# Contextualizing Generated Citation Texts

###### Abstract

Abstractive citation text generation is usually framed as an infilling task,
where a sequence-to-sequence model is trained to generate a citation given a
reference paper and the context window around the target; the generated
citation should be a brief discussion of the reference paper as it relates to
the citing context. However, examining a recent LED-based citation generation
system, we find that many of the generated citations are generic summaries of
the reference paper’s main contribution, ignoring the citation context’s focus
on a different topic. To address this problem, we propose a simple
modification to the citation text generation task: the generation target is
not only the citation itself, but the entire context window, including the
target citation. This approach can be easily applied to any abstractive
citation generation system, and our experimental results show that training in
this way is preferred by human readers and allows the generation model to make
use of contextual clues about what topic to discuss and what stance to take.

Contextualizing Generated Citation Texts

Biswadip Mandal, Xiangci Li, Jessica Ouyang
---
The University of Texas at Dallas
Richardson, TX, USA
{biswadip.mandal, xiangci.li<EMAIL_ADDRESS>

Abstract content

## 1\. Introduction

Citation text generation is the task of generating a short summary of a
reference paper, focused on how it relates to a citing paper. Modern
approaches Xing et al. (2020); Ge et al. (2021); Luu et al. (2021); Chen et
al. (2021); Li et al. (2022) frame this problem as an infilling task: a
context window of up to three sentences before and after the target citation
is extracted, and the target is masked; given this context window and the
reference paper abstract, a sequence-to-sequence (seq2seq) model is trained to
generate the masked target citation.

If we examine the context window of a target citation, we find citations of
similar related works, and the author may criticize them in order to draw
attention to a gap that the citing paper intends to fill; to a human reader,
the context window provides clues indicating the topic the citation will
discuss and the stance the author will take towards the reference paper. Thus,
a citation should not be a generic summary of the reference paper, but rather
a query-focused summary, based on the context. After all, different papers can
cite the same reference paper in completely different ways, depending on
whether they share a task, an approach, or a dataset; even within the same
citing paper, a reference paper can be cited differently in the introduction
versus in the methodology or comparison of results.

It has been observed that neural language models often generate text that is
generic and vague Holtzman et al. (2019). Examining the recent LED-based
citation generation system of Li et al. (2022), we see a similar problem.
Figure 1 shows an example of their LED model’s output. Although the generated
citation is a valid summary of the reference paper, it is too generic,
mentioning only “syntactic information." The context sentence containing the
masked target already describes a constituent-structure approach, making it
clear to a human reader that the target should discuss a corresponding
dependency-structure based approach. However, the LED model produces a generic
summary of the reference paper, which while factually correct, does not fit
the context well and is actually redundant with information given earlier in
the context.

Context: Broadly speaking, prior work on SRL makes use of syntactic
information in two different ways. Carreras and Marquez (2005); Pradhan et al.
(2013) incorporate constituent-structure span-based information, while [MASK]…
Ground truth: Haji et al. (2009) incorporate dependency-structure information.
LED-baseline: Haji et al. (2009) integrate syntactic information into a neural
SRL system. Contextualized: Haji et al. (2009) use dependency-level
information.
---
Figure 1: An infilling-style baseline approach produces a generic summary of
the reference paper that does not fit well in the citation context.

To address this issue, we propose contextualized citation generation, a
context-focused modification to the citation text generation task. In this
version of the task, the model must generate not only the target citation, but
the surrounding context as well. In this way, the context window serves as a
sort of prompt for the decoder. Figure 1 shows how the output of our
contextualized approach correctly identifies dependency parsing as the topic
of the citation, and our experimental results show that human readers prefer
citations generated using our approach over the baseline infilling approach.
Our proposed training method can be easily applied to any seq2seq citation
generation model to produce coherent citations that fit their contexts.

Figure 2: Comparison of the standard infilling-based citation text generation
approach, where the generation target is just the target citation itself, and
our proposed contextualized approach, where the generation target is the
entire context window with the target citation filled in.

## 2\. Related Work

The citation text generation task was proposed by Hoang and Kan (2010). They
and other extractive approaches Hu and Wan (2014); Chen and Zhuge (2019)
selected the most salient sentences from the reference papers to serve as
their citations.

More recently, neural abstractive approaches Xing et al. (2020); Ge et al.
(2021); Luu et al. (2021); Chen et al. (2021); Li et al. (2022) have trained
seq2seq models for the citation infilling task, where the input is the
citation context (with target citation masked) and the reference paper
abstract. Our work aims to condition the generated citation even more strongly
on the context by training the model to generate the entire context window,
not just the target citation. While the above works differ in features and
datasets used, as well as the size and shape of the context window, it is
important to note that our approach changes only the generation target, and so
is compatible with all of them. In our experiments, we use the most recent
approach, Li et al. (2022), as the baseline model.

## 3\. Methodology

We propose a simple modification of the citation text generation task in order
to produce citations that are more coherent with their surrounding contexts.
The task is to generate the entire context window, with the masked target
citation filled in. For ease of evaluation, we add a meta-token [SEP] to
distinguish the context from the target citation. Figure 2 illustrates the
difference between our approach and the standard infilling approach.

### 3.1. Experimental Setup

#### Data.

We use the NLP-domain CORWA citation text generation dataset Li et al. (2022).
Table 1 shows the size of the dataset and its partitions. We merge the human-
annotated train and automatically-labeled distant sets to create a single
large, unified training set.

Partition | Papers | Citations
---|---|---
train | 565 | 2,243
distant | 11,564 | 32,512
test | 362 | 1,322
Table 1: CORWA dataset partitions and statistics.

#### Model.

We use the Longformer Encoder-Decoder (LED; Beltagy et al., 2020) citation
generation model of Li et al. The input to the model is the concatenation of
the citing paper’s introduction section; the paragraph containing the (masked)
target citation; and the citation mark (eg. “Smith et al. (2023)"), title, and
abstract of the reference paper.

## 4\. Results and Analysis

### 4.1. Human Evaluation

We recruit six graduate students from our university’s Computer Science
Department to serve as judges. We split the them into two groups of three,
assigning each group 30 samples. The judges are shown the input to the
generation model and the ground truth, baseline, and contextualized citations;
the order of citations is randomly shuffled to anonymize the models. The
judges are asked to indicate which citation they prefer based on Fluency,
Relevance to the reference paper, Coherence in the citation context, and
Overall quality.

Table 2 shows our human evaluation results. We see that the proposed
contextualized generation model is slightly preferred with respect to
Relevance, Coherence, and Overall, while the baseline model is more fluent.
Comparing the contextualized model with the ground truth, we find that many
samples have indistinguishable (ie. comparable) performance. Interestingly, we
observe that the Relevance of ground truth is judged lower than that of the
contextualized model. This may be because the ground truth citations are
written by human authors who have access to the entire reference paper, while
the generation models and human judges only see the reference paper abstracts.

We achieve moderate inter-annotator agreement for the first group of three
judges, with pairwise Kendall’s $\tau$ of 0.31, 0.17, and 0.35. We had lower
agreement for the second group, with Kendall’s $\tau$ of 0.15, 0.08, and 0.04.
Examining the judges’ scores, we find that one judge from the second group
consistently disagreed with the other two; we also tallied the results
excluding this third judge, but we did not find any significant difference
from the results shown in Table 2.

Model | Flu. | Rel. | Coh. | Overall
---|---|---|---|---
Contextualized vs Baseline
$\heartsuit$Contextualized | 12 | 49 | 41 | 53
$\heartsuit$Baseline | 33 | 44 | 36 | 50
Indistinguishable | 135 | 87 | 103 | 77
Contexualized vs Ground Truth
$\heartsuit$Contextualized | 19 | 38 | 28 | 27
$\heartsuit$Ground Truth | 18 | 13 | 21 | 18
Indistinguishable | 143 | 129 | 131 | 135
Table 2: Human evaluation comparing contextualized and infilling-based
citation generation.

### 4.2. Analyzing Contextualized Citations

Examining the generated citations, we find several common scenarios where the
baseline model ignores clues in the context window that the contextualized
model is able use.

Context: …To the best of our knowledge, [MASK]. Most similar to our work in
spirit, Ding et al. (1997) used Layer-wise Relevance Propagation (LRP; Bach et
al al. 2015), an interpretation method resembling saliency, to interpret …
Ground truth: Li et al. (2016) presented the only work that directly employs
saliency methods to interpret NLP models. Baseline: Li et al. (2016) proposed
three strategies for visualizing compositionality in NMT. Contextualized: Li
et al (2016) is the only work that applied saliency to NMT. They used a simple
recurrent neural network (RNN) to learn a saliency function for each word in
the sentence.
---
Figure 3: Example where the context hints the citation should discuss saliency
(in italics).

