# Improving Language Models via Plug-and-Play Retrieval Feedback

Wenhao Yu♣, Zhihan Zhang♣, Zhenwen Liang♣,
Meng Jiang♣, Ashish Sabharwal♠
♣University of Notre Dame; ♠Allen Institute for Artificial Intelligence
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Large language models (LLMs) exhibit remarkable performance across various NLP
tasks. However, they often generate incorrect or hallucinated information,
which hinders their practical applicability in real-world scenarios. Human
feedback has been shown to effectively enhance the factuality and quality of
generated content, addressing some of these limitations. However, this
approach is resource-intensive, involving manual input and supervision, which
can be time-consuming and expensive. Moreover, it cannot be provided during
inference, further limiting its practical utility in dynamic and interactive
applications. In this paper, we introduce ReFeed, a novel pipeline designed to
enhance LLMs by providing automatic retrieval feedback in a plug-and-play
framework without the need for expensive fine-tuning. ReFeed first generates
initial outputs, then utilizes a retrieval model to acquire relevant
information from large document collections, and finally incorporates the
retrieved information into the in-context demonstration for output refinement,
thereby addressing the limitations of LLMs in a more efficient and cost-
effective manner. Experiments on four knowledge-intensive benchmark datasets
demonstrate our proposed ReFeed could improve over +6.0% under zero-shot
setting and +2.5% under few-shot setting, compared to baselines without using
retrieval feedback.

##  1 Introduction

Large language models (LLMs) have demonstrated exceptional performance in
various NLP tasks, utilizing in-context learning to eliminate the need for
task-specific fine-tuning (Brown et al., 2020; Chowdhery et al., 2022; OpenAI,
2023). Such models are typically trained on extensive datasets, capturing a
wealth of world or domain-specific knowledge within their parameters.

Despite these achievements, LLMs exhibit certain shortcomings, particularly
when confronted with complex reasoning and knowledge-intensive tasks (Zhang et
al., 2023; Yu et al., 2023). One prominent drawback is their propensity to
hallucinate content, generating information not grounded by world knowledge,
leading to untrustworthy outputs and a diminished capacity to provide accurate
information (Yu et al., 2022b; Manakul et al., 2023; Alkaissi and McFarlane,
2023). Another limitation of LLMs is the quality and scope of the knowledge
they store. The knowledge embedded within an LLM may be incomplete or out-of-
date due to the reliability of the sources in the pre-training corpus
(Lazaridou et al., 2022; Shi et al., 2023). The vastness of the information
landscape exacerbates this issue, making it difficult for models to maintain a
comprehensive and up-to-date understanding of world facts. Moreover, LLMs
cannot “memorize” all world information, especially struggling with the long
tail of knowledge from their training corpus (Mallen et al., 2022; Kandpal et
al., 2022). This inherent limitation compels them to balance the storage and
retrieval of diverse and rare knowledge against focusing on more frequently
encountered information, leading to potential inaccuracies when addressing
questions related to less common topics or requiring nuanced understanding.

Existing methods for enhancing the factuality of language models involve
adjusting model outputs based on human feedback, followed by reinforcement
learning-based fine-tuning (Nakano et al., 2021; Campos and Shern, 2022;
Ouyang et al., 2022; Liu et al., 2023). While this approach simulates human-
to-human task learning environments, fine-tuning LLMs, can be exceedingly
costly due to the exponential growth in LLM size and the necessity for
annotators to provide extensive feedback. Over-reliance on positively-rated
data may limit the model’s ability to identify and rectify negative attributes
or errors, potentially hampering its capacity to generalize to unseen data and
novel scenarios. Furthermore, once LLMs are fine-tuned, they are unable to
receive real-time feedback during inference or facilitate immediate error
correction.

In this paper, we aim to provide automatic feedback in a plug-and-play manner
without the need for fine-tuning LLMs. We explore two primary research
questions: First, can we employ a retrieval method to provide feedback on
individual generated outputs without relying on human annotators? Second, can
we integrate the feedback to refine previous outputs in a plug-and-play
manner, circumventing the expensive fine-tuning of language models? With
regards to the two questions posed, we propose a novel pipeline for improving
language model inference through automatic retrieval feedback, named ReFeed,
in a plug-and-play framework. Specifically, the language model generates
initial outputs, followed by a retrieval model using the original query and
generated outputs as a new query to retrieve relevant information from large
document collections like Wikipedia. The retrieved information enables the
language model to reconsider the generated outputs and refine them,
potentially producing a new output (though it may remain the same if no
changes are made).

Notably, compared to retrieve-then-read methods (Lewis et al., 2020; Lazaridou
et al., 2022; Shi et al., 2023), our method benefits from more relevant
documents retrieved from the corpus to directly elucidate the relationship
between query and outputs. Besides, without generating the initial output, the
supporting document cannot be easily retrieved due to the lack of lexical or
semantic overlap with the question. We discuss the comparison in more detail
in related work and experiments.

To further enhance our proposed ReFeed pipeline, we introduce two innovative
modules within this framework. Firstly, we diversify the initial generation
step to produce multiple output candidates, enabling the model to determine
the most reliable answer by analyzing the diverse set of retrieved documents.
Secondly, we adopt an ensemble approach that merges language model outputs
before and after retrieval feedback using a perplexity ranking method, as the
retrieval feedback may occasionally mislead the language model (refer to the
case study in Figure 5 for details).

Overall, our main contributions can be summarized as follows:

  1. 1.

We propose a novel pipeline utilizing retrieval feedback, named ReFeed to
improve large language models in a plug-and-play manner.

  2. 2.

We design two novel modules to advance the ReFeed pipeline, specifically
diversifying generation outputs and ensembling initial and post-feedback
answers.

  3. 3.

Our experiments on three challenging knowledge-intensive tasks demonstrate
that ReFeed can achieve state-of-the-art performance under the few-shot
setting.

