# Exploring Hybrid Question Answering via Program-based Prompting

Qi Shi, Han Cui∗, Haofeng Wang, Qingfu Zhu, Wanxiang Che, Ting Liu
Research Center for Social Computing and Information Retrieval
Harbin Institute of Technology, Harbin, China
<EMAIL_ADDRESS>
Equal Contributions.

###### Abstract

Question answering over heterogeneous data requires reasoning over diverse
sources of data, which is challenging due to the large scale of information
and organic coupling of heterogeneous data. Various approaches have been
proposed to address these challenges. One approach involves training
specialized retrievers to select relevant information, thereby reducing the
input length. Another approach is to transform diverse modalities of data into
a single modality, simplifying the task difficulty and enabling more
straightforward processing. In this paper, we propose HProPro , a novel
program-based prompting framework for the hybrid question answering task.
HProPro  follows the code generation and execution paradigm. In addition,
HProPro  integrates various functions to tackle the hybrid reasoning scenario.
Specifically, HProPro  contains function declaration and function
implementation to perform hybrid information-seeking over data from various
sources and modalities, which enables reasoning over such data without
training specialized retrievers or performing modal transformations.
Experimental results on two typical hybrid question answering benchmarks
HybridQA and MultiModalQA demonstrate the effectiveness of HProPro : it
surpasses all baseline systems and achieves the best performances in the few-
shot settings on both datasets.

Exploring Hybrid Question Answering via Program-based Prompting

Qi Shi††thanks:  Equal Contributions., Han Cui∗, Haofeng Wang, Qingfu Zhu,
Wanxiang Che, Ting Liu Research Center for Social Computing and Information
Retrieval Harbin Institute of Technology, Harbin, China
<EMAIL_ADDRESS>

##  1 Introduction

Question answering systems Pasupat and Liang (2015); Rajpurkar et al. (2016);
Goyal et al. (2017) have attracted significant attention and made considerable
progress in recent years. However, real-world data often exists in diverse
formats and originates from multiple sources. Consequently, researchers turn
their focus to the hybrid question answering (HQA) task Chen et al. (2020b);
Talmor et al. (2020), which necessitates mixed reasoning across various types
of data. The HQA task is challenging due to the vast amount of information and
the organic coupling of heterogeneous data sources. Reasoning over such
diverse data requires the ability to understand multiple data types
simultaneously. For instance, as depicted in Figure 1, the model must engage
in reasoning over both the table and the extensive passages and images linked
in hyperlinks to make accurate predictions.

Figure 1: Example of hybrid question answering task with the corresponding
program.

To tackle these challenges, recent approaches focus on training domain-
specific models to retrieve or rank elements such as table rows, passages, or
images, selecting the most relevant ones to enhance the subsequent reasoning
process Eisenschlos et al. (2021); Kumar et al. (2021); Lei et al. (2023).
Since real-world heterogeneous data is vast and constantly updated, even if
these approaches demonstrate promising performance on their focused datasets,
their applicability to such intricate data is still limited. Furthermore, some
existing approaches tend to transform diverse modalities of data into a single
modality, such as image captioning Cheng et al. (2022); Liu et al. (2023), or
table-to-text generation Li et al. (2021), to reduce the task difficulty.
However, such approaches are constrained by the performance of modal
transformation models, which often result in the loss of information. In a
word, these approaches highly rely on data distribution, and the complexity of
real-world heterogeneous data makes them exorbitant.

In contrast to previous approaches, we argue that the solution of solving the
HQA task should be agnostic to data distribution. Consequently, we advocate
for an optimal solution devising a procedure for determining how to find an
answer, rather than merely generating the answer itself. Noticing that the
program could elucidate the reasoning process employed to arrive at the answer
(as depicted in Figure 1), in the current era of large language models (LLMs),
leveraging a program can serve as an advantageous solution since LLMs are an
excellent program generator. Moreover, the process of program generation
necessitates the incorporation of various functions into the program, enabling
information-seeking across diverse sources and modalities of data.

Based on the aforementioned considerations, in this paper, we introduce a
novel program-based prompting framework HProPro  (Hybrid Program-Based
Prompting) for HQA task. HProPro  considers the solution as a process of code
generation and execution, integrating external customized functions under the
few-shot setting111In this work, we use Python code as the carrier of the
program.. To facilitate the utilization of customized functions, HProPro
incorporates two key components: Function Declaration during the code
generation phase and Function Implementation during the execution phase, which
is shown in Figure 2. During the function declaration stage, HProPro  defines
the function name and formal parameters, utilizing them as prompts to generate
code. Subsequently, in the function implementation stage, HProPro  implements
the declared functions, serving for the direct execution of the generated
code. By defining different functions, HProPro  can support reasoning over
data from various modalities, making it a flexible and scalable framework.
Importantly, HProPro  eliminates the need to convert different modalities of
data into a single modality beforehand. Instead, it acquires information
within the origin modal by the functions themselves. To the best of our
knowledge, HProPro  is the first work to explore the power of LLMs in handling
heterogeneous data without requiring domain-specific retrieval or modal
transformation. Experiments demonstrate that HProPro  significantly
outperforms previous methods.

In summary, our contributions are as follows:

  * •

We introduce HProPro , a program-based prompting framework that enables
reasoning over heterogeneous data without domain-specific retrieval and modal
transformation.

  * •

We implement a few-shot code generation and execution pipeline, calling
various functions by function declaration and implementation to perform
information-seeking across data from different sources and modalities.

  * •

Experiments show the effectiveness that HProPro  achieves the best
performances under the few-shot settings on HybridQA Chen et al. (2020b) and
achieves state-of-the-art performances under all settings on MultiModalQA
Talmor et al. (2020).

Figure 2: Comparison of HProPro  with previous retrieval-based methods.

##  2 Method

###  2.1 HProPro  Framework

#### Task Formulation

In this paper, our focus is on the task of hybrid question answering, which
involves answering questions based on heterogeneous information sources such
as tables, text, and images. The objective is to provide accurate answers to
questions based on the given heterogeneous data. Figure 2 provides the
comparison between retrieval-based methods and our proposed approach HProPro .
Similar to existing program-based prompting approaches, HProPro  follows a
paradigm that involves generating code and executing it to obtain the final
answer. Unlike the previous approaches with a separate retriever, we deal with
the input data with external functions but not the retriever module. As a
result, we introduce two key components: function declaration and function
implementation, which are required during the code generation stage and code
execution stage, respectively. In the following sections, we will delve into
both parts of the framework and discuss their roles and functionalities.

#### Function Declaration

The function declaration process in HProPro  serves the purpose of defining
appropriate functions that can be utilized during the code generation phase.
During this stage, it is necessary to specify the function name and formal
parameters. These declared functions are treated as input prompts for LLMs and
are expected to be leveraged to generate code. In Figure 2, the functions with
different highlight backgrounds on the left represent the declared functions.
Each function has a specific role, which is described briefly alongside the
table and query as the input prompts. These prompts are then fed into LLMs to
generate the corresponding code. The LLMs will attempt to utilize the declared
functions to generate the desired code. By providing function declarations as
prompts, HProPro  enables the LLMs to have a better understanding of the
expected structure and behavior of the code to be generated. This allows for
more accurate and controllable code generation, ultimately facilitating the
HQA task.

#### Function Implementation

The generated code contains formally defined functions, rendering it incapable
of direct execution. Consequently, the process of function implementation aims
to implement the declared functions to make the code able to be executed by
off-the-shelf interpreters. As discussed in Section 1, functions are expected
to interact with data from various sources. However, conventional function
structures cannot be accommodated in some scenarios, such as extracting
information over unstructured texts or images. Therefore, we proceed with the
initial implementation of the declared functions integrated with the ability
of LLMs to ensure that each function encompasses comprehensive functionality.
Specifically, to achieve this process, we pre-design function-related prompts,
which are expected to be fed into LLMs when executing the generated code.

Figure 3: Details of the process of the defined functions and the code
refinement.

###  2.2 Function Instantiation

In this section, we introduce several functions and elaborate on the
declaration and implementation of each function to support HProPro .

#### Extract information from external source

To facilitate reasoning across heterogeneous information sources, we introduce
the function "extract_info". Since data from these sources is often
unstructured, the process of extracting information can be likened to a
reading comprehension task. In the function declaration, "extract_info" is
defined as "extract_info(cell, target_information)". Here, "cell" refers to a
specific cell in a table, and "target_information" represents the information
that is required to be extracted, as shown in Figure 3. The function’s purpose
is to extract the relevant information from the paragraph or image associated
with the "cell" based on the specified "target_information". It should return
the extracted information as a textual string. All the necessary information,
including the function name and its parameters, will be part of the generated
code and are expected to be generated by LLMs. During the function
implementation process, we utilize an automatically constructed dictionary to
locate the corresponding paragraphs or images based on the provided cell.
Subsequently, we construct prompts to invoke LLMs based on the data types,
which can be categorized into text-based extraction and image-based
extraction. The detailed prompts are introduced in Appendix A.

#### Compare two pieces of information

Code often includes rich comparisons between two objects using operators like
">", "<", or "==". However, when dealing with heterogeneous data, the
information extracted from various sources may not adhere to a strict format.
The form of information obtained from functions like "extract_info" cannot be
predetermined. As a result, the traditional comparison operators cannot be
directly applied to compare two objects, such as comparing the values
$"20,000"$ and "ten thousand", or comparing "Beijing" and "the capital of
China". To address this issue, we propose a more flexible function called
"check". In the function declaration process, "check" can be defined as
"check(obj1, obj2, op)". As shown in Figure 3, "obj1" and "obj2" are two
strings representing pieces of information. These strings can be the contents
of table cells, information obtained from other functions, or directly
generated by LLMs based on natural language questions. The "op" parameter
represents one of three operators: ">", "<", or "==". The purpose of the
"check" function is to compare whether "obj1" and "obj2" are semantically
consistent under the specified "op" operator. In other words, it evaluates if
the semantic relationship between the two objects aligns with the given
operator. Similar to the "extract_info" function, all the relevant
information, including the function name and its actual parameters, will be
part of the generated code and are expected to be generated by LLMs. During
the function implementation process, we provide some few-shot cases as prompts
to guide LLMs on how to compare the objects. The detailed prompts are
introduced in Appendix A.

