# Since the Scientific Literature Is Multilingual, Our Models Should Be Too

Abteen Ebrahimi1 Kenneth Church2
1University of Colorado Boulder, 2Northeastern University
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

English has long been assumed the lingua franca of scientific research, and
this notion is reflected in the natural language processing (NLP) research
involving scientific document representation. In this position piece, we
quantitatively show that the literature is largely multilingual and argue that
current models and benchmarks should reflect this linguistic diversity. We
provide evidence that text-based models fail to create meaningful
representations for non-English papers and highlight the negative user-facing
impacts of using English-only models non-discriminately across a multilingual
domain. We end with suggestions for the NLP community on how to improve
performance on non-English documents.

Since the Scientific Literature Is Multilingual, Our Models Should Be Too

Abteen Ebrahimi1 and Kenneth Church2 1University of Colorado Boulder,
2Northeastern University<EMAIL_ADDRESS><EMAIL_ADDRESS>

##  1 Introduction

There has been a noticeable shift in recent years away from the traditional
English-centric focus of traditional natural language processing (NLP)
research. Pretrained models, which drive many state of the art approaches,
have moved from explicitly monolingual language coverage Devlin et al.
(2019a); Radford et al. (2019) to strong multilingual support Brown et al.
(2020); BigScience et al. (2023). Machine translation systems, multilingual by
definition, are now aiming at the long tail of languages, to create usable
translations for languages with minimal resources NLLB et al. (2022); Bapna et
al. (2022). Major NLP conferences and workshops have begun to heavily promote
research involving other languages111ACL ’22, NAACL ’24, particularly those
which are low-resource or endangered222AmericasNLP, Masakhane. This shift is
well motivated; methods which are truly language agnostic and robust to
varying amounts of data are technologically interesting, and from a human-
centered perspective, we must strive to create language technologies which are
more inclusive to speakers of all languages around the world ELRA (2019);
Joshi et al. (2020).

ID |  84779794
---|---
Title |  Relation between ants and aphids in a citrus orchard.
TL;DR |  Five-year-old boy with Down’s syndrome is diagnosed with encephalitis.
Table 1:  Example from Semantic Scholar Wade (2022); Kinney et al. (2023)
where attempts to standardize on English introduce confusions. The paper was
published in Japanese, but the title and tl;dr fields are in English.
Additional examples can be seen in Appendix A.  Figure 1: Main results. Black
dotted line represents the total number of abstracts for each language. Bars
represented the proportion of tokens which are unknown on average for that
language, for a monolingual vs. multilingual model.

In this position piece, we highlight one area of NLP research which we believe
stands to benefit from improving its multilingual support: scientific document
representation. English has been historically considered the lingua franca of
scientific research, to the disadvantage of many non-English speaking
communities Hanauer et al. (2019); Márquez and Porras (2020); Ramírez-
Castañeda (2020); Steigerwald et al. (2022). This belief is also reflected in
prior NLP research which involves scientific literature. By and large, most of
the currently published models and datasets only support English. In this
paper, we show that, while English makes up the dominant proportion of the
literature, the remainder is incredibly multilingual. We further show that
English-only models fail to create meaningful representations for many
languages. Finally, we present the real world effects of the English-only
models and discuss future steps to help better process documents in other
languages.

We note that this work is specific to the NLP community; the broader scope of
issues and solutions surrounding English as the language of research are well-
studied by other fields and far-reaching. We focus on the technical issues
which arise from using English-only models, and consider the larger discussion
out of scope for this work.

##  2 Related Work

Beltagy et al. (2019) present SciBERT, a BERT-style Devlin et al. (2019b)
model trained on papers taken from Semantic Scholar Wade (2022); Kinney et al.
(2023), largely from the computer science and biomedical domains. SciBERT is
the base of document representation models such as Specter Cohan et al.
(2020), through a post-pretraining alignment using the citation graph.
Evaluation benchmarks for document representation include SciDocs Cohan et al.
(2020) and SciRepEval Singh et al. (2023). Relevant to this work, OpenMSD Gao
et al. (2023) presents a benchmark for scientific literature which contains
text from papers spanning 103 languages. Notably, this is the first benchmark
for scientific documents covering multiple languages.

Figure 2: Pseudo-Perplexity (PPPL) per subword for languages with low unknown
token counts.  Figure 3: Relatively small cosines for non-English papers
suggest opportunities for improvement.

##  3 How multilingual is the literature?

To determine the multilinguality of the scientific literature, we take three
samples of abstracts from a bulk download of Semantic
Scholar333https://www.semanticscholar.org/product/api from June 20, 2023. We
use langdetect444https://pypi.org/project/langdetect/ and
langid.py555https://pypi.org/project/langid/ to classify each abstract, and
only keep papers where the two models agree. Across the three sets, the
average sample size is 30.19m abstracts. Filtering abstracts which are too
short for langdetect, or do not meet a minimum language confidence of 0.95,
leaves 29.4m abstracts. Taking the intersection of the two detection
algorithms, the final average sample size is 28.58m with a standard deviation
of 2,385.

We present results in Figure 1. While English is clearly dominant, taking a
85.11% share, we detect a surprisingly large number of other languages – 50
total – in the remaining abstracts. We list the languages along with their
counts in the Appendix.

On the larger end, papers in French, Spanish, Chinese, Portuguese, and
Indonesian make up 10.37% of the samples on average. Of these, the 3rd largest
is Chinese, which while representing a share of over 600k papers, the vast
majority of input tokens, 92.95% and 93.14% on average for cased and uncased
SciBERT respectively, are represented as UNK, effectively leading to random
outputs. For comparison, the average number of unknown characters for a
multilingual model, XLM-R Conneau et al. (2020), is 0.06%. While the remaining
45 languages cumulatively make up less than 5% of the sample, they cannot be
ignored as the scale of the scientific literature must be taken into account;
these languages still account for 1.32m papers.

Although a considerable portion of the Semantic Scholar database consists of
papers written in a language other than English. In contrast, prevalent papers
written about scientific document representation, either models or datasets,
are solely focused on English. More worryingly, is that these papers also fail
the Bender Rule Bender (2019): not only do they not cover other languages, but
they do not mention what languages their experiments or data cover. The use of
English is implied and taken as the default language of research papers.

This lack of evaluation datasets with non-English examples creates a blind
spot in our understanding of how models perform on these papers, and causes a
feedback loop: without an existing dataset, researchers will generally not be
motivated to train models on other languages, for two reasons. First, there
will be no straightforward way to know how well the model works for other
languages, and second, for a fixed model size, introducing other languages
dilutes the English representation of the model, which may lower its
performance on existing benchmarks, even though the model performs better in a
real-world setting. The introduction of OpenMSD represents a strong step in
the right direction, although the creation and release of additional
benchmarks is required for more comprehensive evaluation.

##  4 Implicit Multilingual Evaluation

###  4.1 Unknown Scripts

Models trained with an English-only vocabulary represent abstracts written in
non-Latin scripts with many unknown tokens. If too many input tokens are
unknown, the model will solely rely on parts of the text which are written in
the supported scripts, often numbers or named entities. This can lead to
erroneous embeddings which represent the document based on a few key words
rather than the complete abstract. Even if the abstract includes an English
translation at the end, the UNK tokens from the source script at the beginning
may force the English portions outside the maximum supported sequence length.

To quantify the impact of using only a Latin-script vocabulary, we calculate
the average percentage of UNK tokens contained in the tokenized abstracts for
each language in our sample. Table 2 in the Appendix shows that for languages
such as Chinese, Korean, Japanese, Arabic, and Hindi, more than 60% of the
input tokens are UNK. Overall, 14 total languages, which make up 973k
abstracts or 3.3% of our sample, fall under this constraint. Compared to the
tokenization of a multilingual model, the maximum percentage of unknown tokens
for any given language is 2.98%.

###  4.2 Perplexity on Seen Scripts

To intrinsically evaluate performance on languages without a large number of
UNK tokens, we calculate a pseudo-perplexity (PPPL), or model score Salazar et
al. (2020). We present results in Table 3 (in Appendix) and Figure 2. While
for some languages the PPPL is low, indicating that the model may be capable
of producing reasonable outputs, it rapidly increases as we consider other
languages, indicating deteriorating performance. For comparison, the PPPL of
XLM-R remains consistently low across all languages. This motivates the use of
a vocabulary which covers multiple scripts. Even if the perplexity after
pretraining is high, it still allows for finetuning on task-specific data
without any architectural changes.