##  2 Related Work

###  2.1 Solving Knowledge-intensive Tasks via Retrieve-then-Read Pipeline

Mainstream methods for solving knowledge-intensive NLP tasks employ a
retrieve-then-read model pipeline. Given an input query, a retriever is
employed to search a large evidence corpus (e.g., Wikipedia) for relevant
documents that may contain the answer. Subsequently, a reader is used to
scrutinize the retrieved documents and predict an answer. Recent research has
primarily focused on improving either the retriever (Karpukhin et al., 2020;
Qu et al., 2021; Sachan et al., 2022) or the reader (Izacard and Grave, 2021;
Yu et al., 2022a; Ju et al., 2022), or training the entire system end-to-end
(Singh et al., 2021; Shi et al., 2023). Early retrieval methods largely
employed sparse retrievers, such as BM25 (Chen et al., 2017). Recently, ORQA
(Lee et al., 2019) and DPR (Karpukhin et al., 2020) have revolutionized the
field by using dense contextualized vectors for document indexing, resulting
in superior performance compared to traditional approaches. Recently, several
work proposed to replace the retrieval model with a large language model as
retriever, owing to the powerful knowledge memorization capabilties (Yu et
al., 2023; Sun et al., 2023). However, these methods may still be prone to
hallucination issues and are unable to access up-to-date information. Notably,
compared to retrieve-then-read pipelines like RePLUG (Shi et al., 2023), our
method benefits from more relevant documents retrieved from the corpus to
directly elucidate the relationship between query and outputs. Additionally,
without generating the initial output, the text supporting the output cannot
be easily identified due to the lack of lexical or semantic overlap with the
question.

###  2.2 Aligning Language Model with Instructions via Human Feedback

Human feedback plays a crucial role in evaluating language model performance,
addressing accuracy, fairness, and bias issues, and offering insights for
model improvement to better align with human expectations. Recognizing the
significance of integrating human feedback into language models, researchers
have developed and tested various human-in-the-loop methodologies (Nakano et
al., 2021; Campos and Shern, 2022; Ouyang et al., 2022; Liu et al., 2023;
Scheurer et al., 2023). InstructGPT (Ouyang et al., 2022) was a trailblazer in
this domain, utilizing reinforcement learning from human feedback (RLHF) to
fine-tune GPT-3 to adhere to a wide range of written instructions. It trained
a reward model (RM) on this dataset to predict the preferred model output
based on human labelers’ preferences. The RM then served as a reward function,
and the supervised learning baseline was fine-tuned to maximize this reward
using the PPO algorithm. Liu et al. (2023) proposed converting all forms of
feedback into sentences, which were subsequently used to fine-tune the model.
This approach leveraged the language comprehension capabilities of language
models.

Although these approaches have shown promising results in enhancing language
model performance on specific tasks, they also present several significant
limitations. These methods rely on human-annotated data and positively-rated
model generations for fine-tuning pre-trained language models, which can
involve considerable costs and time investments. Moreover, by relying
exclusively on positively-rated data, the model’s capacity to identify and
address negative attributes or errors may be limited, consequently reducing
its generalizability to novel and unseen data.

###  2.3 Comparative Analysis of Concurrent Related Work

In light of rapid advancements in the field, numerous concurrent works have
adopted a similar philosophy, employing automated feedback to enhance language
model performance (He et al., 2023; Peng et al., 2023). While these studies
jointly validate the efficacy and practicality of incorporating retrieval
feedback, it is crucial to emphasize the differences between our work and
these contemporary investigations. He et al. (2023) initially demonstrated
that retrieval feedback could bolster faithfulness in chain-of-thought
reasoning. However, their study is confined to commonsense reasoning tasks,
and the performance improvement observed is not substantial. In contrast, our
work predominantly targets knowledge-intensive NLP tasks, wherein external
evidence assumes a more critical role in providing valuable feedback to
augment model performance. Peng et al. (2023) proposed utilizing ChatGPT to
generate feedback based on retrieved evidence. In comparison, our research
demonstrates that retrieved documents can be directly employed as feedback to
refine language model outputs, significantly enhancing the efficiency of this
method. Simultaneously, building on the fundamental retrieval feedback
concept, we introduce two novel modules, i.e., diversifying generation outputs
and ensembling initial and post-feedback answers. In comparison to existing
research, our proposed ReFeed methodology offers a distinctive contribution to
the ongoing discourse. As we persist in exploring this avenue of inquiry, we
foresee future studies refining these techniques, ultimately achieving even
greater performance gains.

##  3 Proposed Method

Figure 1:  ReFeed operates by initially prompting a large language model to
generate an answer in response to a given query, followed by the retrieval of
documents from extensive document collections. Subsequently, the pipeline
refines the initial answer by incorporating the information gleaned from the
retrieved documents.

In this section, we provide an in-depth description of our innovative plug-
and-play retrieval feedback (ReFeed) pipeline, specifically designed to tackle
a variety of knowledge-intensive tasks ($\S$3.1). The pipeline operates by
initially prompting a language model (e.g., InstructGPT) to generate an answer
in response to a given query, followed by the retrieval of documents from
extensive document collections, such as Wikipedia. Subsequently, the pipeline
refines the initial answer by incorporating the information gleaned from the
retrieved documents.

Besides, we introduce two novel modules based on our ReFeed framework. The
first module aims to diversify the initial generation step, producing multiple
output candidates. This enables the model to identify the most reliable answer
by examining the broad range of retrieved documents. The second module employs
an ensemble approach that combines language model outputs from both before and
after the retrieval feedback process. This is achieved using a perplexity
ranking method, which mitigates the risk of retrieval feedback inadvertently
misleading the language model.

###  3.1 Proposed Method: ReFeed

Background. Traditional large language models, such as GPT-3.5 based
architectures, have primarily focused on encoding an input query and
predicting the corresponding output answer (Brown et al., 2020; Ouyang et al.,
2022). In this process, the question $q$, when combined with a text prompt,
serves as input to the model, which then generates the answer. This can be
represented as $p(a|q,\theta)$, where $\theta$ denotes the pre-trained model
parameters. In practical scenarios, the maximum a posteriori estimation (MAP)
serves as the final answer, as illustrated by
$\hat{a}=\mathrm{argmax}_{a}p(a|q,\theta)$. However, this direct approach to
eliciting answers from large language models often leads to suboptimal
performance. This is because it does not fully exploit the wealth of
supplementary world knowledge available to the model (Levine et al., 2022). To
address this limitation, recent research has explored methods to improve model
performance by incorporating an additional auxiliary variable, corresponding
to a retrieved document ($d$). This extension modifies the model formulation
to $p(a|q)=\sum_{i}p(a|d_{i},q)p(d_{i}|q)$, marginalizing over all possible
documents. In practice, it is infeasible to compute the sum over all possible
documents ($d$) due to the vast number of potential sources. Consequently, the
most common approach involves approximating the sum over $d$ using the $k$
highest ranked documents, and providing all these documents as part of the
input. We assume, w.l.o.g., that these documents are $d_{1},\ldots,d_{k}$,
yielding $p(a|q)=\sum_{i=1}^{k}p(a|d_{i},q)p(d_{i}|q)$. This technique is
referred to as the retrieve-then-read pipeline (Lazaridou et al., 2022; Shi et
al., 2023).

