar blind analysis in
the STAR collaboration [28] observed significant differences in the
multiplicity and flow harmonics in a given centrality between the two
collision systems, indicating that the magnitude of the non-CME background is
inconsistent between the two species. Many studies suggested that the
discrepancy of CME backgrounds in Ru and Zr collisions was due to the
different geometrical shapes of colliding ions, such as the difference of
deformation and neutron skin thickness for Ru and Zr [26, 29, 30].

In this paper, the CME background difference between Ru and Zr collisions is
studied in A MultiPhase Transport (AMPT) model, with five configurations for
the nuclear structure parameters of Ru and Zr. Our results are compared with
the experimental data to verify which description for nucleon structure of Ru
and Zr is more consistent with the realistic nuclear structure. Further, the
energy dependence of the CME background differences between Ru and Zr is
investigated at various energies, i.e. $\sqrt{s_{NN}}$ = 7.7, 27, 62.4 and 200
GeV, aiming at probing the nature of the background at lower energies and
providing theoretical support for future experiments.

The paper is organized as follows: In Sect. 2, the general setup of modeling
Ru + Ru and Zr + Zr collisions by AMPT is briefly introduced. In Sect. 3, the
numerical results and discussion are presented, and the summary is given in
Sec. 4.

Figure 1: (Upper panels) Distributions of the number of charged hadrons from
the simulation of AMPT in the pseudorapidity window $|\eta|<$ 0.5 in Ru + Ru
and Zr + Zr collisions at $\sqrt{s_{NN}}$ = 200 GeV for five sets of Woods-
Saxon parameters: (a) Case-1, (b) Case-2, (c) Case-3, (d) Case-4, and (e)
Case-5. More details on definition of cases can be found in texts. The STAR
data [4] for isobar collisions at $\sqrt{s_{NN}}$ = 200 GeV are also shown for
comparisons. (Middle panels) The Ru + Ru to Zr + Zr ratio for five sets of
Woods-Saxon parameters at $\sqrt{s_{NN}}$ = 200 GeV, as well as the STAR data
are shown, respectively. (Lower panels) The Ru + Ru to Zr + Zr ratio for five
sets of Woods-Saxon parameters at $\sqrt{s_{NN}}$ = 7.7, 27 and 62.4 GeV,
respectively. Notice that the horizontal axis is zoomed in for clarity in the
lower pannels.

##  II General Setup

###  II.1 The AMPT model

Here we employed the string melting version of the AMPT model [31, 32] to
simulate Ru + Ru and Zr + Zr collisions and analyzed the generated data to
study the CME backgrounds. The model has proven to be effective in describing
collective flow in small and large collision systems at RHIC and LHC [33, 34,
35, 36, 37]. Based on the nonequilibrium transport dynamics, the AMPT model is
composed of four parts: the Heavy-Ion Jet INteraction Generator (HIJING) model
[38, 39] for generating the initial-state information, Zhang’s parton cascade
(ZPC) model [40] for modeling partonic scatterings, the Lund string
fragmentation model or a quark coalescence model for hadrons formation, and a
relativistic transport (ART) model [41] for treating the hadron scatterings.
In the AMPT model with string melting mechanism, the partons freeze-out is
according to local energy density and the hadronization process is simulated
by a naive quark coalescence model, which combines two (three) nearest partons
into a meson (baryon). The method for determining hadronic species is achieved
by the flavor and invariant mass of coalescing partons.

Figure 2: (Upper panels) The mean charge multiplicity $N_{ch}$ within
$|\eta|<$ 0.5 as a function of centrality in Ru + Ru and Zr + Zr collisions at
$\sqrt{s_{NN}}$ = 7.7, 27, 62.4 and 200 GeV. The STAR data [4] for isobar
collisions at $\sqrt{s_{NN}}$ = 200 GeV are also shown for comparison. The
centrality bins are slighted shifted for clarity. (Lower panels) The ratio of
the mean charge multiplicity in Ru + Ru collisions to that in Zr + Zr
collisions in matching centrality and energy. The above data include
statistical uncertainty.