#### The context discusses the historical development of a particular topic.

A coherent citation should focus on the part of the reference paper that is
related to that topic; even if that topic is not especially important in the
reference paper, a good citation generation model should still be able to
identify it as the topic of focus. Figure 3 shows an example where the context
paragraph contains two clues. First, the introductory phrase “to the best of
our knowledge" hints that the citation should claim that few, if any prior
works have addressed the topic, identifying a gap that the citing paper
intends to fill. Second, the following sentences states that the most similar
work uses a “method resembling saliency." The contextualized model is able to
put these clues together, highlighting how “Li et al (2016) is the only work
that applied saliency."

Context: …Previous work on [MASK]. In contrast with their work, we train
models to … Ground truth: curriculum learning for MT (Kocmi and Bojar, 2017;
Wang et al., 2018) proposed methods which feed easier samples to the model
first and later show more complex sentences. However, their focus is on
improving convergence time while providing limited success on improving
translation quality. Baseline: dynamic sentence sampling (Kocmi and Bojar,
2017; Wang et al., 2018) focuses on how to dynamically sample sentences in
order to accelerate the training of NMT. Contextualized: fine-tuning neural
machine translation models (Kocmi and Bojar, 2017; Wang et al., 2018) has
tried to dynamically sample sentences in order to accelerate training.
However, these methods require training models to learn a prior distribution
of similar sentences.
---
Figure 4: Example where the context hints the citation should criticize the
reference paper (in italics).

#### The context implies that the reference paper should be compared or
contrasted.

Teufel et al. (2006) identifies several author intents for citing a given
paper, including contrasting methods or results, claiming the cited work is
worse than or comparable to the current work, or comparing or contrasting with
other cited works. A common pattern is that the context sentence following the
masked target discusses how the current work addresses limitations in the
prior work, making it evident that the target citation criticizes the
reference paper. Figure 4 shows an example where the context sentence
immediately following the target citation begins with “in contrast with their
work."

Context: …Rubinstein et al. (2015) demonstrated that state-of-the-art
distributional semantic models fail to predict attributive properties of
concept words (e.g. the properties is-red and is-round for the word apple) as
accurately as taxonomic properties (e.g. is-a-fruit). Similarly, [MASK] …
Ground truth: Sommerauer and Fokkens (2018) investigated the types of semantic
knowledge encoded within pretrained word embeddings, concluding that some
properties cannot be learned by supervised classifiers. Baseline: Somasomauer
and Fokkens (2018) compared word embeddings to a feature-identification method
based on cosine similarity. Contextualized: Sommerauer and Fokkens (2018)
showed that the properties of concepts are not captured by word embeddings.
---
Figure 5: Example where the context hints the citation should discuss how the
reference paper is similar to a previous citation (in italics).

#### The context implies that the reference paper should be discussed in a
similar way as another work.

Figure 5 shows an example where the sentence containing the target citation
begins with “similarly," hinting that the citation should discuss how the
reference paper is similar to the citation in the previous sentence. The
baseline model again produces a generic summary, while the contextualized
model is able to identify the topic of focus as the failure of word embeddings
to capture information about concept words.

#### The target citation should not be redundant with the context.

We find several instances where the baseline model’s generated citation
repeats information already present in the context. While the citation is
still a valid summary of the reference paper, the baseline model fails to
recognize that the information is redundant and it should focus on something
else, such as a different method or concept. The example in Figure 1 shows a
generic, redundant citation generated by the baseline model that should have
been more specific.

Context: …Rei et al. (2016) extended this model to include character
embeddings in order to capture morphological similarities such as word
endings. [MASK] … Ground truth: Rei (2017) subsequently added a secondary LM
objective to the neural sequence labeling architecture, operating on both word
and character-level embeddings. This was found to be particularly useful for
GED -introducing an LM objective allows the network to learn more generic
features about language and composition. Baseline: Rei (2017) proposed a semi-
supervised approach to the task of GED, using a language modeling objective to
predict surrounding words for every word in the sentence. Contexualized: Rei
(2017) further extended this model with a secondary training objective,
learning to predict the surrounding words in the context.
---
Figure 6: Example where the contextualized model correctly describes one
reference paper as an extension of another (in italics).

#### The reference paper is an extension of another work.

When describing the historical development of a task or topic, it is common to
present works in chronological order, where each work builds on top of the
previous ones. A good citation generation model should recognize when the
reference paper is an extension of another paper cited earlier in the context.
Figure 6 shows an example where the context discusses a paper, and the model
needs to generate a citation for a follow-up work by the same first author.
The contextualized model generates the phrase “further extended" (referring to
the earlier work, which already “extended").

We observe that citations often contain an action verbs, such as extend, and
improve, which are used when the reference paper is dependent on an earlier
work. Compared with the baseline model, our contextualized model more
frequently such verbs.

## 5\. Conclusion

We present a simple reframing of the citation text generation task to make
better use of citing context information. Our approach of generating the
entire citing context window, with the target citation filled in, produces
output that is more appropriate to its context than the existing method of
generating the target citation alone. Our proposed approach changes only the
generation target and is agnostic to any special features or input
representations, so it is straightforward to apply to any existing citation
generation models via retraining. Our human evaluation reveals that readers
are able to judge when a citation better fits its context, and we present a
qualitative analysis of some common shortcomings of citations generated by the
baseline approach that are addressed by our contextualized model.

## Limitations

The standard automatic evaluation metric for text generation, ROUGE, does not
capture coherence well. This is a known problem: small differences in ROUGE do
not correspond to noticeable differences in generation quality Deutsch et al.
(2022), and ROUGE does not capture other aspects of the summary, such as
informativeness Schluter (2017) and factuality Wallace et al. (2021). We
instead present a human evaluation and qualitative analysis to compare the
coherence citations generated using our proposed approach with those of the
baseline model. However, such evaluations are be expensive and time-consuming
to conduct, requiring significant reading and cognitive effort from domain-
expert judges; as a result, we are not able to evaluate a very large number of
samples.

In addition, our work is focused on natural language processing papers
published in the English language. Consequently, the scope of our model may
not encompass the full range of diversity exhibited by papers from other
fields (e.g., biology) or different sub-fields within computer science.

## Ethics Statement

Our work targets the task of citation generation, where plagiarism of the
cited paper is a significant concern. We not attempt to control the level of
extractiveness of our generated citations; as a result, they may copy
extensively from the cited paper abstract, which given as input.

Further, our approach generates citations that criticize the reference paper
when it seems appropriate to do so based on the citation context. Errors or
hallucinations by our model could result in false or unfair criticisms, which
can mislead the reader and negatively impact their perception of the reference
paper.

## 6\. Bibliographical References

##

  * Beltagy et al. (2020) Iz Beltagy, Matthew E. Peters, and Arman Cohan. 2020.  Longformer: The long-document transformer.  _arXiv:2004.05150_.
  * Chen and Zhuge (2019) Jingqiang Chen and Hai Zhuge. 2019.  Automatic generation of related work through summarizing citations.  _Concurrency and Computation: Practice and Experience_ , 31(3):e4261.
  * Chen et al. (2021) Xiuying Chen, Hind Alamro, Mingzhe Li, Shen Gao, Xiangliang Zhang, Dongyan Zhao, and Rui Yan. 2021.  Capturing relations between scientific papers: An abstractive model for related work section generation.  In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pages 6068–6077, Online. Association for Computational Linguistics.
  * Deutsch et al. (2022) Daniel Deutsch, Rotem Dror, and Dan Roth. 2022.  Re-examining system-level correlations of automatic summarization evaluation metrics.  _arXiv preprint arXiv:2204.10216_.
  * Ge et al. (2021) Yubin Ge, Ly Dinh, Xiaofeng Liu, Jinsong Su, Ziyao Lu, Ante Wang, and Jana Diesner. 2021.  BACO: A background knowledge- and content-based framework for citing sentence generation.  In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pages 1466–1478, Online. Association for Computational Linguistics.
  * Hoang and Kan (2010) Cong Duy Vu Hoang and Min-Yen Kan. 2010.  Towards automated related work summarization.  In _Coling 2010: Posters_ , pages 427–435.
  * Holtzman et al. (2019) Ari Holtzman, Jan Buys, Li Du, Maxwell Forbes, and Yejin Choi. 2019.  The curious case of neural text degeneration.  _arXiv preprint arXiv:1904.09751_.
  * Hu and Wan (2014) Yue Hu and Xiaojun Wan. 2014.  Automatic generation of related work sections in scientific papers: An optimization approach.  In _Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 1624–1633, Doha, Qatar. Association for Computational Linguistics.
  * Li et al. (2022) Xiangci Li, Biswadip Mandal, and Jessica Ouyang. 2022.  CORWA: A citation-oriented related work annotation dataset.  In _Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 5426–5440, Seattle, United States. Association for Computational Linguistics.
  * Luu et al. (2021) Kelvin Luu, Xinyi Wu, Rik Koncel-Kedziorski, Kyle Lo, Isabel Cachola, and Noah A. Smith. 2021.  Explaining relationships between scientific documents.  In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pages 2130–2144, Online. Association for Computational Linguistics.
  * Schluter (2017) Natalie Schluter. 2017.  The limits of automatic summarisation according to rouge.  In _Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguistics: Volume 2, Short Papers_ , pages 41–45.
  * Teufel et al. (2006) Simone Teufel, Advaith Siddharthan, and Dan Tidhar. 2006.  Automatic classification of citation function.  In _Proceedings of the 2006 conference on empirical methods in natural language processing_ , pages 103–110.
  * Wallace et al. (2021) Byron C Wallace, Sayantan Saha, Frank Soboczenski, and Iain J Marshall. 2021.  Generating (factual?) narrative summaries of rcts: Experiments with neural multi-document summarization.  _AMIA Summits on Translational Science Proceedings_ , 2021:605.
  * Xing et al. (2020) Xinyu Xing, Xiaosheng Fan, and Xiaojun Wan. 2020.  Automatic generation of citation texts in scholarly papers: A pilot study.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 6181–6190.