###  4.3 Dependencies on Language & Embedding

This section will use the centroid assumption

$vec(v)\approx\sum_{v_{j}\in ref(v)}vec(v_{j})$

to test for possible issues with non-English languages. This assumption
approximates a document vector as the sum of its references.

The boxplots in Figure 3 use cosines to compare documents vectors with the
centroids of their references. The differences between English and other
languages are relatively small for ProNE Zhang et al. (2019), a graph-based
method. In contrast, of the text-based models, Specter clearly produces worse
representations for non-English papers. Other methods, SciNCL Ostendorff et
al. (2022) and Specter2, show clear improvements, but still do not reach the
same performance of ProNE. Our hope is that for a truly multilingual model,
the quality of representations for all languages will approach that of the
graph-based methods.

##  5 Real World Implications

To highlight the real-world implications of deploying English-only models for
a multilingual domain, we lookup papers in our sample to see how they are
displayed on the Semantic Scholar website. We focus on the "tl;dr" section
which, at the time of writing, is based on an English only generation model
Cachola et al. (2020). We sample papers written in Spanish, French, Japanese,
Korean, and Farsi, and present a hand-picked selection in Appendix A666Samples
are picked to highlight cases which both work well and fail. The full sample
will be released.. Extractions for Spanish and French papers are reasonable –
outputs are often directly copied from the abstract but do not present any
other information. For languages with non-Latin script, the performance is
random and nonsensical at best, and potentially biased at worse as in the case
of Example 3.

##  6 Future Directions

Here, we discuss possible future directions to address the current lack of
multilingual support for scientific document representation.

#### Language Detection and Translation

English-only models are often used directly to provide end-users with
recommendations or other suggestions, even though the query paper may be in an
unsupported language. A simple option is to detect the abstract languages and
either not generate outputs, or warn the user that the results may be
unreliable for unsupported languages. For abstracts which have a translation,
the English portion can be detected and used as input. Machine translation
systems can also be used to translate abstracts; while this approach will
introduce noise, we believe the outputs will be more reasonable, particularly
for non-Latin script languages.

#### Using Text Agnostic Models

As shown in § 4.3, graph-based approaches are less sensitive to language. As
these methods will not be directly applicable for papers with no links, a
combination of text-based, and graph-based approaches will likely result in
the best performance.

#### Training Multilingual Models

This solution is the simplest solution which offers the largest potential
improvement. Multilingual models can leverage zero-shot transfer to make a
better than random prediction, even if the language is not heavily trained on.
At the very least, including a vocabulary which covers other scripts would
correct the unknown token problem, and open the door for other users to
continue the model pretraining on other languages.

#### Evaluation Datasets in Other Languages

The direction which we believe is most important, is to create multilingual
benchmark datasets, for three reasons. First, benchmark datasets outlive
specific models. A sufficiently difficult benchmark can be used by multiple
iterations of models. Second, the creation of benchmark dataset can motivate
others to create multilingual models. By creating a readily available way to
evaluate the multilingual ability of models, this may motivate teams which
would otherwise choose to create an English-only model to instead train a
multilingual one. Finally, a benchmark datasets would allow us to evaluate
existing pretrained models on their multilingual ability. As an example,
multilingual large language models have been shown to have strong zero-shot
performance. The creation of a multilingual benchmark would allow us to
evaluate these models on scientific document tasks, potentially revealing
strong performance without requiring additional training.

##  7 Conclusion

In this work, we have highlighted the linguistic diversity found in the
scientific literature, and shown that using English-only models in this domain
leads to poor support for other languages. As the outputs of these models are
often directly user-facing, we hope that this work motivates the community to
create new models and benchmarks which can support document representation for
all languages.

##  8 Ethics

We do not believe there are any ethical concerns which arise from our work. We
rely on commonly used and public APIs, models, and datasets.

##  9 Limitations

Many of the evaluations we present in our work are intrinsic and while we
assume that they are correlated with downstream performance on an extrinsic
task, such as paper recommendation, further experiments are necessary to
confirm this hypothesis. Furthermore, while we have taken steps to reduce
potential noise in our language detection, by using two algorithms, only
keeping papers on which they both agree, and only keeping papers with a large
confidence (>0.95), there is still likely to be some erroneous detections. We
have manually reviewed a sample of the language classifications, and removed
languages which were consistently erroneously. As such we believe our general
findings remain valid, regardless of language detection noise present.

## Acknowledgments

The majority of this work was completed in the 2023 Jelinek Memorial Summer
Workshop on Speech and Language Technologies (JSALT) and was supported with
funds from Johns Hopkins University and ESPERANTO, the EU Horizons 2020
program’s Marie Sklodowska-Curie Grant No 101007666. The authors would like to
thank members of the Better Together team, particularly Peter Vickers and
Shabnam Tafreshi for their help, insights, and feedback on this work.

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##  Appendix A Appendix

Language Counts Mean Prop. Unk ISO Language Script Mean STD Mean Prop. NE Mean
Prop. SciBERT-C SciBERT-U XLM-R en English Latin 24,862k 2.20k 85.11% - 00.08%
00.08% 00.00% fr French Latin 770k .75k 2.64% 17.70% 00.08% 00.08% 00.00% es
Spanish Latin 661k .55k 2.26% 15.20% 00.02% 00.02% 00.00% zh Chinese Chinese
633k 1k 2.17% 14.55% 92.95% 93.14% 00.06% pt Portuguese Latin 488k .52k 1.67%
11.22% 00.02% 00.02% 00.00% id Indonesian Latin 478k .26k 1.64% 11.00% 00.03%
00.02% 00.00% de German Latin 283k .40k 0.97% 6.50% 00.04% 00.04% 00.00% ko
Korean Korean 227k .44k 0.78% 5.22% 81.66% 81.97% 00.11% ru Russian Cyrillic
112k .47k 0.38% 2.57% 10.81% 10.07% 00.00% tr Turkish Latin 94k .08k 0.32%
2.15% 00.07% 00.07% 00.00% ja Japanese Japanese 89k .24k 0.30% 2.04% 86.00%
82.01% 00.03% fa Persian Arabic 66k .23k 0.22% 1.51% 49.71% 24.63% 00.00% sv
Swedish Latin 57k .16k 0.20% 1.32% 00.01% 00.01% 00.00% cs Czech Latin 57k
.14k 0.20% 1.32% 00.03% 00.03% 00.00% hr Croatian Latin 48k .06k 0.16% 1.11%
00.02% 00.02% 00.00% it Italian Latin 46k .13k 0.16% 1.07% 00.06% 00.06%
00.00% uk Ukrainian Cyrillic 40k .21k 0.14% 0.92% 15.54% 14.16% 00.00% nl
Dutch Latin 39k .03k 0.13% 0.90% 00.02% 00.01% 00.00% pl Polish Latin 36k .06k
0.12% 0.83% 00.10% 00.10% 00.00% ar Arabic Arabic 24k .10k 0.08% 0.54% 61.97%
15.78% 00.04% ca Catalan Latin 23k .12k 0.08% 0.54% 00.41% 00.36% 00.00% hu
Hungarian Latin 10k .05k 0.03% 0.23% 00.03% 00.03% 00.00% sl Slovenian Latin
10k .09k 0.03% 0.23% 00.02% 00.02% 00.00% el Modern Greek Greek 9k .06k 0.03%
0.22% 00.75% 00.75% 00.01% vi Vietnamese Latin 8k .05k 0.03% 0.19% 18.08%
18.77% 00.02% no Norwegian Latin 6k .05k 0.02% 0.14% 04.99% 05.11% 00.01% fi
Finnish Latin 5k .09k 0.02% 0.11% 00.01% 00.02% 00.00% ro Romanian Latin 4k
.10k 0.02% 0.10% 00.52% 00.45% 00.01% da Danish Latin 4k .09k 0.02% 0.10%
00.11% 00.09% 00.00% af Afrikaans Latin 4k .06k 0.01% 0.10% 00.06% 00.05%
00.00% th Thai Thai 4k .03k 0.01% 0.09% 49.57% 50.06% 00.02% lt Lithuanian
Latin 4k .06k 0.01% 0.08% 00.18% 00.18% 00.01% sk Slovak Latin 3k .01k 0.01%
0.07% 00.07% 00.08% 00.01% et Estonian Latin 1k .01k 0.00% 0.03% 04.67% 04.69%
00.00% bg Bulgarian Cyrillic 1k .02k 0.00% 0.03% 17.34% 14.88% 00.00% lv
Latvian Latin 1k .02k 0.00% 0.02% 00.01% 00.01% 00.00% mk Macedonian Cyrillic
1k .02k 0.00% 0.02% 22.10% 19.67% 00.01% hi Hindi Devanagari 372 11 0.00%
0.01% 92.59% 92.62% 00.13% sw Swahili Latin 345 18 0.00% 0.01% 00.82% 00.86%
00.00% he Hebrew Hebrew 282 24 0.00% 0.01% 72.35% 72.63% 00.32% ne Nepali
Devanagari 181 7 0.00% 0.00% 93.20% 93.24% 00.00% tl Tagalog Latin 162 4 0.00%
0.00% 00.31% 00.30% 00.00% sq Albanian Latin 151 7 0.00% 0.00% 00.03% 00.03%
00.00% mr Marathi Devanagari 53 8 0.00% 0.00% 87.02% 87.06% 00.00% gu Gujarati
Gujarati 38 8 0.00% 0.00% 87.85% 87.88% 00.85% ta Tamil Tamil 29 3 0.00% 0.00%
66.09% 66.39% 00.00% ml Malayalam Malayalam 13 2 0.00% 0.00% 77.29% 77.53%
02.98% ur Urdu Arabic 13 3 0.00% 0.00% 60.57% 42.69% 00.78% bn Bengali Latin
10 2 0.00% 0.00% 06.96% 06.91% 02.38% pa Panjabi Latin 7 3 0.00% 0.00% 68.58%
69.30% 00.00% kn Kannada Kannada 4 3 0.00% 0.00% 82.67% 82.79% 00.03%