###  2.3 Code Refinement

In HProPro , the final answer is obtained by executing the generated code
using a standard Python interpreter. Any error in the code will terminate the
execution process. However, since the model cannot predict the results
returned by each function during code generation, there is a possibility that
the model may generate code with mismatched processing methods. This can lead
to execution errors or empty results when running the code. Since initial
outputs from LLMs can be improved through iterative feedback and refinement
Madaan et al. (2023), we perform code refinement by re-calling the LLMs and
incorporating error codes and traceback information into the prompts to
generate new code. Figure 3 illustrates the prompts used for code refinement.
By providing the above information to LLMs, the models are expected to
reconsider the code generation process and generate new code that can
alleviate the issues encountered. The detailed prompts are introduced in
Appendix A.

Figure 4: Schematic diagram of query simplification process.

###  2.4 Query Simplification

In the HQA task, code generation is often performed based on the input of a
table and a question since including all relevant data as input would result
in an extensive input length. However, the reasoning process often involves
linked passages or images, which are not directly visible during the code
generation phase. This increases the burden of the code generation process. To
address this issue, we employ query simplification to simplify the question
and establish links between the question and the table cells before conducting
code generation. Figure 4 illustrates the schematic diagram of the query
simplification process. Taking _"Among animated TV shows, who was the original
voice actor on the show whose poster features a character in yellow gloves"_
as an example, we utilize a general retriever222The general retriever stands
for either a naive retriever or a neural retriever trained on a general
corpus, rather than a customized retriever trained on a specific task.
initially to retrieve relevant information from passages or images in the
hyperlinks. Query simplification involves using LLMs that take as input the
retrieved passage or image, the original question, and the table. The goal is
to replace the span in the question (such as _"the show whose poster features
a character in yellow gloves"_) with the corresponding content in the table
cell (such as _"Kick Buttowski: Suburban Daredevil"_). The detailed prompts
are introduced in Appendix A.

##  3 Experiments

###  3.1 Datasets

We conduct experiments on two typical HQA datasets: HybridQAChen et al.
(2020b) and MultiModalQATalmor et al. (2020). Both datasets involve the task
of mixed reasoning over diverse sources of data. HybridQA necessitates
reasoning over hybrid contexts that consist of both tables and texts. On the
other hand, MultiModalQA requires reasoning over tables, texts, and images. To
evaluate HProPro , we follow the official evaluation metrics provided by the
datasets. We report the exact match and F1 score on both HybridQA and
MultiModalQA. For more detailed statistics about the datasets, please refer to
Appendix B.

###  3.2 Experimental Settings

In all our experiments, we utilize different versions of the GPT language
models for different components. Specifically, we use gpt-4-0613 as the
backbone model for code generation, code refinement, and query simplification.
For implementing the function "extract_info", we employ gpt-4-0613 and
gpt-4-vision-preview to extract information in the passages and images
respectively. For implementing the function "check", we employ gpt-3.5-turbo.
In the process of query simplification, for the HybridQA task, we employ a
hybrid retriever that combines TF-IDF and longest-substring matching Chen et
al. (2020b) as the retriever. For the MultiModalQA task, we utilize sentence
transformers Reimers and Gurevych (2020) as the retriever respectively. The
temperature parameter for all models is set to $0$. All few-shot experiments
for code generation are in the settings of $4$ shots. Furthermore, in the
oracle settings of MultiModalQA, where the golden passage and image are
provided, we remove the query simplification module. This allows us to
directly feed all relevant data to the language models without encountering
issues related to excessive input length.

###  3.3 Baseline Systems

We compare HProPro  to various methods on HybridQA and MultiModalQA, which can
be mainly divided into with(w.) and without(w.o.) fine-tuning approaches. For
HybridQA, approaches w. fine-tuning stand for the method that trained on the
training set, including MAFiD Lee et al. (2023), S3HQA Lei et al. (2023), etc,
and approaches w.o. fine-tuning include the Unsupervised-QG Pan et al. (2021)
and End-to-End QA with retriever on GPT-4. For MultiModalQA, baseline methods
consist of approaches w. fine-tuning including SKURGYang et al. (2023),
PReasM-LargeYoran et al. (2022), etc, and approaches w.o. fine-tuning
including BinderCheng et al. (2022), MMHQA-ICL Liu et al. (2023), etc.

Models | Table | Passage | Total
---|---|---|---
Dev | Test | Dev | Test | Dev | Test
EM | F1 | EM | F1 | EM | F1 | EM | F1 | EM | F1 | EM | F1
Approaches w. Fine-tuning
HYBRIDER Chen et al. (2020b) | $54.3$ | $61.4$ | $56.2$ | $63.3$ | $39.1$ | $45.7$ | $37.5$ | $44.4$ | $44.0$ | $50.7$ | $43.8$ | $50.6$
DocHopper Sun et al. (2021) | – | – | – | – | – | – | – | – | $47.7$ | $55.0$ | $46.3$ | $53.3$
MuGER2 Wang et al. (2022) | $60.9$ | $69.2$ | $58.7$ | $66.6$ | $56.9$ | $68.9$ | $57.1$ | $68.6$ | $57.1$ | $67.3$ | $56.3$ | $66.2$
POINTR Eisenschlos et al. (2021) | $68.6$ | $74.2$ | $66.9$ | $72.3$ | $62.8$ | $71.9$ | $62.8$ | $71.9$ | $63.4$ | $71.0$ | $62.8$ | $70.2$
DEHG Feng et al. (2022) | – | – | – | – | – | – | – | – | $65.2$ | $\mathbf{76.3}$ | $63.9$ | $\mathbf{75.5}$
MITQA Kumar et al. (2021) | $68.1$ | $73.3$ | $68.5$ | $74.4$ | $66.7$ | $75.6$ | $64.3$ | $73.3$ | $65.5$ | $72.7$ | $64.3$ | $71.9$
MAFiD Lee et al. (2023) | $69.4$ | $75.2$ | $68.5$ | $74.9$ | $66.5$ | $75.5$ | $65.7$ | $75.3$ | $66.2$ | $74.1$ | $65.4$ | $73.6$
S3HQA Lei et al. (2023) | $\mathbf{70.3}$ | $\mathbf{75.3}$ | $\mathbf{70.6}$ | $\mathbf{76.3}$ | $\mathbf{69.9}$ | $\mathbf{78.2}$ | $\mathbf{68.7}$ | $\mathbf{77.8}$ | $\mathbf{68.4}$ | $75.3$ | $\mathbf{67.9}$ | $\mathbf{75.5}$
Approaches w.o. Fine-tuning
Unsupervised-QG Pan et al. (2021) | – | – | – | – | – | – | – | – | $25.7$ | $30.5$ | – | –
GPT-4 End-to-End QA w. Retriever |  $50.0$† |  $\mathbf{61.8}$† | – | – |  $11.1$† |  $13.3$† | – | – |  $24.5$† |  $30.0$† | – | –
HProPro |  $\mathbf{51.4}$† |  $55.9$† | $\mathbf{52.9}$ | $\mathbf{57.6}$ |  $\mathbf{46.8}$† |  $\mathbf{54.4}$† | $\mathbf{46.5}$ | $\mathbf{57.5}$ |  $\mathbf{48.0}$† |  $\mathbf{54.6}$† | $\mathbf{48.7}$ | $\mathbf{57.7}$

Table 1: Experimental results on HybridQA. † stands for running on $200$
sampled cases from the validation set.

###  3.4 Main Results

#### Results on HybridQA

According to the results presented in Table 1, it is evident that HProPro
outperforms all baseline systems among approaches w.o. fine-tuning. GPT-4 End-
to-End QA w. Retriever stands for leveraging GPT-4 to generate answers
directly along with a retriever. To conduct this experiment, we follow the
retrieval approach proposed by Chen et al. (2020b). In comparison to GPT-4
End-to-End QA w. Retriever, HProPro  achieves more than a $20\%$ improvement
in both EM and F1 scores. This result demonstrates the effectiveness of
HProPro  compared with the approaches relied on retrievers. However, it is
important to note that HProPro  still exhibits a significant performance gap
when compared to the state-of-the-art approaches w. fine-tuning. We argue that
the main reason for this gap is that these methods are fully trained on the
HybridQA dataset. These systems focus on domain-specific training, which
includes training a retriever Wang et al. (2022); Lei et al. (2023), ranker
Kumar et al. (2021), or reasoner Eisenschlos et al. (2021); Lee et al. (2023).
These domain-specific components may lack flexibility and generalization in
handling diverse scenarios.

#### Results on MultiModalQA

Table 2 summarizes the results obtained on the MultiModalQA dataset, where
HProPro  achieves state-of-the-art performances across all experimental
settings. When considering systems w.o. fine-tuning, HProPro  outperforms the
previous system MMHQA-ICL by $4.2\%$ and $0.9\%$ in terms of EM and F1 scores,
respectively. In comparison to the baseline systems Binder and MMHQA-ICL,
which utilize modal transformation modules to convert images into texts,
HProPro  employs various functions to directly extract information from
different modalities. This approach avoids information loss and is more
suitable for real-world scenarios involving heterogeneous data. It is
important to note that the improvements achieved by HProPro  are non-trivial,
considering that MMHQA-ICL leverages domain-specific fine-tuned classifiers
and retrievers to obtain the type and relevant passages of each question
respectively, which heavily relies on the distribution of the targeted
benchmark. In contrast, HProPro  is performed without any supervised signals
from the training set, resulting in a more universal approach.

In the oracle setting which golden passages and images are provided as the
input, HProPro  achieves comparable results to the previous state-of-the-art
system MMHQA-ICL in terms of EM. Demonstrating that regardless of the
retriever (only focus on the reasoning part), the results prove that their
work highly relies on the retrievers to gain the performances. Besides,
compared to their approach, HProPro  follows a code generation and execution
paradigm, which provides enhanced interpretability and generalizability.

Models | EM | F1
---|---|---
Approaches w. Fine-tuning
Implicit-Decomp Talmor et al. (2020) | $48.8$ | $55.5$
AutoRouting Talmor et al. (2020) | $42.1$ | $49.5$
SKURG Yang et al. (2023) | $59.4$ | $63.8$
PReasM-Large Yoran et al. (2022) | $59.0$ | $65.5$
Approaches w.o. Fine-tuning
Binder Cheng et al. (2022) | $51.0$ | $57.1$
MMHQA-ICL Liu et al. (2023) | $54.8$ | $65.8$
HProPro | $\mathbf{59.0}$ | $\mathbf{66.7}$
Approaches in Oracle Setting
Binderoracle Cheng et al. (2022) | $58.1$ | $64.5$
MMHQA-ICLoracle Liu et al. (2023) | $65.0$ | $\mathbf{75.9}$
HProPro | $\mathbf{65.1}$ | $73.1$
Table 2: Experimental results on MultiModalQA.