# Predicting the $\sin\phi_{S}$ Single-Spin-Asymmetry of $\Lambda$ Production
off transversely polarized nucleon in SIDIS

Yongliang Yang<EMAIL_ADDRESS>College of Physics, Qingdao University,
Qingdao 266071, China  Xiaoyu Wang  School of Physics and Microelectronics,
Zhengzhou University, Zhengzhou 450001, China  Zhun Lu<EMAIL_ADDRESS>School of Physics, Southeast University, Nanjing 211189, China

###### Abstract

We investigate the transverse target spin asymmetry $A^{\sin\phi_{S}}_{UT}$
for the unpolarized $\Lambda$ production in semi-inclusive deep inelastic
scattering with the transverse momentum of the final-state lambda hyperon
being integrated out. The asymmetry is contributed by the product of the
transversity distribution function $h_{1}(x)$ of the nucleon and the collinear
twist-3 fragmentation function $\tilde{H}(z)$ of the $\Lambda$ hyperon. The
later one originates from the quark-gluon-quark correlation and is a naive
time-reversal-odd function. We calculate $\tilde{H}$ of the $\Lambda$ hyperon
by adopt a diquark spectator model. Using the numerical result of
$\tilde{H}(z)$ and the available parametrization of $h_{1}(x)$ from SIDIS
data, we predict the $\sin\phi_{S}$ asymmetry in the electroproduction of the
$\Lambda$ hyperon in the kinematical region of EIC, EicC and COMPASS. In the
phenomenological analysis we include the evolution effect of the distribution
functions and the fragmentation functions. The results show that the
asymmetries for the $\Lambda$ production SIDIS process is around 0.1 and may
be accessible at EIC, EicC and COMPASS. We also find that the evolution of
fragmentation function can affect the size of asymmetry.

##  I introduction

Understanding the nonperturbative fragmentation mechanism in hard semi-
inclusive processes is one of the important tasks in hadronic physics.
Fruitful outcomes regarding fragmentation functions have been achieved by
experimental measurements on $e^{+}e^{-}$ annihilation Abe:2005zx ;
Seidl:2008xc ; TheBABAR:2013yha ; Ablikim:2015pta ; Guan:2018ckx , semi-
inclusive deeply inelastic scattering (SIDIS) Airapetian:2004tw ;
Airapetian:2010ds ; Qian:2011py ; Adolph:2014zba  and $pp$ collision
Lesnik:1975my ; Bunce:1976yb ; Adams:2003fx ; Abelev:2008af ; Lee:2007zzh ;
Adamczyk:2012xd , as well as by theoretical studies Collins:1992kk ;
Boer:1997qn ; deFlorian:1997zj ; Boros:1998kc ; Ma:1998pd ; Ma:1999gj ;
Anselmino:2000vs ; Bacchetta:2001di ; Bacchetta:2002tk ; Gamberg:2003eg ;
Bacchetta:2003xn ; Amrath:2005gv ; Xu:2005ru ; Bacchetta:2007wc ; Zhang:2008ez
; Anselmino:2013vqa ; Kang:2014zza ; Kang:2010xv ; Kanazawa:2013uia ;
Gamberg:2018fwy ; Wei:2014pma . Of particular interests are those related to
the spin-orbit correlations which are usually naive time-reversal-odd (T-odd).
A renowned fragmentation function is the Collins function $H_{1}^{\perp}$
Collins:1992kk , which arises from the correlation between the transverse spin
of the quark and the transverse momentum of the fragmented hadron. Another
example is the Sivers-type fragmentation function $D_{1T}^{\perp}$
Anselmino:2000vs ; DAlesio:2020wjq ; Callos:2020qtu  reflecting the
correlation between the transverse polarization of the final state spin-1/2
hadron and the transverse momentum of the quark. Because these functions
describe the asymmetric distribution of hadron inside a fragmenting quark,
they play important roles in the spin or azimuthal asymmetries in various high
energy processes Abe:2005zx ; Seidl:2008xc ; TheBABAR:2013yha ;
Ablikim:2015pta ; Guan:2018ckx ; Airapetian:2004tw ; Airapetian:2010ds ;
Qian:2011py ; Adolph:2014zba ; Lesnik:1975my ; Bunce:1976yb ; Adams:2003fx ;
Abelev:2008af ; Lee:2007zzh ; Adamczyk:2012xd . Furthermore, these functions
contain nontrivial QCD dynamics such as final-state interactions as well as
the Wilson lines which ensure the gauge-invariance of the operator definitions
Metz:2002iz ; Collins:2004nx ; Boer:2003cm ; Yuan:2007nd ; Gamberg:2008yt ;
Meissner:2008yf .

Recently, the twist-3 fragmentation functions arising from multiparton
correlation Qiu:1998ia ; Eguchi:2006qz ; Kang:2010zzb ; Kanazawa:2014dca ;
Koike:2017fxr  also attract a lot of attentions. Particularly, a
phenomenological study Kanazawa:2014tda  on the inclusive pion production in
single transversely polarized $pp$ collision Adams:2003fx ; Abelev:2008af ;
Adamczyk:2012xd ; Lee:2007zzh  shows that, beside the contribution from the
twist three distribution $T_{q}(x,x)$ Efremov:1981sh ; Efremov:1984ip ;
Qiu:1991wg ; Kang:2011hk , the T-odd twist-3 fragmentation functions $\hat{H}$
and $\tilde{H}$ Metz:2012ct  should be included in the analysis in order to
interpret the single spin asymmetry (SSA) in this process in a consistent way.
The fragmentation function $\tilde{H}$ arising from quark-gluon-quark(qgq)
correlation also contributes to the $\sin\phi_{S}$ asymmetry in SIDIS
Bacchetta:2006tn  through the combination $h_{1}(x)\otimes\tilde{H}(z)$, with
$h_{1}(x)$ the tansversity distribution and $\phi_{S}$ the azimuthal angle of
the transverse spin of the nucleon target. Thus, $\tilde{H}(z)$ provides a
unique manner to probe the transversity of the proton via single-hadron
production in the collinear framework. In Ref. Lu:2015wja , the fragmentation
function of the pion meson was calculated by a quark-antiquark-spectator
model. In Ref. Wang:2016tix , the SSA $A_{UT}^{\sin(\phi_{S})}$ for the pion
production at the electron-ion collider (EIC) was predicted.

In this work, we will study the fragmentation function $\tilde{H}$ of the
$\Lambda$ hyperon as well as its role in the SSA of $lp^{\uparrow}\to
l^{\prime}\Lambda X$ process. As the $\Lambda$ hyperon contains the up, down
and strange valence flavors, which is more complicated than the pion meson,
the study of the $\Lambda$ fragmentation function will provide complimentary
information on hadronization mechanism involving spin-orbit correlation. The
investigation could also obtain the flavor dependence Ma:1999gj  of the
fragmentation process. For this purpose, we calculate $\tilde{H}$ of the
$\Lambda$ hyepron for the up, down, and strange quarks using a diquark
spectator model. Previously, the model has been applied to calculate the
fragmentation function $D^{\perp}_{1T}$ and Collins functions of the $\Lambda$
hyperon in Refs. Yang:2017cwi ; Wang:2018wqo . In these cases the spectator
system is diquark, and the contributions from both the scalar diquark and
vector diquark are taken into account. We also consider the gluon rescattering
effect for the qgq correlation functions in the calculation. Based on the
model results, we predict the $\sin\phi_{S}$ asymmetry in the
$lp^{\uparrow}\to l^{\prime}\Lambda X$ process, which can be measured by the
COMPASS as well as the proposed EIC and the EIC in China (EicC). As these
facilities cover different kinematical regions, it is necessary to consider
the QCD evolution effect of $\tilde{H}$ to compare the result at different
energy scales.

The remaining content of the paper is organized as follows. in Sec. II, we
perform the calculation for twist-3 qgq fragmentation function $\tilde{H}$ by
adopting the diquark spectator model and study the evolution effects of
$\tilde{H}$. In Sec. III, we set up the formalism of the $\sin\phi_{S}$
asymmetry in SIDIS process with the transverse momentum of the final-state
hadron being integrated. We present the numerical results of the
$\sin\phi_{S}$ asymmetry in the electroproduction of the $\Lambda$ hyperon at
EIC, EicC and COMPASS, using formalism of the $\sin\phi_{S}$ asymmetry in a
collinear framework. In Sec. IV, we summarize the results of the paper and
give some conclusion.