Table 2:  Main Results ISO | Language | SciBERT-Cased | SciBERT-Uncased | XLM-R | Sample Size
---|---|---|---|---|---
en | English | 05.07 | 04.04 | 04.05 | 4985
es | Spanish | 06.26 | 04.15 | 03.55 | 4988
fr | French | 07.86 | 04.49 | 04.65 | 4985
pt | Portuguese | 08.22 | 03.74 | 03.70 | 4989
de | German | 09.69 | 04.08 | 04.67 | 4965
el | Greek | 17.92 | 10.10 | 02.99 | 4984
it | Italian | 18.26 | 13.34 | 03.83 | 4982
hr | Croatian | 19.19 | 13.83 | 03.40 | 4981
nl | Dutch | 19.89 | 18.30 | 02.95 | 4985
pl | Polish | 25.52 | 15.37 | 03.12 | 4994
tr | Turkish | 28.85 | 12.59 | 06.94 | 4995
id | Indonesian | 28.92 | 29.89 | 03.43 | 4996
ca | Catalan | 33.30 | 26.67 | 03.84 | 4906
sl | Slovenian | 40.05 | 33.40 | 03.54 | 4852
af | Afrikaans | 46.78 | 47.48 | 04.35 | 2576
sk | Slovak | 59.25 | 39.61 | 04.31 | 2149
cs | Czech | 61.92 | 34.87 | 06.46 | 4998
sv | Swedish | 66.19 | 39.59 | 06.55 | 4970
tl | Tagalog | 66.23 | 63.06 | 04.68 | 158
ro | Romanian | 66.86 | 56.35 | 03.36 | 1837
lv | Latvian | 71.15 | 73.97 | 03.86 | 611
fi | Finnish | 73.92 | 64.27 | 03.82 | 3110
da | Danish | 77.21 | 68.22 | 04.59 | 2946
lt | Lithuanian | 85.94 | 92.23 | 03.43 | 3338
sq | Albanian | 101.74 | 119.27 | 03.35 | 133
hu | Hungarian | 148.25 | 88.12 | 05.87 | 4925
Table 3: Perplexity Results

Num. | ISO | CorpusID |  Title |  TL;DR
---|---|---|---|---
Num. | ISO | CorpusID |  Title |  TL;DR
|  |  | Continued on next page
Hand-selected examples which highlight errors in the TL;DR model for papers written in Non-English, Non-Latin scripts. TL;DRs are collected using the Semantic Scholar API on 11/29/23. 1 | es | 90472702 |  Integración productiva ganadero-forestal: evaluación de impacto en dos clubes agrarios en el departamento de Durazno, Uruguay |  El trabajo tuvo como objetivo evaluar el impacto socioeconomico sobre dos clubes agrarios del Movimiento de Juventud Agraria del Acuerdo de Cooperacion with la empresa forestal Montes del Plata en el marco del Programa de Integracion Productiva de la misma, en el departamento de Durazno, Uruguay.
2 | es | 71074825 |  El manejo multidisciplinario de la lesión de vía biliar |  Las fugas por lesiones laterales son susceptibles de tratarse con maniobras that disminuyen the presion ductal (esfinterotomia mas endoprotesis), y mientras exista continuidad de la via biliar, el tratamiento endoscopico esta indicado.
3 | fa | 249368552 |  Effect Of Dimensions Smart Attached Mass And Local Stiffness On Dynamic Response Of Thick Plates |  It is revealed that the number of people affected by the recent Paris attacks has increased by more than 50% in the past year, with the total number of fatalities increasing from 11 to more than 200.
4 | fa | 166356916 |  A comparison between Iran and the Selected Countries on the Efficiency of expenses in Health Sector by Data Envelopment Analysis (1998-2008). |  The DEA has confirmed that it will not be pursuing legal action over the use of the word “toxic” in connection with the death of former DEA agent David Bailey.
5 | fr | 71021028 |  Les hépatites et les atteintes des voies biliaires dues à l’amoxicilline-acide clavulanique |  L’evolution est habituellement favorable and the guerison complete s’observe en quelques semaines, mais plusieurs arguments sont en faveur d’une origine immuno-allergique.
6 | fr | 90599195 |  Sensibilité des races ovines et caprines tchadiennes au virus de la peste des petits ruminants |  Il a ete egalement constate that les moutons de race Sahel paraissent plus sensibles que les chevres deRace Sahel.
7 | fr | 216003344 |  Putting on a show: using computer graphics to train end-users |  Les bibliotheques de l’Universite de Pennsylvanie ont mis au point une formule basee sur un projecteur video connecte a l’ecran d’un ordinateur personnel, pour former leurs utilisateurs a la recherche d’informations en ligne.
8 | ja | 76251764 |  Clinical evaluation of Tranilast for keloid and hyperplasic scar.Double blind comparative test using heparinoid ointment as the control drug. |  It’s time to dust off the sledgehammers and start cleaning up after yourselves.
9 | ja | 31383396 |  [Protective efficacy of antiserum against Bacteroides gingivalis]. |  It is confirmed that Bacteroides gingivalis is a candidate for the Nobel Prize in physiology or medicine.
10 | ja | 87290487 |  Mortality Factors of the Pine-moth, Dendrolimus spectabilis BUTLER in the Suburbs of Chiba City |  It is revealed that, for the first time, a non-European country, the U.S. government has acknowledged homosexuality.
11 | ja | 71808165 |  INTESTINAL OBSTRUCTION CAUSED BY SEVERE OMENTO-MESENTERIC PANNICULITIS FOLLOWING RECTAL CANCER SURGERY |  It’s time to get ready for winter, people!
12 | ja | 84854659 |  Existence of a stem-cell lineage in an infectious venereal tumor of the dog |  The history of stem cell research, Â£1.5bn in research and development, and more.
13 | ja | 84779794 |  Relation between ants and aphids in a citrus orchard. |  Five-year-old boy with Down’s syndrome is diagnosed with encephalitis.
14 | ja | 57713022 |  Lung Cancer Mass Screening with Sputum Cytology in Japan |  The results of famous randomized control studies conducted in 1970s and 1980s in the US and those of Miyagi Prefecture in 1990s showed that higher sensitivity for lung cancer detection and better prognosis in the 1990s, suggesting that lung cancer screening with sputum cytology is important in Japan.
15 | ko | 203819581 |  PPG lul i-yong-han sim-hyel-kwan cil-hwan yey-chuk si-su-theym-uy sel-kyey mich kwu-hyen |  The analyzed results which show the heart rate variability and the distribution of heart rate for before and after using PPG are proposed.


