#  Adapters: A Unified Library for
Parameter-Efficient and Modular Transfer Learning

Clifton Poth 1,4, Hannah Sterz∗1, Indraneil Paul1,
Sukannya Purkayastha1, Leon Engländer1, Timo Imhof1,
Ivan Vulić2, Sebastian Ruder3, Iryna Gurevych1, Jonas Pfeiffer3
1Ubiquitous Knowledge Processing Lab, Technical University of Darmstadt
2Language Technology Lab, University of Cambridge
3Google DeepMind 4Cohere  Authors contributed equally.

###### Abstract

We introduce Adapters, an open-source library that unifies parameter-efficient
and modular transfer learning in large language models. By integrating 10
diverse adapter methods into a unified interface, Adapters offers ease of use
and flexible configuration. Our library allows researchers and practitioners
to leverage adapter modularity through composition blocks, enabling the design
of complex adapter setups. We demonstrate the library’s efficacy by evaluating
its performance against full fine-tuning on various NLP tasks. Adapters
provides a powerful tool for addressing the challenges of conventional fine-
tuning paradigms and promoting more efficient and modular transfer learning.
The library is available via https://adapterhub.ml/adapters.

##  1 Introduction

The ever-increasing size of pretrained large language models (LLMs) (Brown et
al., 2020; Chowdhery et al., 2022) has made the established transfer learning
paradigm of fine-tuning all model parameters on a downstream task (Howard and
Ruder, 2018; Devlin et al., 2019) extremely expensive. Moreover, the
requirement of parameter efficiency at fine-tuning, while definitely
paramount, is not the only shortcoming of the predominant LLM fine-tuning
paradigm. It also suffers from other crucial issues such as negative
interference, lack of positive transfer between tasks in multi-task learning
(McCloskey and Cohen, 1989), catastrophic forgetting (French, 1999), and poor
generalization.

Two closely related lines of research aimed at addressing this set of
challenges have gained significant attention recently. First, parameter-
efficient fine-tuning (Lialin et al., 2023; Sabry and Belz, 2023) focuses on
the aspect of computational efficiency and feasibility by only fine-tuning a
small number of parameters for downstream tasks. Second, modular transfer
learning (Pfeiffer et al., 2023) focuses on the aspect of knowledge transfer
by designing self-contained network modules which can be aggregated for better
multi-task performance and generalization. In practice, these often represent
two sides of the same coin. Methods that devise small components within a
language model for fine-tuning on labeled task data, henceforth generally
denoted as adapters, are both parameter-efficient and modular in nature.

The initial release of AdapterHub (Pfeiffer et al., 2020a) marks the first
attempt to systematically make adapters accessible to researchers and
practitioners in an easy-to-use framework. AdapterHub proposed a framework to
easily integrate, train and use adapters for state-of-the-art Transformer
models with minimal changes. It additionally established an open platform to
share, explore and consume pre-trained adapter modules. While AdapterHub
focused on bottleneck-style adapters (Houlsby et al., 2019) initially, the
field of adapter methods has expanded substantially since (Li and Liang, 2021;
Mahabadi et al., 2021; Hu et al., 2022; He et al., 2022; Liu et al., 2022,
among others).

With increasing interest in adapter methods, new tools and libraries have been
developed. OpenDelta (Hu et al., 2023a), HuggingFace’s PEFT (Mangrulkar et
al., 2022) and LLM-Adapters (Hu et al., 2023b) are recent examples of
libraries which attempt to unify adapter methods in a single code base and
extend their applicability to new model architectures. However, these works
exclusively focus on the parameter-efficiency aspect of adapters, neglecting
the modularity side of these methods.

Contributions. Based on the initial version of AdapterHub, we, therefore,
propose Adapters, a new library aimed at unifying parameter-efficient and
modular transfer learning. Compared to the first AdapterHub iteration and
concurrent libraries, our main contributions can be summarized as follows: 1)
We propose a self-contained library that integrates 10 diverse adapter methods
into a unified interface for easy usage and flexible configuration; 2) we
develop a simple way of leveraging the modularity of adapters by designing
composition blocks that allow flexibly defining complex adapter setups; 3) we
integrate all methods into 20 Transformer-based models spanning NLP, vision,
and multi-modal applications; 4) we evaluate the performance of our adapter
implementations against full fine-tuning on a diverse set of tasks.

##  2 Background

We use the term adapter in a more general sense to refer to a broad family of
transfer learning methods that share the two defining properties: parameter
efficiency and modularity. For a detailed overview of different adapter
architectures, we refer the reader to the recent survey by Pfeiffer et al.
(2023).

###  2.1 Parameter Efficiency

Let the parameters of a language model be composed of a set of pre-trained
parameters $\Theta$ (frozen) and a set of parameters $\Phi$ (where $\Phi$ can
either be newly introduced or $\Phi\subset\Theta$). During fine-tuning,
adapter methods optimize only $\Phi$ according to a loss function $L$ on a
dataset $D$:

$\Phi^{*}\leftarrow\arg\min_{\Phi}L(D;\\{\Theta,\Phi\\})$

Different adapter methods insert parameters $\Phi$ at different locations of a
pre-trained large model. Bottleneck adapters (Rebuffi et al., 2017; Houlsby et
al., 2019), as one of the early methods, introduce bottleneck feed-forward
layers in each layer of a Transformer model. Subsequent designs have adapted a
Transformer model’s self-attentions (Li and Liang, 2021), bias terms (Ben
Zaken et al., 2022), input prompts (Lester et al., 2021) or embeddings
(Pfeiffer et al., 2021b). Complementary lines of work have focused on
optimizing the parameter efficiency (Mahabadi et al., 2021; Liu et al., 2022)
and runtime efficiency (Hu et al., 2022; Lei et al., 2023) of adapters or have
attempted to unify multiple components in a single framework (He et al., 2022;
Mao et al., 2022).

###  2.2 Modularity

A modular deep learning model is composed of modules that each capture a
specific functionality of the full model, such as task or language capacities.
Pfeiffer et al. (2023) propose a taxonomy of modular deep learning methods
covering the dimensions of computation function, routing, aggregation, and
training.

Routing and aggregation are of special interest here as they coordinate the
composition of multiple adapter modules, a key functionality enabled by
modularity. Exemplary existing work includes using stochastic routing through
adapters (Wang et al., 2022), adapter parameter averaging (Friedman et al.,
2021), sequential function aggregation of adapter modules (Pfeiffer et al.,
2022b) as well as weighted (Wang et al., 2021) and attention-based (Pfeiffer
et al., 2021a) output aggregation.

Finally, along the training dimension, the modularity of adapters allows using
pre-trained adapter modules as initialization for further fine-tuning (Poth et
al., 2021; Vu et al., 2022).

##  3 The Adapters Library

| AdapterHub v1 | Adapters
---|---|---
Design |  Fork of Transformers |  Self-contained add-on library
Adapter methods | 2 | 10
Complex configurations | ✗ | ✓
Composition blocks | ✗111V1 already supported stacking and fusing adapter, however without flexibly composable blocks. | ✓ (6)
Model architectures | 3 | 20
AdapterHub.ml / HF Hub integration | ✓ / ✗ | ✓ / ✓

Table 1: Feature comparison between the initial AdapterHub release (Pfeiffer
et al., 2020a) and the proposed Adapters library.

Adapters builds on many design decisions established in the initial AdapterHub
release (Pfeiffer et al., 2020a), but offers substantial extensions both
‘horizontally’ (e.g., extending the support to many more pretrained neural
architectures, extending the coverage of adapter architectures) and
‘vertically’ (e.g., adding new composition and processing capabilities). Table
1 gives an overview of the differences between the initial AdapterHub and
Adapters. The core features adopted from the initial release, facilitating its
ease of use and wider adoption by researchers and practitioners, include: 1)
Tight integration into the widely used HuggingFace Transformers (Wolf et al.,
2020) library; 2) adaptation of pre-existing Transformers fine-tuning scripts
with minimal changes; 3) single-line saving and loading of adapter modules
from a shared community hub.

⬇

import adapters

from transformers import AutoModel

\parmodel = AutoModel.from_pretrained("..")

adapters.init(model)

model.add_adapter("a", config="seq_bn")

model.add_adapter("b", config="seq_bn")

model.train_adapter(Parallel("a", "b"))

Figure 1: Example of adding adapters to an existing Transformers model. After
model instantiation, init() introduces adapter-specific functionality. Two
bottleneck adapters are added via add_adapter() and activated for parallel
training.

###  3.1 Transformers Integration

Unlike the initial AdapterHub, Adapters is designed as a standalone package
that acts as an add-on to the Transformers library. Adapters provides adapter
implementations and management methods that can be injected into pre-trained
Transformer checkpoints without modifying the original model code directly. We
provide two approaches to this end: (i) by attaching to existing models and
(ii) by providing our own, specialized model classes.

Attaching to Models. The `init()` method provides a straightforward solution
for making adapters accessible to pre-existing model classes post-hoc. A model
checkpoint for one of the supported architectures (cf. § 3.5) can be
instantiated via any of the model classes provided by the Transformers
library. An example of this approach is given in Fig. 1. All adapter-related
functionality is injected post-instantiation by passing the model instance to
the `init()` method. Afterwards, all methods provided by Adapters are easily
invokable from the same model instance.

AdapterModel Classes. As an alternative to post-hoc initialization, Adapters
provides a set of built-in model classes optimized for working with adapters.
Following HuggingFace’s naming conventions, these classes are named
`XXXAdapterModel`, where `XXX` is the name of the model architecture. Compared
to models with attached adapters, these classes especially provide more
flexibility with regard to prediction heads. Each model instance can have
multiple named prediction heads targeted towards different tasks loaded
simultaneously. These prediction heads can be associated with adapter modules
by sharing a common name.

Providing this functionality is crucial for enabling features such as quickly
switching between adapters targeted toward different tasks at runtime. It is
also essential for creating composed adapter setups such as parallel inference
on multiple tasks (cf. § 3.4). We, therefore, provide automatic conversion
from HuggingFace’s model classes - typically paired with a single, fixed
prediction head - to our newly introduced classes featuring flexible
prediction heads.

###  3.2 Unified Adapter Interface

Adapters defines a common interface of methods covering the full life cycle of
working with adapters. This includes methods for adding, activating, saving,
releasing, loading, aggregating, and deleting adapter modules. When adding a
new adapter to a model (e.g., via `add_adapter()`), it is given a unique
identifier string. All adapter-related methods then solely use this string to
identify the adapter module an operation should be performed on. Thus, the
adapter interface at the model level can be agnostic to specific adapter
methods.

This interface, as well as adapter implementations at the module level, are
integrated into model classes via Python mixins and dynamically modifying
Python classes at runtime to keep changes to the existing Transformers code
base minimal.

(a) Stack

(b) Fuse

(c) Split

(d) BatchSplit

(e) Parallel

(f) Average

Figure 2: Overview of adapter composition blocks supported in the Adapters
library at the time of writing this paper.

###  3.3 Adapter Methods

Method name | Default config
---|---
Bottleneck adapter (Houlsby et al., 2019) | [double_]seq_bn
Invertible adapter (Pfeiffer et al., 2020b) | seq_bn_inv
Prompt tuning (Lester et al., 2021) | prompt_tuning
Prefix tuning (Li and Liang, 2021) | prefix_tuning
Compacter (Mahabadi et al., 2021) | compacter
LoRA (Hu et al., 2022) | lora
(IA)³ (Liu et al., 2022) | ia3
Parallel adapter (He et al., 2022) | par_bn
Mix-and-Match adapter (He et al., 2022) | mam
UniPELT (Mao et al., 2022) | unipelt

Table 2: Overview of adapter methods supported in the Adapters library at the
time of submission (Aug 2023).

Each adapter method is defined by a configuration object or string which allow
flexible customization of various properties of an adapter module, including
placement, capacity, residual connections, initialization etc. We distinguish
between single methods consisting of one type of adapter module and complex
methods consisting of multiple different adapter module types. Table 2 gives
an overview of all methods currently integrated into Adapters, along with
their configuration strings.

####  3.3.1 Single Methods

Adapters supports single adapter methods that introduce parameters in new
feed-forward modules such as bottleneck adapters (Houlsby et al., 2019),
introduce prompts at different locations such as prefix tuning (Li and Liang,
2021), reparameterize existing modules such as LoRA (Hu et al., 2022) or re-
scale their output representations such as (IA)³ (Liu et al., 2022). Detailed
descriptions for all currently implemented single methods are given in
Appendix A, see also Table 2 again.

####  3.3.2 Complex Methods

While different efficient fine-tuning methods and configurations have often
been proposed as standalone, combining them for joint training has proven to
be beneficial He et al. (2022); Mao et al. (2022). To make this process
easier, Adapters provides the possibility to group multiple configuration
instances using the `ConfigUnion` class. This flexible mechanism allows easy
integration of multiple complex methods proposed in the literature (as the two
examples outlined below), as well as the construction of other, new complex
configurations currently not available nor benchmarked in the literature Zhou
et al. (2023).

Mix-and-Match Adapters (He et al., 2022) was proposed as a combination of
Prefix-Tuning and parallel bottleneck adapters. Using `ConfigUnion`, this
method can be defined as:

⬇

config = ConfigUnion(

PrefixTuningConfig(bottleneck_size=800),

ParallelConfig(),

)

model.add_adapter("name", config=config)

UniPELT (Mao et al., 2022) combines LoRA, Prefix Tuning, and bottleneck
adapters in a single unified setup. It additionally introduces a gating
mechanism that controls the activation of the different adapter modules.
$\mathcal{G}_{m}\leftarrow\sigma(W_{\mathcal{G}_{m}}\cdot x)$.