Figure 2: Rather than generating only one initial answer, we prompt the
language model to sample multiple answers, allowing for a more comprehensive
retrieval feedback based on different answers. Figure 3: By employing an
ensemble method that evaluates initial and refined answers with retrieval
feedback, we enhance the assessment of answer trustworthiness, ensuring a more
accurate output.

####  3.1.1 Basic Pipeline

Contrary to traditional methods mentioned above, ReFeed is designed to offer
feedback via retrieval targeted specifically to individually generated
outputs. It can be formulated as
$p(a|q)=\sum_{i}p(a|d_{i},q,\widehat{a})p(d_{i}|\widehat{a},q)p(\widehat{a}|q)$,
where $\widehat{a}$ represents the initial output, $a$ is the final output,
and $d_{i}$ is conditioned not only on $q$ but also on $\widehat{a}$. Thus,
$d_{i}$ is intended to provide feedback specifically on $\widehat{a}$ as the
output, rather than providing general information to the query $q$. As in the
case of the retrieve-and-read pipeline, we retain only the top $k=10$ highest
ranked documents:
$p(a|q)=\sum_{i=1}^{k}p(a|d_{i},q,\widehat{a})p(d_{i}|\widehat{a},q)p(\widehat{a}|q)$.

This method enables a smooth integration of feedback to refine previous
outputs in a plug-and-play fashion, eliminating the need for costly fine-
tuning. ReFeed takes advantage of a collection of relevant documents retrieved
from an extensive textual corpus, facilitating the direct elucidation of
relationships between queries and outputs. Additionally, without generating an
initial output, it becomes difficult to retrieve text that supports the output
due to the absence of lexical or semantic overlap with the question.
Essentially, ReFeed functions by first prompting a large language model to
produce an answer in response to a given query, followed by the retrieval of
documents from external sources. The pipeline then refines the initial answer
by incorporating information obtained from the retrieved documents. The three-
step process is illustrated in Figure 1 and outlined below.

Step 1: Generate an Initial Answer. In this initial step, our primary
objective is to prompt a language model to generate an answer based on the
given question. To achieve this, various decoding strategies can be employed,
e.g., greedy decoding and sampling methods. In our experiments, we opted for
greedy decoding due to its simplicity and reproducibility, allowing more
consistent performance across multiple runs. This step is essential for
establishing a foundation upon which the following steps can build and refine
the initial answer.

Step 2: Retrieve Supporting Documents. The second step in our pipeline
involves utilizing a retrieval model (e.g., BM25) to acquire a set of document
from an extensive document collection, such as Wikipedia. In our experiments,
we retrieve top-10 documents, which offers a balanced trade-off between
computational efficiency for step 3 inference and the inclusion of sufficient
information. The primary goal of this step is to identify relevant information
that can corroborate or refute the relationship between the question and the
initially generated answer. By extracting pertinent information from these
vast sources, the pipeline can effectively leverage external knowledge to
improve the accuracy and reliability of the generated response.

Step 3: Refine the Previous Answer. The final step of our pipeline focuses on
refining the previously generated answer by taking into account the document
retrieved in step 2. During this stage, the language model evaluates the
retrieved information and adjusts the initial answer accordingly, ensuring
that the response is more accurate. This refinement process may involve
rephrasing, expanding, or even changing the answer based on the newfound
knowledge. By incorporating the insights gleaned from the retrieved document,
the refined answer is better equipped to address the initial query
comprehensively and accurately, resulting in an improved overall performance.
This step is critical for bridging the gap between the initial generation and
the wealth of external knowledge, ultimately producing a high-quality, well-
informed output.

####  3.1.2 Enhanced Modules

Diverse Answer Generation. Diversity in answer generation plays a pivotal role
in the first step of our ReFeed pipeline. Rather than merely generating a
single answer with the highest probability, we implement sampling methods to
produce a set of potential answers. This approach fosters diversity in the
generated outputs and enables a more comprehensive retrieval feedback based on
diverse answers. As a result, a more refined and accurate final response is
produced that effectively addresses the given question.

To elaborate, we input the question $q$ along with a text prompt into the
model, which subsequently samples multiple distinct answers, denoted as
$p(a_{j}|q,\theta)$. We then utilize the $n$ generated answers as input
queries for the retrieval process, i.e., $[q,a_{1}],\cdots,[q,a_{n}]$. This
stage is realized by multiple decoding passes, wherein the input query is fed
into the language model with nucleus sampling. This strategy increases the
probability of obtaining a more diverse set of retrieved documents
encompassing a broader spectrum of relevant information. Formally, it can be
represented as
$p(a|q)=\sum_{i,j}p(a|d_{i,j},q,\widehat{a}_{j})p(d_{i,j}|\widehat{a}_{j},q)p(\widehat{a}_{j}|q)$.

Considering the limitations on the number of documents, or say context length
of language model input, that can be employed in Step 3, we merge all
retrieved documents (across different $\widehat{a}_{j}$), rank them based on
query-document similarity scores, and retain only the top-$k$ documents for
further processing. In our experiments, we use $k=10$, which offers a balanced
trade-off between computational efficiency and the inclusion of diverse
information. Furthermore, to account for the possibility of different queries
retrieving identical documents, we perform de-duplication to ensure that the
set of retrieved documents remains diverse and relevant. Lastly, when
computing the final answer, we provide all $n$ generated answers as well as
the aforementioned top-$k$ documents as part of the prompt. Formally, this can
be represented as By incorporating diversity in answer generation in Step 1,
we effectively broaden the potential answer space, facilitating the
exploration of a wider variety of possible solutions.
$p(a|q)=\sum_{i=1}^{k}\sum_{j=1}^{n}p(a|d_{i,j},q,\widehat{a}_{j})p(d_{i,j}|\widehat{a}_{j},q)p(\widehat{a}_{j}|q)$.