# Connectivity of 2-distance graphs

S.H. Jafaria, S.R. Musawib

aFaculty of Mathematical Sciences, Shahrood University of Technology,
P.O. Box 36199-95161, Shahrood, Iran
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>

bFaculty of Mathematical Sciences, Shahrood University of Technology,
P.O. Box 36199-95161, Shahrood, Iran
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

For a simple graph $G$, the $2$-distance graph, $D_{2}(G)$, is a graph with
the vertex set $V(G)$ and two vertices are adjacent if and only if their
distance is $2$ in the graph $G$. In this paper, we characterize all graphs
with connected 2-distance graph. For graphs with diameter 2, we prove that
$D_{2}(G)$ is connected if and only if $G$ has no spanning complete bipartite
subgraphs. For graphs with a diameter greater than 2, we define a maximal Fine
set and by contracting $G$ on these subsets, we get a new graph $\hat{G}$ such
that $D_{2}(G)$ is connected if and only if $D_{2}(\hat{G})$ is connected.
Especially, $D_{2}(G)$ is disconnected if and only if $\hat{G}$ is bipartite.

Keywords: 2-distance graph, power graph, diameter, connectivity
Mathematics Subject Classification : 05C12, 05C15

##  1 introduction

In this paper, we only consider the undirected finite simple graphs $G=(V,E)$.
The distance between two vertices of $x,y\in V(G)$, $d(x,y)$, is the length of
the shortest path between them. The diameter of $G$, $diam(G)$, is the maximum
distance between vertices of $G$. The graph $G$ is a bipartite graph if $V(G)$
be the union of two disjoint non-empty sets $A$ and $B$ where every edge
connects a vertex of $A$ to a vertex of $B$. It is well known that: A graph
$G$ is bipartite if and only if $G$ has no odd cycle. The contracted graph of
the partition $\\{A_{1},\cdots,A_{t}\\}$ of $V(G)$ is the graph $G_{1}$ with
vertex set $V(G_{1})=\\{A_{1},\cdots,A_{t}\\}$ and two vertices $A_{i}$ and
$A_{j}$ are adjacent if there is an edge $xy\in E(G)$ such that $x\in A_{i}$
and $y\in A_{j}$.

For a positive integer $k$, the $k$-power of $G$, denoted by $G^{k}$, is the
graph with the vertex set $V(G^{k})=V(G)$ and two vertices $x$ and $y$ are
adjacent if and ony if $d(x,y)\leqslant k$. $2$-distance graph of a graph $G$,
$D_{2}(G)$, is a graph with the vertex set $V(G)$ and two vertices are
adjacent if and only their distance is $2$ in the graph $G$. One can see that
for any graph $G$, $E(D_{2}(G))=E(G^{2})-E(G)$.

The idea of 2-distance graphs is a particular case of a general notion of
k-distance graphs, which was first studied by Harary et al. [6]. They
investigated the connectedness of 2-distance graph of some graphs. In the book
by Prisner [9, 157-159], the dynamics of k-distance operator was explored.
Furthermore, Boland et al. [3] extended the k-distance operator to a graph
invariant they call distance-n domination number. To solve one problem posed
by Prisner [9, Open Problem 29], Zelinka [11] constructed a graph that is
r-periodic under the 2-distance operator for each positive even integer r.
Prisner’s problem was completely solved by Leonor in her master’s thesis [8],
wherein she worked with a graph construction different from that of Zelinka’s.
Recently, Azimi and Farrokhi [1] studied all graphs whose 2-distance graphs
have maximum degree 2. They also solved the problem of finding all graphs
whose 2-distance graphs are paths or cycles. More recently, the same authors
[2] determined all finite simple graphs with no square, diamond, or triangles
with a common vertex that are self 2-distance graphs (that is, graphs G
satisfying the equation $D^{2}(G)=G$. They further showed the nonexistence of
cubic self 2-distance graphs. In [5], Ching gives some characterizations of
2-distance graphs and finds all graphs X such that $D_{2}(X)=kP_{2}$ or
$K_{m}\cup K_{n}$, where $k\geqslant 2$ is an integer, $P_{2}$ is the path of
order $2$, and $K_{m}$ is the complete graph of order $m\geqslant 1$. In [7]
Khormali gives some conditions for connectivity $D_{k}(G)$. For $D_{2}(G)$, he
proves that if $G$ has no odd cycle, then $D_{2}(G)$ is disconnected.

In this paper, we charactrize all graphs with connected 2-distance graph. For
graphs with diameter 2, we prove that $D_{2}(G)$ is connected if and only if
$G$ has no spanning complete bipartite subgraphs. For graphs with diameter
greater than 2, we define a maximal Fine set and by contracting $G$ on this
subsets, we get a new graph $\hat{G}$ such that $D_{2}(G)$ is connected if and
only if $D_{2}(\hat{G})$ is connected. Specially, $D_{2}(G)$ is disconnected
if and only if $\hat{G}$ is bipartite.

##  2 Cinnectivity of $D_{2}(G)$

Let $G$ be a graph with $diam(G)=2$, then one can see that
$D_{2}(G)=G^{\prime}$. Therefore, the connectivity of $D_{2}(G)$ come back to
the connectivity of $G^{\prime}$. Thus, we have the following theorem which
charactrized all gaphs with diameter 2 and connected 2-distance graph.

######  Theorem 2.1.

Let $G$ be a graph with $diam(G)=2$, then $D_{2}(G)$ is disconnected if and
only if $G$ has a spannig complete bipartite subgraph.

By above theorem, from now on we focus on connected graphs with diameter at
least $3$. Fiest, we introduce a fmily of graphs with connected $2$-distance
graph which play the main role in our proofs.

######  Definition 2.2.

The family $\mathscr{H}$ of graphs contains all odd cycles with degree greater
than 4 and following three special graphs.

The following results is trivial.

######  Lemma 2.3.

  1. 1.

If $G\in\mathscr{H}$, then $D_{2}(G)$ is connected.

  2. 2.

If $G$ be a connected graph and $|V(G)|\leqslant 4$, then $D_{2}(G)$ is
disconnected.

  3. 3.

Let $G$ be a connected graph and $|V(G)|=5$. $D_{2}(G)$ is connected iff
$G\in\mathscr{H}$.

######  Lemma 2.4.

Let $G$ be a connected graph. If $H\in\mathscr{H}$ be an induced subgraph of
$G$ such that $V(H)\nsubseteq N(x)$ for any vertex $x\in V(G)$, then
$D_{2}(G)$ is connected.

###### Proof.

Since $D_{2}(H)$ is connected and $H$ is an induced subgraph, then for any two
non-adjacent vertices $x,y\in V(H)$, we have $d(x,y)\geqslant 2$ and thus
$D_{2}(H)$ is a subgraph of $D_{2}(G)$. Let $u\in V(G)\setminus V(H)$,
$d(u,V(H))=t$ and $u=u_{0}u_{1}\cdots u_{t}=v$ is a path with $v\in V(H)$.
Since $V(H)\not\subseteq N(u_{t-1})$, then there is $w\in V(H)$ such that
$d(u_{t-1},w)=2$. Consequently, $u$ is connected to $v$ or $w$ in $D_{2}(G)$
as required. ∎

######  Definition 2.5.

Let $A$ and $B$ are two distinct set of vertices. $[A,B]$ denotes the complete
bipartite graph constructed by $A$ and $B$ as the independent sets.

######  Lemma 2.6.

Let $G$ be a graph such that $D_{2}(G)$ is disconnected. If $H$ be a connected
induced subgraph of $G$ and $D_{2}(H)$ is connected, then there is a subgraph
$K$ of $G$ such that $H\leqslant K$, $D_{2}(K)$ is connected and
$[V(K),N_{G}(V(K))\setminus V(K)]$ is a subgraph of $G$.