###  3.4 Adapter Composition

While the modularity and composability aspects of adapters have seen
increasing interest in research, existing open-source libraries (Mangrulkar et
al., 2022; Hu et al., 2023a) have largely overlooked these aspects. Adapters
makes adapter compositions a central and accessible part of working with
adapters by enabling the definition of complex, composed adapter setups. We
define a set of simple composition blocks that each capture a specific method
of aggregating the functionality of multiple adapters. Each composition block
class takes a sequence of adapter identifiers plus optional configuration as
arguments. The defined adapter setup is then parsed at runtime by Adapters to
allow for dynamic switching between adapters per forward pass. Fig. 2 shows
schematic illustrations of all composition blocks supported by Adapters. Fig.
3 shows examples of how the same composition blocks are defined in code:

⬇

Stack("a", "b", "c")

Fuse("d", "e", "f")

Split("g", "h", splits=[64, 64])

BatchSplit("i", "j", batch_sizes=[2, 4])

Parallel("k", "l", "m")

Average("n", "o", weights=[0.3, 0.7])

Stack("p", Parallel("q", "r"))

Figure 3: Code examples of composition blocks supported by Adapters. Strings
represent adapter IDs.

In what follows, we present each supported composition in more detail.

Stack. The Stack block allows stacking multiple adapters sequentially within a
Transformer layer. This type of composition is, e.g., used in the MAD-X
framework for cross-lingual transfer (Pfeiffer et al., 2020b), where language
and task adapters are stacked. In Fig. 3, the input is first passed through a,
the output of a is then inputted to b, and the output of b is finally inputted
to c.

Fuse. The Fuse block can be used to activate an AdapterFusion layer (Pfeiffer
et al., 2021a). AdapterFusion is a non-destructive way to combine the
knowledge of multiple pre-trained adapters on a new downstream task. In Fig.
3, we activate the adapters d, e, and f as well as the fusion layer that
combines the outputs of all three.222Note that this requires a fusion layer to
be added beforehand via add_adapter_fusion().

Split. The Split block can be used to split an input sequence between multiple
adapters. This e.g. enables splitting multimodal input sequences to modality-
specific adapters (Pfeiffer et al., 2022a). In Fig. 3, we split each input
sequence between adapters g and h. All tokens with indices 0 - 63 are
forwarded through g while the next 64 tokens beginning at index 64 are
forwarded through h.

BatchSplit. The BatchSplit block splits inputs along the batch size dimension
between several adapters. That is, different adapters receive different sub-
batches of the full input batch. In Fig. 3, we split the input batch between
adapters i, and j. Adapter i receives two sequences and j receives four
sequences. The sum of all specified sub-batches has to match the batch size of
the input.

Parallel. This block can be used to enable independent parallel training and
inference on different adapters, where each adapter has its own prediction
head. The implementation automatically replicates all inputs at the first
occurrence of parallel adapter modules, sharing the inputs in all lower layers
without parallel modules. This mechanism was first used in Rücklé et al.
(2021). In Fig. 3, we forward all inputs via adapters k, l, and m in parallel.

Average. Following approaches of ensembling full models at inference time for
better generalization, recent work has explored methods of averaging pre-
trained adapters. This includes averaging adapter output representations (Wang
et al., 2021) as well as averaging adapter parameters (Friedman et al., 2021;
Wang et al., 2022; Chronopoulou et al., 2023). Adapters provides built-in
support for both types of inference time-averaging methods. Output averaging
allows to dynamically aggregate the output representations of multiple
adapters in a model forward pass via weighted averaging. This is realized via
the Average composition block. In Fig. 3, two adapters are averaged with the
weights $0.3$ for n and $0.7$ for o. Parameter averaging enables creating a
new adapter via weighted averaging of the parameters of multiple pre-trained
adapters. As this process is typically not done dynamically at runtime,
Adapters provides `average_adapter()` as a dedicated method. Compared to
output averaging, parameter averaging of adapters has the advantage of not
inducing any additional inference time relative to using a single adapter.

Nesting. Finally, it is possible to nest composition blocks within other
composition blocks to create deeper and more complex compositions. Adapters
defines a set of allowed nestings to restrict the users to setups that are
sensible. As an example, we nest a Parallel block within a Stack block in Fig.
3.

###  3.5 Supported Models

At the time of release, Adapters has built-in support for 20 widely adopted
model architectures included in the Transformers library. This covers text
encoder models such as BERT (Devlin et al., 2019) and DeBERTa (He et al.,
2021), text decoder models such as GPT-2 (Radford et al., 2019), sequence-to-
sequence models such as BART (Lewis et al., 2020) and T5 (Raffel et al.,
2020), vision encoder models such as ViT (Dosovitskiy et al., 2021), as well
as multimodal models such as CLIP (Radford et al., 2021).333An up-to-date full
list of supported models can be found at
https://docs.adapterhub.ml/model_overview.html.

While adapter-related implementations mostly can be shared across all
supported models, correctly integrating them into each model implementation
requires manual effort. While it is difficult to standardize this process due
to differences between model architectures, we provide clear guidelines for
integrating new models in the form of shared interfaces and step-by-step
documentation444https://docs.adapterhub.ml/contributing/adding_adapters_to_a_model.html.

Method | Sequence Classification | Regression | Mult. Choice | Extract. QA | Tagging | Avg.
---|---|---|---|---|---|---
CoLA | MNLI | MRPC | QNLI | QQP | RTE | SST-2 | STS-B | Cosmos QA | SQuAD v2 | CoNLL-2003
Dev MCC | Dev Acc. | Dev F1 | Dev Acc. | Dev F1 | Dev Acc. | Dev Acc. | Dev PCC | Dev Acc. | Dev F1 | Test F1
double_seq_bn | 63.58 | 87.61 | 93.31 | 92.84 | 91.58 | 80.87 | 94.73 | 90.85 | 70.99 | 84.89 | 91.92 | 85.74
($\pm$19.19) | ($\pm$26.41) | ($\pm$4.52) | ($\pm$17.17) | ($\pm$36.83) | ($\pm$11.09) | ($\pm$17.51) | ($\pm$27.16) | ($\pm$16.87) | ($\pm$5.52) | ($\pm$17.65) | ($\pm$18.17)
seq_bn | 71.22 | 87.5 | 92.91 | 93.15 | 89.69 | 79.42 | 95.18 | 89.44 | 69.68 | 82.88 | 92.02 | 85.74
($\pm$23.40) | ($\pm$20.39) | ($\pm$4.54) | ($\pm$15.83) | ($\pm$21.31) | ($\pm$9.81) | ($\pm$13.26) | ($\pm$20.33) | ($\pm$16.44) | ($\pm$1.04) | ($\pm$11.48) | ($\pm$14.34)
par_bn | 63.95 | 87.44 | 93.24 | 93.04 | 88.32 | 77.98 | 94.95 | 90.33 | 80.10 | 82.56 | 91.95 | 85.81
($\pm$23.72) | ($\pm$21.66) | ($\pm$4.65) | ($\pm$17.26) | ($\pm$33.14) | ($\pm$10.95) | ($\pm$16.97) | ($\pm$5.64) | ($\pm$18.47) | ($\pm$6.70) | ($\pm$27.60) | ($\pm$16.98)
compacter | 55.52 | 86.10 | 90.43 | 92.42 | 86.68 | 68.59 | 94.15 | 90.06 | 67.91 | 79.20 | 91.27 | 82.03
($\pm$13.67) | ($\pm$1.99) | ($\pm$3.58) | ($\pm$2.68) | ($\pm$2.14) | ($\pm$4.91) | ($\pm$0.81) | ($\pm$23.27) | ($\pm$10.42) | ($\pm$8.87) | ($\pm$8.58) | ($\pm$7.36)
prefix_tuning | 61.62 | 86.98 | 91.06 | 92.46 | 87.07 | 71.12 | 95.18 | 90.13 | 66.13 | 78.16 | 91.46 | 82.85
($\pm$4.93) | ($\pm$18.91) | ($\pm$4.09) | ($\pm$9.55) | ($\pm$15.58) | ($\pm$6.06) | ($\pm$0.54) | ($\pm$29.23) | ($\pm$3.44) | ($\pm$2.41) | ($\pm$2.44) | ($\pm$8.79)
lora | 63.99 | 87.59 | 92.60 | 93.11 | 88.48 | 80.26 | 94.99 | 90.72 | 70.63 | 82.46 | 91.85 | 85.15
($\pm$20.64) | ($\pm$4.29) | ($\pm$4.39) | ($\pm$3.77) | ($\pm$2.57) | ($\pm$9.28) | ($\pm$8.48) | ($\pm$19.31) | ($\pm$8.65) | ($\pm$8.86) | ($\pm$21.68) | ($\pm$10.17)
ia3 | 63.03 | 86.19 | 92.32 | 91.88 | 86.41 | 76.89 | 94.42 | 90.65 | 66.85 | 78.52 | 91.56 | 83.52
($\pm$21.39) | ($\pm$5.08) | ($\pm$3.94) | ($\pm$3.73) | ($\pm$13.46) | ($\pm$7.17) | ($\pm$2.13) | ($\pm$29.16) | ($\pm$9.69) | ($\pm$10.11) | ($\pm$21.94) | ($\pm$11.62)
Full Fine-tuning | 63.66 | 87.63 | 90.20 | 92.81 | 91.92 | 78.77 | 94.81 | 91.20 | 68.87 | 82.91 | 91.33 | 84.91

Table 3: Best performance ($\pm$ std. dev. across all hyper-parameters) of
various supported single adapter methods (cf. §3.3.1) applied to roberta-base,
benchmarked against full fine-tuning. The best results and lowest std. dev.
per task are highlighted in bold.

###  3.6 AdapterHub Ecosystem

Adapters is integrated into the extensive existing open-source ecosystem
introduced by AdapterHub (Pfeiffer et al., 2020a). Most prominently, this
includes AdapterHub.ml as a platform to share and discover pre-trained adapter
modules. Adapters further broadens the possibilities for sharing adapters by
integrating with HuggingFace’s Model Hub, which has emerged as one of the
primary platforms for open-sourcing model checkpoints. The new Hub integration
comes with programmatic methods of discovering and publishing pre-trained
adapter modules, in addition to the previously available methods for
downloading and saving adapters.

At the time of writing, users of Adapters have access to over 700 pre-trained
adapters.

##  4 Adapter Evaluation

In addition to the ease of use aforementioned, we show that the adapter
methods offered by our library are performant across a range of settings. To
this end, we conduct evaluations on the single adapter implementations made
available by Adapters (see §3.3.1). We demonstrate the effectiveness of these
methods against full fine-tuning on a variety of task types: extractive
question answering Rajpurkar et al. (2018), multiple choice classification
Huang et al. (2019), sequence tagging Tjong Kim Sang (2002), sequence to
sequence summarization Narayan et al. (2018), sequence classification and
sequence regression Wang et al. (2019). To make our evaluations reflective of
user experience, we conduct them using the two most commonly used base model
variants on AdapterHub: roberta-base Liu et al. (2019) and bart-base Lewis et
al. (2020).

Setup. We conduct a grid search over a range of common training hyper-
parameters, varying learning rates between $\\{10^{-5},10^{-4},5\cdot
10^{-4},10^{-4},10^{-3}\\}$ and the number of epochs between
$\\{5,10,20,30\\}$. We also augment the grid with a number of adapter-specific
hyper-parameters. These, along with the minimum and maximum trainable
parameters added across the configurations, are detailed in Table 4. The
highest attained performance (and the standard deviation of results across the
grid) for the two chosen base models are outlined in Tables 3 and 4,
respectively.

Results. The obvious takeaway from our evaluations is that all adapter
implementations offered by our framework are competitive with full model fine-
tuning, across all task classes. Approaches that offer more tunable hyper-
parameters (and thus allow for easy scaling) such as Bottleneck adapters,
LoRA, and Prefix Tuning predictably have the highest topline performance,
often surpassing full fine-tuning. However, extremely parameter-frugal methods
like (IA)3, which add $<0.005\%$ of the parameters of the base model, also
perform commendably and only fall short by a small fraction. Finally, the
Compacter is the least volatile among the single methods, obtaining the lowest
standard deviation between runs on the majority of tasks.

##  5 Conclusion

We have presented Adapters, a novel library for research and application of
adapters. Unlike comparable solutions, Adapters equally focuses on the
parameter-efficiency and modularity side of adapters. Our library implements a
diverse set of adapter methods under a unified interface which allows flexible
configuration and mixing of different approaches. We proposed a simple
building block system for leveraging the modularity of adapters to build
complex adapter setups. Adapters tightly integrates into the HuggingFace and
AdapterHub ecosystems and its adapter implementations show performances
competitive to full fine-tuning.

As research on adapters and LLMs continues to advance rapidly, our library
will evolve as well. Its extensibility makes Adapters well prepared for the
integration of new adapter methods and model architectures, both from us and
the community.

## Acknowledgements

This work has been funded by Huawei Technologies (Ireland) Co., Ltd. and
German Research Foundation (DFG) as part of the Research Training Group KRITIS
No. GRK 2222. We thank Martin Hentschel for his help building the demo web app
for our library. We thank Francesco Piccinno and Srini Narayanan for their
helpful comments and suggestions during the initial drafts of this paper. We
also thank the many contributors and users of the AdapterHub open-source
framework.