Models | HybridQA | MultiModalQA
---|---|---
EM | F1 | EM | F1
HProPro | $\mathbf{48.0}$ | $\mathbf{54.6}$ | $\mathbf{56.0}$ | $\mathbf{62.8}$
– "check" | $44.5$ | $52.5$ | $35.5$ | $38.0$
– Question Simplification | $43.0$ | $50.0$ | $37.0$ | $39.4$
– Code Reflection | $43.5$ | $50.9$ | $54.0$ | $60.5$

Table 3: Ablation studies on HybridQA and MultiModalQA. All ablation studies
are performed on $200$ randomly sampled subsets from validation sets.

###  3.5 Ablation Study

#### Effect of the function "check"

The function "check" is designed to compare the semantic relations between two
objects, offering greater flexibility compared to arithmetic operators such as
">", "<", and "==". To demonstrate the effectiveness of the "check" function,
we conduct ablation studies by removing its definition in both the function
declaration and function implementation processes. Table 3 presents the
results of these ablation studies, highlighting the impact of the "check"
function. When the "check" function is removed, there is a noticeable drop of
$3.5\%$ and $2.1\%$ points in terms of EM and F1 scores, respectively, in the
HybridQA dataset. Moreover, the removal of the "check" function has an even
more substantial impact on the MultiModalQA dataset. Specifically, the results
drop by more than $20\%$ for both EM and F1 scores. This is because
constraints from images are weaker than those from passages since LLMs can
copy spans from the passages as the answer, which improves the need for the
"check" function in this set of experiments.

#### Effect of query simplification

The purpose of query simplification is to alleviate the burden of the code
generation process by simplifying the question and establishing links between
the question and the table cells. In Table 3, we present the effectiveness of
query simplification on both the HybridQA and MultiModalQA datasets. When
query simplification is removed, the results demonstrate a decrease of
approximately $2\%$ on the HybridQA dataset and a substantial drop of about
$20\%$ on the MultiModalQA dataset. These findings highlight the effectiveness
of query simplification in the HQA task. It is important to note that the
removal of the query simplification module leads to a significant drop
specifically in the MultiModalQA dataset. We posit that this drop is due to
the presence of passages and images that are necessary to answer the question
but are not linked in the table, which couldn’t be accessed by the model from
the prompt. Therefore, performing query simplification becomes crucial in
handling such scenarios in the MultiModalQA task.

#### Effect of code refinement

The code refinement module aims to enable LLMs to reconsider the code
generation process based on previous execution traceback. In Table 3, we can
observe the impact of removing the code refinement module on both the HybridQA
and MultiModalQA datasets. When the code refinement module is removed, there
is a noticeable decrease in performance. In the HybridQA dataset, the EM and
F1 scores drop by $4.5\%$ and $3.7\%$ respectively. Similarly, in the
MultiModalQA dataset, the EM and F1 scores drop by $2.0\%$ and $2.3\%$
respectively. The drop in performance demonstrates the effectiveness of the
code refinement module in HProPro . By enabling LLMs to refine their code
generation process based on previous execution errors, the code refinement
module plays a vital role in generating accurate code, thereby enhancing the
overall ability of HProPro  in the HQA task.

Figure 5: Error percentage of HProPro  on HybridQA and MultiModalQA.

###  3.6 Error Analysis

We analyze the errors that occurred within randomly selected subsets of $200$
cases from the validation sets of HybridQA and MultiModalQA. Our examination
reveals that the main errors can be classified into four distinct types, with
the corresponding percentages depicted in Figure 5.

The first type of error involves predicted answers that possess similar
meanings to the golden answers but differ in their expressions ($25.0\%$ for
HybridQA and $26.1\%$ for MultiModalQA). For instance, an instance may present
the predicted answer as "the Southeastern Conference (SEC)", while the correct
answer is simply "Southeastern Conference". From a technological perspective,
we contend that such cases have already been resolved, as the underlying code
solution is entirely accurate.

The second type of error observed is related to execution failure ($11.5\%$
for HybridQA and $5.7\%$ for MultiModalQA). This error arises due to the
inherent complexity of the heterogeneous data, which lacks a standardized
format and therefore cannot be effectively addressed using a uniform solution.

The third type of error pertains to failures in the information-seeking from
heterogeneous data sources ($37.5\%$ for HybridQA and $29.5\%$ for
MultiModalQA). These errors occur when the "extract_info" function fails to
produce a valid result. This may be attributed to a mismatch between the
generated code and the expected solution for answering the given question, or
it could be indicative of instability in the implementation of the
"extract_info" function.

The last type of error involves wrong predicted answers ($26.0\%$ for HybridQA
and $38.7\%$ for MultiModalQA). Due to the similarity between contents in
different columns, the model encounters difficulty in discerning the
appropriate location to locate the answer when generating code solely based on
the provided table. Addressing this challenge remains a topic for future
research.

For the detailed visualization results of this analysis, please refer to
Appendix C.

##  4 Related Work

###  4.1 Hybrid Question Answering

The first line of our related work introduces the HQA task, which focuses on
answering questions that require reasoning over diverse information sources.
Currently, HQA can be broadly categorized into three subtasks based on the
nature of the information sources: table-text question answering Chen et al.
(2020b, a); Zhu et al. (2021), image-text question answering Reddy et al.
(2022); Singh et al. (2021), and table-image-text question answering Hannan et
al. (2020); Talmor et al. (2020). Numerous approaches have been explored for
reasoning over heterogeneous data in the context of HQA. Many of these methods
primarily focus on supervised fine-tuning over specific benchmarks. This
includes training dedicated retrievers Wang et al. (2022); Kumar et al.
(2021); Lei et al. (2023), rankers Kumar et al. (2021), reasoners Wang et al.
(2022); Kumar et al. (2021); Eisenschlos et al. (2021); Lee et al. (2023); Lei
et al. (2023), or transforming different modalities of data into a unified
modality Cheng et al. (2022); Liu et al. (2023); Li et al. (2021). In contrast
to existing works, HProPro  performs reasoning over heterogeneous data without
relying on domain-specific retriever and modal transformation modules.
Instead, it integrates various functions to facilitate information-seeking
across data from different sources and modalities.

###  4.2 Program-based Prompting

The second line of our related work focuses on the program-based prompting
strategy, with two closely related works: Program-of-Thought-Prompting Chen et
al. (2022); Gao et al. (2023) and Binder Cheng et al. (2022). Program-of-
Thought-Prompting Chen et al. (2022); Gao et al. (2023) generates code and
executes it using an interpreter. However, their approach is not designed to
handle heterogeneous data. In contrast, HProPro  integrates function
declaration and implementation to specify different functions, enabling
effective handling of heterogeneous data. On the other hand, Binder Cheng et
al. (2022) converts images into passages and pre-retrieves relevant passages.
These passages, along with the table and question, are then fed into LLMs to
generate SQL and Python code for solving the question. In comparison, HProPro
does not rely on a modal transformation module or a retriever. Instead, it
utilizes various functions to directly interact with data from different
sources and modalities.

##  5 Conclusion

In this work, we propose HProPro , a novel program-based prompting framework
for HQA tasks, which does not require domain-specific retriever and modal
transformation, but integrates various functions to interact with
heterogeneous data instead. Experimental results on two typical HQA benchmarks
HybridQA and MultiModalQA show the effectiveness of HProPro  that HProPro
achieves the best performances under the few-shot settings. For future work,
we hope to further utilize the coding capabilities of the LLMs, allowing the
model to judge and self-create more customized functions based on different
scenarios.

## Limitations

The main limitation of this paper is that the performance of HProPro  relies
on the abilities of LLMs, which vary according to the different choices of
LLMs. Model updates and server status may affect our experimental results. In
addition, the existing benchmarks only focus on heterogeneous data containing
tables, passages, and images. More types of data including knowledge graphs
and charts are expected to be explored in the future.

## Ethics Statement

In this paper, we propose HProPro , a program-based prompting framework for
the HQA task. We conduct experiments on two benchmarks, namely, HybridQA and
MultiModalQA. Both datasets are free and open for research use, which means no
ethics issues.