Ensembling Initial and Post-Feedback Answers. Retrieval feedback serves as a
crucial component in obtaining relevant information to validate the accuracy
of initially generated answers. Nonetheless, there may be instances where the
retrieved documents inadvertently mislead the language model, causing a
correct answer to be revised into an incorrect one (see examples in Figure 5).
To address this challenge, we introduce an ensemble technique that considers
both the initial answers and the revised answers post-retrieval feedback,
ultimately improving the overall generation performance.

In ensemble process, we utilize average negative log-likelihood to rank the
generated answers before (i.e.,
$\mathrm{LL}_{\mathrm{before}}(a|q)=\frac{1}{t}\sum_{i=1}^{t}p(x_{i}|x_{<i},q)$)
and after incorporating retrieved documents (i.e.,
$\mathrm{LL}_{\mathrm{after}}(a|q)=\frac{1}{t}\sum_{i=1}^{t}p(x_{i}|x_{<i},q,\widehat{a},d)$).
If the log-likelihood of an answer before retrieval feedback is higher than
that after retrieval feedback, we retain the initially generated answer. On
the other hand, if the log-likelihood is lower after retrieval feedback, we
choose the refined answer. This strategy allows for a more informed assessment
of the trustworthiness of answer before and after retrieval feedback, ensuring
a more accurate final response.

##  4 Experiments

Models | NQ | TriviaQA | HotpotQA | WoW
---|---|---|---|---
EM | F1 | EM | F1 | EM | F1 | F1 | R-L
*close book methods without using retriever |  |  |
$\mathrm{TD}$-$\mathrm{003}$ (Ouyang et al., 2022) | 29.9 | 35.4 | 65.8 | 73.2 | 26.0 | 28.2 | 14.2 | 13.3
GenRead (Yu et al., 2023) | 32.5 | 42.0 | 66.2 | 73.9 | 36.4 | 39.9 | 14.7 | 13.5
*open book methods with using retriever |  |  |
Retrieve-then-Read | 31.7 | 41.2 | 61.4 | 67.4 | 35.2 | 38.0 | 14.6 | 13.4
ReFeed (Ours) | 39.6 | 48.0 | 68.9 | 75.2 | 41.5 | 45.1 | 15.1 | 14.0
Table 1: ReFeed achieves SoTA performance on three zero-shot knowledge
intensive NLP tasks. The backbone model is
$\mathrm{Text}$-$\mathrm{Davinci}$-$\mathrm{003}$
($\mathrm{TD}$-$\mathrm{003}$), which is trained to follow human instructions.

In this section, we conduct comprehensive experiments on three knowledge-
intensive NLP tasks, including single-hop QA (i.e., NQ (Kwiatkowski et al.,
2019), TriviaQA (Joshi et al., 2017)), multi-hop QA (i.e., HotpotQA (Yang et
al., 2018)) and dialogue system (i.e., WoW (Dinan et al., 2019)). In single-
hop QA datasets, we employ the same splits as previous approaches (Karpukhin
et al., 2020; Izacard and Grave, 2021). With regard to the HotpotQA and WoW
datasets, our approach involves the usage of dataset splits provided by the
KILT challenge (Petroni et al., 2021).

To thoroughly assess the performance of our model, we employ a variety of
evaluation metrics, taking into consideration the professional standards
established in the field. For the evaluation of open-domain QA, we use exact
match (EM) and F1 score for evaluating model performance (Zhu et al., 2021).
For EM score, an answer is deemed correct if its normalized form – obtained
through the normalization procedure delineated by Karpukhin et al. (2020) –
corresponds to any acceptable answer in the provided list. Similar to EM
score, F1 score treats the prediction and ground truth as bags of tokens, and
compute the average overlap between the prediction and ground truth answer.
Besides, we also incorporate Recall@K (R@K) as an intermediate evaluation
metric, which is calculated as the percentage of top-K retrieved or generated
documents containing the correct answer. This metric has been widely adopted
in previous research (Karpukhin et al., 2020; Sachan et al., 2022; Yu et al.,
2023), thereby establishing its credibility within the field. When evaluating
open-domain dialogue systems, we adhere to the guidelines set forth by the
KILT benchmark (Petroni et al., 2021), which recommends using a combination of
F1 and Rouge-L (R-L) scores as evaluation metrics. This approach ensures a
comprehensive and rigorous assessment of our model’s performance, aligning
with professional standards and best practices.

Models | NQ | TriviaQA | HotpotQA | WoW
---|---|---|---|---
EM | F1 | EM | F1 | EM | F1 | F1 | R-L
Backbone Language Model: $\mathrm{Text}$-$\mathrm{Davinci}$-$\mathrm{003}$
($\mathrm{TD}$-$\mathrm{003}$)
*close book methods without using retriever |  |  |
$\mathrm{TD}$-$\mathrm{003}$ (Ouyang et al., 2022) | 36.5 | 46.3 | 71.2 | 76.5 | 31.2 | 37.5 | 14.1 | 13.3
GenRead (Yu et al., 2023) | 38.2 | 47.3 | 71.4 | 76.8 | 36.6 | 47.5 | 14.7 | 14.1
*open book methods with using retriever |  |  |
Retrieve-then-Read | 34.3 | 45.6 | 66.5 | 70.6 | 35.2 | 46.8 | 14.5 | 13.8
ReFeed (Ours) | 40.1 | 50.0 | 71.8 | 77.2 | 41.5 | 54.2 | 15.1 | 14.3
Backbone Language Model: $\mathrm{Code}$-$\mathrm{Davinci}$-$\mathrm{002}$
($\mathrm{Codex}$)
*close book methods without using retriever |  |  |
$\mathrm{Codex}$ (Ouyang et al., 2022) | 41.6 | 52.8 | 73.3 | 79.2 | 32.5 | 42.8 | 16.9 | 14.7
GenRead (Yu et al., 2023) | 44.2 | 55.2 | 73.7 | 79.6 | 37.5 | 48.8 | 17.2 | 15.1
*open book methods with using retriever |  |  |
Retrieve-then-Read | 43.9 | 54.9 | 75.5 | 81.7 | 41.5 | 53.7 | 17.0 | 14.9
ReFeed (Ours) | 46.4 | 57.0 | 76.6 | 82.7 | 43.5 | 56.5 | 17.6 | 15.5
Table 2: ReFeed achieved SoTA performance on three few-shot knowledge
intensive NLP tasks.