## References

  * Ben Zaken et al. (2022) Elad Ben Zaken, Yoav Goldberg, and Shauli Ravfogel. 2022.  BitFit: Simple parameter-efficient fine-tuning for transformer-based masked language-models.  In _Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)_ , pages 1–9, Dublin, Ireland.
  * Brown et al. (2020) Tom B. Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel M. Ziegler, Jeffrey Wu, Clemens Winter, Christopher Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. 2020.  Language models are few-shot learners.  In _Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual_.
  * Chowdhery et al. (2022) Aakanksha Chowdhery, Sharan Narang, Jacob Devlin, Maarten Bosma, Gaurav Mishra, Adam Roberts, Paul Barham, Hyung Won Chung, Charles Sutton, Sebastian Gehrmann, Parker Schuh, Kensen Shi, Sasha Tsvyashchenko, Joshua Maynez, Abhishek Rao, Parker Barnes, Yi Tay, Noam Shazeer, Vinodkumar Prabhakaran, Emily Reif, Nan Du, Ben Hutchinson, Reiner Pope, James Bradbury, Jacob Austin, Michael Isard, Guy Gur-Ari, Pengcheng Yin, Toju Duke, Anselm Levskaya, Sanjay Ghemawat, Sunipa Dev, Henryk Michalewski, Xavier Garcia, Vedant Misra, Kevin Robinson, Liam Fedus, Denny Zhou, Daphne Ippolito, David Luan, Hyeontaek Lim, Barret Zoph, Alexander Spiridonov, Ryan Sepassi, David Dohan, Shivani Agrawal, Mark Omernick, Andrew M. Dai, Thanumalayan Sankaranarayana Pillai, Marie Pellat, Aitor Lewkowycz, Erica Moreira, Rewon Child, Oleksandr Polozov, Katherine Lee, Zongwei Zhou, Xuezhi Wang, Brennan Saeta, Mark Diaz, Orhan Firat, Michele Catasta, Jason Wei, Kathy Meier-Hellstern, Douglas Eck, Jeff Dean, Slav Petrov, and Noah Fiedel. 2022\.  PaLM: Scaling Language Modeling with Pathways.  _ArXiv preprint_.
  * Chronopoulou et al. (2023) Alexandra Chronopoulou, Matthew Peters, Alexander Fraser, and Jesse Dodge. 2023.  AdapterSoup: Weight averaging to improve generalization of pretrained language models.  In _Findings of the Association for Computational Linguistics: EACL 2023_ , pages 2054–2063, Dubrovnik, Croatia.
  * Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019.  BERT: Pre-training of deep bidirectional transformers for language understanding.  In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , pages 4171–4186, Minneapolis, Minnesota.
  * Dosovitskiy et al. (2021) Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. 2021.  An image is worth 16x16 words: Transformers for image recognition at scale.  In _9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021_.
  * French (1999) Robert M. French. 1999.  Catastrophic forgetting in connectionist networks.  _Trends in Cognitive Sciences_ , (4):128–135.
  * Friedman et al. (2021) Dan Friedman, Ben Dodge, and Danqi Chen. 2021.  Single-dataset experts for multi-dataset question answering.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pages 6128–6137, Online and Punta Cana, Dominican Republic.
  * He et al. (2022) Junxian He, Chunting Zhou, Xuezhe Ma, Taylor Berg-Kirkpatrick, and Graham Neubig. 2022.  Towards a unified view of parameter-efficient transfer learning.  In _The Tenth International Conference on Learning Representations, ICLR 2022, Virtual Event, April 25-29, 2022_.
  * He et al. (2021) Pengcheng He, Xiaodong Liu, Jianfeng Gao, and Weizhu Chen. 2021.  Deberta: decoding-enhanced bert with disentangled attention.  In _9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021_.
  * Houlsby et al. (2019) Neil Houlsby, Andrei Giurgiu, Stanislaw Jastrzebski, Bruna Morrone, Quentin de Laroussilhe, Andrea Gesmundo, Mona Attariyan, and Sylvain Gelly. 2019.  Parameter-efficient transfer learning for NLP.  In _Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA_ , Proceedings of Machine Learning Research, pages 2790–2799.
  * Howard and Ruder (2018) Jeremy Howard and Sebastian Ruder. 2018.  Universal language model fine-tuning for text classification.  In _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 328–339, Melbourne, Australia.
  * Hu et al. (2022) Edward J. Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen. 2022.  Lora: Low-rank adaptation of large language models.  In _The Tenth International Conference on Learning Representations, ICLR 2022, Virtual Event, April 25-29, 2022_.
  * Hu et al. (2023a) Shengding Hu, Ning Ding, Weilin Zhao, Xingtai Lv, Zhen Zhang, Zhiyuan Liu, and Maosong Sun. 2023a.  OpenDelta: A plug-and-play library for parameter-efficient adaptation of pre-trained models.  In _Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 3: System Demonstrations)_ , pages 274–281, Toronto, Canada.
  * Hu et al. (2023b) Zhiqiang Hu, Yihuai Lan, Lei Wang, Wanyu Xu, Ee-Peng Lim, Roy Ka-Wei Lee, Lidong Bing, and Soujanya Poria. 2023b.  Llm-adapters: An adapter family for parameter-efficient fine-tuning of large language models.  _ArXiv preprint_.
  * Huang et al. (2019) Lifu Huang, Ronan Le Bras, Chandra Bhagavatula, and Yejin Choi. 2019.  Cosmos QA: Machine reading comprehension with contextual commonsense reasoning.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pages 2391–2401, Hong Kong, China.
  * Lei et al. (2023) Tao Lei, Junwen Bai, Siddhartha Brahma, Joshua Ainslie, Kenton Lee, Yanqi Zhou, Nan Du, Vincent Y. Zhao, Yuexin Wu, Bo Li, Yu Zhang, and Ming-Wei Chang. 2023.  Conditional adapters: Parameter-efficient transfer learning with fast inference.  _ArXiv preprint_.
  * Lester et al. (2021) Brian Lester, Rami Al-Rfou, and Noah Constant. 2021.  The power of scale for parameter-efficient prompt tuning.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pages 3045–3059, Online and Punta Cana, Dominican Republic.
  * Lewis et al. (2020) Mike Lewis, Yinhan Liu, Naman Goyal, Marjan Ghazvininejad, Abdelrahman Mohamed, Omer Levy, Veselin Stoyanov, and Luke Zettlemoyer. 2020.  BART: Denoising sequence-to-sequence pre-training for natural language generation, translation, and comprehension.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 7871–7880, Online.
  * Li and Liang (2021) Xiang Lisa Li and Percy Liang. 2021.  Prefix-tuning: Optimizing continuous prompts for 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 4582–4597, Online.
  * Lialin et al. (2023) Vladislav Lialin, Vijeta Deshpande, and Anna Rumshisky. 2023.  Scaling Down to Scale Up: A Guide to Parameter-Efficient Fine-Tuning.  _ArXiv preprint_.
  * Liu et al. (2022) Haokun Liu, Derek Tam, Mohammed Muqeeth, Jay Mohta, Tenghao Huang, Mohit Bansal, and Colin A Raffel. 2022.  Few-shot parameter-efficient fine-tuning is better and cheaper than in-context learning.  In _Advances in Neural Information Processing Systems 35: Annual Conference on Neural Information Processing Systems 2022, NeurIPS 2022, Nov 28 - Dec 9, 2022, virtual and New Orleans, LA, USA_ , pages 1950–1965.
  * Liu et al. (2019) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019.  Ro{bert}a: A robustly optimized {bert} pretraining approach.  _ArXiv preprint_.
  * Mahabadi et al. (2021) Rabeeh Karimi Mahabadi, James Henderson, and Sebastian Ruder. 2021.  Compacter: Efficient low-rank hypercomplex adapter layers.  In _Advances in Neural Information Processing Systems 34: Annual Conference on Neural Information Processing Systems 2021, NeurIPS 2021, December 6-14, 2021, virtual_ , pages 1022–1035.
  * Mangrulkar et al. (2022) Sourab Mangrulkar, Sylvain Gugger, Lysandre Debut, Younes Belkada, and Sayak Paul. 2022.  Peft: State-of-the-art parameter-efficient fine-tuning methods.  https://github.com/huggingface/peft.
  * Mao et al. (2022) Yuning Mao, Lambert Mathias, Rui Hou, Amjad Almahairi, Hao Ma, Jiawei Han, Scott Yih, and Madian Khabsa. 2022.  UniPELT: A unified framework for parameter-efficient language model tuning.  In _Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 6253–6264, Dublin, Ireland.
  * McCloskey and Cohen (1989) Michael McCloskey and Neal J Cohen. 1989.  Catastrophic interference in connectionist networks: The sequential learning problem.  In _Psychology of learning and motivation_ , volume 24, pages 109–165.
  * Narayan et al. (2018) Shashi Narayan, Shay B. Cohen, and Mirella Lapata. 2018.  Don’t give me the details, just the summary! topic-aware convolutional neural networks for extreme summarization.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 1797–1807, Brussels, Belgium.
  * Pfeiffer et al. (2022a) Jonas Pfeiffer, Gregor Geigle, Aishwarya Kamath, Jan-Martin Steitz, Stefan Roth, Ivan Vulić, and Iryna Gurevych. 2022a.  xGQA: Cross-lingual visual question answering.  In _Findings of the Association for Computational Linguistics: ACL 2022_ , pages 2497–2511, Dublin, Ireland.
  * Pfeiffer et al. (2022b) Jonas Pfeiffer, Naman Goyal, Xi Lin, Xian Li, James Cross, Sebastian Riedel, and Mikel Artetxe. 2022b.  Lifting the curse of multilinguality by pre-training modular transformers.  In _Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 3479–3495, Seattle, United States.
  * Pfeiffer et al. (2021a) Jonas Pfeiffer, Aishwarya Kamath, Andreas Rücklé, Kyunghyun Cho, and Iryna Gurevych. 2021a.  AdapterFusion: Non-destructive task composition for transfer learning.  In _Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume_ , pages 487–503, Online.
  * Pfeiffer et al. (2020a) Jonas Pfeiffer, Andreas Rücklé, Clifton Poth, Aishwarya Kamath, Ivan Vulić, Sebastian Ruder, Kyunghyun Cho, and Iryna Gurevych. 2020a.  AdapterHub: A framework for adapting transformers.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_ , pages 46–54, Online.
  * Pfeiffer et al. (2023) Jonas Pfeiffer, Sebastian Ruder, Ivan Vuli’c, and Edoardo M. Ponti. 2023.  Modular Deep Learning.  _ArXiv preprint_.
  * Pfeiffer et al. (2020b) Jonas Pfeiffer, Ivan Vulić, Iryna Gurevych, and Sebastian Ruder. 2020b.  MAD-X: An Adapter-Based Framework for Multi-Task Cross-Lingual Transfer.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 7654–7673, Online.
  * Pfeiffer et al. (2021b) Jonas Pfeiffer, Ivan Vulić, Iryna Gurevych, and Sebastian Ruder. 2021b.  UNKs everywhere: Adapting multilingual language models to new scripts.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pages 10186–10203, Online and Punta Cana, Dominican Republic.
  * Poth et al. (2021) Clifton Poth, Jonas Pfeiffer, Andreas Rücklé, and Iryna Gurevych. 2021.  What to pre-train on? Efficient intermediate task selection.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pages 10585–10605, Online and Punta Cana, Dominican Republic.
  * Radford et al. (2021) Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, Gretchen Krueger, and Ilya Sutskever. 2021.  Learning transferable visual models from natural language supervision.  In _Proceedings of the 38th International Conference on Machine Learning, ICML 2021, 18-24 July 2021, Virtual Event_ , Proceedings of Machine Learning Research, pages 8748–8763.
  * Radford et al. (2019) Alec Radford, Jeff Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. 2019.  Language models are unsupervised multitask learners.  _Technical report_.
  * Raffel et al. (2020) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu. 2020.  Exploring the limits of transfer learning with a unified text-to-text transformer.  _J. Mach. Learn. Res._ , pages 140:1–140:67.
  * Rajpurkar et al. (2018) Pranav Rajpurkar, Robin Jia, and Percy Liang. 2018.  Know what you don’t know: Unanswerable questions for SQuAD.  In _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)_ , pages 784–789, Melbourne, Australia.
  * Rebuffi et al. (2017) Sylvestre-Alvise Rebuffi, Hakan Bilen, and Andrea Vedaldi. 2017.  Learning multiple visual domains with residual adapters.  In _Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA_ , pages 506–516.
  * Rücklé et al. (2021) Andreas Rücklé, Gregor Geigle, Max Glockner, Tilman Beck, Jonas Pfeiffer, Nils Reimers, and Iryna Gurevych. 2021.  AdapterDrop: On the efficiency of adapters in transformers.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pages 7930–7946, Online and Punta Cana, Dominican Republic.
  * Sabry and Belz (2023) Mohammed Sabry and Anya Belz. 2023.  Peft-ref: A modular reference architecture and typology for parameter-efficient finetuning techniques.  _ArXiv preprint_.
  * Tjong Kim Sang (2002) Erik F. Tjong Kim Sang. 2002.  Introduction to the CoNLL-2002 shared task: Language-independent named entity recognition.  In _COLING-02: The 6th Conference on Natural Language Learning 2002 (CoNLL-2002)_.
  * Vu et al. (2022) Tu Vu, Brian Lester, Noah Constant, Rami Al-Rfou’, and Daniel Cer. 2022.  SPoT: Better frozen model adaptation through soft prompt transfer.  In _Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 5039–5059, Dublin, Ireland.
  * Wang et al. (2019) Alex Wang, Amanpreet Singh, Julian Michael, Felix Hill, Omer Levy, and Samuel R. Bowman. 2019.  GLUE: A multi-task benchmark and analysis platform for natural language understanding.  In _7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019_.
  * Wang et al. (2021) Xinyi Wang, Yulia Tsvetkov, Sebastian Ruder, and Graham Neubig. 2021.  Efficient test time adapter ensembling for low-resource language varieties.  In _Findings of the Association for Computational Linguistics: EMNLP 2021_ , pages 730–737, Punta Cana, Dominican Republic.
  * Wang et al. (2022) Yaqing Wang, Sahaj Agarwal, Subhabrata Mukherjee, Xiaodong Liu, Jing Gao, Ahmed Hassan Awadallah, and Jianfeng Gao. 2022.  AdaMix: Mixture-of-adaptations for parameter-efficient model tuning.  In _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , pages 5744–5760, Abu Dhabi, United Arab Emirates.
  * Wolf et al. (2020) Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Remi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander Rush. 2020.  Transformers: State-of-the-art natural language processing.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_ , pages 38–45, Online.
  * Zhou et al. (2023) Han Zhou, Xingchen Wan, Ivan Vulić, and Anna Korhonen. 2023.  AutoPEFT: Automatic configuration search for parameter-efficient fine-tuning.  _ArXiv preprint_.

##  Appendix A Description of Adapter Methods

Bottleneck Adapters introduce bottleneck modules in each layer of a
Transformer model. Generally, these adapter modules consist of a down-
projection matrix $W_{down}$ that projects into a lower dimension
$d_{bottleneck}$, a non-linearity $f$, an up-projection $W_{up}$ that projects
back into the original hidden layer dimension and a residual connection $r$,
i.e.: $h\leftarrow W_{up}\cdot f(W_{down}\cdot h)+r$.