##  II Model calculation of fragmentation function in qgq correlator

In this section, we present the model calculation of the twist-3 transverse
momentum dependent fragmentation function $\tilde{H}(z,k_{T})$ utilizing the
diquark spectator model. The fragmentation function can be obtained from the
following trace of the transverse correlator
$\tilde{\Delta}^{\alpha}_{A}(z,k_{T})$

$\displaystyle{z\over
4M_{\Lambda}}\operatorname*{Tr}\nolimits[(\tilde{\Delta}_{A\alpha}(z,k_{T};S_{\Lambda})+\tilde{\Delta}_{A\alpha}(z,k_{T};-S_{\Lambda}))\sigma^{\alpha-}]$
$\displaystyle=\tilde{H}(z,k_{T})+i\tilde{E}(z,k_{T})$ (1)

where $M_{\Lambda}$ is the mass of $\Lambda$ hyperon, and the twist-3 qgq
fragmentation correlator $\tilde{\Delta}_{A\alpha}$ can be expressed as
Gamberg:2006ru ; Bacchetta:2006tn

$\displaystyle\tilde{\Delta}_{A}^{\alpha}(z,k_{T};S_{\Lambda})$
$\displaystyle=\sum_{X}\int\;\frac{1}{2z}\int\frac{d\xi^{+}d^{2}\bm{\xi}_{T}}{(2\pi)^{3}}\int
e^{ik\cdot\xi}\langle
0|\int^{\xi^{+}}_{\pm\infty^{+}}d{\eta^{+}}\mathcal{U}^{\bm{\xi}_{T}}_{(\infty^{+},\eta^{+})}$
$\displaystyle\times
gF^{-\alpha}_{\perp}(\eta)\mathcal{U}^{\bm{\xi}_{T}}_{(\eta^{+},\xi^{+})}\psi(\xi)|P_{\Lambda},S_{\Lambda};X\rangle\langle
P_{\Lambda},S_{\Lambda};X|\bar{\psi}(0)\mathcal{U}^{\bm{0}_{T}}_{(0^{+},\infty^{+})}\mathcal{U}^{\infty^{+}}_{(\bm{0}_{T},\bm{\xi}_{T})}|0\rangle\bigg{|}_{\begin{subarray}{l}\eta^{+}=\xi^{+}=0\\\
\eta_{T}=\xi_{T}\end{subarray}}\,.$ (2)

with ${\cal U}^{c}_{(a,b)}$ the Wilson line (gauge link) running along the
direction from $a$ to $b$ at the fixed position $c$. The detailed discussion
on the Wilson line $\cal U$ has been given in Refs. Bacchetta:2006tn ;
Bacchetta:2007wc ; Ji:2002aa ; Belitsky:2002sm . The gauge invariant of
correlator is guaranteed by the antisymmetric field strength tensor
$F^{\mu\nu}$ of the gluon. The state $|P_{\Lambda},S_{\Lambda}\rangle$
represents final-state lambda hyperon with the momentum of $P_{\Lambda}$ and
the spin of $S_{\Lambda}$. In the diquark spectator model Nzar:1995wb ;
Jakob:1997wg , the fragmentation function $\tilde{H}$ can be calculated from
Fig. 1, where the contribution to the T-odd fragmentation function also
originates from the imaginary part of the one-loop diagram Yang:2016mxl ;
Lu:2015wja . Specifically, the quark-diquark-hyperon vertex
$\langle\,P_{\Lambda},S_{\Lambda};X|\,\bar{\psi}(0)|0\rangle$ appearing in the
r.h.s of Eq.(2) has the following form:

$\displaystyle\langle\,P_{\Lambda},S_{\Lambda};X|\,\bar{\psi}(0)|0\rangle=\begin{cases}\bar{U}(P_{\Lambda},S_{\Lambda})\,{\Upsilon}_{s}\,\displaystyle{\frac{i}{\kern
1.99997ptk\kern-4.49997pt/-m_{q}}}&\textrm{scalar diquark,}\\\
\bar{U}(P_{\Lambda},S_{\Lambda}\,){\Upsilon}^{\mu}_{v}\,\displaystyle{\frac{i}{\kern
1.99997ptk\kern-4.49997pt/-m_{q}}}\,\varepsilon_{\mu}\,&\textrm{axial-vector
diquark.}\end{cases}$ (3)

where $k$ denotes the parent quark momentum. $\varepsilon_{\mu}$ is the
polarization vector of the axial-vector diquark. The summation over all
polarizations states of the axial-vector diquark can be expressed as
$d_{\mu\nu}=\sum_{\lambda}\varepsilon^{*(\lambda)}_{\mu}\varepsilon^{(\lambda)}_{\nu}$,
which has the form as
$d_{\mu\nu}=-g_{\mu\nu}+{\frac{P_{\Lambda\mu}\,P_{\Lambda\nu}}{M_{\Lambda}^{2}}}$
Jakob:1997wg . The scalar and axial-vector coupling vertex of the quark-
diquark-hyperon can be expressed by ${\Upsilon}_{s}=g_{s}(k^{2})$ and
${\Upsilon}^{\mu}_{v}={g_{v}(k^{2})\over\sqrt{3}}\gamma_{5}(\gamma^{\mu}+{P_{\Lambda}^{\mu}\over
M_{\Lambda}})$, respectively. In this work we assume that $g_{s}$ and $g_{v}$
have the same form $g_{s}=g_{v}=g_{qh}$ as Gaussian form denoted as $g_{qh}$:

$\displaystyle g_{qh}(k^{2})~{}\mapsto{g_{D}\over
z}\,e^{-{k^{2}\over\Lambda^{2}}}\,,$ (4)

where $\Lambda^{2}$ has the general form
$\Lambda^{2}=\lambda^{2}z^{\alpha}(1-z)^{\beta}$ and $g_{D}$, $\lambda$,
$\alpha$ and $\beta$ are the model parameters.

Figure 1: The Feynman diagram that is relevant to the calculation of the
quark-gluon-quark correlator in the diquark model. The notation ${\Upsilon}$
and $\Gamma$ describe the quark-diquark-hyperon and gluon-diquark coupling
vertex, respectively.

Analogously, we also provide the expression for the vertex $\langle
0|gF^{-\alpha}_{\perp}(\eta)\psi(\xi)|P_{\Lambda},S_{\Lambda};X\rangle$ in Eq.
(2), in which the Feynman rules corresponding to the gluon field strength
tensor $F^{\alpha\beta}$ is given by the factor
$i\,(l^{\alpha}g_{T}^{\rho\beta}-l_{T}^{\alpha}g^{\beta\rho})$ , as denoted by
the open circle in Fig. 1 Lu:2015wja . With the above Feynman rules, the
contribution of Fig. 1 to the correlator $\tilde{\Delta}^{\alpha}_{A}$ is
given by:

$\displaystyle\tilde{\Delta}^{\alpha}_{A\,s}(z,k_{T},S_{\Lambda})$
$\displaystyle=-i{C_{F}\alpha_{S}\over 2(2\pi)^{2}(1-z)P_{\Lambda}^{-}}{1\over
k^{2}-m^{2}}\int{d^{4}l\over(2\pi)^{4}}$
$\displaystyle\times{(l^{-}g_{T}^{\alpha\rho}-l^{\alpha}\,n_{+}^{\rho})(\kern
1.99997ptk\kern-4.49997pt/-\kern
1.99997ptl\kern-5.0pt/+m)\bar{\Upsilon}_{s}U(P_{\Lambda},S_{\Lambda}\,)\bar{U}(P_{\Lambda},S_{\Lambda}\,)\Upsilon_{s}(\kern
1.99997ptk\kern-4.49997pt/+m)\over((k-l)^{2}-m^{2})(l^{2}-i\epsilon)((k-l-
P_{\Lambda})^{2}-m_{s}^{2})(-l^{-}-i\epsilon)}\bar{\Gamma}_{\rho}$ (5)
$\displaystyle\tilde{\Delta}^{\alpha}_{A\,v}(z,k_{T},S_{\Lambda})$
$\displaystyle=i{C_{F}\alpha_{S}\over 2(2\pi)^{2}(1-z)P_{\Lambda}^{-}}{1\over
k^{2}-m^{2}}\int{d^{4}l\over(2\pi)^{4}}$
$\displaystyle\times{(l^{-}g_{T}^{\alpha\rho}-l^{\alpha}\,n_{+}^{\rho})(\kern
1.99997ptk\kern-4.49997pt/-\kern
1.99997ptl\kern-5.0pt/+m)\bar{\Upsilon}_{v}^{\nu}U(P_{\Lambda},S_{\Lambda}\,)\bar{U}(P_{\Lambda},S_{\Lambda}\,)\Upsilon_{v}^{\mu}(\kern
1.99997ptk\kern-4.49997pt/+m)\over((k-l)^{2}-m^{2})(l^{2}-i\epsilon)((k-l-
P_{\Lambda})^{2}-m_{s}^{2})(-l^{-}-i\epsilon)}d^{i\mu}d^{j\nu}\bar{\Gamma}_{i\,j\rho}$
(6)

where $\tilde{\Delta}^{\alpha}_{A\,s}$ and $\tilde{\Delta}^{\alpha}_{A\,v}$
represent scalar and axial vector diquark form of qgq correlator,
respectively. Here, the light-cone coordinates $a^{\pm}=a\cdot
n_{\pm}=(a^{0}\pm a^{3})/\sqrt{2}$ is applied, and
$k^{-}={P_{\Lambda}^{-}/{z}}$. In Fig. 1, the notation $\Gamma$ describe the
gluon-diquark coupling vertex. In order to explicitly calculate the
correlator, we choose the following form for the vertex between the gluon and
the scalar diquark ($\Gamma_{s}$) and the axial vector diquark ($\Gamma_{v}$):