###### Proof.

If $[V(H),N_{G}(V(H))\setminus V(H)]$ is a subgraph of $G$, then $K=H$.
Otherwise, there is $x\in N_{G}(V(H))\setminus V(H)$ such that
$V(H)\not\subseteq N_{G}(x)$. Similar to proof of Lemma 2.4, $D_{2}(\langle
V(H)\cup{x}\rangle)$ is connected. By setting $H_{1}=\langle
V(H)\cup{x}\rangle$, we can continue this process by replacing $H_{1}$ to $H$.
Since $G$ is finite there $H_{t}$ such that $H_{t}\neq G$ and for any $x\in
N_{G}(V(H_{t}))\setminus V(H_{t})$, $V(H_{t})\subseteq N_{G}(x)$. By setting
$K=H_{t}$, $[V(K),N_{G}(V(K))\setminus V(K)]$ is a subgraph of $G$, which
completes the proof. ∎

Due to the above lemma, we introduce the following definition:

######  Definition 2.7.

Let $G$ be a graph and $A\subseteq V(G)$. The set $A$ is called Fine if
$[V(A),N_{G}(V(A))\setminus V(A)]$ is a subgraph of $G$, or equivalently for
any $x\in A$ , $N(x)\setminus A=N(A)\setminus A\neq\emptyset$ .

######  Lemma 2.8.

For any $x\in V(G)$ with $diam(G)\geqslant 3$, there is a unique maximal Fine
set $A$ such that $x\in A$.

###### Proof.

For $x\in V(G)$, $\\{x\\}$ is Fine. Since $G$ is finite, there are some
maximal Fine sets containing $x$. Let $A$ and $B$ two maximal Fine sets
containing $x$. we claim that $A\cup B$ is Fine. By hypothesis $N(x)\setminus
A=N(A)\setminus A$ and $N(x)\setminus B=N(B)\setminus B$.
First, we prove that $N(A\cup B)\setminus(A\cup B)\neq\emptyset$.
If $N(A\cup B)\subseteq(A\cup B)$, since $G$ is connected, $A\cup B=V(G)$ and
$N(B)\setminus B\subseteq A$ and $N(A)\setminus A\subseteq B$. Let $y\in
N(B)\setminus B$, then $y\in A$, and since $B$ is Fine, then $B\subseteq
N(y)\subseteq N(A)$. Thus, $B\setminus A\subseteq N(A)\setminus A$ and
therefore $B\setminus A=N(A)\setminus A$. Therefore $diam(G)=2$ because $A$ is
Fine, a contradiction.

Since $x\in A\cup B$ then it is trivial that $N(x)\setminus(A\cup B)\subseteq
N(A\cup B)\setminus(A\cup B)$. Now let $y\in N(A\cup B)\setminus(A\cup B)$
then $y\not\in A\cup B$ and $y\in N(A\cup B)$. So $y\in N(A)$ or $y\in N(B)$.
Without loss of generality, let $y\in N(A)$ then $y\in N(A)\setminus
A=N(x)\setminus A$, hence $y\in N(x)\setminus(A\cup B)$. Therefore,
$N(x)\setminus(A\cup B)=N(A\cup B)\setminus(A\cup B)$. On the other hand, for
any $a\in A$, $N(a)\setminus A=N(A)\setminus A=N(x)\setminus N(A)$ and so
$N(a)\setminus(A\cup B)=N(x)\setminus(A\cup B)=N(A\cup B)\setminus(A\cup B)$.
Similarly, for any $b\in B$, $N(b)\setminus(A\cup B)=N(A\cup B)\setminus(A\cup
B)$. Consequently, $A\cup B$ is a Fine set and by maximality $A$ and $B$,
$A=A\cup B=B$. ∎

######  Remark 2.9.

For a connected graph with diameter 2, it may be the maximal Fine set
containing a vertex do not be unique. For example, in Cycle graph $C_{4}$, the
maximal Fine set corresponding to each vertex is unique.

$d$$a$$b$$c$$\\{a,c\\}\text{ is the unique maximal Fine set containing
}a\text{ or }c.$$\\{b,d\\}\text{ is the unique maximal Fine set containing
}b\text{ or }d.$

But in the following graph, the maximal containing a vertex is not unique.

$d$$a$$b$$c$$\\{a,b,c\\},\\{a,c,d\\}\text{ are two maximal Fine sets
containing }a\text{ or }c.$$\\{b,d\\},\\{a,b,c\\}\text{ are two maximal Fine
sets containing }b.$$\\{b,d\\},\\{a,c,d\\}\text{ are two maximal Fine sets
containing }d.$

######  Remark 2.10.

It seems that if a Fine set is not maximal, we can add a vertex to find a new
Fine set contaning the previous Fine set. It is not true.
For example, in the graph shown in the following graph, $\\{a,b\\}$ is Fine
but it is not a maxmal Fine, because $\\{a,b,c,x,y\\}$ is maximal Fine
containing both $a$ and $b$. But none of the 3-element sets contaning
$\\{a,b\\}$ is not Fine.

$c$$a$$b$$d$$y$$x$$u$$v$

By the above lemma we have the following:

######  Corollary 2.11.

The maximal Fine sets partition the vertex set of a graph with diameter
greater than $2$.

######  Definition 2.12.

Let $G$ be a graph with $diam(G)\geqslant 3$. $\hat{G}$ be the graph with

$V(\hat{G})=\\{\hat{x}:\hat{x}\text{ is the maximal Fine set of }G\text{
containing }x\\}$

and two distinct vertices $\hat{x},\hat{y}\in V(\hat{G})$ are adjacent if and
only if $x$ and $y$ are adjacent in $G$.

######  Lemma 2.13.

Let $G$ be a graph with $diam(G)\geqslant 3$ and $\hat{x},\hat{y}\in
V(\hat{G})$ are distinct vertices, where $x,y\in V(G)$. Then
$d(x,y)=d(\hat{x},\hat{y})$.

###### Proof.

Let $\hat{x}=\hat{x_{0}}\hat{x_{1}}\hat{x_{2}}\cdots\hat{x_{t}}=\hat{y}$ be a
$\hat{x}\hat{y}$-path. By the properties of Fine sets $x=x_{0}x_{1}x_{2}\cdots
x_{t}=y$ is a $xy$-path. Then $d(x,y)\leqslant d(\hat{x},\hat{y})$. On the
other hand, if $x=x_{0}x_{1}x_{2}\cdots x_{t}=y$ be a $xy$-path, then
$\hat{x}=\hat{x_{0}}\hat{x_{1}}\hat{x_{2}}\cdots\hat{x_{t}}=\hat{y}$ is a
$\hat{x}\hat{y}$-walk and then contains a $\hat{x}\hat{y}$-path such that its
length is less than $t$. Consequently $d(\hat{x},\hat{y})\leqslant d(x,y)$ and
then $d(\hat{x},\hat{y})=d(x,y)$. ∎

One can see that if $x,y\in V(G)$ and $\hat{x}=\hat{y}$, then $d(x,y)\leqslant
2$ and the following is hold:

######  Corollary 2.14.

Let $G$ be a graph with $diam(G)\geqslant 3$. Then, $diam(G)=diam(\hat{G})$.

######  Remark 2.15.

Let $G$ be a graph with $diam(G)\geqslant 3$. Then, $diam(\hat{G})\geqslant
3$. Thus we can consider $\hat{\hat{G}}$. But one can see that if
$S\subseteq\hat{G}$ be a Fine set, then $S_{1}=\\{a:\hat{a}\in S\\}$ is a Fine
set of $G$, which mean that $S$ has a unique element an then
$\hat{\hat{G}}=\hat{G}$.

######  Theorem 2.16.

Let $G$ be a graph with $diam(G)\geqslant 3$. Then, $D_{2}(G)$ is connected if
and only if $D_{2}(\hat{G})$ is connected.