###  4.1 Backbone Language Model

$\mathrm{\mathbf{Codex}}$: OpenAI Codex, i.e.,
$\mathrm{code}$-$\mathrm{davinci}$-$\mathrm{002}$, a sophisticated successor
to the GPT-3 model, has undergone extensive training utilizing an immense
quantity of data. This data comprises not only natural language but also
billions of lines of source code obtained from publicly accessible
repositories, such as those found on GitHub. As a result, the Codex model
boasts unparalleled proficiency in generating human-like language and
understanding diverse programming languages.

Models | NQ | TriviaQA | HotpotQA | WoW
---|---|---|---|---
EM | F1 | EM | F1 | EM | F1 | F1 | R-L
ReFeed (Ours) | 46.4 | 57.0 | 76.6 | 82.7 | 43.5 | 56.5 | 17.6 | 15.5
$\vdash$ w/o diversifying generation | 45.1 | 56.2 | 75.9 | 82.1 | 42.1 | 54.8 | 17.0 | 14.8
$\vdash$ w/o ensemble before & after | 45.5 | 56.5 | 76.1 | 82.4 | 42.5 | 55.3 | 17.1 | 14.9
Table 3: Ablation Study. Our proposed ensemble method and diversifying
generation in ReFeed can improve model performance on four benchmark datasets.
The backbone model is $\mathrm{Code}$-$\mathrm{Davinci}$-$\mathrm{002}$
($\mathrm{Codex}$).

$\mathrm{\mathbf{Text}}$-$\mathrm{\mathbf{davinci}}$-$\mathrm{\mathbf{003}}$:
Building on the foundation laid by previous InstructGPT models, OpenAI’s
$\mathrm{text}$-$\mathrm{davinci}$-$\mathrm{003}$ represents a significant
advancement in the series. This cutting-edge model showcases considerable
progress in multiple areas, including the ability to generate superior quality
written content, an enhanced capacity to process and execute complex
instructions, and an expanded capability to create coherent, long-form
narratives.

After careful consideration, we ultimately decided against employing ChatGPT
and GPT-4 as the foundational models for our project. The primary reason for
this decision is OpenAI’s announcement that both models will be subject to
ongoing updates in their model
parameters111https://platform.openai.com/docs/models/gpt-3-5. These continual
modifications would lead to non-reproducible experiments, potentially
compromising the reliability of our research outcomes.

###  4.2 Baseline Methods

In our comparative analysis, we assess our proposed model against two distinct
groups of baseline methodologies. The first group encompasses closed-book
models, including InstructGPT (Ouyang et al., 2022) and GenRead (Yu et al.,
2023), which operate without the assistance of any external supporting
documents during the inference process. Each of these baseline methods adheres
to a uniform input format, specifically utilizing the structure: [prompt
words; question].

The second group of models adheres to a retrieve-then-read pipeline (Lazaridou
et al., 2022; Shi et al., 2023), which entails a two-stage process. In the
initial stage, a retriever component is employed to identify and extract a
select number of relevant documents pertaining to a given question from an
extensive corpus, such as Wikipedia. Subsequently, a reader component is
tasked with inferring a conclusive answer based on the content gleaned from
the retrieved documents. Similar to the first group, all baseline methods
within this second group adhere to a standardized input format, which is
defined as: [prompt words; passage; question]. As RePLUG are not a open-source
model, we implemented it independently, which may result in slightly different
outcomes compared to the performance reported in their respective papers.

###  4.3 Experimental Analysis

####  4.3.1 Zero/Few-shot Question Answering and Dialogue Evaluation

In the zero-shot setting, there is no training question-answer pairs and
conversational input-output pairs for the models. Consequently, all models are
expected to generate answers solely based on the input test question provided,
without the benefit of prior training data to guide their responses.

For the purposes of our experiments, we utilized
$\mathrm{text}$-$\mathrm{davinci}$-$\mathrm{003}$ as the backbone model due to
its remarkable performance in zero-shot scenarios. This model excels in
situations where no training question-answer pairs are available, as it is
adept at generating accurate and relevant answers based solely on the input
test question. As demonstrated in Table 1, our proposed ReFeed outperforms
baseline method by effectively leveraging retrieval feedback. In particular,
ReFeed exhibits a significant improvement in EM scores by +7.7 on two open-
domain QA benchmarks in comparison to the original
$\mathrm{text}$-$\mathrm{davinci}$-$\mathrm{003}$. We also observe a similar
trend in the context of multi-hop QA tasks and dialogue systems, in which our
proposed ReFeed consistently surpasses the baseline model.

On the other hand, when juxtaposed with methods that directly retrieve or
generate documents, our proposed ReFeed demonstrates a markedly superior
performance. This can be attributed to the fact that alternative methods often
struggle to retrieve relevant passages when there is an absence of lexical
overlap between the query and the source text. Our proposed ReFeed offers a
more robust and accurate solution for knowledge-intensive tasks, outpacing
baseline methods across various benchmarks and experimental settings.

In the few-shot setting, as shown in Table 2, we observed a similar pattern to
the zero-shot setting, further reinforcing the effectiveness of our proposed
ReFeed . This consistency across various settings underscores the model’s
versatility and adaptability, illustrating its potential to deliver superior
results across a wide range of question-answering and dialogue evaluation
tasks.

Figure 4: Recall@K on test sets, measured as the percentage of top-K documents
that contain the answer. The “Baseline” refers to direct retrieval based on
the input query, where the “ReFeed-O” represents generating only one answer,
and the “ReFeed-D” represents diverse answer generation.