$\displaystyle\Gamma^{\rho,a}_{s}=$ $\displaystyle i\,T^{a}(2k-2P-l)^{\rho}\,$
(7) $\displaystyle\Gamma^{\rho\mu\nu,a}_{v}=$
$\displaystyle-i\,T^{a}[(2k-2P-l)^{\rho}\,g^{\mu\nu}-(k-P-l)^{\mu}g^{\nu\rho}-(k-P)^{\nu}g^{\rho\mu}];$
(8)

Where $T^{a}$ is the Gell-Mann matrix.

Similar to the calculation of the T-odd quark-gluon-quark fragmentation
function $\tilde{G}^{\perp}$ in Ref. Yang:2016mxl , we obtain the imaginary
part of the correlator usig the Cutkosky cut rules to put the gluon and quark
lines on the mass shell. This corresponds to the following replacements on the
propagators by using the Dirac delta functions

$\displaystyle{1\over l^{2}+i\varepsilon}\rightarrow-2\pi
i\delta(l^{2}),~{}~{}~{}~{}~{}~{}~{}{1\over(k-l)^{2}-m^{2}+i\varepsilon}\rightarrow-2\pi
i\delta((k-l)^{2}-m^{2})\,.$ (9)

Using the cut rules in Eq. (9), we perform the trace and integration over the
loop momentum $l$, we first give the scalar diquark component of
$\tilde{H}(z,k_{T}^{2})$ for $\Lambda$ hyperon as follows

$\displaystyle\tilde{H}^{s}(z,\bm{k}_{T}^{2})={\alpha_{S}\,g_{qh}^{2}\,C_{F}\over\,(2\pi)^{4}M_{\Lambda}(1-z)}{1\over
k^{2}-m^{2}}\tilde{H}_{1s}(z,\bm{k}_{T}^{2})$ (10)

where

$\displaystyle\tilde{H}_{1s}(z,\bm{k}_{T}^{2})$ $\displaystyle={1\over
z}[z\mathcal{A}(z^{2}\bm{k}^{2}_{T}m+k^{2}(z-1)(zm+M_{\Lambda}))+\mathcal{B}M_{\Lambda}(z(z-1)mM_{\Lambda}-z^{2}\bm{k}^{2}_{T}+(z-1)M_{\Lambda}^{2})].$
(11)

Here $k^{2}=z\bm{k_{T}}^{2}/(1-z)+m_{s}^{2}/(1-z)+m_{h}^{2}/z$.

Similarly, using the gluon-diquark vertex given in Eq. (8), we can also
calculate the expression for $\tilde{H}$ from the axial vector diquark
component

$\displaystyle\tilde{H}^{v}(z,k_{T}^{2})={\alpha_{S}\,g_{qh}^{2}\,C_{F}\over
4(2\pi)^{4}M_{\Lambda}(1-z)}{1\over
k^{2}-m^{2}}(\tilde{H}^{v}_{s}(z,\bm{k}_{T}^{2})+\tilde{H}^{v}_{0}(z,\bm{k}_{T}^{2})+\tilde{H}^{v}_{1}(z,\bm{k}_{T}^{2})+\tilde{H}^{v}_{2}(z,\bm{k}_{T}^{2}))$
(12)

where the four terms in the r.h.s. of Eq. (12) are given by

$\displaystyle\tilde{H}^{v}_{s}$ $\displaystyle={4\over
z}[(z\mathcal{A}(z^{2}\bm{k}^{2}_{T}m+k^{2}(z-1)(zm+M_{\Lambda}))+\mathcal{B}M_{\Lambda}(z(z-1)mM_{\Lambda}-z^{2}\bm{k}^{2}_{T}+(z-1)M_{\Lambda}^{2})]\,,$
(13) $\displaystyle\tilde{H}_{0}$ $\displaystyle=-2z{\bm{k}_{T}^{2}\over
3M_{\Lambda}}[(2\mathcal{CC}k^{-}k\cdot
P+2\mathcal{CD}k^{-}M_{\Lambda}^{2}+2z\mathcal{CE}k^{-}k^{-}-\mathcal{C}k^{-}k^{2}z+\mathcal{C}k^{-}m^{2}z)]\,,$
(14) $\displaystyle\tilde{H}_{1}$ $\displaystyle={2\over
3M_{\Lambda}^{2}z}\bigg{\\{}zk\cdot
P[z(\bm{k}^{2}_{T}(\mathcal{AA}mz-2\mathcal{A}(mz+M_{\Lambda}))+\mathcal{AA}k^{2}(mz+M_{\Lambda})$
$\displaystyle-2\mathcal{A}k^{2}mz-5\mathcal{A}k^{2}M_{\Lambda}-3\mathcal{A}m^{2}M_{\Lambda}-2\mathcal{B}mM_{\Lambda}^{2})+\mathcal{AB}M_{\Lambda}^{2}(mz-3M_{\Lambda})]$
$\displaystyle-2m_{h}z(\mathcal{AA}-2\mathcal{A})(k\cdot
P)^{2}+M_{\Lambda}[z(M_{\Lambda}(\mathcal{AB}k^{2}mz^{2}+\mathcal{AB}k^{2}M_{\Lambda}z+\mathcal{BB}M_{\Lambda}^{2}(mz-
M_{\Lambda})$
$\displaystyle-2\bm{I}_{2}(k^{2}-m^{2})(mz+M_{\Lambda})+m\mathcal{W}_{1}z-M_{\Lambda}\mathcal{W}_{1})+\bm{k}^{2}_{T}(\mathcal{AB}mM_{\Lambda}z^{2}+\mathcal{A}(z^{2}(k^{2}+m^{2})-4M_{\Lambda}^{2}))$
$\displaystyle+\mathcal{A}k^{2}(z^{2}(k^{2}+m^{2})+4mM_{\Lambda}z+4M_{\Lambda}^{2}))+2\mathcal{B}M_{\Lambda}^{2}(-z^{2}(\bm{k}^{2}_{T}+m^{2})+2mM_{\Lambda}z+2M_{\Lambda}^{2})]\bigg{\\}}\,,$
(15) $\displaystyle\tilde{H}_{2}$ $\displaystyle={4\over
3M_{\Lambda}}[M_{\Lambda}(\mathcal{A}+\mathcal{B}z)(M_{\Lambda}(2\bm{k}^{2}_{T}+k^{2}+m^{2})+2mk\cdot
P)-(k\cdot P+mM_{\Lambda})((\mathcal{AA}+\mathcal{AB}z)k\cdot P$
$\displaystyle+(\mathcal{AB}+z\mathcal{BB})M_{\Lambda}^{2}+z\mathcal{W}_{1}))]\,.$
(16)

Here the number of the subscript for $\tilde{H}$ denote the number of the
factor $l^{-}$ in the numerator of Eq. (1) after the trace calculation is
performed. $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ and $\bm{I}_{2}$ are
functions of $k^{2}$, $m$, $m_{D}$ and $M_{\Lambda}$, and can be found in Ref.
Yang:2017cwi . The functions $\mathcal{AA}$, $\mathcal{AB}$, $\mathcal{BB}$,
$\mathcal{CC}$, $\mathcal{CD}$, $\mathcal{CE}$ and $\mathcal{W}_{1}$ are come
from the following double-$l$ integrals

$\displaystyle\int{d^{4}l}$
$\displaystyle{l^{\mu}\,l^{\nu}\,\delta(l^{2})\delta((k-l)^{2}-m^{2})\over((k-P_{h}-l)^{2}-m_{s}^{2})}=\mathcal{AA}k^{\mu}\,k^{\nu}+\mathcal{BB}P^{\mu}\,P^{\nu}+\mathcal{AB}(k^{\mu}\,P^{\nu}+P^{\mu}\,k^{\nu})+g^{\mu\nu}\mathcal{W}_{1}\,,$
(17) $\displaystyle\int{d^{4}l}$
$\displaystyle{l^{\mu}\,l^{\nu}\,\delta(l^{2})\delta((k-l)^{2}-m^{2})\over((k-P_{h}-l)^{2}-m_{s}^{2})(-l\cdot
n_{+}+i\epsilon)}=\mathcal{CC}_{f}k^{\mu}\,k^{\nu}+\mathcal{DD}_{f}P^{\mu}\,P^{\nu}+\mathcal{EE}_{f}n_{+}^{\mu}\,n_{+}^{\nu}$
$\displaystyle+\mathcal{CD}_{f}(k^{\mu}\,P^{\nu}+P^{\mu}\,k^{\nu})+\mathcal{CE}_{f}(k^{\mu}\,n_{+}^{\nu}+n_{+}^{\mu}\,k^{\nu})+\mathcal{DE}_{f}(n_{+}^{\mu}\,P^{\nu}+P^{\mu}\,n_{+}^{\nu})+g^{\mu\nu}\mathcal{W}_{2}\,.$
(18)

Note that a main difference between the calculation for the $\Lambda$ hyperon
and the one for the pion Lu:2015wja  is that there are double $l$ in the
numerators in Eqs. (17) and (18), which need to be evaluated carefully. We
provide the complete expressions for these functions in the Appendix.