Depending on the specific configuration, these layers can be introduced at
different locations and in sequential or parallel order relative to the
adapted Transformer layer. Adapters provides pre-defined configurations for
the sequential configurations of Houlsby et al. (2019) and Pfeiffer et al.
(2021a) as well as the parallel configuration of He et al. (2022).

Invertible Adapters were proposed as part of MAD-X (Pfeiffer et al., 2020b) to
learn language-specific transformations. Embedding outputs are passed through
an invertible adapter module in the forward direction before entering the
first Transformer layer and in the inverse direction after leaving the last
Transformer layer.

Prompt Tuning (Lester et al., 2021) is an approach to condition language
models on a task-specific soft prompt. While hard prompts consist of fixed
textual descriptions prepended to the model’s input (Brown et al., 2020),
these soft prompts are continuously optimized towards the target task via
gradient descent. Prompt tokens are prepended to the embedded input sequence.

Prefix Tuning (Li and Liang, 2021) is an extension of prompt tuning which
prepends trainable prefix vectors $P^{K}$ and $P^{V}$ to the keys and values
of the multi-head attention block inputs. The prefix vectors have a
configurable length and are not optimized directly but reparameterized via a
bottleneck MLP in the built-in default configuration, following the original
implementation.

Compacter (Mahabadi et al., 2021) exchanges the linear down- and up-projection
of a bottleneck adapter for a PHM layer555Parametrized hypercomplex
multiplication layer.. This PHM layer constructs its weight matrix from two
smaller matrices, which reduces the number of parameters needed for the
adapters. These matrices can be factorized and shared between all adapter
layers. Adapters provides pre-defined configurations for the Compacter and
Compacter++ variants.

LoRA (Hu et al., 2022) injects trainable low-rank decomposition matrices into
the layers of a pre-trained model. For any model layer expressed as a matrix
multiplication of the form $h=W_{0}x$, it performs a reparameterization such
that: $h=W_{0}x+\frac{\alpha}{r}BAx$. Here, $A\in\mathbb{R}^{r\times k}$ and
$B\in\mathbb{R}^{d\times r}$ are the decomposition matrices and $r$ is the
low-dimensional rank of the decomposition. With Adapters, LoRA modules can be
configured to be placed into the self-attention, intermediate, or output
components of a Transformer layer. Following Hu et al. (2022), Adapters
provides a built-in method of merging LoRA modules with the original pre-
trained weights of a model for inference without additional latency.

(IA)³ (Liu et al., 2022) introduces trainable vectors $l_{W}$ into different
components of a Transformer model, which perform element-wise rescaling of
inner model activations. For any model layer expressed as a matrix
multiplication of the form $h=Wx$, it therefore performs an element-wise
multiplication with $l_{W}$, such that: $h=l_{W}\odot Wx$.

Method | Sequence Classification | Regression | Extract. QA | Seq2Seq | Avg.
---|---|---|---|---|---
CoLA | MRPC | QNLI | RTE | SST-2 | STS-B | SQuAD v2 | XSum
Dev MCC | Dev F1 | Dev Acc. | Dev Acc. | Dev Acc. | Dev PCC | Dev F1 | Rouge 2
double_seq_bn | 61.80 | 91.62 | 92.18 | 73.65 | 94.04 | 89.89 | 79.37 | 17.49 | 75.00
($\pm 21.47$) | ($\pm$3.73) | ($\pm$15.35) | ($\pm$8.26) | ($\pm$13.89) | ($\pm$23.18) | ($\pm$12.17) | ($\pm$1.51) | ($\pm$10.86)
seq_bn | 53.29 | 91.16 | 92.11 | 75.45 | 93.46 | 89.44 | 79.47 | 17.76 | 74.02
($\pm 18.82$) | ($\pm$3.59) | ($\pm$6.29) | ($\pm$6.84) | ($\pm$1.09) | ($\pm$20.38) | ($\pm$10.41) | ($\pm$1.60) | ($\pm$7.55)
par_bn | 55.58 | 90.75 | 92.12 | 75.09 | 94.08 | 89.58 | 79.63 | 18.26 | 74.39
($\pm$19.70) | ($\pm$3.46) | ($\pm$14.42) | ($\pm$7.91) | ($\pm$14.39) | ($\pm$10.86) | ($\pm$8.19) | ($\pm$ 1.19) | ($\pm$10.01)
compacter | 44.81 | 87.44 | 91.45 | 69.68 | 93.35 | 87.62 | 74.33 | 13.55 | 70.28
($\pm$12.06) | ($\pm$4.30) | ($\pm$3.14) | ($\pm$3.85) | ($\pm$1.18) | ($\pm$14.09) | ($\pm$9.81) | ($\pm$5.29) | ($\pm$6.72)
prefix_tuning | 48.57 | 92.31 | 91.61 | 75.95 | 93.58 | 90.27 | 80.45 | 16.81 | 73.69
($\pm$8.23) | ($\pm$3.65) | ($\pm$6.97) | ($\pm$5.63) | ($\pm$1.92) | ($\pm$13.72) | ($\pm$8.73) | ($\pm$4.94) | ($\pm$6.73)
lora | 55.94 | 91.61 | 92.53 | 76.05 | 94.03 | 89.57 | 78.81 | 17.21 | 74.47
($\pm$21.93) | ($\pm$4.17) | ($\pm$4.31) | ($\pm$8.67) | ($\pm$1.31) | ($\pm$33.03) | ($\pm$10.47) | ($\pm$1.13) | ($\pm$10.63)
ia3 | 50.65 | 89.68 | 90.80 | 72.92 | 94.22 | 89.05 | 72.46 | 13.51 | 71.66
($\pm$13.64) | ($\pm$2.79) | ($\pm$7.29) | ($\pm$5.83) | ($\pm$6.04) | ($\pm$21.27) | ($\pm$8.46) | ($\pm$4.08) | ($\pm$8.68)
Full Fine-tuning | 62.80 | 90.4 | 94.9 | 87.0 | 96.6 | 91.2 | 89.2 | 17.73 | 86.84

Table 4: Best performance ($\pm$ std. dev. across all hyper-parameters) of
various supported single adapter methods (refer §3.3.1) applied to
facebook/bart-base, benchmarked against full fine-tuning. The best results and
lowest std. dev. per task are highlighted in bold.

PEFT Method | Attribute Name | Range | Added Params.
---|---|---|---
|  | Min | Max
double_seq_bn | reduction_factor | $\\{2,16,64\\}$ | 461,088 | 14,183,424
seq_bn | reduction_factor | $\\{2,16,64\\}$ | 230,544 | 7,091,712
par_bn | reduction_factor | $\\{2,16,64\\}$ | 230,544 | 7,091,712
compacter | reduction_factor | $\\{4,16\\}$ | 58,816 | 69,184
phm_dim | $\\{4,8\\}$
prefix_tuning | bottleneck_size | $\\{32,128,512\\}$ | 636,704 | 10,002,944
prefix_length | $\\{5,50,200\\}$
lora | r | $\\{4,8,16,64,200\\}$ | 147,456 | 7,372,800
ia3 | $-$ | $-$ | 55,296 | 55,296

Table 5: Adapter-specific attributes (name as in framework) and their values
that we used for grid-search along with the minimum and maximum possible
parameters added over the resultant grid. Extreme values are highlighted in
bold.

##  Appendix B Adapter Evaluation Results

Best task performance for all task–adapter method combinations are presented
in Table 3 with RoBERTa as base and model and in Table 4 with BART as base
model. All adapter methods provided by Adapters are competitive to full fine-
tuning. Table 5 gives an overview of the evaluated adapter configurations.
Fig. 4 plots best performing learning rate–capacity combinations for evaluated
single adapter methods. Lower capacity adapters perform better with higher
learning rates.

Fig. 5 plots mean task performance by learning rate for evaluated single
adapter methods. For bottleneck adapters, the best-performing learning rate is
$10^{-4}$. Compacter performs best with a learning rate of $10^{-3}$, prefix
tuning with $3\cdot 10^{-4}$, LoRA with $5\cdot 10^{-4}$, and (IA)³ with
$5\cdot 10^{-3}$. These results align with the learning rates proposed by the
respective adapter method authors.

Figure 4: Best performing learning rate–capacity combinations for evaluated
single adapter methods. Lower capacity adapters perform better with higher
learning rates. Figure 5: Mean task performance by learning rate for evaluated
single adapter methods.


# Unveiling the $a_{0}(1710)$ nature in the process
$J/\psi\to{\bar{K}}^{0}K^{+}\rho^{-}$

Yan Ding  School of Physics and Microelectronics, Zhengzhou University,
Zhengzhou, Henan 450001, China  En Wang<EMAIL_ADDRESS>School of Physics
and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China
Guangxi Key Laboratory of Nuclear Physics and Nuclear Technology, Guangxi
Normal University, Guilin 541004, China  De-Min Li<EMAIL_ADDRESS>School of
Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001,
China  Li-Sheng Geng<EMAIL_ADDRESS>School of Physics, Beihang
University, Beijing 102206, China  Beijing Key Laboratory of Advanced Nuclear
Materials and Physics, Beihang University, Beijing, 102206, China  Peng Huanwu
Collaborative Center for Research and Education, Beihang University, Beijing
100191, China  Southern Center for Nuclear-Science Theory (SCNT), Institute of
Modern Physics, Chinese Academy of Sciences, Huizhou 516000, Guangdong
Province, China  Ju-Jun Xie<EMAIL_ADDRESS>Southern Center for
Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of
Sciences, Huizhou 516000, Guangdong Province, China  Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 730000, China  School of Nuclear
Sciences and Technology, University of Chinese Academy of Sciences, Beijing
101408, China

###### Abstract

We have investigated the process $J/\psi\to{\bar{K}}^{0}K^{+}\rho^{-}$ by
taking into account the $S$-wave ${K^{*}\bar{K}^{*}}$, $\rho\omega$, and
$\rho\phi$ final state interactions within the unitary coupled-channel
approach, where the scalar meson $a_{0}(1710)$ is dynamically generated. In
addition, the contributions from the intermediate resonances
$K_{1}(1270)^{-}(\to{\bar{K}}^{0}\rho^{-})$ and $K_{1}(1270)^{0}(\to
K^{+}\rho^{-})$ are also considered. We find a significant peak structure
around 1.8 GeV, associated to the $a_{0}(1710)$, in the ${{\bar{K}}^{0}K^{+}}$
invariant mass distribution, and the clear peaks of the $K_{1}(1270)$ in the
${\bar{K}}^{0}\rho^{-}$ and $K^{+}\rho^{-}$ invariant mass distributions. A
precise measurement of this process by the BESIII and BelleII Collaborations
and at the planed super tau charm facility in the future could shed light on
the nature of $a_{0}(1710)$.

##  I Introduction

In the last two decades, it has been accumulated experimentally many states
whose properties can not be well described by the $q\bar{q}$ mesons and the
$qqq$ baryons within the conventional quark model. Some exotic explanations
are proposed for their nature, such as tetraquark, pentaquark, hybrid,
glueball, kinematic effects, and the mixing of different components. It is
difficult to distinguish between those explanations, especially for states
with the quantum numbers the conventional quark model allows.

Recently, the $BABAR$ Collaboration reported the scalar resonance
$a_{0}(1710)$ in the $\pi^{\pm}\eta$ invariant mass spectrum of the process
$\eta_{c}\to\eta\pi^{+}\pi^{-}$ BaBar:2021fkz . The $a_{0}(1710)$ state was
also observed by the BESIII Collaboration in the $K_{S}^{0}K_{S}^{0}$
invariant mass spectrum of the process $D_{s}^{+}\to
K_{S}^{0}K_{S}^{0}\pi^{+}$ BESIII:2021anf  and in the $K_{S}^{0}K^{+}$
invariant mass spectrum of the process $D_{s}^{+}\to K_{S}^{0}K^{+}\pi^{0}$
BESIII:2022npc . We have tabulated the experimental masses and widths of
$a_{0}(1710)$ in Table 1. It should be noted that, in Ref. BESIII:2021anf ,
BESIII did not distinguish between the $a_{0}(1710)$ and $f_{0}(1710)$ in the
process $D_{s}^{+}\to K_{S}^{0}K_{S}^{0}\pi^{+}$, and denoted the combined
state as $S(1710)$, while in Ref. BESIII:2022npc  the $a_{0}(1710)$ was
renamed as $a_{0}(1817)$ because of the different fitted Breit-Wigner mass of
this state.

Table 1:  Experimental measurements on the mass [$M_{a_{0}(1710)}$] and width [$\Gamma_{a_{0}(1710)}$] of the scalar state $a_{0}(1710)$. The first error is statistical, and the second one is systematic. All values are in units of MeV. Collaboration | $M_{a_{0}(1710)}$ | $\Gamma_{a_{0}(1710)}$ | Reference
---|---|---|---
BABAR | $1704\pm 5\pm 2$ |  $110\pm 15\pm 11$ |  BaBar:2021fkz
BESIII | $1723\pm 11\pm 2$ |  $140\pm 14\pm 4$ |  BESIII:2021anf
BESIII | $1817\pm 8\pm 20$ |  $97\pm 22\pm 15$ |  BESIII:2022npc

Before the observation of $a_{0}(1710)$, there have been many theoretical
studies about the $a_{0}(1710)$ and its isospin partner $f_{0}(1710)$ from
various perspectives Nagahiro:2008bn ; Branz:2009cv ; Geng:2010kma ;
Xie:2014gla ; MartinezTorres:2012du ; Wang:2021jub ; Garcia-Recio:2010enl ;
Garcia-Recio:2013uva ; Close:2005vf ; Gui:2012gx ; Janowski:2014ppa ;
Fariborz:2015dou ; Chen:2005mg . In Refs. Geng:2008gx ; Du:2018gyn , the
$f_{0}(1710)$, as a well-established state according to the Review of Particle
Physics (RPP) ParticleDataGroup:2022pth , could be dynamically generated from
the vector-vector interactions, and one isovector scalar state $a_{0}$ with a
mass around 1770 MeV was also predicted, the picture of which remains
essentially the same when the pseudoscalar-pseudoscalar coupled-channels were
taken into account Wang:2022pin . Based on the SU(6) spin-flavor symmetry, the
$f_{0}(1710)$ is mostly a $K^{*}\bar{K}^{*}$ bound state, and a $a_{0}$ state
with a pole position of $\sqrt{s_{R}}=(1760,-12)$ was predicted to couple
strongly to $K^{*}\bar{K}^{*}$ and $\phi\rho$ in Ref. Garcia-Recio:2010enl .
In addition, a scalar state $a_{0}$ with a mass of 1744 MeV is also predicted
in the approach of the Regge trajectories Wang:2017pxm . In Ref. Close:2005vf
, it was suggested that the $f_{0}(1710)$ wave function contains a large
$s\bar{s}$ component, while in Refs. Gui:2012gx ; Janowski:2014ppa ;
Fariborz:2015dou ; Chen:2005mg , the $f_{0}(1710)$ was regarded as the
candidate of a scalar glueball. Recently, it was shown that the $a_{0}(1710)$
as a $K^{*}\bar{K}^{*}$ molecule plays an important role in the three-body
interactions of $\eta K^{*}\bar{K}^{*}$, which could dynamically generate the
$\pi(2070)$ Shen:2023uus .