## References

  * Chen et al. (2020a) Wenhu Chen, Ming-Wei Chang, Eva Schlinger, William Yang Wang, and William W Cohen. 2020a.  Open question answering over tables and text.  In _International Conference on Learning Representations_.
  * Chen et al. (2022) Wenhu Chen, Xueguang Ma, Xinyi Wang, and William W Cohen. 2022.  Program of thoughts prompting: Disentangling computation from reasoning for numerical reasoning tasks.  _arXiv preprint arXiv:2211.12588_.
  * Chen et al. (2020b) Wenhu Chen, Hanwen Zha, Zhiyu Chen, Wenhan Xiong, Hong Wang, and William Yang Wang. 2020b.  Hybridqa: A dataset of multi-hop question answering over tabular and textual data.  In _Findings of the Association for Computational Linguistics: EMNLP 2020_ , pages 1026–1036.
  * Cheng et al. (2022) Zhoujun Cheng, Tianbao Xie, Peng Shi, Chengzu Li, Rahul Nadkarni, Yushi Hu, Caiming Xiong, Dragomir Radev, Mari Ostendorf, Luke Zettlemoyer, et al. 2022.  Binding language models in symbolic languages.  In _The Eleventh International Conference on Learning Representations_.
  * Eisenschlos et al. (2021) Julian Eisenschlos, Maharshi Gor, Thomas Mueller, and William Cohen. 2021.  Mate: Multi-view attention for table transformer efficiency.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pages 7606–7619.
  * Feng et al. (2022) Yue Feng, Zhen Han, Mingming Sun, and Ping Li. 2022.  Multi-hop open-domain question answering over structured and unstructured knowledge.  In _Findings of the Association for Computational Linguistics: NAACL 2022_ , pages 151–156.
  * Gao et al. (2023) Luyu Gao, Aman Madaan, Shuyan Zhou, Uri Alon, Pengfei Liu, Yiming Yang, Jamie Callan, and Graham Neubig. 2023.  Pal: Program-aided language models.  In _International Conference on Machine Learning_ , pages 10764–10799. PMLR.
  * Goyal et al. (2017) Yash Goyal, Tejas Khot, Douglas Summers-Stay, Dhruv Batra, and Devi Parikh. 2017\.  Making the v in vqa matter: Elevating the role of image understanding in visual question answering.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pages 6904–6913.
  * Hannan et al. (2020) Darryl Hannan, Akshay Jain, and Mohit Bansal. 2020.  Manymodalqa: Modality disambiguation and qa over diverse inputs.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 34, pages 7879–7886.
  * Kumar et al. (2021) Vishwajeet Kumar, Saneem Chemmengath, Yash Gupta, Jaydeep Sen, Samarth Bharadwaj, and Soumen Chakrabarti. 2021.  Multi-instance training for question answering across table and linked text.  _arXiv preprint arXiv:2112.07337_.
  * Lee et al. (2023) Sung-Min Lee, Eunhwan Park, Daeryong Seo, Donghyeon Jeon, Inho Kang, and Seung-Hoon Na. 2023.  Mafid: Moving average equipped fusion-in-decoder for question answering over tabular and textual data.  In _Findings of the Association for Computational Linguistics: EACL 2023_ , pages 2337–2344.
  * Lei et al. (2023) Fangyu Lei, Xiang Li, Yifan Wei, Shizhu He, Yiming Huang, Jun Zhao, and Kang Liu. 2023.  S3HQA: A three-stage approach for multi-hop text-table hybrid question answering.  _arXiv preprint arXiv:2305.11725_.
  * Li et al. (2021) Xiao Li, Yawei Sun, and Gong Cheng. 2021.  Tsqa: tabular scenario based question answering.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 35, pages 13297–13305.
  * Liu et al. (2023) Weihao Liu, Fangyu Lei, Tongxu Luo, Jiahe Lei, Shizhu He, Jun Zhao, and Kang Liu. 2023.  Mmhqa-icl: Multimodal in-context learning for hybrid question answering over text, tables and images.  _arXiv preprint arXiv:2309.04790_.
  * Madaan et al. (2023) Aman Madaan, Niket Tandon, Prakhar Gupta, Skyler Hallinan, Luyu Gao, Sarah Wiegreffe, Uri Alon, Nouha Dziri, Shrimai Prabhumoye, Yiming Yang, et al. 2023\.  Self-refine: Iterative refinement with self-feedback.  _arXiv preprint arXiv:2303.17651_.
  * Pan et al. (2021) Liangming Pan, Wenhu Chen, Wenhan Xiong, Min-Yen Kan, and William Yang Wang. 2021\.  Unsupervised multi-hop question answering by question generation.  In _Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 5866–5880.
  * Pasupat and Liang (2015) Panupong Pasupat and Percy Liang. 2015.  Compositional semantic parsing on semi-structured tables.  In _Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pages 1470–1480.
  * Rajpurkar et al. (2016) Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and Percy Liang. 2016.  Squad: 100,000+ questions for machine comprehension of text.  In _Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing_ , pages 2383–2392.
  * Reddy et al. (2022) Revant Gangi Reddy, Xilin Rui, Manling Li, Xudong Lin, Haoyang Wen, Jaemin Cho, Lifu Huang, Mohit Bansal, Avirup Sil, Shih-Fu Chang, et al. 2022.  Mumuqa: Multimedia multi-hop news question answering via cross-media knowledge extraction and grounding.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 36, pages 11200–11208.
  * Reimers and Gurevych (2020) Nils Reimers and Iryna Gurevych. 2020.  Making monolingual sentence embeddings multilingual using knowledge distillation.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing_. Association for Computational Linguistics.
  * Singh et al. (2021) Hrituraj Singh, Anshul Nasery, Denil Mehta, Aishwarya Agarwal, Jatin Lamba, and Balaji Vasan Srinivasan. 2021.  Mimoqa: Multimodal input multimodal output question answering.  In _Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 5317–5332.
  * Sun et al. (2021) Haitian Sun, William W Cohen, and Ruslan Salakhutdinov. 2021.  End-to-end multihop retrieval for compositional question answering over long documents.  _arXiv preprint arXiv:2106.00200_.
  * Talmor et al. (2020) Alon Talmor, Ori Yoran, Amnon Catav, Dan Lahav, Yizhong Wang, Akari Asai, Gabriel Ilharco, Hannaneh Hajishirzi, and Jonathan Berant. 2020.  Multimodalqa: complex question answering over text, tables and images.  In _International Conference on Learning Representations_.
  * Wang et al. (2022) Yingyao Wang, Junwei Bao, Chaoqun Duan, Youzheng Wu, Xiaodong He, and Tiejun Zhao. 2022.  Muger2: Multi-granularity evidence retrieval and reasoning for hybrid question answering.  In _Findings of the Association for Computational Linguistics: EMNLP 2022_ , pages 6687–6697.
  * Yang et al. (2023) Qian Yang, Qian Chen, Wen Wang, Baotian Hu, and Min Zhang. 2023.  Enhancing multi-modal multi-hop question answering via structured knowledge and unified retrieval-generation.  In _Proceedings of the 31st ACM International Conference on Multimedia_ , pages 5223–5234.
  * Yoran et al. (2022) Ori Yoran, Alon Talmor, and Jonathan Berant. 2022.  Turning tables: Generating examples from semi-structured tables for endowing language models with reasoning skills.  In _Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 6016–6031.
  * Zhu et al. (2021) Fengbin Zhu, Wenqiang Lei, Youcheng Huang, Chao Wang, Shuo Zhang, Jiancheng Lv, Fuli Feng, and Tat-Seng Chua. 2021.  Tat-qa: A question answering benchmark on a hybrid of tabular and textual content in finance.  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 3277–3287.

##  Appendix A Detail Prompts of of HProPro

The system prompt and detail prompts of the function "extract_info", function
"check", code refinement, and query simplification are shown in Figure 7,
Figure 8, Figure 9 and Figure 10.

##  Appendix B Data Statistics for Each Dataset

The statistics of HybridQA and MultiModalQA are presented in Table 4 and Table
5.

##  Appendix C Error Analysis

The error analysis results are presented in Figure 6.

Figure 6: Details of the system prompt. Figure 7: Details of the prompt of "extract_info" function. Figure 8: Details of the prompt of "check" function. Figure 9: Details of the prompt of the code refinement process. Figure 10: Details of the prompt of the query simplification process. Split | Train | Dev | Test | Total
---|---|---|---|---
In-Passage | 35,215 | 2,025 | 20,45 | 39,285 (56.4%)
In-Table | 26,803 | 1,349 | 1,346 | 29,498 (42.3%)
Computed | 664 | 92 | 72 | 828 (1.1%)
Total | 62,682 | 3,466 | 3,463 | 69,611
Table 4: Data Statistics for HybridQA.

_Measurement_ | _Value_
---|---
# Distinct Questions | 29,918
Train multimodal questions | 34.6%
Dev.+test multimodal questions | 40.1%
Train compositional questions | 58.8%
Dev.+test compositional questions | 62.3%
Average question length (words) | 18.2
Average # of answers per question | 1.16
List answers | 7.4%
List answers per intermediate question | 18.9%
Average answer length (words) | 2.1
# of distinct words in questions | 49,649
# of distinct words in answers | 20,820
# of distinct context tables | 11,022

Table 5: Data statistics for MultiModalQA.

Type |  Question |  Generated Code |  Result Comparison
---|---|---|---
EM Not Match |  What conference does the team won in 2016 belong to? |  |  The predicted answer is "the Southeastern Conference ( SEC )", while the golden answer is "Southeastern Conference". The code has the completely correct solution but has different expressions with the golden answer.
Execution Error |  Who was the older person involved in writing the book from 2000? |  |  The golden answer is "Sally Jenkins", while the existence of function "split" and "int" make the code fail to execute since the heterogeneous data is complex.
Failed Information Seeking |  What shipbuilder designed the ship that the delivery voyage was the longest by a patrol boat? |  |  The golden answer is "Brooke Marine", while the code cannot locate the answer since the "extract_info" returns "None".
Wrong Predicted Answer |  At which air force base did the pilot who graduated in 1968 serve ? |  |  The prediction answer is "Hurlburt Field", while the golden answer is "Hickam". The model cannot distinguish where to find the answer between the columns "ClassYear" and "Notability" since the information in the two columns is similar.

Table 6: Main error types of HProPro  in HybridQA and MultiModalQA.


LHAASO J2226+6057 as a pulsar wind nebula

Agnibha De Sarkar1,2,
Wei Zhang2,4,
Jonatan Martín3,
Diego F. Torres2,4,5,
Jian Li 6,7,
Xian Hou 8,9

Raman Research Institute, Bengaluru 560080, Karnataka, India
Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Magrans s/n, E-08193 Barcelona, Spain
INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy
Institut d’Estudis Espacials de Catalunya (IEEC), E-08034 Barcelona, Spain
Institució Catalana de Recerca i Estudis Avanc̨ats (ICREA), Barcelona, Spain
CAS Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Hefei, China
School of Astronomy and Space Science, University of Science and Technology of China, Hefei, China
Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, China
Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650216, China

The Large High Altitude Air Shower Observatory has reported the detection of cosmic-ray sources in Milky Way that can accelerate particles up to PeV (= 10$^{15}$ eV) energies. These sources, so called “PeVatrons”, are mostly unidentified. Several classes of sources, such as supernova remnants, pulsar wind nebula, or young stellar clusters can potentially be the counterparts of these PeVatrons.

The aim of this work is to study a pulsar wind nebula interpretation of one of these PeVatrons, LHAASO J2226+6057, which has a relatively well covered multi-frequency spectrum.

We have performed a leptonic, time-dependent modeling of the pulsar wind nebula (PWN) associated with PSR J2229+6114 considering a time-energy-dependent diffusion-loss equation. Injection, energy losses, as well as escape of particles were considered to balance the time-dependent lepton population. We have also included the dynamics of the PWN and the associated supernova remnant (SNR) and their interaction via the reverse shock to study the reverberation phase of the system.

We have considered different values of braking index ($n$) and true age ($t_{age}$) for the fitting of the multi-wavelength (MWL) spectral energy distribution (SED) of LHAASO J2226+6057. The best-fit PWN model parameters and their 1$\sigma$ confidence intervals were evaluated. We have also demonstrated the impact of reverberation on the MWL SED with increasing time. Additionally, we have discussed the resultant large radius and low magnetic field associated with the PWN in question, as caveats for the possible physical connection of the pulsar as the origin of this high energy source.

LHAASO J2226+6057
De Sarkar et al.