###  II.2 Description of ${}^{96}_{44}$Ru and ${}^{96}_{40}$Zr

The spatial distribution of nucleons in the rest frame of ${}^{96}_{44}$Ru and
${}^{96}_{40}$Zr can be described by the following 2-parameter Fermi mass
density of the Woods-Saxon (WS) form [26, 42, 43]:

$\rho(r,\theta)=\frac{\rho_{0}}{1+\operatorname{exp}\left[\frac{r-R_{0}\left(1+\beta_{2}Y_{2}^{0}(\theta)+\beta_{3}Y_{3}^{0}(\theta\right)}{a}\right]},$
(1)

where r is radial position and $\theta$ is polar angle in spherical
coordinates, $\rho_{0}$ = 0.16 fm-3 is the nuclear saturation density, $R_{0}$
and $a$ represent the “radius” of the nucleus and the surface diffuseness
parameter, respectively, and the deformation of the nucleus is denoted by the
most relevant axial symmetric quadrupole deformation $\beta_{2}$ and octupole
deformation $\beta_{3}$. Since the $\beta_{2}$ and $\beta_{3}$ values of
${}^{96}_{44}$Ru and ${}^{96}_{40}$Zr are not accurately known at present [26,
44, 45], here we take five sets of WS parameters which are suggested and used
in heavy-ion collisions from recent references [46, 47, 48, 49, 50, 51], as
shown in Table 1, to investigate the effect of nuclear structures on CME
backgrounds for isobar collisions.

The parameters are arranged by the following way: Case-1, Ru ($\beta^{Ru}_{2}$
= 0.13) has large quadrupole deformation than Zr ($\beta^{Zr}_{2}$ = 0.06);
Case-2, Ru ($\beta^{Ru}_{2}$ = 0.03), in contrast, has a smaller quadrupole
deformation than Zr($\beta^{Zr}_{2}$ = 0.18); Case-3 is based on the latest
calculations of the energy density functional theory (DFT) [49, 30], which
assumes that the nucleus is spherical ($\beta_{2}$ = 0). According to the
calculation of proton and neutron distribution, it shows that the overall size
of Ru is smaller than that of Zr because the neutron skin of Zr is much
thicker. Case-4 is also derived from the DFT result, but with recent research
findings [50, 51] for $\beta_{2}$ and $\beta_{3}$, which gives a description
of nuclear structure contained both the deformation effect and the neutron
skin effect for ${}^{96}_{44}$Ru and ${}^{96}_{40}$Zr. Case-5 is from the
recent result [50], which also includes the deformation effect and the neutron
skin effect. It should be noted that the neutron skin effect comes from the
different density distribution of proton and neutron, but in our computation,
proton and neutron are identical in Case-3, Case-4 and Case-5, and the total
nucleon density (the sum of the proton and neutron) is used to configure the
initial nucleon coordinates. It has been proved in previous research [49] that
the Ru+Ru/Zr+Zr ratio is almost the same for charged multiplicity distribution
$N_{ch}$ and the eccentricity $\varepsilon_{2}$ which were calculated by the
total nucleon density and the DFT density. This makes it possible to study the
neutron skin effect by using the WS parameters of the total nucleon density.
It has been proved in a recent work [50] that the effects of WS parameters on
elliptic flow ratio of $v_{2,Ru}$/$v_{2,Zr}$ are as follows: 1) the $v_{2}$
ratio is mainly dominated by $\beta_{2}$ and to a minor extent by $\beta_{3}$
in the most-central collisions; 2) the $v_{2}$ ratio is influenced by a
positive contribution from $\beta_{2}$ and a larger negative contribution from
$\beta_{3}$ in the near-central collisions; 3) the influence of $\Delta
a=a_{Ru}-a_{Zr}$ is manifested in the mid-central and peripheral collisions;
4) and the impact of $\Delta R_{0}=R_{0,Ru}-R_{0,Zr}$ only exists in the
central collisions.

Table 1: The Woods-Saxon parameters used in the AMPT model. |  ${}^{96}_{44}$Ru |  ${}^{96}_{40}$Zr
---|---|---
Nucleus |  $R$(fm) |  $a$(fm) | $\beta_{2}$ | $\beta_{3}$ |  $R$(fm) |  $a$(fm) | $\beta_{2}$ | $\beta_{3}$
Case-1 | 5.13 | 0.46 | 0.13 | 0 | 5.06 | 0.46 | 0.06 | 0
Case-2 | 5.13 | 0.46 | 0.03 | 0 | 5.06 | 0.46 | 0.18 | 0
Case-3 | 5.067 | 0.5 | 0 | 0 | 4.965 | 0.556 | 0 | 0
Case-4 | 5.065 | 0.485 | 0.154 | 0 | 4.961 | 0.544 | 0.062 | 0.202
Case-5 | 5.09 | 0.46 | 0.162 | 0 | 5.02 | 0.52 | 0.06 | 0.20