####  4.3.2 Ablation Study on Ensemble Method and Diverse Generation

Ensemble Method: As shown in Table 3, it is evident that the performance of
ReFeed declines by an average of 0.8 EM score across three QA datasets when
the ensemble method is not employed. This finding highlights the importance of
implementing an ensemble strategy both prior to and following retrieval
feedback in order to bolster the model’s predictive accuracy and overall
effectiveness. The ensemble method effectively utilizes the language model’s
inherent beliefs in conjunction with the retrieval feedback. It is worth
noting that in several instances, the language model is already capable of
determining the correct answer to a given question. However, the inclusion of
retrieved documents may inadvertently mislead the model. The ensemble method
addresses this issue by selecting the predicted answer with a higher log-
probability, thus mitigating the negative impact of the retrieved documents on
the model’s overall performance.

Diverse Generation: As shown in Table 3, the performance of ReFeed experiences
a decline by an average of 1.1 EM score across three QA datasets when diverse
generation is not utilized. This observation underscores the significance of
incorporating diverse generation, as it can lead to multiple, distinct
answers, thereby resulting in a more diverse set of documents retrieved during
subsequent stages. Diverse generation plays a crucial role in enhancing the
retrieval process by diversifying the range of retrieved documents.
Consequently, this increased coverage improves the overall quality and
relevance of the information obtained during retrieval. As shown in Figure 4,
the use of diverse generation through sampling techniques brings a positive
improvement on the answer hit ratio, which is a consistent finding with that
in the self-consistency paper (Wang et al., 2023). Incorporating diverse
generation into the ReFeed framework offers several benefits, including the
ability to explore a wider array of potential answers and the capacity to
retrieve more comprehensive and diverse documents. With this approach, the
model is better equipped to handle complex questions, ultimately leading to
more accurate predictions and improved performance across various
applications.

####  4.3.3 Analysis on Chain-of-thought Reasoning on Multi-hop QA

Models | HotpotQA
---|---
EM | F1
No Retriever, QA Prompt | 32.5 | 42.8
No Retriever, CoT Prompt | 35.0 | 46.8
Retrieve-Read with CoT Prompt | 42.1 | 54.8
ReFeed with CoT Prompt | 44.2 | 57.4

Table 4: ReFeed can be applied to chain-of-thought (CoT) reader as well, on
multi-step reasoning task.

Chain-of-thought reasoning entails the generation of a sequence of
intermediate reasoning steps, as described in recent literature (Wei et al.,
2022). We demonstrate that ReFeed can be effectively integrated with chain-of-
thought reasoning to address complex tasks, as opposed to merely relying on
the language model to generate answers directly. As illustrated in Table 4, we
implemented ReFeed in conjunction with chain-of-thought reasoning by
generating intermediate reasoning steps prior to arriving at the final answer.
Following this, we utilized the answer to retrieve documents for feedback and
subsequently generated another chain-of-thought reasoning to refine the
previously generated response. This approach led to a significant improvement
on complex QA scenarios in the HotpotQA, when compared to employing
straightforward QA prompts.

Figure 5: Case Studies. The first two examples illustrate how utilizing
retrieval feedback leads to the generation of correct answers, while the final
example demonstrates a negative outcome where the language model is misled by
the retrieved document, resulting in an incorrect response.

To summarize, our proposed ReFeed methodology can be seamlessly integrated
with chain-of-thought reasoning, thereby showcasing their complementary
nature. The successful combination of ReFeed and chain-of-thought reasoning
enables the model to handle more intricate tasks and exhibits its potential
for tackling real-world challenges that demand complex problem-solving
capabilities.

####  4.3.4 Case Study

In our case studies, we present three examples depicted in Figure 5 to
illustrate the impact of retrieval feedback on the language model’s ability to
refine answers. The initial two instances showcase favorable outcomes, as the
language model effectively refines answers into accurate ones by utilizing
retrieval feedback. Conversely, the third instance is unfavorable, as the
response is misguided by the documents after retrieval, culminating in an
inaccurate response.

In the first instance, both the question-answering (QA) prompt with
$\mathrm{text}$-$\mathrm{davinci}$-$\mathrm{003}$ and the retrieve-and-read
model produce “June 1, 2018” as answer, which is erroneous. Upon investigating
the retrieved document, it discloses that the film’s release date was amended
to “May 18, 2018”. However, the document (Deadpool 2) is not retrieved when
solely employing the query for retrieval. This occurs because, although the
generated answer “June 1, 2018” is inaccurate, it augments the lexical overlap
between the input query and candidate documents, complicating the model’s
capacity to pinpoint the accurate information. In the subsequent instance,
both the QA prompt with $\mathrm{text}$-$\mathrm{davinci}$-$\mathrm{003}$ and
the retrieve-and-read model generate “Tom Waits” as the response, which is
incorrect. The retrieved document elucidates that “Tom Waits” is the composer,
rather than the vocalist. This distinction diminishes the generation
likelihood of the name, facilitating the model to produce the accurate answer,
“Steve Earle”, following retrieval feedback. It is crucial to emphasize that
this informative document is retrieved irrespective of whether the generated
answer is employed as a component of the query for retrieval. In the last
instance, both the QA prompt with
$\mathrm{text}$-$\mathrm{davinci}$-$\mathrm{003}$ and the retrieve-and-read
model can generate the accurate answer. Regrettably, when incorporating the
generated answer as a component of the query for retrieval, a document
containing extraneous information is retrieved. This document is not retrieved
in the retrieve-then-read pipeline. Alas, this document misdirects the
language model, ultimately yielding an inaccurate answer.

##  5 Conclusion

In conclusion, this paper presents a novel pipeline, ReFeed, designed to
improve large language models’ performance in a plug-and-play framework,
effectively addressing the challenges arising from knowledge-intensive tasks.
By employing a retrieval method to provide automatic feedback on generated
outputs and integrating this feedback to refine the outputs without the need
for expensive fine-tuning, ReFeed offers a practical and efficient solution.
We introduce two innovative modules within the ReFeed pipeline: diverse answer
generation and an ensemble approach. These two modules further enhance ReFeed
to produce more reliable and accurate answers by considering a wider array of
retrieved documents and mitigating the risk of misleading retrieval feedback.
Our extensive experiments on four challenging knowledge-intensive benchmarks
demonstrate the effectiveness of ReFeed in achieving state-of-the-art
performance under the few-shot setting. We believe by continuing to refine and
optimize the ReFeed pipeline, we can unlock its full potential and expand its
applicability across a diverse range of scenarios and applications.