§ INTRODUCTION

Recent observations by state-of-the-art observatories such as the
Large High Altitude Air Shower Observatory (LHAASO), Tibet AS$\gamma$, the
High Altitude Water Cherenkov (HAWC), and others, have paved the way for the detection of multiple Galactic ultra-high energy (UHE; E$_\gamma \ge$ 100 TeV) gamma ray sources <cit.>.

Upcoming observatories such as the Cherenkov Telescope Array <cit.> and the Southern Wide-field Gamma-ray Observatory <cit.> will be of importance to identify and characterize these PeVatrons: Galactic CR sources that accelerate particles up to PeV energies.

LHAASO is a state-of-the-art dual-task facility designed for CRs and gamma ray studies at few hundred GeV to few PeV, located at 4410 m above sea level in China <cit.>. The recent data reported by the LHAASO observatory shows the existence of 12 significantly detected sources ($>$ 7$\sigma$) that emit gamma rays with energies above several hundred TeVs <cit.>.

Most of the sources reported by LHAASO have significantly extended gamma ray emission
regions up to $\sim$ 1$^\circ$. The very high energy (VHE; 100 GeV $\le$ E$_\gamma$ $\le$ 100 TeV) counterparts of these sources residing in the Galactic plane have been associated with pulsar wind nebulae (PWNe), based on the spatial proximity with highly energetic pulsars and typically extended morphological features <cit.>.

It has already been posited that UHE gamma ray emission spatially coincident or in very close proximity of energetic pulsars with high spin-down luminosity ($\Dot{E} >$ 10$^{36}$ erg s$^{-1}$) may be a universal feature, see e.g., <cit.>. Moreover, the Crab nebula, associated with pulsar PSR B0531+21, was confirmed to be a PeVatron by recent LHAASO observations <cit.>. Bearing all of this in mind, it is natural to consider PWNe as  possible Galactic PeVatrons, from which UHE gamma rays are detected.

PWNe, considered to be one of the most efficient lepton accelerators in Galaxy, are powered by highly energetic pulsars.

Pulsars dissipate most of their rotational energy via the injection of ultrarelativistic electron-positron pairs, which form a cold, ultrarelativistic wind of particles. Since the bulk velocity of this ultrarelativistic wind is supersonic with respect to the ambient medium, this wind creates a termination shock. Injected particles can be accelerated to very high energies at this termination shock.

The accelerated leptons can then interact with the ambient matter, photon fields, and the magnetic field through Bremsstrahlung, inverse-Compton (IC) and synchrotron processes. The cooling of the accelerated leptons results in a MWL spectrum ranging from radio to gamma ray energies.

Here, we take LHAASO J2226+6057 as our source of interest, as this is the only UHE gamma ray source for which data have been observed across radio, X-ray, and gamma ray energy ranges.

It has been observed that LHAASO J2226+6057 is situated in a complex morphological region. Due to the close spatial proximity of Boomerang PWN, as well as SNR G106.3+2.7 and associated molecular clouds (MCs), it is hard to confirm the exact source responsible for UHE gamma ray emission observed. From the observations of Imaging Air Cherenkov Telescopes (IACTs) such as MAGIC and VERITAS, it was found that the emission region was divided into two morphological regions; the head and the tail.

The head region contains Boomerang PWN and PSR J2229+6114, and the tail region contains VER J2227+608, which is likely to be associated with SNR G106.3+2.7 and the MCs in the region. Faint and diffuse radio and X-ray emission were also observed from the tail region. It is possible that the emission from Boomerang PWN illuminates the head region, whereas hadronic interaction occurring between the SNR and the MCs are responsible for the gamma ray emission in the tail.

Although it is difficult to confirm which source is actually responsible for the UHE gamma ray emission, comprehensive studies are needed to explore both scenarios. New data from upcoming observations with high angular resolution by MAGIC and VERITAS will be crucial to shed new light on this source.

We discuss the features of LHAASO J2226+6057, as well as the previous works done for this source in Sec. <ref>. Then we comment on the PWN model used in this work in Sec. <ref>, and present the results in Sec. <ref>. We finally conclude in Sec. <ref>.

§ LHAASO J2226+6057 FEATURES

LHAASO J2226+6057 was detected at RA = 336.75$^\circ$ and Decl. = 60.95$^\circ$ with a significance of 13.6$\sigma$ above 100 TeV.

Its gamma ray spectrum reaches up to a maximum energy of 0.57 $\pm$ 0.19 PeV <cit.>. This source is spatially associated with the supernova remnant SNR G106.3+2.7, as well as the pulsar J2229+6114 and its wind nebula, known as the “Boomerang” nebula <cit.>.

PSR J2229+6114 is a bright gamma ray pulsar with a spin period of 51.6 ms, a characteristic age of 10460 yr and a spin-down luminosity of 2.2 $\times$ 10$^{37}$ erg s$^{-1}$ <cit.>.

Pulsed GeV gamma ray emission from this pulsar was detected by Fermi-LAT <cit.>.

In the VHE range, SNR G106.3+2.7 was observed by VERITAS as VER J2227+608, located 0.4$^{\circ}$ away from PSR J2229+6114 <cit.>. GeV gamma rays <cit.>, diffuse non-thermal X-rays <cit.>, as well as radio data <cit.> have also been observed from the source region.

The distance of the source suffers from great uncertainty. It was estimated to be 7.5 kpc <cit.>, 3 kpc <cit.>, and 0.8 kpc <cit.>. In this work we consider the distance to be 3 kpc, similar to <cit.> and <cit.>.

The possible connection between LHAASO J2226+6057 and PSR J2229+6114 has previously been studied by <cit.>, <cit.> and <cit.>.

<cit.> has performed a time-independent one-zone treatment, a steady-state leptonic scenario from the PWN, to explain only the highest energy gamma ray data. The evolution of the source was neglected, as well as MWL data were not used.

<cit.> have used a time-dependent one-zone leptonic scenario from the PWN. However, the authors did not consider the effect of escape for LHAASO J2226+6057, which they accounted only for LHAASO J1908+0621, nor explored the effect of age and braking index on the evolution of the injected leptonic population. The impact of SNR reverse shock and its effects on PWN radius evolution were not taken into account, assuming that such effects are important only if the age of the PWN is greater than 10 kyr, which is not necessarily the case, see, e.g., <cit.>.

<cit.> have also performed a similar study. The authors had argued that a distorted nebula, created due to the impact of SNR reverse shock, is responsible for the GeV gamma ray emission observed from the source region by <cit.>. Their results are similar to those of <cit.>.

Our PWN model here intends to test their conclusions after relaxing assumptions or adding additional physical details.

A recent paper by <cit.> proposed a hadronic origin of the LHAASO source based on spatial proximity of a molecular cloud (MC) with the gamma ray centroid of the source. However, as pointed out by <cit.>, the associated SNR is quite old to produce the observed hard gamma ray spectrum at the highest energies. Consequently, a hadronic scenario from a SNR+MC association would need a peculiar modeling to explain the observed UHE gamma ray spectrum observed from the source. A novel approach was explored in <cit.> to explain the hadronic origin of LHAASO J1908+0621 and perhaps, a similar approach may play a role here as well.

§ BRIEF DESCRIPTION OF THE MODEL

For this work, we have used the code , for which earlier applications can be found in, e.g., <cit.>.

The numerical code solves the evolution of leptonic pair distribution in the PWN, as a function of Lorentz factor $\gamma$ at time $t$ described by the equation,

\begin{equation}\label{eq1}
\frac{\partial N(\gamma, t)}{\partial t} = Q(\gamma, t) - \frac{\partial}{\partial \gamma} [\dot{\gamma}(\gamma, t) N(\gamma, t)] - \frac{N(\gamma, t)}{\tau(\gamma, t)}
\end{equation}

The left hand side of equation <ref> describes the variation of lepton distribution in time. The first term on the right hand side is the lepton injection function Q($\gamma$, t), which is usually assumed as a broken power law

\begin{equation}
\label{eq2}
\begin{split}
Q(\gamma, t) = Q_0(t)
\begin{cases}
(\gamma/\gamma_b)^{-\alpha_1}&  \gamma \le \gamma_b,\\
(\gamma/\gamma_b)^{-\alpha_2} &  \gamma > \gamma_b,\\
\end{cases}
\end{split}
\end{equation}

The second and third terms on the right hand side take into account radiative losses such as synchrotron, IC, Bremsstrahlung, adiabatic losses or heating, as well as escape of the particles (we assume Bohm diffusion) respectively (see <cit.> for the incorporated formulae).

The normalization factor $Q_0$(t) is calculated using the spin down luminosity L(t) of the pulsar through the equation,
(1 - \eta)L(t) = \int^{\gamma_{max}}_{\gamma{min}}\gamma m_e c^2 Q(\gamma, t) d\gamma
where $\eta$ is the fraction of spin down power that goes on to power the magnetic field of the PWN. The magnetic field varies in time as a result of the balance between the magnetic field power and adiabatic gains or losses of the field due to the contraction or the expansion of the PWN <cit.>.
The reverberation phase of the PWN included in the model, has been considered to be the same as that in <cit.>. The variation of the PWN radius is calculated by taking into account the age, SNR explosion energy, ambient medium density, expansion velocity of the nebula and pressure profiles of SNR at the position of the PWN. The model takes into account the change in pressure profiles depending on whether the PWN shell is surrounded by unshocked ejecta (i.e. R $<$ R$_{RS}$), or the shocked ejecta (i.e. R$_{RS}$ $<$ R $<$ R$_{SNR}$), where R$_{RS}$ and R$_{SNR}$ are the radii of the SNR reverse shock and the SNR respectively. After reverberation, an assumed Sedov expansion follows when the PWN reaches the pressure of the SNR. For more details, please see <cit.>. For a more detailed accounting of the reverberation period, we need to account for the fact that the
ejecta pressure is not constant <cit.>. This is beyond the scope of this work, as for the moment, a prescription to deal with this fact
in the context of radiative PWN is unavailable. Given the likely young age of the source, a PWN would be
before the time of largest compression (see below),
and thus we expect the current approach to be acceptable.

§ RESULTS

§.§ Braking index and true age exploration

In the works of <cit.> and <cit.> the true age and the braking index were chosen to be 7000 years and 3, respectively.   Since the choice of the true age is essential for a time evolutionary model of PWNe, we explore other values here as well.