Figure 1 shows the distributions of charged hadron numbers from the AMPT model
within $|\eta|<$ 0.5 in Ru + Ru and Zr + Zr collisions at $\sqrt{s_{NN}}$ =
7.7, 27, 62.4 and 200 GeV for above five sets of Woods-Saxon parameters. The
multiplicity $P(N_{ch})$ of five cases perfectly matches the STAR multiplicity
after multiplying by 1.21 (the same treatment was done in the work [51]) at
$\sqrt{s_{NN}}$ = 200 GeV, and the ratio of the Case-4 and the Case-5 give the
best description of the RHIC-STAR data [4]. It presents similar distribution
pattern at other energies in different $N_{ch}$ region. Based on the
multiplicity distribution, the collision system is divided into different
centralities for each set of the WS parameters. It’s obvious that the $N_{ch}$
interval of centralities class becomes more sensitive to the distribution for
low multiplicity (peripheral collisions) at lower energy. Here the
multiplicity $N_{ch}$ is the number of particles at mid-pseudorapidity
$\left(|\eta|<0.5\right)$ in the collisions.

Figure 3:  (Upper panels) Elliptic flow $v_{2}\\{2\\}$ measurements for five
cases in isobar collisions at $\sqrt{s_{NN}}$ = 7.7, 27, 62.4 and 200 GeV as a
function of centrality by using two-particle correlations. The solid and open
symbols represent measurements for Ru + Ru and Zr + Zr collisions,
respectively. The STAR data [4] for isobar collisions at $\sqrt{s_{NN}}$ = 200
GeV are also shown for comparison. The data points are shifted along the $x$
axis for clarity. (Lower panels) The ratios of $v_{2}$ in Ru + Ru over Zr + Zr
collisions. The statistical uncertainties are represented by lines.

##  III RESULTS AND DISCUSSION

In this section, the simulation results from the AMPT model for five nuclear
density parameters of Ru + Ru and Zr + Zr collisions at $\sqrt{s_{NN}}$ = 7.7,
27, 62.4 and 200 GeV are presented. We will show the predictions for the mean
multiplicity $\left\langle N_{ch}\right\rangle$, the elliptic flow $v_{2}$ and
the triangular flow $v_{3}$ of all particles as a function of centrality in
the two isobaric collision systems. The effect of initial geometry on CME
background will also be discussed according to the eccentricities in Ru + Ru
and Zr + Zr collisions. The numbers of events we simulated for all cases for
both ${}^{96}_{44}$Ru and ${}^{96}_{40}$Zr are, respectively, 5M (7.7 GeV), 2M
(27 GeV), 2M (62.4 GeV) and 2.5M (200 GeV).

###  III.1 Mean charge multiplicity $\left\langle N_{ch}\right\rangle$

The upper panels of Fig. 2 show the mean charge multiplicity $\left\langle
N_{ch}\right\rangle$ at mid-pseudorapidity $\left(|\eta|<0.5\right)$ as a
function of centrality, from the string melting mode of AMPT, for five cases
of the parameter settings of isobar collision systems at $\sqrt{s_{NN}}$ =
7.7, 27, 62.4 and 200 GeV. The Ru+Ru/Zr+Zr ratio of the mean charge
multiplicities is shown in the lower panels of Fig. 2. The STAR data [4] for
isobar collision systems at $\sqrt{s_{NN}}$ = 200 GeV are also shown for
comparison. Our simulation results show that the mean charge multiplicity
$\left\langle N_{ch}\right\rangle$ of the two isobar systems for five cases is
close to each other, which is perfectly consistent with the data of the STAR
Collaboration at different centralities.

The Ru+Ru/Zr+Zr ratio in the bottom panel of Fig. 2 gives a clear illustration
of the difference between these two isobar collisions. The ratios for the five
cases are different in shape, with Case-1 and Case-2 remaining close the unity
for all centralities at different energies, Case-3, Case-4 and Case-5 rising
almost directly in proportion to the centrality at $\sqrt{s_{NN}}$ = 200 GeV,
which are also rising in a zig-zag pattern at low energy. The shapes of the
ratio in the Case-3, Case-4 and Case-5 are the closest to the STAR
measurements at $\sqrt{s_{NN}}$ = 200 GeV. This indicates that the neutron
skin effect has the major contribution to the ratio of $\left\langle
N_{ch}\right\rangle$.