As shown in Table 1, there is not yet a consensus on the mass of the
$a_{0}(1710)$ experimentally, which could complicate the understanding of its
nature. For instance, $a_{0}(1710)$ [or $a_{0}(1817)$] and $X(1812)$ have been
explained as the $3^{3}P_{0}$ $q\bar{q}$ state by assuming $a_{0}(980)$ and
$f_{0}(980)$ as $1^{3}P_{0}$ $q\bar{q}$ states Guo:2022xqu . However,
$X(1812)$ was observed in the process $J/\psi\to\gamma\phi\omega$ by the
BESIII Collaboration BES:2006vdb ; BESIII:2012rtd , and the enhancement near
the $\phi\omega$ threshold, associated to $X(1812)$, could be described by the
reflection of $f_{0}(1710)$, as discussed in Ref. MartinezTorres:2012du .
Regarding the $a_{0}(1710)$ as a $K^{*}\bar{K}^{*}$ molecular state, Refs.
Zhu:2022wzk ; Zhu:2022guw ; Dai:2021owu ; Oset:2023hyt ; Wang:2023aza  have
successfully described the invariant mass distributions of the processes
$D_{s}^{+}\to K_{S}^{0}K_{S}^{0}\pi^{+}$ and $D_{s}^{+}\to
K_{S}^{0}K^{+}\pi^{0}$ measured by the BESIII Collaboration BESIII:2021anf ;
BESIII:2022npc .

Since the peak positions of the $a_{0}(1710)$ in the $K\bar{K}$ invariant mass
distributions of the processes $D_{s}^{+}\to
K_{S}^{0}K_{S}^{0}\pi^{+},K_{S}^{0}K^{+}\pi^{0}$ observed by the BESIII
Collaboration are very close to the boundary region of the $K\bar{K}$
invariant mass, we have suggested to measure its properties in the process
$\eta_{c}\to\bar{K}^{0}K^{+}\pi^{-}$ in Ref. Ding:2023eps , and predicted a
dip structure around 1.8 GeV, associated with the $a_{0}(1710)$, in the
$\bar{K}^{0}K^{+}$ invariant mass distribution Ding:2023eps , which is
consistent with the $BABAR$ measurements BaBar:2015kii . In addition, the
photo-production process is also proposed to search for the $a_{0}(1710)$ in
Ref. Wang:2023lia .

The BESIII Collaboration has accumulated $(10.09\pm 0.04)\times 10^{9}$
$J/\psi$ events at the BEPCII collider BESIII:2021cxx , and the super tau
charm facility (STCF) project under development in China is expected to
accumulate $3.4\times 10^{12}$ $J/\psi$ events per year Achasov:2023gey .
Since the dominant decay mode of the $a_{0}(1710)$ state is $K\bar{K}$ within
the molecular picture Geng:2008gx ; Wang:2022pin , it is natural to search for
the scalar $a_{0}(1710)$ in the process $J/\psi\to
a_{0}(1710)^{+}\rho^{-}\to\bar{K}^{0}K^{+}\rho^{-}$. It should be stressed
that this process has been measured by the $BABAR$ Collaboration, and the
branching fraction is $\mathcal{B}(J/\psi\to
K_{S}^{0}K^{\pm}\rho^{\mp})=(1.87\pm 0.18\pm 0.34)\times 10^{-3}$
BaBar:2017pkz . However, the $K_{S}^{0}K^{\pm}$ mass spectrum was not reported
by the $BABAR$ Collaboration BaBar:2017pkz .

In this work, we investigate the process $J/\psi\to
a_{0}(1710)^{+}\rho^{-}\to\bar{K}^{0}K^{+}\rho^{-}$, by regarding the
$a_{0}(1710)$ as a dynamically generated state from the vector-vector
interactions in coupled channels. On the other hand, the interactions of
vector meson-pseudoscalar meson within the unitary approach could dynamically
generate the resonance $K_{1}(1270)$ with a two-pole structure Geng:2006yb ;
Wang:2020pyy ; Wang:2019mph , where the lower pole mainly couples to the
$K^{*}\pi$ channel, and the higher one couples strongly to the $K\rho$
channel. Thus, we also take into account the contribution from the
intermediate resonance $K_{1}(1270)$ in this work, and precise measurements of
the process $J/\psi\to\bar{K}^{0}K^{+}\rho^{-}$ could also shed light on the
two-pole structure of $K_{1}(1270)$, which is crucial for us to understand the
hadron-hadron interactions Xie:2023jve ; Xie:2023cej .

The paper is organized as follows. In Sec. II, we present the theoretical
formalism for studying the $J/\psi\to\bar{K}^{0}K^{+}\rho^{-}$ decay, and in
Sec. III, we show our numerical results and discussions, followed by a summary
in the last section.

##  II Formalism

First, we present the theoretical formalism for studying the process
$J/\psi\to\bar{K}^{0}K^{+}\rho^{-}$ via the $K^{*}\bar{K}^{*}$, $\omega\rho$,
and $\phi\rho$ final state interactions in coupled channels, which dynamically
generate the scalar resonance $a_{0}(1710)$ in Subsect. II.1. Next, we show
the formalism for the process $J/\psi\to
K_{1}(1270)^{-}K^{+}[K_{1}(1270)^{0}\bar{K}^{0}]$ with
$K_{1}(1270)^{-}\to\bar{K}^{0}\rho^{-}$ [$K_{1}(1270)^{0}\to K^{+}\rho^{-}$]
in Subsect. II.2. At last, the formalism of the double differential widths for
this process is given in Subsect. II.3.

###  II.1 Mechanism for the intermediate $a_{0}(1710)$

Figure 1: Diagrammatic expression of $\textless VVV\textgreater$,
$\quad\textless VV\textgreater\textless V\textgreater$, $\quad\textless
V\textgreater\textless V\textgreater\textless V\textgreater$ terms in Eq. (1).

As done in Refs. Zhu:2022guw ; Zhu:2022wzk ; Ding:2023eps , the scalar meson
$a_{0}(1710)$ is regarded as a vector-vector molecular state Geng:2008gx ;
Du:2018gyn . To study the role of $a_{0}(1710)$ in the
$J/\psi\to\rho^{-}K^{+}\bar{K}^{0}$ decay, one needs to firstly produce the
meson $\rho^{-}$ and a vector-vector pair, then the final state interactions
of the vector-vector pair will produce the $a_{0}(1710)$, which decays into
$K^{+}\bar{K}^{0}$ in the final state. Considering that the $J/\psi$ is a
flavor singlet, we could introduce the combination modes in the primary vertex
Ikeno:2019grj ; Jiang:2019ijx ; Sakai:2019uig , whose diagrammatic expression
is depicted in Fig. 1,

$\displaystyle\textless VVV\textgreater,\quad\textless VV\textgreater\textless
V\textgreater,\quad\textless V\textgreater\textless V\textgreater\textless
V\textgreater,$ (1)

where $V$ is the matrix of the SU(3) vector mesons Ikeno:2019grj ;
Jiang:2019ijx ; Sakai:2019uig ; Zhang:2023nnn ; Wang:2021naf ; Duan:2021pll ,

$\displaystyle
V=\left(\begin{matrix}\frac{{\rho}^{0}}{\sqrt{2}}+\frac{{\omega}}{\sqrt{2}}&\rho^{+}&K^{*+}\\\
\rho^{-}&-\frac{{\rho}^{0}}{\sqrt{2}}+\frac{{\omega}}{\sqrt{2}}&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\end{matrix}\right),$ (2)

with the symbol ‘<…>’ stands for the trace of the SU(3) matrices. Since no
term contains $\rho^{-}$ in $\textless V\textgreater\textless
V\textgreater\textless V\textgreater$, we do not take this combination in our
work. One could obtain the relevant contributions by isolating the terms
containing $\rho^{-}$, as follows,

$\displaystyle\textless
VVV\textgreater:\alpha\times\left[\frac{\rho^{+}\rho^{0}}{\sqrt{2}}+3\sqrt{2}\omega\rho^{+}+3\bar{K}^{*0}K^{\ast+}\right]\rho^{-},$
(3) $\displaystyle\textless VV\textgreater\textless
V\textgreater:\beta\times\left[2\sqrt{2}\omega\rho^{+}+2\phi\rho^{+}\right]\rho^{-}.$
(4)

Here, two parameters $\alpha$ and $\beta$ are introduced to account for the
weights of $\textless VVV\textgreater$ and $\textless VV\textgreater\textless
V\textgreater$ structures, respectively. It is concluded in Refs.
Abreu:2023xvw ; Geng:2008gx ; Molina:2019wjj  that the $\textless
VVV\textgreater$ one was favored and the best ratio $\beta/\alpha=0.32$ was
obtained by fitting to the experimental measurements of the process
$J/\psi\to\phi VV$ Geng:2008gx ; Molina:2019wjj , thus we will use this
finding and take $\alpha=1$ and $\beta=0.32$ in this work.

Figure 2: Diagram for the process $J/\psi\to VV\rho^{-}\to
a_{0}(1710)^{+}\rho^{-}\to K^{+}\bar{K}^{0}\rho^{-}$ with $VV$ stands for
$\bar{K}^{*0}K^{\ast+}$, $\omega\rho^{+}$ and $\phi\rho^{+}$.

In the molecular picture, the $a_{0}(1710)$ is dynamically generated from the
$S$-wave $K^{*}\bar{K}^{*}$, $\omega\rho$, and $\phi\rho$ interactions in
coupled channels Geng:2008gx ; Du:2018gyn  111It should be noted that, the
conservation of $G$-parity forbids the coupling of $a_{0}(1710)$ to the
channel $\rho\rho$., and then decays into the final state $\bar{K}^{0}K^{+}$,
as depicted in Fig. 2. The decay amplitude of Fig. 2 can be written as,

$\displaystyle\mathcal{M}_{a}$ $\displaystyle=$ $\displaystyle
V_{p}\times\left[3\alpha
G_{\bar{K}^{*0}K^{\ast+}}t_{\bar{K}^{*0}K^{\ast+}\to\bar{K}^{0}K^{+}}\right.$
(5)
$\displaystyle\left.+(2\sqrt{2}\beta+3\sqrt{2}\alpha)G_{\omega\rho^{+}}t_{\omega\rho^{+}\to\bar{K}^{0}K^{+}}\right.$
$\displaystyle\left.+2\beta\phi\rho^{+}G_{\phi\rho^{+}}t_{\phi\rho^{+}\to\bar{K}^{0}K^{+}}\right],$

where $V_{p}$ is a global factor, and
$t_{\bar{K}^{*0}K^{\ast+}\to\bar{K}^{0}K^{+}}$,
$t_{\omega\rho^{+}\to\bar{K}^{0}K^{+}}$, and
$t_{\phi\rho^{+}\to\bar{K}^{0}K^{+}}$ are the transition amplitudes.
${G}_{\bar{K}^{*0}K^{\ast+}}$, ${G}_{\omega\rho^{+}}$, and $G_{\phi\rho^{+}}$
are the loop functions for the $\bar{K}^{*0}{K}^{*+}$, $\omega\rho^{+}$, and
$\phi\rho^{+}$ channels, respectively, and read Geng:2008gx ; Molina:2008jw ,

$\displaystyle{G}_{i}(M_{\bar{K}^{0}K^{+}})$ $\displaystyle=$
$\displaystyle\int_{m_{1-}^{2}}^{m_{1+}^{2}}\int_{m_{2-}^{2}}^{m_{2+}^{2}}d\tilde{m}_{1}^{2}d\tilde{m}_{2}^{2}\times$
(6)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\omega(\tilde{m}_{1}^{2})\omega(\tilde{m}_{2}^{2})\tilde{G}(M_{\bar{K}^{0}K^{+}},\tilde{m}_{1}^{2},\tilde{m}_{2}^{2}),$

where

$\displaystyle\omega(\tilde{m}_{i}^{2})={\frac{1}{N}}\left(-\frac{1}{\pi}\right)\text{Im}\left[\frac{1}{\tilde{m}_{i}^{2}-m_{V_{i}}^{2}+i\Gamma(\tilde{m}_{i}^{2})\tilde{m}_{i}}\right],$
(7) $\displaystyle
N=\int_{\tilde{m}_{i-}^{2}}^{\tilde{m}_{i+}^{2}}d\tilde{m}_{i}^{2}\left(-\frac{1}{\pi}\right)\text{Im}\left[\frac{1}{\tilde{m}_{i}^{2}-m_{V_{i}}^{2}+i\Gamma(\tilde{m}_{i}^{2})\tilde{m}_{i}}\right],$
(8)
$\displaystyle\Gamma(\tilde{m}_{i}^{2})=\Gamma_{V_{i}}\frac{\tilde{k}^{3}}{k^{3}}\Theta(\tilde{m}-m_{P_{1}}-m_{P_{2}}),$
(9)
$\displaystyle\tilde{k}=\frac{\lambda^{\frac{1}{2}}(\tilde{m}_{i}^{2},m_{P_{1}}^{2},m_{P_{2}}^{2})}{2\tilde{m}_{i}},~{}k=\frac{\lambda^{\frac{1}{2}}(m_{V_{i}}^{2},m_{P_{1}}^{2},m_{P_{2}}^{2})}{2m_{V_{i}}},$
(10)

with the K${\ddot{a}}$llen function
$\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2xz-2yz$. Here, we consider the decay
channels $\pi\pi$ and $K\pi$ for the vector mesons $\rho$ and $K^{*}$,
respectively, and neglect the small widths of $\omega$ ($\Gamma_{\omega}=8.68$
MeV) and $\phi$ ($\Gamma_{\phi}=4.249$ MeV). Taking the vector $K^{\ast}$ for
example, $m_{1+}^{2}=\left(m_{K^{\ast}}+2\Gamma_{K^{\ast}}\right)^{2}$ and
$m_{1-}^{2}=\left(m_{K^{\ast}}-2\Gamma_{K^{\ast}}\right)^{2}$. Similarly, one
can obtain $m_{1+}^{2}$ and $m_{1-}^{2}$ for the $\rho$ meson. The masses,
widths, and spin-parities of the involved particles are taken from the RPP
ParticleDataGroup:2022pth , as listed in Table 2.