On the other hand, if the pulsar is assumed to be a dipole in vacuum, and considered to be spinning down by emitting only magnetic dipole radiation (MDR), then the corresponding braking index associated with the spin down is calculated to be 3 <cit.>.

Alternatively, the spin down of a pulsar driven entirely by a particle wind would result in a braking index of 1 <cit.>. A combination of magnetic dipole radiation and wind braking would result in a braking index with a value in between 1 and 3 <cit.>. Most of the observed pulsar braking indices falls within this range (<cit.>, and references therein). Although there are exceptions, for example PSR J1640–4631, which has a braking index of n = 3.15 $\pm$ 0.03 <cit.> and PSR J1734–3333 has a braking index of n = 0.9 $\pm$ 0.2 <cit.>.

For this work, we only consider the braking index range $1 < n < 3$. Since the choice of the braking index affects the characteristics of the pulsar spin down, it is also important to explore whether the variation of $n$ affects the SED.

The LHAASO J2226+6057 MWL SEDs along with the calculated model flux (top row), present lepton spectrum (middle row) and magnetic field evolution with time (bottom row) are given for $t_{age}$ = 1000 years (left column), 4000 years (middle column) and 7000 years (right column), for a fixed braking index n = 2.5. The radio data (green) and X-ray data (royalblue) are taken from <cit.> and <cit.> respectively. Fermi-LAT (brown), VERITAS (teal), Tibet AS$\gamma$ (turquoise), MILAGRO (orange) and LHAASO (crimson) data are taken from <cit.>, <cit.>, <cit.>, <cit.> and <cit.>, respectively.

MWL SEDs of LHAASO J2226+6057 for $n = 1.5, 2.0, 2.5$, and 3.0. The color scheme of the data points is same as given in Figure <ref>.

To explore the effect of $n$ and $t_{age}$ on the MWL SED, we have chosen $t_{age}$ = 1000, 4000 and 7000 years, and $n = 1.5, 2.0, 2.5$ and 3.0. Given the characteristic age $\tau_c$ $\approx$ 10.5 kyr and present day spin down luminosity $L(t_{age}) \approx 2.25 \times 10^{37}$ erg/s, the initial spin down age ($\tau_0$) and initial spin down luminosity ($L_0$) were calculated using the relations,

\begin{equation}
\label{eq4}
L (t) = L_0 \left(1 + \frac{t}{\tau_0}\right)^{-\frac{n+1}{n-1}},
\end{equation}

\begin{equation}
\label{eq5}
\tau_0 = \frac{2 \tau_c}{n - 1} - t,
\end{equation}

for a specific choice of $n$ and $t_{age}$.

We also assume the values of the following parameters are the same throughout the work; minimum Lorentz factor $\gamma_{min}$ = 1, SN explosion energy E$_{SN}$ = 10$^{51}$ erg, interstellar medium (ISM) ambient density $\rho_{ISM}$ = 0.1 cm$^{-3}$, SNR core density index = 0, SNR envelope density index = 9 (we assume a type II SN as the progenitor, as in <cit.>), PWN adiabatic index = 1.333 and SNR adiabatic index = 1.667, containment factor $\epsilon$ (ratio of the Larmor radius of particles to the radius of the termination shock) = 0.5 and magnetic compression ratio $\kappa$ = 3 (strong shock condition).

The ejected mass of the progenitor SNR was also fixed at $M_{ej}$ = 8$M_{\odot}$ for the $n-t_{age}$ exploration.

For the target radiation fields for the IC interaction, we consider the cosmic microwave background,

Far-Infrared (FIR) and Near-Infrared (NIR) radiation fields at the source position with temperatures of $T_{FIR}$ = 25 K and $T_{NIR}$ = 5000 K, respectively. The energy densities associated with FIR and NIR field were taken from <cit.> and fixed at those values for $n-t_{age}$ exploration. It has been observed (see <cit.>) that almost always the energy densities of FIR/NIR fields required to explain the IC emission from PWNe, differ from the interstellar radiation fields reported by <cit.>.

<cit.> has also argued that an enhancement of the radiation field is needed to explain the hard gamma ray spectrum observed at highest energies.
However, since the aim of this subsection is to compare the effect of $n$ and $t_{age}$ on the computed SED, we have fixed the energy densities of FIR/NIR fields at the values given by <cit.>, i.e. $\omega_{FIR}$ = 0.29 eV cm$^{-3}$ and $\omega_{NIR}$ = 0.45 eV cm$^{-3}$, so there are less free parameters at play.

The maximum Lorentz factor of the lepton distribution was fixed by the most restrictive condition between the synchrotron limit <cit.> or gyroradius limit <cit.> during the evolution. For a specific choice of $n$ and $t_{age}$, we vary the injection function parameters, i.e. low energy index $\alpha_1$, high energy index $\alpha_2$, energy break $\gamma_b$, as well as the magnetic fraction $\eta$, to describe the MWL SED of the source.

First, we fix the braking index at the value of $n = 2.5$, and consider three different cases of $t_{age}$. The MWL SED along with the computed model flux, injected lepton spectrum and magnetic field at the considered $t_{age}$ are shown in Figure <ref>. From the figure, it can be seen that the fit corresponding to $t_{age}$ = 7000 years is the best of the three ages considered.

For $t_{age}$ = 1000 years and $t_{age}$ = 4000 years, the X-ray and radio data can be adequately explained. However, for $t_{age}$ = 1000 years, the IC emission is not significant enough to explain the high-energy data. Also for $t_{age}$ = 4000 years, the IC emission only partially explains it. From this exploration, it is apparent that the age of the PWN lies within 4000 and 7000 years and we consider $t_{age}$ = 7000 years to compare with <cit.> and <cit.>, albeit these authors did not explore other options. We also discuss the results of considering the true age as a free parameter at a later subsection.

After limiting the age of the system, we move on to explore the effect of braking index on the evolution of the source. Assuming a fixed $t_{age}$ = 7000 years, we change the braking index, and try to describe the MWL SED. $L_0$ and $\tau_0$ have been calculated using equations <ref> and <ref>. We again change $\alpha_1$, $\alpha_2$, $\gamma_b$ and $\eta$, similarly to the previous case.

The calculated spectra for $n = 1.5, 2.0, 2.5,$ and 3.0 are given in Figure <ref>.  $n$ does not significantly affect the computed SEDs and we have selected $n = 2.5$ for further study.

§.§ $\chi^2$ fitting of the MWL SED

We have considered typical PWN parameters as initial input, and, using the  code <cit.>, subsequently varied them to solve equation <ref> and compute the best $\chi^2$ fitted model spectrum.

To find the best-fit spectra, we have used $t_{age}$ = 7000 years and $n = 2.5$. Similar to the previous discussion, $L_0$ and $\tau_0$ have been fixed at the value calculated by equation <ref> and <ref>.

The PWN parameters that have been fixed already in the above discussion are also fixed in this calculation as well, except for the cases of ejected mass and FIR/NIR energy densities (between 0.01 eV cm$^{-3}$ to 5 eV cm$^{-3}$). Since the ejected mass of the progenitor SNR directly affects the size, as well as the magnetic field of the PWN, it was left free within the typical ejected mass range of 7M$_{\odot}$ to 15M$_{\odot}$.

Apart from these changes, similarly to the previous discussion, $\alpha_1$, $\alpha_2$, $\gamma_b$ and $\eta$ were left free to vary to find the best-fit MWL spectrum. The parameters used in the model are reported in Table <ref>, in which we have divided the parameters among measured or assumed, derived, and fitted values. The resulting plots are given in Figure <ref>. The time evolution of the calculated MWL spectrum and injection spectrum are given in Figure <ref>.

From the top row of Figure <ref>, it can be seen that the computed MWL spectrum using the model along with the 1$\sigma$ confidence interval, matches well with the observed MWL data. The goodness of fit can also be seen from the bottom residual plot. The systematic uncertainty associated with the model is  0.32, and the $\chi^2$/D.O.F. for the given fit is 35.65/30. Also from the figure, as well as from Table <ref>, it can be seen that FIR and NIR radiation fields do not contribute to the IC emission needed to explain the VHE-UHE data, rather it was found the CMB is most likely solely responsible as the target photon field required.

A recent work by <cit.> also considered CMB photons as the most relevant target for IC scattering.

However, such result is somewhat different from what was assumed by <cit.>, in which they have considered radiation fields 1.5-3.0 times that of CMB to fit the data.

From the bottom row of Figure <ref> and Table <ref>, it can be seen that the PWN would be  very extended at the present age according to the model, and concurrently the associated magnetic field would be very low, close to the Galactic average value.

The large radius of the PWN may agree with the large extension measured by LHAASO, but it contradicts with the radio and X-ray sizes observed for the Boomerang PWN <cit.>, as these sizes are much smaller compared to the calculated PWN radius. The required magnetic field to fit the data is also uncomfortably low (we discuss this below), and begs the question how are the particles confined in such diluted PWN.

Physical parameters used by and resulting from the fit. The bracketed terms in the fitted parameters section signify the lower and  upper bounds of 1$\sigma$ confidence interval respectively.