One of the puzzling things about the $\left\langle N_{ch}\right\rangle$ ratio
of STAR’s data at $\sqrt{s_{NN}}$ = 200 GeV is that there is a zig-zag pattern
after 50% of the centrality, which is interestingly similar to how Case-3,
Case-4 and Case-5 behaves at low energy. The zig-zag ratio after 50% of the
centrality is actually due to the fact that in low multiplicity events (i.e.,
in peripheral collisions) the $N_{ch}$ interval for a specific centrality has
a little difference between ${}^{96}_{44}$Ru and ${}^{96}_{40}$Zr [51]. In
other word, the events in ${}^{96}_{44}$Ru and ${}^{96}_{40}$Zr collisions
within the same centrality region have different multiplicity region. This
little multiplicity difference can result in the zig-zag fluctuations for some
observables. As shown in Fig. 1, the ratio with zig-zag fluctuations appears
before 50% of the centrality at low energy collisions, because low energy
collisions contain more low multiplicity events. In the work of Ref. [51], it
points out that the ratio of multiplicity $\left\langle N_{ch}\right\rangle$
depends on centrality significantly, the ratio of $v_{2}$ dose not.

###  III.2 Harmonic flow

The harmonic flow is investigated for the two isobaric collision systems,
which contributes as a major background of the CME-sensitive observable, such
as $\Delta\gamma_{112}$. The harmonic flow coefficient $v_{n}\\{2\\}$ is
calculated by the two-particle correlation method in the following form [4]:

$\centering v^{2}_{n=2,3}\\{2\\}=\left\langle
cos(n\phi_{1}-n\phi_{2})\right\rangle.\@add@centering$ (2)

Here we use all particles and larger pseudo-rapidity range ($|\eta|<2$) to
calculate $v^{2}_{n}\\{2\\}$ for higher statistics, and the $\Delta\eta_{1,2}$
is set to 0.05 for $v^{2}_{n}\\{2\\}$ calculation. The non-flow effect which
can be suppressed by using a larger $\Delta\eta_{1,2}$ cut does not affect the
ratio essentially [51].

The upper panels of Fig. 3 present the AMPT results for the centrality
dependence of elliptic flow $v_{2}${2} at mid-pseudorapidity ($|\eta|<$2) with
five geometry settings of isobaric collisions. It can be seen that $v_{2}${2}
for Ru + Ru and Zr + Zr are similar to each other at different energies, and
the $v_{2}${2} with five setting parameters are also close to each other. Our
results are consistent with the STAR data at $\sqrt{s_{NN}}$ = 200 GeV, but
slightly higher than that of the STAR data in central collisions.

Figure 4:  (a) Elliptic flow $v_{2}\\{2\\}$ measurements for five cases in the
most central isobar collisions as a function of energy by using two-particle
correlations. (b) The ratios of $v_{2}\\{2\\}$ in Ru + Ru over Zr + Zr
collisions in the most central collisions. (c) Elliptic flow $v_{2}\\{2\\}$
measurements for five cases in mid-central isobar collisions as a function of
energy by using two-particle correlations. (d) The ratios of $v_{2}\\{2\\}$ in
Ru + Ru over Zr + Zr collisions in mid-central collisions. For (a) and (b),
the STAR data [4] for isobar collisions at $\sqrt{s_{NN}}$ = 200 GeV are also
shown for comparison. The data points are shifted along the $x$ axis for
clarity, and the statistical uncertainties are represented by lines.

The lower panels of Fig. 3 give the $v_{2}${2} ratios between Ru + Ru and Zr +
Zr collisions. The centrality dependences of the ratios for the five settings
are very different at $\sqrt{s_{NN}}$ = 200 GeV, for Case-1 (-2) the ratio
decreases (increases) from central to peripheral collisions until about 20%
centrality staying close to unit, and for Case-3 it shows as a bow above unit,
and for Case-4 and Case-5 it shows the almost same shape as the STAR data. As
seen from Case-1 and Case-2, the quadrupole deformation has a major effect on
the $v_{2}${2} ratios in central (less than 20% centrality) isobaric collision
systems, which means that the larger quadrupole deformation $\beta_{2}$ of
nucleon distribution is, the larger its $v_{2}${2}. Comparing with the STAR
experimental results [4], Case-1 is more consistent with the data in the most
central collisions, Case-3 is more consistent with it in mid-central
collisions, while Case-4 and Case-5, which take account of both the
deformation and the neutron skin effect, are almost consistent with the STAR
data in all centralities. It can be seen that the influences of setting of the
parameters are consistent with the conclusion in Ref. [50] and this also
provokes the investigation of the beam energy dependence of the effect from
the initial geometry properties.