Table 2: Masses, widths, and spin-parities of the involved particles in this work. All values are in units of MeV. state | mass | width | spin-parity ($J^{P}$)
---|---|---|---
$J/\psi$ | 3096.900 | 0.0926 | $1^{-}$
$\rho^{\pm,0}$ | 775.26 | 149.1 | $1^{-}$
$\bar{K}^{0}$ | 497.611 | — | $0^{-}$
$K^{\pm}$ | 493.677 | — | $0^{-}$
$K^{*}$ | 893.6 | 49.1 | $1^{-}$
$\omega$ | 782.66 | 8.68 | $1^{-}$
$\phi$ | 1019.461 | 4.249 | $1^{-}$
$K_{1}(1270)$ | 1284 | 146 | $1^{+}$

The loop function $\tilde{G}$ in Eq. (6) is for stable particles, and in the
dimensional regularization scheme, it can be written as Lyu:2023ppb ,

$\displaystyle\tilde{G}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\Bigg{\\{}a_{\mu}+\text{In}\frac{m_{1}^{2}}{\mu^{2}}+\frac{m_{2}^{2}-m_{1}^{2}+s}{2s}\text{In}\frac{m_{2}^{2}}{m_{1}^{2}}$
(11)
$\displaystyle\frac{p}{\sqrt{s}}\bigg{[}\text{In}\left(s-\left(m_{2}^{2}-m_{1}^{2}\right)+2p\sqrt{s}\right)$
$\displaystyle+\text{In}\left(s+\left(m_{2}^{2}-m_{1}^{2}\right)+2p\sqrt{s}\right)$
$\displaystyle-\text{In}\left(-s+\left(m_{2}^{2}-m_{1}^{2}\right)+2p\sqrt{s}\right)$
$\displaystyle-\text{In}\left(-s-\left(m_{2}^{2}-m_{1}^{2}\right)+2p\sqrt{s}\right)\bigg{]}\Bigg{\\}},$

with

$\displaystyle p=\frac{\lambda^{1/2}(s,m_{1}^{2},m_{2}^{2})}{2\sqrt{s}},$ (12)

where $a_{\mu}$ is the subtraction constant, $\mu$ is the dimensional
regularization scale, and $s=M^{2}_{\bar{K}^{0}K^{+}}$. In this work, we take
$a_{\mu}=-1.726$ and $\mu=1000$ MeV as used in Ref. Geng:2008gx . It is worth
mentioning that any change in $\mu$ could be reabsorbed by a change in
$a_{\mu}$ through $a_{\mu^{\prime}}-a_{\mu}={\rm ln}(\mu^{\prime 2}/\mu^{2})$,
which implies that the loop function $\tilde{G}$ is scale independent
Duan:2022upr .

To show the influence of the widths of vector mesons on the loop functions, we
have calculated the loop functions $G_{\phi\rho^{+}}$ and
$\tilde{G}_{\phi\rho^{+}}$ as functions of the $\bar{K}^{0}K^{+}$ invariant
mass 222One can find the invariant mass distributions of the loop functions
$\tilde{G}_{\bar{K}^{*0}K^{\ast+}}$ and $\tilde{G}_{\omega\rho}$ in Refs.
Zhu:2022wzk ; Ding:2023eps ., as presented in Fig. 3. The blue long-dashed and
red dot-dashed curves correspond to the real and imaginary parts of the loop
function $G$, considering the width of $\rho$. In contrast, the green solid
and purple dotted curves correspond to the real and imaginary parts of the
loop function $\tilde{G}$ without the contribution from the $\rho$ width,
respectively. One can find that the loop functions $G$, considering the width
of the vector meson, become smoother around the threshold of the $\phi\rho$.

Figure 3: Real and imaginary parts of the loop functions $G_{\phi\rho}$ and $\tilde{G}_{\phi\rho}$ as a function of the $\bar{K}^{0}K^{+}$ invariant mass. The blue long-dashed and red dot-dashed curves correspond to the real and imaginary parts of the loop function $G$, considering the width of $\rho$. Meanwhile, the green solid and purple dotted curves correspond to the real and imaginary parts of the loop function $\tilde{G}$ without the contribution from the $\rho$ width, respectively. Table 3: Mass, width, and coupling constants of the scalar $a_{0}(1710)$ Geng:2008gx . All values are in units of MeV. parameters | value
---|---
mass | 1777
width | 148
$\Gamma_{K\bar{K}}$ | 36
$g_{K\bar{K}}$ | 1966
$g_{\rho\rho}$ | 0
$g_{K^{*}{\bar{K}}^{*}}$ | (7525, -i1529)
$g_{\omega\rho}$ | (-4042, i1391)
$g_{\phi\rho}$ | (4998, -i1872)
$g_{K\rho}$ | (4804, i395)

On the other hand, the transition amplitudes $t_{i\to\bar{K}^{0}K^{+}}$ in Eq.
(5) can be written as,

$\displaystyle t_{i\to\bar{K}^{0}K^{+}}={\frac{g_{i}\times
g_{K\bar{K}}}{M_{\bar{K}^{0}K^{+}}^{2}-M_{a_{0}}^{2}+iM_{a_{0}}\Gamma_{a_{0}}}},$
(13)

where $M_{a_{0}}$ and $\Gamma_{a_{0}}$ are the mass and width of the
$a_{0}(1710)$, respectively, and we take their values from Refs. Geng:2008gx ;
Geng:2009gb , which are tabulated in Table 3. $g_{i}$ are the coupling
constants of $a_{0}(1710)$ to $K^{*}{\bar{K}}^{*}$, $\omega\rho$, and
$\phi\rho^{+}$, whose values are determined in Ref. Geng:2008gx , while the
coupling $g_{K\bar{K}}$ is determined from the partial decay width of
$a_{0}(1710)\to K\bar{K}$,

$\displaystyle\Gamma_{K\bar{K}}=\frac{g^{2}_{K\bar{K}}}{8\pi}\frac{|{\vec{p}}_{K}|}{M^{2}_{a_{0}}},$
(14)

where ${\vec{p}}_{K}$ is the three-momentum of the $K$ or $\bar{K}$ meson in
the $a_{0}(1710)$ rest frame,

$\displaystyle|{\vec{p}}_{K}|=\frac{\lambda^{1/2}(M_{a_{0}}^{2},m_{\bar{K}}^{2},m_{K}^{2})}{2M_{a_{0}}}.$
(15)

With the partial decay width $\Gamma_{K\bar{K}}=36$ MeV predicted by Ref.
Geng:2008gx , one can only obtain the absolute value of the coupling constant,
but not the phase, thus by assuming that $g_{K\bar{K}}$ is real and positive
we take $g_{K\bar{K}}=1966$ MeV, as done in Refs. Zhu:2022wzk ; Zhu:2022guw .

###  II.2 Mechanism for the intermediate $K_{1}(1270)$

In Fig. 4, we show the Dalitz plot for the process
$J/\psi\rightarrow\bar{K}^{0}K^{+}\rho^{-}$, and one can find that the
contribution from the $K_{1}(1270)$ (the green band) could interfere with the
one from the $a_{0}(1710)$ (the red band) close to the $K\rho$ threshold.
Since the $K_{1}(1270)$ state could be dynamically generated from the
interaction of vector meson-pseudoscalar meson Geng:2006yb ; Wang:2020pyy ;
Wang:2020pyy , the $K^{+}\rho^{-}$ and $\bar{K}^{0}\rho^{-}$ could undergo the
$S$-wave final state interaction, which will generate the $K_{1}(1270)$ state,
followed by the decay $K_{1}(1270)\to K\rho$, as depicted in Fig. 5.

Figure 4: The Dalitz plot for the $J/\psi\rightarrow\bar{K}^{0}K^{+}\rho^{-}$. The red band stands for the region of $M_{a_{0}}\pm\frac{1}{2}\Gamma_{a_{0}}$ where the predicted $a_{0}(1710)$ state lies. The green band stands for the region of $M_{K_{1}}\pm\frac{1}{2}\Gamma_{K_{1}}$ where the $K_{1}(1270)$ state lies. Figure 5: Diagrams for $J/\psi\rightarrow\bar{K}^{0}K^{+}\rho^{-}$ via the intermediate $K_{1}(1270)^{0}$ (a) and $K_{1}(1270)^{-}$ (b), followed by the decay $K_{1}(1270)^{0,-}\to K^{+}\rho^{-}/\bar{K}^{0}\rho^{-}$ . Table 4: Pole positions and coupling constants of the two poles of the $K_{1}(1270)$ Geng:2006yb . All values are in units of MeV. | 1st pole | 2nd pole
---|---|---
pole position $\sqrt{s_{0}}$ | $1195-i123$ | $1284-i73$
$g_{K\rho}$ | $-1671+i1599$ | $4804+i395$

The decay amplitude for
$J/\psi\rightarrow\bar{K}^{0}K_{1}(1270)^{-}\to\bar{K}^{0}K^{+}\rho^{-}$ of
Fig. 5(a) can be written as,

$\displaystyle\mathcal{M}_{b}$ $\displaystyle=$ $\displaystyle
V_{p}^{\prime}\times G_{K^{+}\rho^{-}}t_{K^{+}\rho^{-}\to K^{+}\rho^{-}},$
(16)

where $V^{\prime}_{p}$ stands for the weight of contribution from the
$K_{1}(1270)^{0}$, and the $t_{K^{+}\rho^{-}\to K^{+}\rho^{-}}$ is the
transition amplitude, which can be written as

$\displaystyle t_{K^{+}\rho^{-}\to K^{+}\rho^{-}}$ $\displaystyle=$
$\displaystyle{\frac{g_{K^{+}\rho^{-}}g_{K^{+}\rho^{-}}}{M_{K^{+}\rho^{-}}^{2}-M_{K_{1}}^{2}+iM_{K_{1}}\Gamma_{K_{1}}}},$
(17)

where $M_{K^{+}\rho^{-}}$ is the invariant mass of the $K^{+}\rho^{-}$ system,
and $g_{K^{+}\rho^{-}}$ denotes the coupling constant. In this work, we adopt
the value $g_{K^{+}\rho^{-}}=(4804+i395)$ MeV of Table 4, where the axial-
vector meson $K_{1}(1270)$ is dynamically generated from the pseudoscalar
meson-vector meson interactions. Since the higher pole of the $K_{1}(1270)$
mainly couples to the $K\rho$ channel, we can relate the mass and width of the
$K_{1}(1270)$ with the higher pole position of Table 4, i.e. $M_{K_{1}}={\rm
Re}\sqrt{s_{0}}$ and $\Gamma_{K_{1}}=2{\rm Im}\sqrt{s_{0}}$. Similarly, the
amplitude of the process $J/\psi\to K^{+}{K}_{1}(1270)^{-}\to
K^{+}\bar{K}^{0}\rho^{-}$, as depicted in Fig. 5(b), can be expressed as,

$\displaystyle\mathcal{M}_{c}$ $\displaystyle=$ $\displaystyle
V_{p}^{\prime}\times
G_{\bar{K}^{0}\rho^{-}}t_{\bar{K}^{0}\rho^{-}\to\bar{K}^{0}\rho^{-}},$ (18)
$\displaystyle t_{\bar{K}^{0}\rho^{-}\to\bar{K}^{0}\rho^{-}}$ $\displaystyle=$
$\displaystyle{\frac{g_{\bar{K}^{0}\rho^{-}}g_{\bar{K}^{0}\rho^{-}}}{M_{\bar{K}^{0}\rho^{-}}^{2}-M_{K_{1}}^{2}+iM_{K_{1}}\Gamma_{K_{1}}}},$
(19)

where $M_{\bar{K}^{0}\rho^{-}}$ is the $\bar{K}^{0}\rho^{-}$ invariant mass,
and $g_{\bar{K}^{0}\rho^{-}}$ is the coupling constant,
$g_{\bar{K}^{0}\rho^{-}}=g_{K^{+}\rho^{-}}=(4804+i395)$ MeV in this work. We
take the same weight $V^{\prime}_{p}$ for the contributions from the
$K_{1}(1270)^{0}$ and $K_{1}(1270)^{-}$.

According to Eqs. (6) and (11), we have also calculated the loop function
$G_{K^{+}\rho^{-}}$ / $G_{\bar{K}^{0}\rho^{-}}$ and
$\tilde{G}_{K^{+}\rho^{-}}$ / $\tilde{G}_{\bar{K}^{0}\rho^{-}}$ as functions
of the $K^{+}\rho^{-}$ and $\bar{K}^{0}\rho^{-}$ invariant masses,
respectively, as presented in Figs. 6 and 7. One can find that the loop
functions $G$ become smoother around the threshold when considering the width
of the $\rho$.