###### Proof.

First, let $D_{2}(\hat{G})$ is connected and $x,y\in V(G)$. We consider the
following two cases:
Case 1: $\hat{x}\neq\hat{y}$. There is a $\hat{x}\hat{y}$-path,
$\hat{x}=\hat{x_{1}}\hat{x_{2}}\cdots\hat{x_{t}}=\hat{y}$ in $D_{2}(\hat{G})$.
Since $\hat{x_{i}}\neq\hat{x_{i+1}}$, by Lemma 2.13,
$d_{G}(x_{i},x_{i+1})=d_{\hat{G}}(\hat{x_{i}},\hat{x_{i+1}})=2$, and then
$x=x_{1}x_{2}\cdots x_{t}=y$ is a $xy$-path in $D_{2}(G)$.
Case 2: $\hat{x}=\hat{y}$. Since $diam(G)\geqslant 3$, there is $z\in V(G)$
such that $\hat{x}\neq\hat{z}$. Then, $\hat{y}\neq\hat{z}$ and by Case 1,
there are some paths between $x$ and $y$ to $z$ in $D_{2}(G)$.
Consequently, $D_{2}(G)$ is connected.

Now, assume that $D_{2}(G)$ is connected and $\hat{x}\neq\hat{y}\in
V(\hat{G})$ and $x=x_{1}x_{2}\cdots x_{t}=y$ is a $xy$-path in $D_{2}(G)$.
Since $\hat{x_{i}}=\hat{x_{i+1}}$ or
$d_{\hat{G}}(\hat{x_{i}},\hat{x_{i+1}})=2$, then
$\hat{x}=\hat{x_{1}}\hat{x_{2}}\cdots\hat{x_{t}}=\hat{y}$ is a
$\hat{x}\hat{y}$-walk and contains a $\hat{x}\hat{y}$-path in
$D_{2}(\hat{G})$. Thus $D_{2}(\hat{G})$ is connected. ∎

By similarly argument, one can see that:

######  Corollary 2.17.

Let $G$ be a graph with $diam(G)\geqslant 3$. Then,
$diam(D_{2}(G))=diam(D_{2}(\hat{G}))$.

Now, we are ready to state and prove the main result of the paper.

######  Theorem 2.18.

If $G$ be a connected graph and $diam(G)\geqslant 3$, then $D_{2}(G)$ is
disconnected if and only if $\hat{G}$ is a bipartite graph.

###### Proof.

If $\hat{G}$ is a bipartite graph, then it is trivial that $D_{2}(\hat{G})$ is
disconnected and by Theorem 2.16, $D_{2}(G)$ is disconnect.
Now, by the contrary, let $D_{2}(G)$ is disconnected but $\hat{G}$ contains an
odd cycle. Let $\hat{H}\leqslant\hat{G}$ where $\hat{H}$ is a non-trival
induced connected subgraph with
$V(\hat{H})=\\{\hat{x}_{1},\hat{x}_{2},\cdots,\hat{x}_{t}\\}$ and
$D_{2}(\hat{H})$ is connected. Obviously,
$H=\langle\\{x_{1},x_{2},\cdots,x_{t}\\}\rangle$ is a connected induced
subgraph of $G$ and $D_{2}(H)$ is connected too. By the proof of Lemma 2.6,
there is an induced subgraph $K$ which containing $H$ and
$[V(K),N_{G}(V(K))\setminus V(K)]$ is a subgraph of $G$. Then, $V(K)$ is Fine
and then there is $x\in V(G)$ such that $V(H)\subseteq\hat{x}$, a
contradiction.

Thus, $\hat{G}$ has no connected induced subgraph $\hat{H}$ with connected
2-distance graph. Consequently, by Lemma 2.3, $\hat{G}$ has no induced
subgraph belongs to $\mathscr{H}$, specially, $\hat{G}$ has no odd cycle of
length greater than $3$.
Let $\hat{G}$ has a triangle.
Claim 1: For any triangle $\hat{a}\hat{b}\hat{c}\hat{a}$ in $\hat{G}$,
$\langle N(\hat{a})\rangle$ is connected.
Proof: By a contrary, let $\langle N(\hat{a})\rangle$ is disconnected and $H$
is the connected component of $\langle N(\hat{a})\rangle$ containing the edge
$\hat{b}\hat{c}$. Let $\hat{y}\in N(\hat{a})\setminus V(H)$. If
$N(V(H))\subseteq N(\hat{a})$, then $N(V(H))\setminus V(H)=\\{\hat{a}\\}$.
Therefore, $V(H)$ is Fine and containing $\hat{b}$ and $\hat{c}$. By Remark
2.15, we have $\hat{b}=\hat{c}$, a contradiction. Then, we assume that
$\hat{u}\in N(V(H))\setminus N(\hat{a})$. If $V(H)\not\subseteq N(\hat{u})$
then since $H$ is connected, there are $\hat{v},\hat{w}\in V(H)$ such that
$\hat{v}\hat{w}\in H$ and $\hat{v}\in N(\hat{u})$ and $\hat{w}\notin
N(\hat{u})$. Consequently, one can see that
$\langle\\{\hat{y},\hat{a},\hat{v},\hat{w},\hat{u}\\}\rangle\in\mathscr{H}$, a
contradiction. Thus, $V(H)\subseteq N(\hat{u})$ for all $\hat{u}\in
N(V(H))\setminus N(\hat{a})$ and again $V(H)$ is Fine, which contradict to
Remark 2.15.
Claim 2: Any vertex of $\hat{G}$ is on a triangle.
Proof: By contrary, there is a vertex $\hat{a}$ lies on no triangle. Since
$\hat{G}$ is connected and has at least a triangle, then there is a vertex
$\hat{b}$ and triangle $\hat{x}\hat{y}\hat{z}\hat{x}$ such that $\hat{b}\in
N(\hat{x})$ and $\hat{G}$ has no triangle containing $\hat{b}$, that is,
$N(\hat{b})\cap N(\hat{x})=\emptyset$. Now, $\langle N(\hat{x})\rangle$ and
consequently $\langle N(x)\rangle$ is not connected, a contradiction by Claim
1.
Claim 3: $\hat{G}$ has a subgraph $\hat{H}$ with vertex set
$V(\hat{H})=\\{\hat{a},\hat{b},\hat{x},\hat{c},\hat{d}\\}$ and
$E(\hat{H})=\\{\hat{a}\hat{b},\hat{a}\hat{x},\hat{b}\hat{x},\hat{c}\hat{x},\hat{b}\hat{c},\hat{c}\hat{d}\\}$
where $d(\hat{a},\hat{d})=3$.

$\hat{a}$$\hat{x}$$\hat{b}$$\hat{c}$$\hat{d}$

Proof: Since $diam(\hat{G})\geqslant 3$, there is a path
$\hat{a}\hat{b}\hat{c}\hat{d}$ with $d(\hat{a},\hat{d})=3$. By Claim 2,
$\hat{b}$ lies on a triangle and by claim 1, $\langle N(\hat{b})\rangle$ is
connected. Then there is $\hat{z}\in N(\hat{b})\cap N(\hat{c})$. If
$\hat{z}\notin N(\hat{a})\cup N(\hat{d})$, then
$\langle\\{\hat{a},\hat{b},\hat{z},\hat{c},\hat{d}\\}\rangle\in\mathscr{H}$, a
contradiction. Therefore, $z\in N(\hat{a})\cup N(\hat{d})$ and by symmetry, we
can assume $\hat{z}$ is adjacent to $\hat{a}$ and the proof is completed.

Now we have all the necessary information to continue the argument

Let $H$ be the subgraph constructed in Claim 3. Let $K$ be a connected
component of $\langle N(\hat{a})\cap N(\hat{c})\rangle$ containing the edge
$bx$. Let $S_{1}=N(\hat{a})\cap N(\hat{c})\setminus V(K)$ and
$S_{2}=N(\hat{a})\setminus N(\hat{c})$. We claim that $V(K)$ is Fine. If
$\hat{y}\in N(V(K))\setminus N(\hat{a})$ and $V(K)\not\subseteq N(\hat{y})$,
Since $K$ is connected, then there is $\hat{u},\hat{v}\in V(K)$ such that
$\hat{u}\hat{v},\hat{u}\hat{y}\in E(\hat{G})$ but $\hat{v}\notin N(\hat{y})$.