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# Probing the shape of the quark-gluon plasma droplet via event-by-event QGP
tomography

Bithika Karmakar  Institute of Physics Belgrade, University of Belgrade,
Serbia  Incubator of Scientific Excellence—Centre for Simulations of
Superdense Fluids, University of Wrocław, Poland  Dusan Zigic  Institute of
Physics Belgrade, University of Belgrade, Serbia  Pasi Huovinen  Incubator of
Scientific Excellence—Centre for Simulations of Superdense Fluids, University
of Wrocław, Poland  Marko Djordjevic  Faculty of Biology, University of
Belgrade, Serbia  Magdalena Djordjevic  Institute of Physics Belgrade,
University of Belgrade, Serbia  Jussi Auvinen111E-mail:
<EMAIL_ADDRESS>University of Jyvaskyla, Finland

###### Abstract

This study investigates Quark-Gluon Plasma (QGP) in heavy-ion collisions
through two avenues: high-$p_{\perp}$ frameworks and hydrodynamic modeling.
Using the T${}_{\text{R}}$ENTo model, we find that IP-Glasma mimicking $p=0$
value aligns well with high-$p_{\perp}$ data, in agreement with Bayesian
analysis of the low-$p_{\perp}$ regime. While adjusting $p$ values may improve
a fit to a particular high-$p_{\perp}$ observable, it does not permit an
earlier onset of transverse expansion.

###### pacs:

12.38.Mh; 24.85.+p; 25.75.-q

##  I Introduction

A new state of matter, called Quark-Gluon Plasma (QGP), consisting of
deconfined but interacting quarks, antiquarks, and gluons, is formed in ultra-
relativistic heavy ion collisions at the Relativistic Heavy Ion Collider
(RHIC) and the Large Hadron Collider (LHC) experiments QGP1 ; QGP2 ; QGP3 ;
QGP4 . These collisions create an expanding fireball of quarks and gluons,
which thermalizes to form QGP and eventually hadronizes. Extracting useful
information about the properties of QGP formed in these collisions from the
measured particle distributions, requires comparing theoretical predictions
with the data, using phenomenological models describing the evolution of the
collision system. In particular fluid-dynamical models have been very
successful in describing a large amount of the low-$p_{\perp}$ data.

Fluid-dynamical models do not provide the initial state of the fluid-dynamical
evolution, but it must be provided by some other model, or parametrization.
This ambiguity causes significant uncertainty in the properties of QGP
extracted from the data (see e.g. Ref Song:2010mg .) Recently, Bayesian
analyses of the data have significantly reduced this uncertainty by showing
that the data strongly favors initial entropy deposition proportional to the
geometric mean of the local nuclear thickness function Bernhard:2019bmu ;
Nijs:2020roc . Such proportionality corresponds to the value of the $p$
parameter of the T${}_{\text{R}}$ENTo model Moreland:2014oya  being
approximately zero, and is consistent with the state-of-the-art IP-Glasma
Schenke:2012wb ; Schenke:2012hg  and EKRT Eskola:1999fc ; Paatelainen:2012at ;
Paatelainen:2013eea  models for initial particle production. However, these
analyses are based on low-$p_{\perp}$ particles, most of which leave the
system at the very end of the evolution, and thus reflect its properties at
the late stage. High-$p_{\perp}$ particles formed in the primary nucleus-
nucleus collisions, on the other hand, leave the system during the first few
fm/$c$ of its evolution, and probe the initial stages of the system more
directly than low-$p_{\perp}$ particles. Therefore, it is important to study
whether the high-$p_{\perp}$ observables, $R_{AA}$ and flow harmonics $v_{2}$,
$v_{3}$, and $v_{4}$, favor the same initial shape of the fireball as the
low-$p_{\perp}$ particles.

Furthermore, in our previous works Stojku:2020wkh ; Stojku:2023mze  we found
that the high-$p_{\perp}$ observables can be reproduced only if the transverse
expansion is very weak (or nonexistent) until $\tau\approx 1$ fm, but the
Bayesian analyses using the T${}_{\text{R}}$ENTo model usually assume
immediate transverse expansion after the primary collisions in terms of free-
streaming Bernhard:2019bmu ; Nijs:2020roc . This brings into question whether
delaying transverse expansion until $\tau_{0}=1$ fm will change the shape
favored by the data. As well, the requirement of late onset of transverse
expansion was based on conventionally shaped initial states corresponding to
the $p\approx 0$ value. Thus, it is possible that if the initial state was way
more anisotropic than commonly assumed, transverse expansion could begin
earlier, and high-$p_{\perp}$ anisotropies would still be reproduced.

In this work, we address these two issues: We modulate the initial shape of
the system by altering the T${}_{\text{R}}$ENTo parameter $p$ while adjusting
the other parameters to reproduce a subset of the low-$p_{\perp}$ data. We do
not allow transverse expansion before the onset of fluid-dynamical evolution
at $\tau_{0}=1$ fm, calculate the high-$p_{\perp}$ observables for each value
of $p$, and compare with the data to see which value, and thus shape, the data
favor. We also systematically decrease the $p$-value down to $p=-3$ to select
more anisotropic initial distribution and carry out calculations using
$\tau_{0}=0.6$ fm and $\tau_{0}=0.2$ fm to see if highly anisotropic initial
state would lead to large enough high-$p_{\perp}$ anisotropies even if the
early transverse expansion diluted a large part of the spatial anisotropy by
the time the jets become sensitive to it.