Figure 2: Result of $z\tilde{H}^{\Lambda/s}(z)$ (left panel) and
$z\tilde{H}^{\Lambda/u(d)}(z)$ (right panel) at the model scale $Q^{2}\
=0.23\,\mathrm{GeV}^{2}$ (red solid lines) and the evolved results at
$Q^{2}=100\,\mathrm{GeV}^{2}$ (blue dashed lines).

We will focus on the favored quark contribution to the fragmentation function
of $\Lambda$, i.e., $u\rightarrow\Lambda$ or $s\rightarrow\Lambda$, while the
unfavored quark contribution is zero. Assuming an SU(6) spin-flavor symmetry,
the fragmentation functions of the $\Lambda$ hyperon for light flavors satisfy
the following relations between different quark flavors and diquark types
VanRoyen:1967nq ; Jakob:1993th

$\displaystyle
D^{\textrm{u}\rightarrow\Lambda}=\,D^{\textrm{d}\rightarrow\Lambda}={1\over
4}D^{(s)}+{3\over
4}D^{(v)}\,,~{}~{}D^{\textrm{s}\rightarrow\Lambda}=D^{(s)}\,,$ (19)

where $u$, $d$ and $s$ denote the up, down and strange quarks, respectively.
In this study we assume that the relation in Eq. (19) holds for all
fragmentation functions. Neglecting the mass differences between the up, down
and strange quarks ($m=0.36$ GeV), we obtain the light flavors fragmentation
function $\tilde{H}$ as follows

$\displaystyle\tilde{H}^{\textrm{u}\rightarrow\Lambda}=\tilde{H}^{\textrm{d}\rightarrow\Lambda}={1\over
4}\tilde{H}^{(s)}+{3\over
4}\tilde{H}^{(v)}\,,~{}~{}\tilde{H}^{\textrm{s}\rightarrow\Lambda}=\tilde{H}^{(s)}\,,$
(20)

In order to obtain the numerical result for $\tilde{H}$, we should choose the
values for the parameters as Ref. Yang:2017cwi

$\displaystyle
g_{D}=1.983,~{}~{}~{}m_{D}=0.745\textrm{GeV},~{}~{}~{}\lambda=5.967\textrm{GeV},~{}~{}~{}\alpha=0.5(\textrm{fixed}),~{}~{}~{}\beta=0(\textrm{fixed})\,,$
(21)

the parameters were obtained by fitting the model result of
$D_{1}^{\Lambda}(z)$ in the same model to the deFlorian-Stratmann-Vogelsang
(DSV) parametrization deFlorian:1997zj  at the model scale
$Q^{2}=0.23\,\text{GeV}^{2}$.

Using the model parameters, we calculate the collinear twist-3 fragmentation
function $\tilde{H}(z)$ of $\Lambda$ by integrating over the transverse
momentum:

$\displaystyle\tilde{H}(z)=z^{2}\int
d^{2}\bm{k_{T}}\tilde{H}(z,\bm{k_{T}}^{2}).$ (22)

Since the energy scale in the experiments covers a wide range of $Q$, which is
much higher than the model scale, it is necessary to include the QCD evolution
effects of the fragmentation functions. There are studies on the evolution of
several twist-3 fragmentation functions in the literature Kang:2010xv ;
Kang:2010zzb ; Kang:2014zza ; Ma:2017upj . However, the DGLAP evolution kernel
for $\tilde{H}$ is still unknown. In this work, we assume the evolution kernel
of $\tilde{H}$ is same as the homogenous terms in the kernel of
$\hat{H}^{(3)}(z)$ in Ref. Kang:2010xv , as has been done in Ref. Kang:2015msa
. This kernel has the same form as that of transversity distribution function.
In Ref. Wang:2016tix , the evolution kernel for the pion $\tilde{H}(z)$ has
also been adopted as same as that for the transversity distribution function
$h_{1}$, which was motivated by the fact that $\tilde{H}$ is also a chiral odd
fragmentation function. To do this, we apply the QCDNUM package Botje:2010ay
and customize the package to include the kernel of transversity to perform the
evolution.

In the left and right panels of Fig. 2, we plot the $z-$dependence of the
collinear twist-3 fragmentation function $z\tilde{H}(z)$ of $\Lambda$ for $s$
and $u(d)$ quark of the twist-3 fragmentation functions at the model scale
$Q^{2}=0.23\mathrm{GeV}^{2}$ (the solid lines) and the evolved results at
$Q^{2}=100\mathrm{GeV}^{2}$ (the dotted lines), respectively. Fig. 2 shows
that the magnitude of $z\tilde{H}(z)$ for $u(d)\rightarrow\Lambda$ increases
with increasing $z$ when $0<z<0.9$, while it decreases with increasing $z$
after $z>0.9$ at the model scale, while the peak is around $z=0.5$ for the $s$
quark. The evolved fragmentation functions at $Q^{2}=100\mathrm{GeV}^{2}$ show
the relatively strong impact of the evolution effects. The $z$-dependences of
the $u(d)$ and $s$ quark for $z\tilde{H}$ is obvious different in the entire
$z$ region. We can see that the evolution from low $Q$ to higher $Q$ increases
the sizes of $u(d)\rightarrow\Lambda$ in the region $z<0.4$, while the region
for strange quark is $z<0.25$. This is because $s$ quark only come from the
contribution of the scalar diquark component in Eq. (20).

##  III Prediction on the $\sin\phi_{S}$ transverse SSAs of $\Lambda$ hyperon
production in SIDIS

Figure 3: The definition of the azimuthal angles in SIDIS. $P_{h}$ stands for
the momentum of the $\Lambda$ hyperon hadron, $S_{\perp}$ is the transverse
component of the spin vector $S$ with respect to the virtual photon momentum
Bacchetta:2006tn .

The process under study is the unpolarized $\Lambda$ production in SIDIS
process with an unpolarized lepton beam colliding on a transversely polarized
nucleon beam (or target):

$l(\ell)+N^{\uparrow}(P)\longrightarrow
l(\ell^{\prime})+{\Lambda}(P_{\Lambda})+X(P_{X}),$ (23)

where $l$ and $l^{\prime}$ stand for the momenta of the incoming and outgoing
leptons, $P$ and $P_{\Lambda}$ denote the momenta of the target nucleon and
the final-state $\Lambda$ hyperon, respectively. The momentum of the exchanged
virtual photon is defined as $q=l-l^{\prime}$ and $Q^{2}=-q^{2}$. The
reference frame in this work are adopted as Fig. 3, in which we will consider
the case the polarization of detected $\Lambda$ hyperon is not measured. The
azimuthal angle $\phi_{S}$ stands for the angle between the lepton scattering
plane and the direction of the transverse spin of the nucleon target.

We introduce the invariant variables to express the differential cross section
as follows Bacchetta:2006tn

$\displaystyle x$ $\displaystyle=\frac{Q^{2}}{2P\cdot q}\,,\quad
y=\frac{P\cdot q}{P\cdot l}\,,\quad z=\frac{P\cdot P_{h}}{P\cdot q}\,,$
$\displaystyle\gamma$ $\displaystyle=\frac{2Mx}{Q}\,,\quad
W^{2}=(P+q)^{2}\,,\quad s=(P+l)^{2}\,.$

where $M$ denotes the mass of the target nucleon. With the invariant
variables, the differential cross section of the process for unpolarized
$\Lambda$ production in SIDIS off an transverse polarized target can be
expressed as Wang:2016tix ; Bacchetta:2006tn

$\displaystyle\frac{d^{6}\sigma}{dxdydzd\phi_{h}d\phi_{S}dP_{hT}^{2}}=\frac{\alpha^{2}}{xyQ^{2}}\frac{y^{2}}{2(1-\varepsilon)}(1+\frac{\gamma^{2}}{2x})$
$\displaystyle\times\left\\{F_{UU,L}(x,z,P_{T})+|S_{\perp}|[\sin\phi_{S}\,F^{\sin\phi_{S}}_{UT}(x,z,P_{T})\right.$
$\displaystyle+\sin(2\phi_{h}-\phi_{S})\,F^{\sin(2\phi_{h}-\phi_{S})}_{UT}(x,z,P_{T})$
$\displaystyle+\left.\textrm{leading twist terms}]\right\\}\,,$ (24)

where $\varepsilon$ is the ratio of the longitudinal and transverse photon
flux

$\varepsilon=\frac{1-y-\frac{1}{4}\gamma^{2}y^{2}}{1-y+\frac{1}{2}y^{2}+\frac{1}{4}\gamma^{2}y^{2}}.$
(25)