Figure 6: Real and imaginary parts of the loop functions $G_{K^{+}\rho^{-}}$
and $\tilde{G}_{K^{+}\rho^{-}}$ as a function of the $K^{+}\rho^{-}$ invariant
mass. The notations of the curves are the same as those of Fig. 3. Figure 7:
Real and imaginary parts of the loop functions $G_{\bar{K}^{0}\rho^{-}}$ and
$\tilde{G}_{\bar{K}^{0}\rho^{-}}$ as a function of the $\bar{K}^{0}\rho^{-}$
invariant mass. The notations of the curves are the same as those of Fig. 3.

###  II.3 Invariant mass distributions

With the amplitudes obtained above, we can write down the total decay
amplitude of $J/\psi\to\bar{K}^{0}K^{+}\rho^{-}$ as follows,

$\displaystyle\mathcal{M}=\mathcal{M}_{a}+\mathcal{M}_{b}+\mathcal{M}_{c},$
(20)

and the double differential widths of the process
$J/\psi\to\bar{K}^{0}K^{+}\rho^{-}$ are

$\displaystyle\frac{d^{2}\Gamma}{dM_{\bar{K}^{0}K^{+}}{dM_{K^{+}\rho^{-}}}}=\frac{M_{\bar{K}^{0}K^{+}}M_{K^{+}\rho^{-}}}{128\pi^{3}m_{J/\psi}^{3}}|\mathcal{M}|^{2},$
(21)
$\displaystyle\frac{d^{2}\Gamma}{dM_{\bar{K}^{0}K^{+}}{dM_{\bar{K}^{0}\rho^{-}}}}=\frac{M_{\bar{K}^{0}K^{+}}M_{\bar{K}^{0}\rho^{-}}}{128\pi^{3}m_{J/\psi}^{3}}|\mathcal{M}|^{2}.$
(22)

Furthermore, one can easily obtain ${d\Gamma}/{dM_{\bar{K}^{0}K^{+}}}$,
${d\Gamma}/{dM_{\bar{K}^{0}{\rho}^{-}}}$, and
${d\Gamma}/{dM_{K^{+}{\rho}^{-}}}$ by integrating over each of the invariant
mass variables with the limits of the Dalitz plot given in the RPP
ParticleDataGroup:2022pth . However, since the final meson $\rho^{-}$ has a
large width ($\sim$149.1 MeV), and the $K_{1}(1270)$ mass is very close to the
$K\rho$ threshold (as shown by Fig. 4), one needs to take into account its
finite width by folding with the vector-meson $\rho^{-}$ spectral function for
the invariant mass distributions Wang:2019mph , as follows,

$\displaystyle\frac{d\tilde{\Gamma}}{{d{M}_{12}}}=\int_{m_{\rho^{-}}-2\Gamma_{\rho^{-}}}^{m_{\rho^{-}}+2\Gamma_{\rho^{-}}}d\hat{m}_{\rho^{-}}\left[\frac{d\Gamma}{d{M}_{12}}\times\omega(\hat{m}^{2}_{\rho^{-}})\right],$
(23)

where the mass $m_{\rho^{-}}$ in ${d\Gamma}/dM_{12}$ should be replaced by
$\hat{m}_{\rho^{-}}$. For example, the differential width
${d\tilde{\Gamma}}/{dM_{\bar{K}^{0}K^{+}}}$ of the process
$J/\psi\to\bar{K}^{0}K^{+}\rho^{-}$ becomes,

$\displaystyle\frac{d\tilde{\Gamma}}{dM_{\bar{K}^{0}K^{+}}}$ $\displaystyle=$
$\displaystyle\int_{m_{\rho^{-}}-2\Gamma_{\rho^{-}}}^{{\rm
min}\left(m_{\rho^{-}}+2\Gamma_{\rho^{-}},m_{J/\psi-
M_{\bar{K}^{0}K^{+}}}\right)}d\hat{m}_{\rho^{-}}$ (24)
$\displaystyle\times\int_{{M_{K^{+}\rho^{-}}^{\rm
min}}}^{M_{K^{+}\rho^{-}}^{\rm max}}dM_{K^{+}\rho^{-}}$
$\displaystyle\times\frac{M_{\bar{K}^{0}K^{+}}M_{K^{+}\rho^{-}}}{128\pi^{3}m_{J/\psi}^{3}}|\mathcal{M}|^{2}\times\omega(\hat{m}^{2}_{\rho^{-}}),$

where the range of $M_{\bar{K}^{0}K^{+}}$ is,

$\displaystyle
m_{\bar{K}^{0}}+m_{K^{+}}<M_{\bar{K}^{0}K^{+}}<m_{J/\psi}-m_{\rho^{-}}+2\Gamma_{\rho^{-}},$
(25)

and the upper and lower limits for $M_{K^{+}\rho^{-}}$ are,

$\displaystyle\left(M_{K^{+}\rho^{-}}^{\rm max}\right)^{2}$ $\displaystyle=$
$\displaystyle\left(E_{K^{+}}^{\ast}+E_{\rho^{-}}^{\ast}\right)^{2}-$
$\displaystyle\left(\sqrt{E_{K^{+}}^{\ast
2}-m_{K^{+}}^{2}}-\sqrt{E_{{\rho^{-}}}^{\ast
2}-\hat{m}_{{\rho^{-}}}^{2}}\right)^{2}$
$\displaystyle\left(M_{K^{+}\rho^{-}}^{\rm min}\right)^{2}$ $\displaystyle=$
$\displaystyle\left(E_{K^{+}}^{\ast}+E_{{\rho^{-}}}^{\ast}\right)^{2}-$
$\displaystyle\left(\sqrt{E_{K^{+}}^{\ast
2}-m_{K^{+}}^{2}}+\sqrt{E_{\rho^{-}}^{\ast
2}-\hat{m}^{2}_{\rho^{-}}}\right)^{2},$

where $E_{K^{+}}^{\ast}$ and $E_{\rho^{-}}^{\ast}$ are the energies of $K^{+}$
and $\rho^{-}$ in the $\bar{K}^{0}\rho^{-}$ rest frame, respectively,

$\displaystyle
E_{K^{+}}^{\ast}=\frac{M_{\bar{K}^{0}K^{+}}^{2}-m_{\bar{K}^{0}}^{2}+m_{K^{+}}^{2}}{2M_{\bar{K}^{0}K^{+}}},$
$\displaystyle
E_{\rho^{-}}^{\ast}=\frac{m_{J/\psi}^{2}-M_{\bar{K}^{0}K^{+}}^{2}-\hat{m}_{\rho^{-}}^{2}}{2M_{\bar{K}^{0}K^{+}}}.$
(26)

##  III Results and Discussion

Figure 8: $\bar{K}^{0}K^{+}$ invariant mass distribution of the process
$J/\psi\rightarrow\bar{K}^{0}K^{+}\rho^{-}$. The results are obtained with Eq.
(23), where the width of final $\rho^{-}$ is considered. The red-solid curve
stands for the total contribution, while the blue-dashed curve, the green-dot-
dashed curve, and purple-dotted curve correspond to the contributions from the
$a_{0}(1710)$ state, the intermediate $K_{1}(1270)^{0}$, and
$K_{1}(1270)^{-}$, respectively.

Figure 9: $K^{+}\rho^{-}$ (a) and $\bar{K}^{0}\rho^{-}$ (b) invariant mass
distributions of the process $J/\psi\rightarrow\bar{K}^{0}K^{+}\rho^{-}$. The
explanations of the curves are the same as those of Fig. 8.

Figure 10: The $\bar{K}^{0}K^{+}$ (a), $K^{+}\rho^{-}$ (b) and
$\bar{K}^{0}\rho^{-}$ (c) invariant mass distributions of the process
$J/\psi\rightarrow\bar{K}^{0}K^{+}\rho^{-}$ with the parameters of lower pole
of $K_{1}(1270)$. The explanations of the curves are the same as those of Fig.
8.

Figure 11:  $\bar{K}^{0}K^{+}$, $K^{+}\rho^{-}$, and $\bar{K}^{0}\rho^{-}$
invariant mass distributions of the process
$J/\psi\rightarrow\bar{K}^{0}K^{+}\rho^{-}$ with
$V_{p}^{\prime}/V_{p}=0.5,1.0,1.5$.

In our formalism, there are two unknown parameters, $V_{p}$ for the weight of
the $a_{0}(1710)$ contribution and $V_{p}^{\prime}$ for the one of the
intermediate $K_{1}(1270)$ contribution. Since there are some similarities
between the processes $J/\psi\to VVV$ and $J/\psi\to VPP$, it is expected that
$V_{p}$ and $V^{\prime}_{p}$ are of the same order of magnitude. Thus, we
first take $V_{p}=V^{\prime}_{p}$ and discuss the influence on our results of
different values of $V_{p}$ and $V^{\prime}_{p}$.

In Fig. 8, we show the $\bar{K}^{0}K^{+}$ invariant mass distribution of the
process $J/\psi\to\bar{K}^{0}K^{+}\rho^{-}$. The red-solid curve stands for
the total contributions from the $a_{0}(1710)$ state and the axial-vector
$K_{1}(1270)$ meson, while the blue-dashed curve corresponds to the
contribution from the $a_{0}(1710)$ state. Moreover, the green-dot-dashed and
purple-dotted curves correspond to the contributions from the intermediate
$K_{1}(1270)^{0}$ and $K_{1}(1270)^{-}$, respectively. One can find a clear
peak structure around 1.8 GeV, which could be associated with the scalar
$a_{0}(1710)$. The intermediate resonances $K_{1}(1270)^{0}$ and
$K_{1}(1270)^{-}$ give the smooth contributions in the region of $1.4\sim 2.4$
GeV, which is due to that the $K_{1}(1270)$ couples to $K\rho$ in $S$-wave.

Next, we have predicted the $K^{+}\rho^{-}$ and $\bar{K}^{0}\rho^{-}$
invariant mass distributions of the process
$J/\psi\rightarrow\bar{K}^{0}K^{+}\rho^{-}$ in Figs. 9(a) and 9(b),
respectively. One can see the clear peaks of the $K_{1}(1270)^{0}$ and
$K_{1}(1270)^{-}$ around 1.3 GeV. It should be stressed that the axial vector
$K_{1}(1270)$ is predicted to have a two-pole structure, the lower pole is
around 1200 MeV, coupled strongly to the $K^{*}\pi$ channel, which the higher
pole is around 1280 MeV, coupled strongly to the $K\rho$ channels Geng:2006yb
.

As we discussed above, since the higher pole of $K_{1}(1270)$ mainly couples
to the $K\rho$ channel, we have adopted the pole position and coupling
constant of the higher pole in Eq. (19). To show the difference between the
two poles of the $K_{1}(1270)$, we have calculated the $\bar{K}^{0}K^{+}$,
$K^{+}\rho^{-}$, and $\bar{K}^{0}\rho^{-}$ invariant mass distributions for
the process $J/\psi\rightarrow\bar{K}^{0}K^{+}\rho^{-}$ with the pole position
and coupling constant of the lower pole of Table 4 in Eq. (19), as shown in
Fig. 10. Because the coupling of the lower pole to $K\rho$ is smaller and the
width is larger, the contribution from the lower pole of the $K_{1}(1270)$ is
very small. For the contributions from the lower pole and the higher pole to
have the same order of magnitude, we take $V^{\prime}_{p}/{V_{p}}=8.0$ in the
results of Fig. 10. In the $\bar{K}^{0}K^{+}$ invariant mass distribution of
Fig. 10(a), one can find the clear peak structure around 1.8 GeV. However, in
the $\bar{K}^{0}\rho^{-}$ and $K^{+}\rho^{-}$ invariant mass distributions of
Figs. 10(b) and 10(c), only an enhancement structure appears. Thus, future
measurements of $\bar{K}^{0}\rho^{-}$ and $K^{+}\rho^{-}$ invariant mass
distributions could shed light on the two-pole structure of the $K_{1}(1270)$.

In addition, we take different ratios of $V_{p}^{\prime}/V_{p}=0.5,1.0,1.5$,
and show the $\bar{K}^{0}K^{+}$, $K^{+}\rho^{-}$, and $\bar{K}^{0}\rho^{-}$
invariant mass distributions for the process
$J/\psi\rightarrow\bar{K}^{0}K^{+}\rho^{-}$ in Figs. 11(a), 11(b), and 11(c),
respectively. One can find that the peak structure of the $a_{0}(1710)$ in the
$\bar{K}^{0}K^{+}$ invariant mass distribution remains similar for different
values of $V_{p}^{\prime}/V_{p}$, and the peak structures of the $K_{1}(1270)$
become clearer for larger values of $V_{p}^{\prime}/V_{p}$.

##  IV Summary

Assuming the $a_{0}(1710)$ as a $K^{*}\bar{K}^{*}$ molecular state, we have
investigated the process $J/\psi\rightarrow\bar{K}^{0}K^{+}\rho^{-}$ by taking
into account the contribution from the $S$-wave $\omega\rho^{+}$,
$\bar{K}^{*0}K^{*+}$, and $\phi\rho$ interactions, as well as the contribution
from the intermediate resonance $K_{1}(1270)$.

We have predicted one peak structure around 1.8 GeV in the $\bar{K}^{0}K^{+}$
invariant mass distribution, which could be associated with the scalar meson
$a_{0}(1710)$. Furthermore, we have also predicted the $K^{+}\rho^{-}$ and
$\bar{K}^{0}\rho^{-}$ invariant mass distributions of the process
$J/\psi\rightarrow\bar{K}^{0}K^{+}\rho^{-}$, and find clear peaks of the
resonance $K_{1}(1270)^{0,-}$. Considering the two-pole structure of the
$K_{1}(1270)$, we have also calculated the results adopting parameters of the
lower pole and find an enhancement structure near 1.3 GeV in the
$K^{+}\rho^{-}$ and $\bar{K}^{0}\rho^{-}$ invariant mass distributions, which
implies that future measurements of the $K^{+}\rho^{-}$ and
$\bar{K}^{0}\rho^{-}$ invariant mass distributions could shed light on the
two-pole structure of the $K_{1}(1270)$.