The subgraph
$\langle\\{\hat{a},\hat{c},\hat{d},\hat{u},\hat{v},\hat{y}\\}\rangle$ can be
one of the four graphs shown in the below.

$\hat{a}$$\hat{u}$$\hat{v}$$\hat{c}$$\hat{d}$$\hat{y}$

$\hat{a}$$\hat{u}$$\hat{v}$$\hat{c}$$\hat{d}$$\hat{y}$

$\hat{a}$$\hat{u}$$\hat{v}$$\hat{c}$$\hat{d}$$\hat{y}$

$\hat{a}$$\hat{u}$$\hat{v}$$\hat{c}$$\hat{d}$$\hat{y}$

One can see that in each case,
$\langle\\{\hat{a},\hat{c},\hat{d},\hat{u},\hat{v},\hat{y}\\}\rangle$ contains
an induced subgraph of $\hat{G}$ which belongs to $\mathscr{H}$, a
contradiction.

If $\hat{y}\in S_{2}\cap N(V(K))$ and $V(K)\not\subseteq N(\hat{y})$ , then
there are $\hat{u}\hat{v}\in E(K)$ such that $\hat{u}\hat{y},\hat{u}\hat{v}\in
E(\hat{G})$ but $\hat{v}\notin N(\hat{y})$. Then,
$\langle\\{\hat{y},\hat{u},\hat{v},\hat{c},\hat{d}\\}\rangle\in\mathscr{H}$, a
contradiction. Thus, $V(K)$ is Fine, again is a contradiction. Therefore
$\hat{G}$ has no triangle which complete the proof. ∎

## References

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# New Results on Hard Probes in Heavy-Ion Collisions with ALICE

Institut für Kernphysik, Westfälische Wilhelms-Universität Münster, Wilhelm-
Klemm-Str. 9, 48149 Münster, Germany
ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung,
Planckstr. 1, 64291 Darmstadt, Germany
E-mail For the ALICE Collaboration and dedicated to Oliver Busch (1976–2018).

###### Abstract:

Hard probes – final state particles related to an interaction with large
momentum transfer or mass scale – play a distinguished role in the discovery
and the study of the Quark-Gluon Plasma (QGP), a phase of deconfined quarks
and gluons reached at high temperatures in heavy ion collisions. In heavy ion
collisions, parton scatterings with a large momentum transfer ($1/Q\ll 1$
fm/$c$) occur prior to QGP formation and thus provide a source of coloured
probe particles for the QGP created in the later stage of the reaction. The
hard scattered partons and the subsequent parton shower interact strongly with
the QGP and its constituents via elastic and radiative processes before
hadronization into jets of observable particles. Thus, the comparison to jet
and high-$\pt$ observables in pp (_vacuum_) potentially probes their
modification due to medium effects.

One of the key observables in the discovery and investigation of these jet
modifications has been the nuclear modification factor $\raa$, for which new
results on charged particle production in different colliding systems are
presented and the question of apparant suppression in peripheral collisions is
addressed. For more differential studies of the jet sub-structure and hence
the parton shower evolution in the medium, recent results on jet grooming in
heavy ion colisions are presented.

##  1 Introduction

The nuclear modification factor $\raa$ compares the particle production in
nucleus-nucleus reaction ($AA$) to pp collisions scaled with the number of
binary collision ($N_{\mathrm{coll}}$) or the increased parton flux ($T_{AA}$)
in nuclear collisions

$\raa=\frac{\mathrm{d}N_{X}^{AA}/\mathrm{d}\pt}{N_{\mathrm{coll}}\cdot\mathrm{d}N_{X}^{pp}/\mathrm{d}\pt}=\frac{\mathrm{d}N_{X}^{AA}/\mathrm{d}\pt}{T_{AA}\cdot\mathrm{d}\sigma_{X}^{pp}/\mathrm{d}\pt},$
(1)

where $X$ can be any reconstructed final state object: charged particles,
specific hadrons, leptons, gauge bosons or jets. In the absence of any strong
medium effects in the initial and final state, $\raa$ is expected to be unity
for single particles in the $\pt$ region where hard processes are the dominant
source of particle production.

For reconstructed jets, the interpretation and comparison of $\raa$
measurements is more sophisticated. In principle, a jet algorithm aims to
recover the full kinematic information on the initial parton. If this also
holds in $AA$ collisions, one would expect a $\raa$ of unity, i.e. the full
energy is recovered and all medium modification is visible only in a change of
the jet structure. In practice, jet reconstruction in heavy ion collisions is
hindered by the large soft background, unrelated to the initial hard
scatterings. This background can be subtracted with an event-wise average of
the background density and is in general reduced for jets with smaller radius
and larger constituent-$\pt$ thresholds. Different experimental choices on
$\pt$ thresholds and jet radii, together with given detector-specific methods
for background corrections and input for jet reconstruction, make a direct
comparison between different experiments difficult. However, in jet $\raa$
measurements a general trend is observed by all LHC experiments: the nuclear
modification factor in central collisions appears to approach an asymptotic
value of $\approx 0.5$ at high $\pt$ similar to the measurements of single
charged hadrons.

The data presented here have been collected with the ALICE experiment during
various runs of the Large Hadron Collider at CERN. In particular, the precise
measurement of charged particle tracks at central (pseudo-)rapidity
($\left|\eta\right|<0.9$) in high multiplicity environment and down to low
$\pt\approx 0.15$ GeV allows for a detailed quantification of the jet
structure and the characterisation of the underlying background [1].

##  2 System Size Dependence and Centrality Biases

Figure 1:  Left: Nuclear modification factor for charged particles in three
$\pt$ ranges for different colliding systems and energies [2] Right: Nuclear
modification in p-Pb as well as in peripheral and central [3, 4]

In the measurement of single charged particles at high $\pt$ the ALICE
collaboration recently published the results for collisions at the highest
collision energy so far, $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV, with significantly
improved systematic uncertainties and a $\pt$ reach up to 50 GeV [3]. The
nuclear modification factor $\raa$ shows only little variation from the
measurements at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV, which can be understood
when considering that the expected larger parton energy loss is compensated by
a harder parton spectrum by which the energy loss is filtered. The harder
spectrum at LHC leads in general to an increased importance of the subleading
fragments already in the single particle spectrum compared to RHIC. This
directly affects the description of $\raa$, where parton energy loss models
beyond leading particles perform better at the LHC [5]. In addition, it is
already visible in the comparison of high-$\pt$ identified particles in pp
collisions at large $\sqrt{s}$ that leading order and next-to-leading order
(LO and NLO) Monte Carlo Models with parton showering provide a better
description of the data than NLO perturbative QCD calculation with one
dimensional fragmentation functions [6] extracted at a lower collisions
energy.

Figure 2: Left: Nuclear modification factor for charged particles over the
full centrality range [7] Right: Comparison of the apparent suppression to the
expectation of the Hg-Pythia model [8]

The comparison of the charged particle production in at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV [3] to recent ALICE results obtained with
lighter Xe-ions at similar energies ($\sqrt{s_{\mathrm{NN}}}=5.44$ TeV, [2])
is important to disentangle effects of the collision geometry. In particular
this can be done by studying the nuclear modification factor in three distinct
$\pt$ ranges as shown in Fig. 1 (left). The lowest $\pt$ range is dominated by
effects induced by the collective expansion of the medium (flow). In the
intermediate range, pronounced differences between baryon and mesons have been
observed, which could be explained by quark coalescence, while in the highest
$\pt$ range above 10 GeV the effects of parton energy loss dominate. Plotted
as function of the charged particle multiplicity as measure for the energy
density, it is remarkable that all data points agree beyond
$\mathrm{d}N/\mathrm{d}\eta\approx 500$. Irrespective of the ion size and
colliding energy, the driving mechanism in all three $\pt$ ranges appears to
be energy density, even though the ranges are dominated by different physics
processes. At low $\mathrm{d}N/\mathrm{d}\eta$ the nuclear modification
factors for Xe and Pb data deviate. At the same $\mathrm{d}N/\mathrm{d}\eta$,
Xe-Xe collisions are more asymmetric, suggesting that the collision geometry
now becomes an additional factor. Interestingly, these deviations are similar
in all three $\pt$ ranges, which highlights the importance of geometry as a
connection between the various processes involved in particle production in
heavy ion collisions.