Definition     Parameter     Value

Measured or assumed parameters:

Age     $t_{age}$ [kyr]     7

Characteristic age     $\tau_c$ [kyr]     10.5

Braking index     n     2.5

Present day spin down luminosity     L($t_{age}$) [erg s$^{-1}$]     2.25 $\times$ 10$^{37}$

Distance     D [kpc]     3

Minimum energy at injection     $\gamma_{min}$     1

SN explosion energy     E$_{SN}$ [erg]     10$^{51}$

ISM density     $\rho_{ISM}$ [cm$^{-3}$]     0.1

SNR core density index     w$_{core}$     0

SNR envelope density index     w$_{env}$     9

PWN adiabatic index     $\gamma_{PWN}$     1.333

SNR adiabatic index     $\gamma_{SNR}$     1.667

Containment factor     $\epsilon$     0.5

Magnetic compression ratio     $\kappa$     3

CMB temperature     T$_{CMB}$ [K]     2.73

CMB energy density     $\omega_{CMB}$ [eV cm$^{-3}$]     0.25

FIR temperature     T$_{FIR}$ [K]     25

NIR temperature     T$_{NIR}$ [K]     5000

Derived parameters:

Initial spin down luminosity     L$_0$ [erg s$^{-1}$]     1.13 $\times$ 10$^{38}$

Initial spin down age     $\tau_0$ [kyr]     7

Fitted parameters:

Energy break at injection     $\gamma_b$     3338.00 (2082.91, 10597.30)

Low energy index at injection     $\alpha_1$     1.4522 (1.0000, 1.6432)

High energy index at injection     $\alpha_2$     2.3727 (2.3316, 2.3890)

Ejected mass     $M_{ej}$ [$M_{\odot}$]     8.8927 (8.1735, 9.3202)

Magnetic fraction     $\eta$     0.0033 (0.0026, 0.0060)

FIR energy density     $\omega_{FIR}$ [eV cm$^{-3}$]     0.0100 (0.0100, 0.4611)

NIR energy density     $\omega_{NIR}$ [eV cm$^{-3}$]     0.0100 (0.0100, 5.0000)

Resulting features:

PWN radius     R$_{PWN}$ (t$_{age}$) [pc]     9.33

SNR forward shock radius     R$_{FS}$ (t$_{age}$) [pc]     16.23

SNR reverse shock radius     R$_{RS}$ (t$_{age}$) [pc]     8.98

PWN magnetic field     B$_{PWN}$ (t$_{age}$) [$\mu$G]     1.91

§.§ Possible impact of reverberation

From Figure $\ref{fig3}$, and from the resulting values given in Table <ref>, it can be seen that the radius of the SNR reverse shock is smaller than the PWN radius, which means that the reverse shock has just reached the position of PWN shell, at the onset of the reverberation phase of the PWN. If the age of the system is increased further from the considered age, the effect of reverberation will be apparent on the MWL spectrum. Since the PWN has just started to contract at the considered age of 7000 years, it is unlikely that it is heavily distorted.

We consider $t_{age}$ = 7000, 8000, 9000 and 10500 years to study the effect of reverberation on the MWL spectrum. The braking index was chosen to be 2.5, and $L_0$ and $\tau_0$ were calculated using equation <ref> and <ref> for a specific choice of $t_{age}$.

Since we are only interested in comparing the effect of reverberation on the MWL spectrum based on the assumption of different ages of the PWN, we assume the values of <cit.> for the FIR/NIR energy densities. Similarly to subsection <ref>, $\alpha_1$, $\alpha_2$, $\gamma_b$ and $\eta$ were varied to find the most adequately fitted MWL spectrum for each cases of $t_{age}$. The resulting plot is given in the left panel of Figure <ref>.

The MWL spectrum gradually deviates away from the observed MWL SED, and consequently the fit worsens. Since the magnetic field increases due to the compression of the PWN during the reverberation phase, the chosen input of the magnetic fraction $\eta$ was decreased with increasing considered age,
to control the fraction of spin down luminosity that goes on to power the magnetic field of the nebula.

Nevertheless, due to the compression of the nebula and high synchrotron burning, the power law index at high energy gradually softens with age,
indicating efficient cooling of the injected high energy electrons to lower energies during reverberation. This in turn affects the spectrum of the resultant synchrotron and IC photons produced. This fact is apparent from the figure, as the calculated MWL spectrum is unable to explain the observed X-ray and the VHE-UHE data present in the SED.

We have also considered $t_{age}$ = 6500, 6800, 7200 and 7500 years and fitted the SED to study how the magnetic field and radius change. The corresponding plot is given in the right panel of Figure <ref>. The final PWN radius increases with increasing age, and also corresponds to the onset of the PWN contraction due to the reverberation in all cases. The PWN must be at the beginning stage of the contraction if the observed SED is to be explained.
In all cases, the magnetic field is still low and close to, or lower than the Galactic average value.

§.§ $t_{age}$ as a free parameter

From the above discussion, it is apparent that the true age of the PWN can not be much greater than 7000 years, due to reverberation kicking in afterwards. As discussed in subsection <ref>, the true age can not also be lower than 4000 years. Thus, we have used , leaving the true age parameter free.

Apart from $t_{age}$, the free parameters considered in this case are same as that discussed in subsection <ref>.

The best-fit $t_{age}$ from the fitting of the MWL SED is 4880.3 (3871.4, 6060.4) years. This value is consistent with the range of the said parameter discussed above. The best fitted model spectrum is similar to that shown in Figure <ref>. Although the best-fit values of the free parameters, apart from t$_{age}$, are not exactly the same to those obtained assuming  $t_{age}$ = 7000 years, the 1$\sigma$ confidence intervals of these free parameters for the two cases, overlap with each other  (see subsection <ref> and Table <ref>). Both solutions are essentially the same within uncertainties. So we do not report the best fit results obtained in this case. We have also obtained a comparatively large PWN radius ($R_{PWN}$ = 8.26 pc) and low magnetic field ($B_{PWN}$ = 1.87 $\mu$G) associated with the PWN at the best fitted $t_{age}$, similar to the case discussed in subsection <ref> for t$_{age}$ = 7000 years, what we discuss next.

§ CONCLUDING REMARKS: LARGE RADIUS/LOW MAGNETIC FIELD ISSUE

In this paper, we have provided a detailed, time-dependent, one-zone model to explain the UHE gamma ray emission observed from the direction of LHAASO J2226+6057 using the emission from PWN associated with PSR J2229+6114.
We summarize the main points obtained in this work, and compare them with previous studies.

* The effects due to the variation of the true age and braking index were not explored in previous studies done for this source.

From our study, we found that a true age between 4000 and 7000 years is most suitable for the fitting of the SED of the source. Considering $t_{age}$ as a free parameter during the fitting of the MWL SED also revealed a compatible result. Our study found that no significant effects can be seen on the MWL spectrum if we take different braking indices.

* We have used the  code to fit the observed MWL SED by computing the best $\chi^2$ fit model spectrum.

From the fit, we found that CMB is the target photon field responsible for IC cooling of the injected leptons from the PWN, whereas the effects of FIR/NIR radiation fields are  negligible in this case.

Additionally we have found that the PWN at current age must be extended ($R_{PWN} \sim$ 10 pc), as are the extended source regions observed by VERITAS ($\sim$ 14 pc (0.27$^{\circ}$)) <cit.> and LHAASO ($\sim$ 25.6 pc (0.49$^{\circ}$)) <cit.>. The obtained PWN radius is larger when compared to e.g., X-ray radii observed for the Boomerang PWN, which is normal in one-zone models.

* We have taken the effect of reverberation into account in our modeling. It was found that the PWN is at the onset of compression due to the impact of reverse shock hitting the shell of the PWN. The effect of reverberation on the MWL spectrum was also explored by gradually increasing the true age of the PWN to the characteristic age. The MWL description worsened with increasing age. The true age can not be much larger than the considered age of 7000 years if a PWN is responsible for the gamma ray emission detected. Since reverberation does not provide  a better fit, its inclusion does not solve the large radius/low magnetic field issue.

* We have also found that a very low magnetic field ($\sim$ 2 $\mu$G) is needed to explain the MWL SED of the source, which is comparable with the Galactic average magnetic field value. A similarly low magnetic field (a few $\mu$G) was found in previous studies as the one needed to describe this source <cit.>. Moreover, a low magnetic field was found for other LHAASO detected PeVatron cadidates as well <cit.>, which is an a priori obvious outcome of requesting a leptonically-generated
high energy emission. Further complications of the model, as we discussed, do not significantly alleviate this.

As discussed earlier, an uncertainty regarding the distance of the source remains. We have explored this uncertainty by considering two very different values $D = 800$ pc and $D = 7.5$ kpc. For $D = 7.5$ kpc, we have found the fit is comparatively worse compared to that discussed in subsection <ref>, with $\chi^2$/D.O.F. = 36.78/30 and systematic uncertainty of 0.41.

On the other hand, the fit in the case of $D = 800$ pc is comparable to that discussed in subsection <ref>, with $\chi^2$/D.O.F. = 35.88/30 and systematic uncertainty of 0.32.  There is no clear evidence to overrule one distance over the other, so we only report in detail the case of $D = 3$ kpc to directly compare with the works of <cit.> and <cit.>. It is to be noted that power law lepton injection spectrum is more favoured in case of $D = 800$ pc, rather than a broken power law spectrum. However, the issue of large radius/low magnetic field remains in this case as well.

The large estimated PWN size appears to be a caveat of the PWN interpretation of the LHAASO source. It is likely that both the head and the tail region contribute to the total observed emission from the source. In light of such a complicated source morphology, a one-zone treatment of the source region considered in the model proves to be a simplistic take on the same. The large PWN radius obtained from the calculation could be a consequence of such simplistic assumption. Nevertheless, our model tends to be the most definitive PWN approach considered thus far to explain the MWL emission from the LHAASO source, complete with reverberation consideration and PWN true age estimation. It is to be noted that the PWN radius obtained by <cit.> ($\sim$ 3.1 pc) is different from that obtained in this work.  Although one zone model has been considered in both cases, the  differences in the formalism adopted to compute the radius evolution in presence of the background SNR may be the reason behind this. In any case, further investigation will be important to properly address the complexity of the source morphology.

The Galactic magnetic field (GMF) plays a important role in the cosmic ray propagation. The intensity and orientation of GMF are constrained by several methods such as Zeeman splitting observations <cit.>, infrared, synchrotron and starlight polarization studies <cit.>, and Faraday rotation measures <cit.>. The GMF model typically has three components, namely the Disc ($B_{Disc}$), Halo ($B_{Halo}$) and Turbulent ($B_{Turb}$) contributions.

Typical values for $B_{Disc}$ and $B_{Halo}$ lie within the range of 2-11 $\mu$G <cit.>, although the value of the turbulent component depends on the halo height of the Galaxy <cit.>, e.g. for a typical halo height of 8 kpc, the value of $B_{Turb}$ comes out to be $\approx$ 6 $\mu$G <cit.>. Although the exact structure of the small scale GMF is not known yet, from the models given by <cit.> and <cit.>, and also from observed secondary-to-primary ratio (<cit.>, and references therein), the large scale, average GMF can be estimated to be in the range 2-6 $\mu$G. The PWN magnetic field ($\sim$ 2 $\mu$G) would be marginally close to, or even lower than, any of the estimates of the average GMF value.

We see two possible ways out of this. On the one side, the local environment of the PWN could have been vacuated by the explosion of the supernova, and/or by earlier explosions, so that the local field in the vicinity is actually much lower than a few $\mu$G, allowing for a magnetic field contrast to appear between the PWN and its environment.

On the other hand, there is still a chance that the approximate representation of reverberation we have adopted here is still misleading. We know that assuming the ejecta pressure as constant (as all PWNe models have so far done) is in fact an oversimplification <cit.>. A better treatment of the ejecta pressure could plausibly change the reverberation results, and we shall explore this in the future.