After integrating over $\bm{P}_{hT}$, the differential cross section in Eq.
(24) turns to the form

$\displaystyle\frac{d^{4}\sigma}{dxdydzd\phi_{S}}=\frac{2\alpha^{2}}{xyQ^{2}}\,\frac{y^{2}}{2(1-\varepsilon)}\,\left(1+\frac{\gamma^{2}}{2x}\right)$
$\displaystyle\times\,\left\\{F_{UU,L}(x,z)+|S_{\perp}|\,\sqrt{2\varepsilon(1+\varepsilon)}\sin\phi_{S}\,F^{\sin\phi_{S}}_{UT}\left(x,z\right)+\cdots\right\\}\,.$
(26)

Here, the non-vanishing integrated structure functions are as follows
Boer:1997nt ; Bacchetta:2006tn

$\displaystyle F_{UU}(x,z)$
$\displaystyle=x\sum_{q}e_{q}^{2}f_{1}^{q}(x)D_{1}^{q}(z)\,,$ (27)
$\displaystyle F^{\sin\phi_{S}}_{UT}\left(x,z\right)$
$\displaystyle=-x\sum_{q}e_{q}^{2}\frac{2M_{\Lambda}}{Q}h_{1}^{q}(x)\frac{\tilde{H}^{q}(z)}{z}\,,$
(28)

and only the convolution of the transversity and the twist-3 collinear
fragmentation function $\tilde{H}$ remains in structure function
$F^{\sin\phi_{S}}_{UT}$. Following Eqs. (27) and (28), we can obtain the $z$
dependent-$\sin\phi_{S}$ asymmetry:

$\displaystyle A_{UT}^{\sin\phi_{s}}(z)$ $\displaystyle=\frac{\int dx\int
dy\frac{\alpha^{2}}{xyQ^{2}}\frac{y^{2}}{2(1-\epsilon)}(1+\frac{\gamma^{2}}{2x})\sqrt{2\epsilon(1+\epsilon)}F^{\sin\phi_{s}}_{UT}(x,z)}{\int
dx\int
dy\frac{\alpha^{2}}{xyQ^{2}}\frac{y^{2}}{2(1-\epsilon)}(1+\frac{\gamma^{2}}{2x})F_{UU}(x,z)}\,.$
(29)

In a similar way, $x$ dependent $\sin\phi_{S}$ asymmetry can be written as

$\displaystyle A_{UT}^{\sin\phi_{S}}(x)$ $\displaystyle=$
$\displaystyle\frac{\int dy\int
dz\frac{\alpha^{2}}{xyQ^{2}}\frac{y^{2}}{2(1-\epsilon)}(1+\frac{\gamma^{2}}{2x})\sqrt{2\epsilon(1+\epsilon)}F^{\sin\phi_{s}}_{UT}(x,z)}{\int
dy\int
dz\frac{\alpha^{2}}{xyQ^{2}}\frac{y^{2}}{2(1-\epsilon)}(1+\frac{\gamma^{2}}{2x})F_{UU}(x,z)}\,,$
(30)

In order to estimate the $\sin\phi_{S}$ asymmetry, we need the information of
the transversity distribution $h_{1}^{q}(x)$, for which we adopt the
parametrization from Ref. Anselmino:2013vqa :

$h_{1}^{q}(x)=\frac{1}{2}\mathcal{N}_{q}^{T}(x)[f_{1}^{q}(x)+g_{1}^{q}(x)]\,,$
(31)

with

$\mathcal{N}_{q}^{T}(x)=N_{q}^{T}\,x^{\alpha}(1-\beta)^{\beta}\frac{(\alpha+\beta)^{\alpha+\beta}}{\alpha^{\alpha}\beta^{\beta}}\,.$
(32)

where values of the parameters
$N_{u}^{T}=0.36,\,N_{d}^{T}=-1.00,\,\alpha=1.06$, and $\beta=3.66$ are taken
from Ref. Anselmino:2013vqa . The parametrization for the unpolarized
distribution $f_{1}^{q}(x)$ is from Ref. Lai:2010vv  and that for the helicity
distribution $g_{1}^{q}(x)$ is from Ref. deFlorian:2008mr , respectively. We
note that currently there is no available parametrization on $h_{1}^{q}(x)$
for the sea quarks, therefore, in this calculation we will not consider the
contribution from the transversity of the sea quarks.

The kinematical region of the EIC adopted in our calculation is
Accardi:2012qut

$\displaystyle 0.001<x<0.4,\quad 0.01<y<0.95,\quad 0.2<z<0.8,$ $\displaystyle
Q^{2}>1\mathrm{GeV}^{2},\quad\sqrt{s}=45\ \mathrm{GeV},\quad W>5\
\mathrm{GeV}.$ (33)

where $W$ is invariant mass of the virtual photon-nucleon system and
$W^{2}\approx\frac{1-x}{x}Q^{2}$. As for the EicC, we adopt the following
kinematical cuts eicc ; Anderle:2021wcy

$\displaystyle 0.005<x<0.5,\quad 0.07<y<0.9,\quad 0.2<z<0.7,$ $\displaystyle
Q^{2}>1\mathrm{GeV}^{2},\quad\sqrt{s}=16.7\ \mathrm{GeV},\quad
W>2\mathrm{GeV}.$ (34)

As the kinematics at EIC and EicC cover a wide range of $Q$, the QCD evolution
effect of the distribution and fragmentation functions are also considered.

Figure 4: Transverse SSA $A_{UT}^{\sin\phi_{S}}$ of $\Lambda$ hyperon
production in SIDIS at EIC for $\sqrt{s}=45$ GeV. The left and the right
panels show the $x$-dependent and the z-dependent asymmetry, respectively.

Figure 5: Transverse SSA $A_{UT}^{\sin\phi_{S}}$ of $\Lambda$ hyperon
production in SIDIS at EicC for $\sqrt{s}=16.7$ GeV. The left and the right
panels show the $x$-dependent and the z-dependent asymmetry, respectively.

The numerical results of the $\sin\phi_{S}$ asymmetries for $\Lambda$ hyperon
production at EIC and EicC are shown in Fig. 4 and Fig. 5, respectively. The
left panel and the right panel plot the asymmetries as functions of $x$ and
$z$. The dashed lines depict the results simultaneously evolving the
fragmentation function $\tilde{H}$ and distribution function $h_{1}(x)$. The
solid lines denote the asymmetries without considering the evolution of the
fragmentation functions $\tilde{H}$. We find that the $\sin\phi_{S}$
asymmetries for the $\Lambda$ hyperon production are negative. The magnitude
of the asymmetry at EIC and EicC is around 0.1, which is quite sizable. We
also find that the $x$-dependent asymmetries have a peak at the intermediate
$x$ region, around $x\approx 0.2$, and the magnitude of the $z$-dependent
asymmetry increases with increasing $z$. Compared our results with the the
same $\sin\phi_{S}$ asymmetry in pion production Wang:2016tix , we find that
the signs of asymmetry are both are negative; However, the magnitude of the
asymmetry in $\Lambda$ production is several times larger than the one in pion
production. The difference may be caused by the factor $M_{h}\over Q$
appearing in Eq. (28) as $M_{\Lambda}$ is almost one order of magnitude larger
than $M_{\pi}$. Therefore, there could be a great opportunity to measure the
$\sin\phi_{S}$ asymmetry in $\Lambda$ production at a future EIC and EicC.

An other important observation is that the evolution effect for the
$\sin\phi_{S}$ asymmetry is substantial in certain kinematical region both at
EIC and EicC. First of all, as shown by the dashed lines in Figs. 4 and 5, the
magnitudes of the $x$-dependent and $z$-dependent asymmetries for $\Lambda$
hyperon has changed substantially by the QCD evolution effect. Secondly, at
small-$x$ region ($x<0.04$ at EIC) and ($x<0.07$ at EicC) the evolution does
not affect the $x$-dependent asymmetries, while the evolution effect is
sizable in the intermediate-$x$ region. For the $z$-dependence asymmetries,
the evolution effect may be observed in the all $z$ region, where the
evolution effect of size is smaller than non-evolution result in the region
$z>0.5$ at EIC and EicC, thereby it should not be neglected. Nevertheless, the
evolution almost does not change the signs and the shapes of the asymmetries.