Figure 5: (Upper panels) Triangular flow $v_{3}${2} for five cases in isobar
collisions at $\sqrt{s_{NN}}$ = 200 GeV as a function of centrality by using
two-particle correlations. The STAR data [4] for isobar collisions at
$\sqrt{s_{NN}}$ = 200 GeV are also shown for comparison. The data points are
shifted along the $x$ axis for clarity. (Lower panels) The ratios of $v_{2}$
in Ru + Ru over Zr + Zr collisions. The statistical uncertainties are
represented by the bars on dots. Figure 6:  (Upper panels) Eccentricity
$\varepsilon_{2}$ for three cases in isobar collisions at $\sqrt{s_{NN}}$ =
7.7, 27, 62.4 and 200 GeV as a function of centrality. The solid and open
symbols represent measurements for Ru + Ru and Zr + Zr collisions,
respectively. The data points are shifted along the $x$ axis for clarity.
(Lower panels) The ratios of $\varepsilon_{2}$ in Ru + Ru over Zr + Zr
collisions. The statistical uncertainties are represented by bars on dots.

It was also observed that the ratios of $v_{2}${2} is enhanced at lower
energies in isobar collisions, but the five settings have different
performances at different centralities. As shown in Fig. 4 (b), in the central
collisions (0-5%), the absolute value of $v_{2}${2} ratios decreases with
energy for Case-1, Case-2, Case-4 and Case-5，meaning that the effect of
$\beta_{2}$ to $v_{2}${2} ratio, which mainly dominates in the most-central
collisions, is magnified at lower energy. Comparing the ratios of these cases,
with a larger absolute value of $\Delta\beta_{2,case2}$ ($\beta_{2,Ru}$ \-
$\beta_{2,Zr}$ = -0.15), the ratios of Case-2 drop more strongly in central
region than other case, which indicates that a larger $\Delta\beta_{2}$
results in a larger $v_{2}${2} ratio in central collisions. In addition, due
to the negative contribution to $v_{2}${2} ratio from $\beta_{3}$ in the near-
central collisions, although the $\Delta\beta_{2}$ of Case-4
($\Delta\beta_{2,case4}$ = 0.092) and Case-5 ($\Delta\beta_{2,case5}$ = 0.102)
is larger than that of Case-1($\Delta\beta_{2,case1}$ = 0.07), their
$v_{2}${2} ratio is still smaller than that of Case-1. It also can be seen
that the ratio of Case-3, which does not include the deformation description
in WS parameters, remains at unit with energy.

Figure 4(d) shows the ratios of $v_{2}${2} with energy in the mid-central
collisions (20-50%). It can be seen that the ratio of Case-3 decreases with
energy, while the other cases show a non-monotonic energy dependence of
ratios. Since the neutron skin effect (i.e. $\Delta a$) is mainly manifested
in the mid-central collisions, the behaviour of Case-3 in Fig. 4(d) indicate
that the influence of neutron skin effect to the ratio of $v_{2}${2} is
enhanced at low energy in the mid-central collisions. For Case-4 and Case-5,
the ratios of $v_{2}${2} shows a non-monotonic energy dependence in this
collision energy region because both sets of WS parameters contain not only
the neutron skin effect but also octupole deformation $\beta_{3}$. Since in
the near-central collisions $\beta_{3}$ has a negative contribution to
$v_{2,Ru}$/$v_{2,Zr}$ and in the mid-central collisions $\Delta a$ has a
positive contribution, the two factors conspire to produce the non-monotonic
energy dependence of ratios in this centrality region (20-50%). The ratios of
$v_{2}${2} present a minor difference for Case-1 and Case-2 at different
energy in this centrality region, because the quadrupole deformation
$\beta_{2}$ has little effect on $v_{2}${2} ratios in the mid-central
collisions.