The search for the system size and/or energy density dependence of parton
energy loss can also be viewed from a different perspective, when comparing
peripheral with p-Pb at similar $\mathrm{d}N/\mathrm{d}\eta$. Indeed, as seen
in Fig. 2 (left), even for peripheral reactions the nuclear modification
factor at high $\pt$ does not increase beyond 0.8 [7]. This apparent
contradiction to the observation of $\raa\approx 1$ in p-Pb in Fig. 1 (right),
can be explained when considering that indeed not the number of binary
nucleon-nucleon collisions is the relevant scaling assumption, but the number
of _hard_ collisions. This number depends on the probability for multiple
parton interactions (MPI), which is not uniform across the nucleus as assumed
in geometric Glauber models. As discussed in [8], the average distance between
two colliding nucleons is larger on average in peripheral collisions compared
to central, which leads to fewer MPI per $N_{\mathrm{coll}}$. The geometry
bias is further enhanced by the experimental centrality selection on
multiplicity, which in turn is increasing with the number of MPIs. These
biases have been evaluated in a hybrid model, coupling the impact parameter
dependent number of MPIs from Hijing with Pythia. As shown in Fig. 2 (right),
the apparent suppression can be very well reproduced in this Hg-Pythia-Model,
without the need to invoke any other initial or final state effects.

Figure 3: Left: Jet spectra opposite of a trigger hadron with given $\pt$ and
difference spectrum between them Right: Comparison of difference spectra in
p-Pb collisions with different event activity [9]

An unbiased measure for parton energy loss, or redistribution of jet energy,
has been introduced by the ALICE collaboration with the measurement of jet
distributions with respect to a recoiling hadron [10]. In these measurements a
deliberate bias is put on the momentum transfer $Q^{2}$ in the reaction by
requiring a certain hadron trigger $\pt$. The jet distribution is
reconstructed opposite in $\varphi$ to this trigger and is shown as an example
in Fig. 3 (left): a clear hardening with increasing trigger $\pt$ is observed.
Now the difference of the spectra can be compared for various multiplicities
in p-Pb. Since this is a _per-trigger_ quantity (i.e. per hard collision) it
is independent of the biases discussed above. It is seen that the energy
loss/spectrum shift in p-Pb is compatible with no effect and smaller than 0.4
GeV at 90% confidence level [9].

##  3 Differential Jet Observables

The differential study of jet structure has recently seen the development of a
wealth of observables, well beyond cross sections ratios for different radii
or transverse momentum distributions of jet constituents. One particular
development is to facilitate the widely used sequential recombination
algorithms to _decluster_ a found jet and essentially rewind the QCD
splitting. This can be done e.g. with the Cambridge–Aachen algorithm, which is
based on angular proximity only. Each step in the procedure splits a cluster
into two sub-clusters (sub-jets), separated by $\Delta R$ and with momentum
fractions $z$ and $1-z$, respectively. The $z$ and $\Delta R$ values can be
used to populate a _Lund diagram_ , which in principle maps the phase space of
all splittings and allows for the isolation of different regions for medium
effects. Certain regions of interest can also be amplified or filtered by
_grooming_ methods, e.g. the _soft drop_ [11] algorithm unwinds the jets,
following the largest $z$ (dropping the soft sub-jet), until
$z>z_{\mathrm{cut}}\Delta R^{\beta}$. Preliminary results on the $z$
distribution of the first splitting identified with this procedure are shown
in Fig. 4 (left), using $z_{\mathrm{cut}}=0.1,\beta=0$ in central collisions.
A constituent subtraction scheme has been applied to take into account the
background (see also [12] and references therein). The normalization to the
total number of jets and the comparison to Pythia jets embedded into central
events reveals that there is no sign of enhanced large angle $(\Delta R>0.1)$
splitting at any $z$. Further studies in different $\Delta R$ regions [12],
show that there is an indication for overall enhancement of collimated
splitting and suppression of large angle splittings. This behaviour is also
visible in the Lund representation of the difference between data splittings
and Pythia embedded level splittings in Fig. 4 (right).

Figure 4: Left: Inclusive measurement of $z_{g}$ distribution in central
collisions at $\sqrt{s}$ = 2.76 TeV with $\Delta R$ cut $>0.1$ Right: Lund
representation of the difference between Pb-Pb data splittings and Pythia
embedded level splittings in the 10% most central collisions at
$\sqrt{s}=2.76$ TeV. [12]

##  4 Summary and Outlook

In summary, the $\raa$ observable alone cannot reflect the complexity of jet
medium modification but is still highly popular. A new detailed picture of the
parton shower is provided by the Lund diagram and groomed jets have been used
as a tool to select jet substructure. Both methods show a suppression of large
angle splittings and enhancement of collinear splittings compared to embedded
Pythia. These results, together with the wealth of ALICE hard probes and jet
measurements building on the inclusion of low $\pt$ constituents, provide an
important benchmark for the test and development of Monte Carlo frameworks
coveringr pp, p$A$ and $AA$ reactions, including the modeling of parton shower
modification and the underlying event.

This contribution is dedicated to Oliver Busch (1976 – 2018); a dear friend
and long-term collaborator who has driven many jet analysis within ALICE and
whose work will have a lasting impact on the jet programme to come.

## References

  * [1] ALICE collaboration, B. Abelev et al., _Performance of the ALICE Experiment at the CERN LHC_ , _Int. J. Mod. Phys._ A29 (2014) 1430044 [1402.4476].
  * [2] ALICE collaboration, S. Acharya et al., _Transverse momentum spectra and nuclear modification factors of charged particles in Xe-Xe collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.44 TeV_, 1805.04399.
  * [3] ALICE collaboration, S. Acharya et al., _Transverse momentum spectra and nuclear modification factors of charged particles in pp, p-Pb and Pb-Pb collisions at the LHC_ , 1802.09145.
  * [4] ALICE collaboration, B. Abelev et al., _Transverse Momentum Distribution and Nuclear Modification Factor of Charged Particles in $p$-Pb Collisions at $\sqrt{s_{NN}}=5.02$ TeV_, _Phys. Rev. Lett._ 110 (2013) 082302 [1210.4520].
  * [5] ALICE collaboration, B. Abelev et al., _Centrality Dependence of Charged Particle Production at Large Transverse Momentum in Pb–Pb Collisions at $\sqrt{s_{\mathrm{{NN}}}}=2.76$ TeV_, _Phys. Lett._ B720 (2013) 52 [1208.2711].
  * [6] ALICE collaboration, S. Acharya et al., _$\pi^{0}$ and $\eta$ meson production in proton-proton collisions at $\sqrt{s}=8$ TeV_, _Eur. Phys. J._ C78 (2018) 263 [1708.08745].
  * [7] ALICE collaboration, S. Acharya et al., _Analysis of the apparent nuclear modification in peripheral Pb-Pb collisions at 5.02 TeV_ , 1805.05212.
  * [8] C. Loizides and A. Morsch, _Absence of jet quenching in peripheral nucleus–nucleus collisions_ , _Phys. Lett._ B773 (2017) 408 [1705.08856].
  * [9] ALICE collaboration, S. Acharya et al., _Constraints on jet quenching in p-Pb collisions at $\mathbf{\sqrt{s_{NN}}}$ = 5.02 TeV measured by the event-activity dependence of semi-inclusive hadron-jet distributions_, _Phys. Lett._ B783 (2018) 95 [1712.05603].
  * [10] ALICE collaboration, J. Adam et al., _Measurement of jet quenching with semi-inclusive hadron-jet distributions in central Pb-Pb collisions at ${\sqrt{{s}_{\mathrm{NN}}}}$ = 2.76 TeV_, _JHEP_ 09 (2015) 170 [1506.03984].
  * [11] A. J. Larkoski, S. Marzani, G. Soyez and J. Thaler, _Soft Drop_ , _JHEP_ 05 (2014) 146 [1402.2657].
  * [12] ALICE collaboration, H. A. Andrews, _Exploring the Phase Space of Jet Splittings at ALICE using Grooming and Recursive Techniques_ , _Submitted to: Nucl. Phys._ (2018) [1807.06439].