Finally, we also estimate the transverse asymmetries for $\Lambda$ at COMPASS
which applies a $160\mathrm{GeV}$ muon beam scattering off the nucleon target.
In this calculation, we adopt the following kinematical cults Alekseev:2010rw

$\displaystyle Q^{2}>1\mathrm{GeV}^{2},\quad 0.004<x<0.7,\quad 0.1<y<0.9,$
(35) $\displaystyle z>0.2,\quad W>5\ \mathrm{GeV},\quad
E_{h}>1.5~{}\mathrm{GeV}\,.$

The results of the $x$ and $z$-dependent asymmetries are depicted in the left
panel and right panels in Fig. 6, respectively. The solid lines denote the
asymmetries without considering the evolution of fragmentation functions
$\tilde{H}$ in Eqs. (III) and (III). The dashed lines correspond to the
evolution of $\tilde{H}^{q}(z)$ as $h_{1}$. We find that the overall tendency
of the asymmetries at the COMPASS are similar to that at EIC and EicC. The
evolution effect for $x$-dependent asymmetry may be observed in the region
$x>0.05$, and $z$-dependent asymmetry is larger than that of EIC and EicC.

Some comments are in order. Firstly, in the model calculation we have
calculated not only the $\tilde{H}^{\Lambda/q}$ for the up and down quarks,
but also that for the strange quark, while in the phenomenological analysis of
the $\sin\phi_{S}$ asymmetry we only have only considered the contributions
from the up and down quark. This is because currently there is no available
information for the transversity of the sea quarks. In several
phenomenological studies Lu:2011cw ; Xue:2020xba , the transversity of the sea
quarks were included in the calculation through model assumptions, such as
assuming the sea quark transversity is proportional to that of the valence
quarks. In this work we refrain to do so since at this stage our result is a
rough estimate of the asymmetry. The $\tilde{H}^{\Lambda/s}$ could be
measurable provided the transversity of the strange quark is sizable.
Secondly, in this work we consider the $\Lambda$ production in SIDIS process,
which is normally more difficult to be measured than the meson production.
However, in this process it is not necessary to measure the polarization of
the $\Lambda$. Besides, as our estimate indicates, the asymmetry is quite
sizable at EIC, EicC and COPMASS due to the factor $M_{\Lambda}/Q$ is much
larger than $M_{\pi}/Q$. Thus, we expect that the $\sin\phi_{S}$ asymmetry of
$\Lambda$ production in SIDIS can be measured with the help of high statistic
of the future $ep$ facilities.

Figure 6: Transverse SSA $A_{UT}^{\sin\phi_{S}}$ of $\Lambda$ hyperon
production in SIDIS at COMPASS for $E_{\mu}=160$ GeV. The left and the right
panels show the $x$-dependent and the $z$-dependent asymmetry, respectively.

##  IV Conclusion

In this work, we have studied the single-spin ${\sin\phi_{S}}$ asymmetry of
the $\Lambda$ production in SIDIS off an transversely polarized proton target.
We have calculated the twist-3 T-odd quark-gluon-quark fragmentation function
$\tilde{H}$ of the $\Lambda$ hyperon with two different types of diquark
spectator model by considering both scalar and axial-vector diquarks
Jakob:1997wg ; Bacchetta:2008af . The relation between the quark flavors and
diquark types for the fragmentation functions, motivated by the SU(6)
symmetric wave functions of the $\Lambda$ hyperon, has been taken into account
to provide results for different flavors. In addition, we have included the
leading-order evolution effects for the fragmentation functions. Using the
numerical results of $\tilde{H}(z)$, we have estimated the SSA
$A_{\mathrm{UT}}^{\sin\phi_{S}}$ in SIDIS at the kinematics of EIC, EicC and
COMPASS. Our calculation shows that the estimated $\sin\phi_{S}$ asymmetry for
the $\Lambda$ production in SIDIS is sizable, around 0.1. The sign of the
asymmetry is negative in the entire $x$ and $z$ region. We also find that the
inclusion of the evolution effects of $\tilde{H}$ can change the shape and
size of the asymmetry at the intermediate region of $x$ and $z$. The evolution
effects should be important for the interpretation of future experimental
data. In conclusion, the $\sin\phi_{S}$ asymmetries of $\Lambda$ production in
SIDIS may be measured at the kinematics of EIC, EicC and COMPASS, which
provide a feasible way to access the twist-3 collinear $\tilde{H}(z)$ of the
$\Lambda$ hyperon via the $\sin\phi_{S}$ within the collinear framework.

## Acknowledgements

This work is partially supported by the Shandong Provincial Natural Science
Foundation, China(Grants No. ZR2020QA081) and National Natural Science
Foundation of China (Grants No. 11575043,11905187,11847217). X. Wang is
supported by the China Postdoctoral Science Foundation under Grant No.
2018M640680 and the Academic Improvement Project of Zhengzhou University.

###  IV.1 Appendix: Double $l$ integrals

$\displaystyle\int{d^{4}l}{l^{\mu}\,l^{\nu}\,\delta(l^{2})\delta((k-l)^{2}-m^{2})\over((k-P_{h}-l)^{2}-m_{s}^{2})}=\mathcal{AA}k^{\mu}\,k^{\nu}+\mathcal{BB}P^{\mu}\,P^{\nu}+\mathcal{AB}(k^{\mu}\,P^{\nu}+P^{\mu}\,k^{\nu})+g^{\mu\nu}\mathcal{W}_{1}$
(36)

where

$\displaystyle\mathcal{AA}$
$\displaystyle=-{(k^{2}-m^{2})[k^{2}M^{2}(3\mathcal{A}+\mathcal{B})-2(\mathcal{B}-\mathcal{A})(k\cdot
P)^{2}-\mathcal{B}M^{2}k\cdot P]\over k^{2}\lambda(m_{s},M)}$
$\displaystyle\mathcal{BB}$
$\displaystyle=-{(k^{2}-m^{2})((\mathcal{A}+3\mathcal{B})k^{2}-2\mathcal{B}k\cdot
P)\over\lambda(m_{s},M)}$ $\displaystyle\mathcal{AB}$
$\displaystyle=-{(k^{2}-m^{2})(2\mathcal{B}M^{2}-(\mathcal{A}+3\mathcal{B})k\cdot
P)\over\lambda(m_{s},M)}$ $\displaystyle\mathcal{W}_{1}$
$\displaystyle=-{(\mathcal{A}+\mathcal{B})(k^{2}-m^{2})\over 4}$

Note that $4k^{2}M^{2}-4(k\cdot
P)^{2}=(4k^{2}M^{2}-(k^{2}+M^{2}-m_{s}^{2})^{2})=-\lambda(m_{s},M)$.

$\displaystyle\int{d^{4}l}$
$\displaystyle{l^{\mu}\,l^{\nu}\,\delta(l^{2})\delta((k-l)^{2}-m^{2})\over((k-P_{h}-l)^{2}-m_{s}^{2})(-l\cdot
n_{+}+i\epsilon)}=\mathcal{CC}_{f}k^{\mu}\,k^{\nu}+\mathcal{DD}_{f}P^{\mu}\,P^{\nu}+\mathcal{EE}_{f}n_{+}^{\mu}\,n_{+}^{\nu}$
(37)
$\displaystyle+\mathcal{CD}_{f}(k^{\mu}\,P^{\nu}+P^{\mu}\,k^{\nu})+\mathcal{CE}_{f}(k^{\mu}\,n_{+}^{\nu}+n_{+}^{\mu}\,k^{\nu})+\mathcal{DE}_{f}(n_{+}^{\mu}\,P^{\nu}+P^{\mu}\,n_{+}^{\nu})+g^{\mu\nu}\mathcal{W}_{2}$
(38)

where

$\displaystyle\mathcal{CC}_{f}k^{-}=$
$\displaystyle-{2z(\mathcal{A}-\mathcal{B})k\cdot
P-2\mathcal{A}k^{2}z-\mathcal{A}M_{\Lambda}^{2}+\mathcal{B}M_{\Lambda}^{2}z+\mathcal{C}k^{-}(z-1)z(k^{2}-m^{2})\over
z^{2}\bm{k}_{T}^{2}}$ (39) $\displaystyle\mathcal{CD}_{f}k^{-}=$
$\displaystyle{\mathcal{A}k^{2}z-2\mathcal{A}k^{2}-2\mathcal{B}k\cdot
P+\mathcal{B}M_{\Lambda}^{2}+\mathcal{C}k^{-}(z-1)(k^{2}-m^{2})\over
z^{2}\bm{k}_{T}^{2}}$ (40) $\displaystyle\mathcal{CE}_{f}k^{-}k^{-}=$
$\displaystyle-{2(k\cdot
P(\mathcal{B}(k^{2}z^{2}+M_{\Lambda}^{2}(z-1))-\mathcal{A}k^{2}z^{2})+k^{2}(\mathcal{A}(k^{2}z^{2}+M_{\Lambda}^{2}(z-1))-\mathcal{B}M_{\Lambda}^{2}z^{2}))\over
2z^{3}\bm{k}_{T}^{2}}$ (41)
$\displaystyle-{\mathcal{C}k^{-}(k^{2}-m^{2})(k^{2}z^{2}-2z^{2}k\cdot
P+M_{\Lambda}^{2}(2z-1))\over 2z^{3}\bm{k}_{T}^{2}}$ (42)

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