The trend of $v_{2}${2} ratios at different energies with different centrality
reconfirms that the effect of quadrupole deformation $\beta_{2}$ on $v_{2}${2}
ratios mainly occurs in the most-central collisions, the octupole deformation
$\beta_{3}$ dominates in the near-central collisions, and the impact of
neutron skin effect on $v_{2}${2} ratios mainly happens in the mid-central
collisions.

The above results tell us that each nuclear structural factor could result in
a larger difference of CME background at low energy. However, due to the
negative contribution from $\beta_{3}$ to $v_{2,Ru}$/$v_{2,Zr}$ ratio in the
near-central collisions, the monotonic energy dependence of $v_{2}$ ratio is
weakened at 0-5% and 20-50%. Therefore, we suggest that the energy dependence
of the ratio for isobar collisions can be studied in the extremely ultra-
central collisions (0-1%) [51] in the future, which can avoid the offset of
the $\beta_{3}$ to $v_{2}$ ratio to obtain a stronger monotonic energy
dependence. At the same time, it can solve the problem that the centrality
classes of two colliding nuclei are not completely aligned at low energy.

On the other hand, perhaps the lower energy collisions are more suitable for
us to study the nuclear structure of ${}^{96}_{44}$Ru and ${}^{96}_{44}$Zr
because of stronger signal than that at higher energy. In other words, it is
worth performing a beam energy scan of isobaric collisions to constrain the
nuclear structure parameters or to disclose the CME background from nuclear
structure.

We also investigate the effect of the quadrupole deformation $\beta_{2}$ and
octupole deformation $\beta_{3}$ to the triangular flow $v_{3}${2} at mid-
pseudorapidity ($|\eta|<$ 2) as a function of centrality at $\sqrt{s_{NN}}$ =
200 GeV, as shown in Fig. 5. From the upper panels of Fig. 5, it is seen that
the calculated $v_{3}${2} are close to the STAR data [4] for five settings,
and the lower panels of Fig. 5 showing the results of Case-4 and Case-5, which
include the octupole deformation $\beta_{3}$, are most similar to STAR data in
central collisions. The above results are consistent with the conclusion in
Ref. [29] that the $\beta_{2}$ has an effect on $v_{2}$ but does not on
$v_{3}$, and $\beta_{3}$ has a significant effect on $v_{3}${2} in central
collisions. On the other hand, there exists a similar effect of the chain or
triangle structure for the $\alpha$-clustering nucleus like 12C on $v_{2}$ and
$v_{3}$, which was observed in our previous work [52], i.e. the chain
structure has an effect on $v_{2}$ but does not on $v_{3}$, but triangle
structure has a significant effect on $v_{3}$. Actually, both phenomena
induced by the above deformations and clustering structures are essentially
related to the transformation from initial-state geometric structure and
fluctuation [53] to final-state momentum space.

###  III.3 The initial geometry

Next, we want to find whether the differences in final state measurements due
to the effects of nuclear structure are already implicit in the early stages
after isobar collisions. The initial geometry of a nucleus-nucleus collision
can be characterized by eccentricity, which represents the initial geometric
anisotropy of the collision zone in the transverse plane (perpendicular to the
beam direction). The definition of eccentricity for $n$-th harmonics in the
coordinate space of the initial partons [54] for a single collision event is
given by following form [55, 56]

$\varepsilon_{n}\\{P\\}=\frac{\sqrt{\langle
r^{n}cos(n\varphi)\rangle^{2}+\langle
r^{n}\operatorname{sin}(n\varphi)\rangle^{2}}}{\langle r^{n}\rangle},$ (3)

where $r$ and $\varphi$ are the position and azimuthal angle of each initial
parton in the transverse plane. In practice, the event averaged eccentricity
coefficients $\langle\varepsilon_{n}\\{P\\}\rangle$ are used to characterize
the initial geometry asymmetry, and we mainly focus on the $\varepsilon_{2}$
in this study.

The upper panels of Fig. 6 show the $\sqrt{\langle\varepsilon^{2}_{2}\rangle}$
for five cases in isobar collisions at $\sqrt{s_{NN}}$ = 7.7, 27, 62.4 and 200
GeV as a function of centrality, and the
$\sqrt{\langle\varepsilon^{2}_{2}\rangle}$ for five settings are close to each
other at different centralities. The lower panels of Fig. 6 present the
$\sqrt{\langle\varepsilon^{2}_{2}\rangle}$ ratios between Ru + Ru and Zr + Zr
collisions. As expected, the shape of
$\sqrt{\langle\varepsilon^{2}_{2}\rangle}$ ratios is similar to the above
presented flow ratios for the five settings at different centralities, i.e.
the effect of deformation on eccentricity mainly occurs in the central
collisions, while the effect of neutron skin on eccentricity mainly dominates
in mid-central collisions. This illustrates that part of the contribution to
the difference on CME backgrounds in Ru + Ru and Zr + Zr collisions originates
from the effect of nuclear structure on the initial geometry after collisions.