Of course, it is also possible that the PWN explanation of the source is not realized at all, and/or that a hadronic component plays a dominant role.

For the moment, a conflicting low value for the PWN magnetic field (not solved by age, braking indices, or currently assumed behavior of the reverberation process)
leaves the PWN origin of LHASSO J2226+6057, and other similar sources,
in search for further observational tests.

The calculated best-fit MWL spectrum is shown at the top row, along with the MWL data points. The color scheme of the data points are same as that in Figure <ref>. In the bottom pannel of the figure, the residuals are also plotted. The color scheme of the residuals are same as the data points. The middle row shows the timescales of radiative losses, adiabatic losses and the escape of particles considered in the model (left) and the injected lepton spectrum at the present age (right). Also the time evolution of the magnetic field (left), and SNR forward shock, SNR reverse shock and PWN radius (right) are given in the bottom row.

The time evolution of calculated MWL spectrum (left), as well as the injected lepton spectrum (right) are shown in the figure, assuming the parameters given in Table <ref>.

Impact of reverberation on the PWN MWL spectrum with increasing age is shown (left). Time evolution of PWN magnetic field (red) and radius (blue) is shown (right) for $t_{age}$ = 6500 years (dashed), 6800 years (dotted), 7000 years (solid), 7200 years (dot-dashed) and 7500 years (long-dashed).

This work has been supported by grant
DFT acknowledges as well USTC and the Chinese Academy of Sciences Presidential International Fellowship Initiative 2021VMA0001.
This work was also supported by the Spanish program Unidad de Excelencia María de
Maeztu CEX2020-001058-M. ADS acknowledges hospitality provided by the Institute of Space Sciences, where majority of this work was carried out. ADS also acknowledges RRI HPC2020 cluster for providing facilities to perform numerical computations reported in this work. ADS thanks Nayantara Gupta for helpful comments.

Example below of non-structurated natbib references
To use the v8.3 macros with this form of composition of bibliography,
the option "bibyear" should be added to the command line

[1] Baker, N. 1966,
in Stellar Evolution,
ed. R. F. Stein,& A. G. W. Cameron
(Plenum, New York) 333

[2] Balluch, M. 1988,
A&A, 200, 58

[3] Cox, J. P. 1980,
Theory of Stellar Pulsation
(Princeton University Press, Princeton) 165

[4] Cox, A. N.,& Stewart, J. N. 1969,
Academia Nauk, Scientific Information 15, 1

[5] Mizuno H. 1980,
Prog. Theor. Phys., 64, 544

[6] Tscharnuter W. M. 1987,
A&A, 188, 55

[7] Terlevich, R. 1992, in ASP Conf. Ser. 31,
Relationships between Active Galactic Nuclei and Starburst Galaxies,
ed. A. V. Filippenko, 13

[8] Yorke, H. W. 1980a,
A&A, 86, 286

[9] Zheng, W., Davidsen, A. F., Tytler, D. & Kriss, G. A.
1997, preprint

Examples for figures using graphicx
A guide "Using Imported Graphics in LaTeX2e"  (Keith Reckdahl)
is available on a lot of LaTeX public servers or ctan mirrors.
The file is : epslatex.pdf

Vibrational stability equation of state
$S_{\mathrm{vib}}(\lg e, \lg \rho)$.
$>0$ means vibrational stability.

Vibrational stability equation of state
$S_{\mathrm{vib}}(\lg e, \lg \rho)$.
$>0$ means vibrational stability.

Vibrational stability equation of state
$S_{\mathrm{vib}}(\lg e, \lg \rho)$.
$>0$ means vibrational stability.

Vibrational stability equation of state
$S_{\mathrm{vib}}(\lg e, \lg \rho)$.
$>0$ means vibrational stability.

Vibrational stability equation of state
$S_{\mathrm{vib}}(\lg e, \lg \rho)$.
$>0$ means vibrational stability.

Vibrational stability equation of state
$S_{\mathrm{vib}}(\lg e, \lg \rho)$.
$>0$ means vibrational stability.

Nonlinear Model Results

HJD     $E$     Method#2     Method#3

1     50     $-837$     970

2     47     877        230

3     31     25         415

4     35     144        2356

5     45     300        556

Nonlinear Model Results

HJD     $E$     Method#2     4cMethod#3

1     50     $-837$     970     65     67     78

2     47     877        230     567    55     78

3     31     25         415     567    55     78

4     35     144        2356    567    55     78

5     45     300        556     567    55     78

Spectral types and photometry for stars in the

Star   Spectral type   RA(J2000)   Dec(J2000)

69              B1 V        09 15 54.046     $-$50 00 26.67

49              B0.7 V      *09 15 54.570    $-$50 00 03.90

LS 1267 (86)    O8 V        09 15 52.787   11.07

24.6            7.58         1.37    0.20

LS 1262         B0 V        09 15 05.17   11.17

MO 2-119        B0.5 V      09 15 33.7    11.74

LS 1269         O8.5 V      09 15 56.60   10.85

The top panel shows likely members of Pismis 11. The second
panel contains likely members of Alicante 5. The bottom panel
displays stars outside the clusters.

Spectral types and photometry for stars in the

Star   Spectral type   RA(J2000)   Dec(J2000)

69              B1 V        09 15 54.046     $-$50 00 26.67

49              B0.7 V      *09 15 54.570    $-$50 00 03.90

LS 1267 (86)    O8 V        09 15 52.787   11.07a

24.6            7.581   1.37a      0.20a

LS 1262         B0 V        09 15 05.17   11.17b

MO 2-119        B0.5 V      09 15 33.7    11.74c

LS 1269         O8.5 V      09 15 56.60   10.85d

The top panel shows likely members of Pismis 11. The second
panel contains likely members of Alicante 5. The bottom panel
displays stars outside the clusters.

aPhotometry for MF13, LS 1267 and HD 80077 from
Dupont et al.
bPhotometry for LS 1262, LS 1269 from
Durand et al.
cPhotometry for MO2-119 from
Mathieu et al.

[]List of nearby SNe used in this work.

SN name

(with respect to $B$ maximum)

1981B       0     UBV     1

1986G        $-$3, $-$1, 0, 1, 2     BV      2

1989B       $-$5, $-$1, 0, 3, 5     UBVRI      3, 4

1990N       2, 7     UBVRI      5

1991M       3     VRI      6

4c SNe 91bg-like

1991bg       1, 2     BVRI      7

1999by       $-$5, $-$4, $-$3, 3, 4, 5     UBVRI      8

4c SNe 91T-like

1991T       $-$3, 0     UBVRI       9, 10

2000cx      $-$3, $-$2, 0, 1, 5     UBVRI      11

(1) <cit.>;
(2) <cit.>; (3) <cit.>; (4) <cit.>;
(5) <cit.>; (6) <cit.>; (7) <cit.>;
(8) <cit.>; (9) <cit.>; (10) <cit.>;
(11) <cit.>.

Summary for ISOCAM sources with mid-IR excess
(YSO candidates).

ISO-L1551     $F_{6.7}$ [mJy]     $\alpha_{6.7-14.3}$
YSO type$^{d}$     Status     Comments

6cNew YSO candidates

1     1.56 $\pm$ 0.47     –        Class II$^{c}$     New     Mid

2     0.79:               0.97:     Class II ?         New

3     4.95 $\pm$ 0.68     3.18      Class II / III     New

5     1.44 $\pm$ 0.33     1.88      Class II           New

6cPreviously known YSOs

61     0.89 $\pm$ 0.58     1.77     Class I     HH 30     Circumstellar disk

96     38.34 $\pm$ 0.71     37.5    Class II    MHO 5              Spectral type

Summary for ISOCAM sources with mid-IR excess
(YSO candidates).

ISO-L1551     $F_{6.7}$ [mJy]     $\alpha_{6.7-14.3}$
YSO type$^{d}$     Status     Comments

6cNew YSO candidates

1     1.56 $\pm$ 0.47     –        Class II$^{c}$     New     Mid

2     0.79:               0.97:     Class II ?         New

3     4.95 $\pm$ 0.68     3.18      Class II / III     New

5     1.44 $\pm$ 0.33     1.88      Class II           New

6cPreviously known YSOs

61     0.89 $\pm$ 0.58     1.77     Class I     HH 30     Circumstellar disk

96     38.34 $\pm$ 0.71     37.5    Class II    MHO 5              Spectral type

Sample stars with absolute magnitude

Catalogue    $M_{V}$     Spectral     Distance     Mode     Count Rate

Catalogue    $M_{V}$     Spectral     Distance     Mode     Count Rate

Gl 33        6.37     K2 V     7.46     S     0.043170

Gl 66AB      6.26     K2 V     8.15     S     0.260478

Gl 68        5.87     K1 V     7.47     P     0.026610

H     0.008686

Gl 86
[Source not included in the HRI catalog. See Sect. 5.4.2 for details.]
5.92     K0 V     10.91    S     0.058230

Sample stars with absolute magnitude

Catalogue    $M_{V}$     Spectral     Distance     Mode     Count Rate

Catalogue    $M_{V}$     Spectral     Distance     Mode     Count Rate

Gl 33        6.37     K2 V     7.46     S     0.043170

Gl 66AB      6.26     K2 V     8.15     S     0.260478

Gl 68        5.87     K1 V     7.47     P     0.026610

H     0.008686

Gl 86
[Source not included in the HRI catalog. See Sect. 5.4.2 for details.]
5.92     K0 V     10.91    S     0.058230

§ BACKGROUND GALAXY NUMBER COUNTS AND SHEAR NOISE-LEVELS

Because the optical images used in this analysis...

Shown in greyscale is a...

In this case....

Plotted above...

Because the optical images...

§ TITLE OF SECOND APPENDIX.....

These studies, however, have faced...

Complexes characterisation.

Complex     $F_{60}$     8.6      No. of

The second method produces...

§ BACKGROUND GALAXY NUMBER COUNTS AND SHEAR NOISE-LEVELS

[1]
Line data and abundances ...

Def     mol     Ion     $\lambda$     $\chi$     $\log gf$     N     e      rad     $\delta$     $\delta$
red     References

Def     mol     Ion     $\lambda$     $\chi$     $\log gf$     B     C      rad     $\delta$     $\delta$
red     References

A     CH     1    3638     0.002     $-$2.551                    $-$150     150      Jorgensen et al. (1996)

[1]