Finally, considering different weights ratio
$V_{p}^{\prime}/V_{p}=0.5,1.0,1.5$ of contributions from $a_{0}(1710)$ and
$K_{1}(1270)$, we have shown the $\bar{K}^{0}K^{+}$, $K^{+}\rho^{-}$ and
$\bar{K}^{0}\rho^{-}$ invariant mass distributions of the process
$J/\psi\rightarrow\bar{K}^{0}K^{+}\rho^{-}$, and find that the peak structure
of $a_{0}(1710)$ remains essentially the same. We hope that our theoretical
predictions could be tested by the BESIII and BelleII experiments and the
planned STCF in the future, and precise measurements of the process
$J/\psi\rightarrow\bar{K}^{0}K^{+}\rho^{-}$ could shed light on the nature of
the scalar $a_{0}(1710)$ and the axial-vector $K_{1}(1270)$.

## Acknowledgments

This work is supported by the Natural Science Foundation of Henan under Grant
No. 222300420554 and No. 232300421140. This work is partly supported by the
National Natural Science Foundation of China under Grants Nos. 12075288,
11975041, 11961141004, 12361141819, and 12192263, the Project of Youth
Backbone Teachers of Colleges and Universities of Henan Province
(2020GGJS017), and the Open Project of Guangxi Key Laboratory of Nuclear
Physics and Nuclear Technology, No. NLK2021-08. It is also partly supported by
the Youth Innovation Promotion Association CAS.

## References

  * (1) J. P. Lees et al. [BaBar], Phys. Rev. D 104 (2021) no.7, 072002 doi:10.1103/PhysRevD.104.072002 [arXiv:2106.05157 [hep-ex]].
  * (2) M. Ablikim et al. [BESIII], Phys. Rev. D 105 (2022) no.5, L051103 doi:10.1103/PhysRevD.105.L051103 [arXiv:2110.07650 [hep-ex]].
  * (3) M. Ablikim et al. [BESIII], Phys. Rev. Lett. 129 (2022) no.18, 18 doi:10.1103/PhysRevLett.129.182001 [arXiv:2204.09614 [hep-ex]].
  * (4) H. Nagahiro, L. Roca, E. Oset and B. S. Zou, Phys. Rev. D 78 (2008), 014012 doi:10.1103/PhysRevD.78.014012 [arXiv:0803.4460 [hep-ph]].
  * (5) T. Branz, L. S. Geng and E. Oset, Phys. Rev. D 81 (2010), 054037 doi:10.1103/PhysRevD.81.054037 [arXiv:0911.0206 [hep-ph]].
  * (6) L. S. Geng, F. K. Guo, C. Hanhart, R. Molina, E. Oset and B. S. Zou, Eur. Phys. J. A 44 (2010), 305-311 doi:10.1140/epja/i2010-10971-5 [arXiv:0910.5192 [hep-ph]].
  * (7) J. J. Xie and E. Oset, Phys. Rev. D 90 (2014) no.9, 094006 doi:10.1103/PhysRevD.90.094006 [arXiv:1409.1341 [hep-ph]].
  * (8) A. Martinez Torres, K. P. Khemchandani, F. S. Navarra, M. Nielsen and E. Oset, Phys. Lett. B 719 (2013), 388-393 doi:10.1016/j.physletb.2013.01.036 [arXiv:1210.6392 [hep-ph]].
  * (9) Z. L. Wang and B. S. Zou, Phys. Rev. D 104 (2021) no.11, 114001 doi:10.1103/PhysRevD.104.114001 [arXiv:2107.14470 [hep-ph]].
  * (10) C. Garcia-Recio, L. S. Geng, J. Nieves and L. L. Salcedo, Phys. Rev. D 83 (2011), 016007 doi:10.1103/PhysRevD.83.016007 [arXiv:1005.0956 [hep-ph]].
  * (11) C. García-Recio, L. S. Geng, J. Nieves, L. L. Salcedo, E. Wang and J. J. Xie, Phys. Rev. D 87 (2013) no.9, 096006 doi:10.1103/PhysRevD.87.096006 [arXiv:1304.1021 [hep-ph]].
  * (12) F. E. Close and Q. Zhao, Phys. Rev. D 71 (2005), 094022 doi:10.1103/PhysRevD.71.094022 [arXiv:hep-ph/0504043 [hep-ph]].
  * (13) L. C. Gui et al. [CLQCD], Phys. Rev. Lett. 110, 021601 (2013). doi:10.1103/PhysRevLett.110.021601 [arXiv:1206.0125 [hep-lat]].
  * (14) S. Janowski, F. Giacosa and D. H. Rischke, Phys. Rev. D 90, 114005 (2014). doi:10.1103/PhysRevD.90.114005 [arXiv:1408.4921 [hep-ph]].
  * (15) A. H. Fariborz, A. Azizi and A. Asrar, Proximity of $f_{0}$(1500) and $f_{0}$(1710) to the scalar glueball, Phys. Rev. D 92, 113003 (2015).
  * (16) Y. Chen, A. Alexandru, S. J. Dong, T. Draper, I. Horvath, F. X. Lee, K. F. Liu, N. Mathur, C. Morningstar and M. Peardon, et al. Phys. Rev. D 73 (2006), 014516 doi:10.1103/PhysRevD.73.014516 [arXiv:hep-lat/0510074 [hep-lat]].
  * (17) L. S. Geng and E. Oset, Phys. Rev. D 79 (2009), 074009 doi:10.1103/PhysRevD.79.074009 [arXiv:0812.1199 [hep-ph]].
  * (18) M. L. Du, D. Gülmez, F. K. Guo, U. G. Meißner and Q. Wang, Eur. Phys. J. C 78 (2018) no.12, 988 doi:10.1140/epjc/s10052-018-6475-8 [arXiv:1808.09664 [hep-ph]].
  * (19) R. L. Workman et al. [Particle Data Group], PTEP 2022 (2022), 083C01 doi:10.1093/ptep/ptac097
  * (20) Z. L. Wang and B. S. Zou, Eur. Phys. J. C 82 (2022) no.6, 509 doi:10.1140/epjc/s10052-022-10460-4 [arXiv:2203.02899 [hep-ph]].
  * (21) G. Y. Wang, S. C. Xue, G. N. Li, E. Wang and D. M. Li, Phys. Rev. D 97 (2018) no.3, 034030 doi:10.1103/PhysRevD.97.034030 [arXiv:1712.10180 [hep-ph]].
  * (22) Q. H. Shen, X. Zhang, X. Liu and J. J. Xie, Phys. Rev. D 109 (2024) no.1, 014012 doi:10.1103/PhysRevD.109.014012 [arXiv:2310.07258 [hep-ph]].
  * (23) D. Guo, W. Chen, H. X. Chen, X. Liu and S. L. Zhu, Phys. Rev. D 105 (2022) no.11, 114014 doi:10.1103/PhysRevD.105.114014 [arXiv:2204.13092 [hep-ph]].
  * (24) M. Ablikim et al. [BES], Phys. Rev. Lett. 96 (2006), 162002 doi:10.1103/PhysRevLett.96.162002 [arXiv:hep-ex/0602031 [hep-ex]].
  * (25) M. Ablikim et al. [BESIII], Phys. Rev. D 87 (2013) no.3, 032008 doi:10.1103/PhysRevD.87.032008 [arXiv:1211.5668 [hep-ex]].
  * (26) X. Zhu, D. M. Li, E. Wang, L. S. Geng and J. J. Xie, Phys. Rev. D 105 (2022) no.11, 116010 doi:10.1103/PhysRevD.105.116010 [arXiv:2204.09384 [hep-ph]].
  * (27) X. Zhu, H. N. Wang, D. M. Li, E. Wang, L. S. Geng and J. J. Xie, Phys. Rev. D 107 (2023) no.3, 034001 doi:10.1103/PhysRevD.107.034001 [arXiv:2210.12992 [hep-ph]].
  * (28) L. R. Dai, E. Oset and L. S. Geng, Eur. Phys. J. C 82 (2022) no.3, 225 doi:10.1140/epjc/s10052-022-10178-3 [arXiv:2111.10230 [hep-ph]].
  * (29) E. Oset, L. R. Dai and L. S. Geng, Sci. Bull. 68 (2023), 243-246 doi:10.1016/j.scib.2023.01.011 [arXiv:2301.08532 [hep-ph]].
  * (30) Z. Y. Wang, Y. W. Peng, J. Y. Yi, W. C. Luo and C. W. Xiao, Phys. Rev. D 107 (2023) no.11, 116018 doi:10.1103/PhysRevD.107.116018
  * (31) Y. Ding, X. H. Zhang, M. Y. Dai, E. Wang, D. M. Li, L. S. Geng and J. J. Xie, Phys. Rev. D 108, no.11, 114004 (2023) doi:10.1103/PhysRevD.108.114004 [arXiv:2306.15964 [hep-ph]].
  * (32) J. P. Lees et al. [BaBar], Phys. Rev. D 93 (2016), 012005 doi:10.1103/PhysRevD.93.012005 [arXiv:1511.02310 [hep-ex]].
  * (33) X. Y. Wang, H. F. Zhou and X. Liu, Phys. Rev. D 108 (2023) no.3, 034015 doi:10.1103/PhysRevD.108.034015 [arXiv:2306.12815 [hep-ph]].
  * (34) M. Ablikim et al. [BESIII], Chin. Phys. C 46 (2022) no.7, 074001 doi:10.1088/1674-1137/ac5c2e [arXiv:2111.07571 [hep-ex]].
  * (35) M. Achasov, X. C. Ai, L. P. An, R. Aliberti, Q. An, X. Z. Bai, Y. Bai, O. Bakina, A. Barnyakov and V. Blinov, et al. Front. Phys. (Beijing) 19 (2024) no.1, 14701 doi:10.1007/s11467-023-1333-z [arXiv:2303.15790 [hep-ex]].
  * (36) J. P. Lees et al. [BaBar], Phys. Rev. D 95 (2017) no.9, 092005 doi:10.1103/PhysRevD.95.092005 [arXiv:1704.05009 [hep-ex]].
  * (37) L. S. Geng, E. Oset, L. Roca and J. A. Oller, Phys. Rev. D 75 (2007), 014017 doi:10.1103/PhysRevD.75.014017 [arXiv:hep-ph/0610217 [hep-ph]].
  * (38) G. Y. Wang, L. Roca, E. Wang, W. H. Liang and E. Oset, Eur. Phys. J. C 80 (2020) no.5, 388 doi:10.1140/epjc/s10052-020-7939-1 [arXiv:2002.07610 [hep-ph]].
  * (39) G. Y. Wang, L. Roca and E. Oset, Phys. Rev. D 100 (2019) no.7, 074018 doi:10.1103/PhysRevD.100.074018 [arXiv:1907.09188 [hep-ph]].
  * (40) J. M. Xie, J. X. Lu, L. S. Geng and B. S. Zou, [arXiv:2312.17287 [hep-ph]].
  * (41) J. M. Xie, J. X. Lu, L. S. Geng and B. S. Zou, Phys. Rev. D 108 (2023) no.11, L111502 doi:10.1103/PhysRevD.108.L111502 [arXiv:2307.11631 [hep-ph]].
  * (42) N. Ikeno, J. M. Dias, W. H. Liang and E. Oset, Phys. Rev. D 100 (2019) no.11, 114011 doi:10.1103/PhysRevD.100.114011 [arXiv:1909.11906 [hep-ph]].
  * (43) S. J. Jiang, S. Sakai, W. H. Liang and E. Oset, Phys. Lett. B 797 (2019), 134831 doi:10.1016/j.physletb.2019.134831 [arXiv:1904.08271 [hep-ph]].
  * (44) S. Sakai, W. H. Liang, G. Toledo and E. Oset, Phys. Rev. D 101 (2020) no.1, 014005 doi:10.1103/PhysRevD.101.014005 [arXiv:1909.08888 [hep-ph]].
  * (45) H. Zhang, B. C. Ke, Y. Yu and E. Wang, Chin. Phys. C 47 (2023) no.6, 063101 doi:10.1088/1674-1137/acc642 [arXiv:2302.10541 [hep-ph]].
  * (46) J. Y. Wang, M. Y. Duan, G. Y. Wang, D. M. Li, L. J. Liu and E. Wang, Phys. Lett. B 821 (2021), 136617 doi:10.1016/j.physletb.2021.136617 [arXiv:2105.04907 [hep-ph]].
  * (47) M. Y. Duan, G. Y. Wang, E. Wang, D. M. Li and D. Y. Chen, Phys. Rev. D 104 (2021) no.7, 074030 doi:10.1103/PhysRevD.104.074030 [arXiv:2109.00731 [hep-ph]].
  * (48) L. M. Abreu, W. F. Wang and E. Oset, Eur. Phys. J. C 83 (2023) no.3, 243 doi:10.1140/epjc/s10052-023-11384-3 [arXiv:2301.08058 [hep-ph]].
  * (49) R. Molina, L. R. Dai, L. S. Geng and E. Oset, Eur. Phys. J. A 56 (2020) no.6, 173 doi:10.1140/epja/s10050-020-00176-y [arXiv:1909.10764 [hep-ph]].
  * (50) R. Molina, D. Nicmorus and E. Oset, Phys. Rev. D 78 (2008), 114018 doi:10.1103/PhysRevD.78.114018 [arXiv:0809.2233 [hep-ph]].
  * (51) W. T. Lyu, Y. H. Lyu, M. Y. Duan, D. M. Li, D. Y. Chen and E. Wang, Phys. Rev. D 109 (2024) no.1, 014008 doi:10.1103/PhysRevD.109.014008 [arXiv:2306.16101 [hep-ph]].
  * (52) M. Y. Duan, D. Y. Chen and E. Wang, Eur. Phys. J. C 82 (2022) no.10, 968 doi:10.1140/epjc/s10052-022-10948-z [arXiv:2207.03930 [hep-ph]].
  * (53) L. S. Geng, E. Oset, R. Molina and D. Nicmorus, PoS EFT09 (2009), 040 doi:10.22323/1.069.0040 [arXiv:0905.0419 [hep-ph]].