Figure 7:  (Upper panels) The ratios of
$\sqrt{\langle\varepsilon^{2}_{2}\rangle}$ in Ru + Ru over Zr + Zr collisions
in the most central collisions. (Lower panels) The ratios of
$\sqrt{\langle\varepsilon^{2}_{2}\rangle}$ in Ru + Ru over Zr + Zr collisions
in mid-central collisions. The statistical uncertainties are represented by
bars on dots.

In addition, we can see that the shape of
$\sqrt{\langle\varepsilon^{2}_{2}\rangle}$ ratio have no significant changes
at different energy as shown in Fig. 7 for the centrality ranges of 0-5% and
20-50% with five settings of nuclear structure parameters. The energy
independence of $\sqrt{\langle\varepsilon^{2}_{2}\rangle}$ ratios is opposite
to the previous situation of $v_{2}$ ratios, suggesting that it is energy
dependent for the transition efficiency from initial geometry asymmetry to
final momentum space. And the energy dependence of the $v_{2}$ ratios via the
transport model, such as AMPT, indicates the hydrodynamical-like evolution
mechanism in the collisions [57, 58, 59].

##  IV Conclusion and Outlook

The recent STAR measurement of the final state observables confirmed the
differences in nuclear structure between Ru and Zr systems. By comparing the
simulation results in Ru + Ru and Zr + Zr collisions from the AMPT model with
the STAR data, we found that these differences can be explained by different
quadrupole deformation $\beta_{2}$, octupole deformation $\beta_{3}$, as well
as the neutron skin. These results are consistent with previous studies [30,
26, 29, 60, 50, 51]. Our results reconfirm the centrality dependence of the
difference in final state observables between Ru + Ru and Zr + Zr collisions
on nuclear structure [50], i.e., the positive contribution on
$v_{2,Ru}$/$v_{2,Zr}$ of the quadrupole deformation $\beta_{2}$ and the
neutron skin effect occurs in the most-central collisions and the mid-central
collisions, respectively, and the negative contribution on
$v_{2,Ru}$/$v_{2,Zr}$ of octupole deformation $\beta_{3}$ happens in the near-
central collisions. This conclusion was supported more strongly by subsequent
investigation on the energy dependence of $v_{2}$ ratios in Ru + Ru / Zr + Zr
in the central collisions and the mid-central collisions, respectively, which
is shown that each nuclear structure factor magnifies the difference of
$v_{2}$ in lower energy in the collision region where they dominate. The
results of the energy dependence of $v_{2}$ ratios also indicate that larger
CME backgrounds can be generated in lower energy from Ru + Ru and Zr + Zr
collisions.

The study of eccentricity ratios of two systems tells us that part of the CME
background difference is originated from the difference of initial geometry in
Ru + Ru and Zr + Zr collisions. The fact that eccentricity ratios are
independent of energy shows that the energy dependence of the difference in
$v_{2}$ ratio is also influenced by the dynamical evolution of the collision
zone.

If background differences due to the different nuclear structures can be more
accurately predicted, we might be able to isolate them and get a clean signal
of the CME effect. Fortunately, more precise deformation description has been
tested in some work and seems successful in describing new data of RHIC and
LHC experiments [29, 61, 62, 63]. Our analysis also shows that the background
difference caused by each nuclear structural factor in isobar collisions can
be magnified at lower energy. As Ref. [29] pointed out that isobar collisions
can be used as a precision tool to measure the shape of nuclei, this may be
easier to do at lower energy.

###### Acknowledgements.

This work was supported in part by the National Natural Science Foundation of
China under contract Nos. 11875066, 11890710, 11890714, 12061141008, 11925502,
11975078 and 12147101, National Key R&D Program of China under Grant No.
2018YFE0104600 and 2016YFE0100900, the Strategic Priority Research Program of
CAS under Grant No. XDB34000000, and the Guangdong Major Project of Basic and
Applied Basic Research No. 2020B0301030008.

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