##  II Methods

###  II.1 Bulk medium evolution

To generate the event-by-event fluctuating initial entropy profiles, we use
the phenomenological model T${}_{\text{R}}$ENTo Moreland:2014oya , which
produces the initial profiles without assuming any specific mechanism for
entropy production. Instead, the starting point is the so-called thickness
function of the colliding nuclei $A$ and $B$

$\displaystyle\tilde{T}_{A,B}(x,y)=\int dz\rho_{A,B}^{\text{part}}(x,y,z),$
(1)

where $\rho_{A,B}^{\text{part}}$ is the density of the nuclear matter that
participates in the inelastic collisions. In event-by-event case it is
calculated by sampling the Woods-Saxon distribution to create the positions of
nucleons, which are described as Gaussian distributions. For this process, we
use the parameters from Ref. Karmakar:2023bxq : width $w=0.5$ fm, minimum
nucleon-nucleon distance $d=0.5$ fm, inelastic nucleon-nucleon cross section
$\sigma_{NN}=7$ fm2 and fluctuation parameter $k=1.19$. The initial entropy
density is related to the thickness functions as

$\displaystyle
s(x,y)\propto\tilde{T}_{R}(p;\tilde{T}_{A},\tilde{T}_{B})\equiv\left(\frac{\tilde{T}_{A}^{p}+\tilde{T}_{B}^{p}}{2}\right)^{1/p},$
(2)

where $p$ is a continuous parameter which can have values from $-\infty$ to
$\infty$. As known, T${}_{\text{R}}$ENTo can be used to mimic and interpolate
between various initial state models; the values of $p$ $-2/3,0$ and $1$
corresponding to KLN, EKRT/IP-Glasma, and wounded nucleon models,
respectively. In this study, however, we consider $p$ a free parameter that
allows us to test different shapes of the initial state. In particular, we use
$p\in\\{1/3,0,-1/3,-2/3,-1,-3\\}$ to see which value the high-$p_{\perp}$ data
favors and whether higher initial anisotropy would allow the earlier onset of
transverse expansion.

We do not allow any pre-equilibrium evolution (free-streaming or otherwise).
In the first part of the study, we use $\tau_{0}=1$ fm as the initial time for
fluid-dynamical evolution with zero transverse velocity since the reproduction
of the high-$p_{\perp}$ observables does not allow strong transverse expansion
earlier Stojku:2020wkh . In the second part of the study, we use
$\tau_{0}=0.6$ fm and $\tau_{0}=0.2$ fm to test whether large anisotropy would
allow the earlier onset of transverse expansion.

We evolve the event-by-event fluctuating initial distributions using the
version of VISHNew Song:2007ux ; Song:2008si  used in Refs. Bernhard:2016tnd ;
Bernhard:2018hnz ; Bernhard:2019bmu 222Code available at Jonahs_github . This
code solves the dissipative fluid-dynamical equations in 2+1-dimensions, i.e.,
assuming boost invariance. Shear stress and bulk pressure are taken as
dynamical variables and evolved according to the Israel-Stewart type equations
Israel:1979wp . We use an Equation of State (EoS) Bernhard:2016tnd  that
combines the lattice QCD-based EoS of the HotQCD collaboration HotQCD:2014kol
at large temperatures to a hadron resonance gas EoS at low temperatures. The
fluid is particlized on an isothermal hypersurface of $T_{\mathrm{sw}}=151$
MeV using the Cooper-Frye prescription Cooper:1974mv , and the particles fed
to the UrQMD hadron cascade Bass:1998ca ; Bleicher:1999xi , which describes
the evolution of the hadronic stage of the system.

The temperature dependence of the specific bulk viscosity is parameterized as
a Cauchy distribution Bernhard:2018hnz  with the maximum value
$(\zeta/s)_{\text{max}}$ = 0.03, and the other parameters characterizing the
bulk viscosity being the same as in Ref. Karmakar:2023bxq . Instead of
allowing temperature-dependent specific shear viscosity, we use constant
$\eta/s$, since in our earlier study Karmakar:2023bxq  we found that the
high-$p_{\perp}$ observables could be equally well reproduced using constant
$\eta/s$ as temperature dependent $\eta/s$.

| $\tau_{0}=1$ fm |  $\tau_{0}=0.6$ fm |  $\tau_{0}=0.2$ fm
---|---|---|---
$p$ | 1/3 | 0 | -1/3 | -2/3 | -1 | -3 | 0 | -3 | 0 | -3
$\eta/s$ |  0.03 | 0.15 | 0.18 | 0.20 | 0.25 | 0.29 |  0.17 | 0.33 |  0.20 | 0.60
Table 1: $\eta/s$ values used for each value of $p$ and initial time
$\tau_{0}$

We keep all the other parameters fixed but adjust the T${}_{\text{R}}$ENTo
normalization parameter and $\eta/s$ for each value of the $p$ parameter to
reproduce the charged particle elliptic flow $v_{2}\\{4\\}$, and proton, pion
and kaon multiplicities in each centrality class. The required values of
$\eta/s$ are shown in Table 1.

We generate $10^{4}$ events for each T${}_{\text{R}}$ENTo $p$ value. These
events are then sorted into centrality bins according to the number of
participants in the collision. We analyze the events in four centrality
classes, 10-20%, 20-30%, 30-40%, and 40-50%, where the low-$p_{\perp}$
observables are calculated for each event and averaged over the events in the
class. Subsequently, the temperature profiles in each centrality class are
used as input to the generalized DREENA-A to calculate the high-$p_{\perp}$
observables in each class.

###  II.2 Generalized DREENA-A framework

In our study, the high-$p_{\perp}$ energy loss has been computed using the
DREENA-A (Dynamical Radiative and Elastic ENergy loss Approach) framework
where $`$A’ stands for Adaptive, i.e., arbitrary temperature profile can be
included as input in this framework. This has been further optimized to
incorporate event-by-event fluctuations in the temperature profiles. The
framework is based on generalized hard thermal loop perturbation theory
Kapusta:1989tk  in which the infrared divergences are naturally regulated
Djordjevic:2006tw ; Blagojevic:2018nve . It has several important features
that make it a reliable tool for QGP tomography: i) The system is considered
as finite size dynamical medium i.e., it consists of moving partons, ii) Both
the radiative Djordjevic:2009cr ; Djordjevic:2008iz  and collisional
Djordjevic:2006tw  energy loss have been computed within the same framework,
iii) It is generalized to include the running coupling Djordjevic:2013xoa ,
finite magnetic mass Djordjevic:2011dd , and beyond soft-gluon approximation
Blagojevic:2018nve . iv) It does not have fitting parameters in the energy
loss, i.e., all the parameters are set to standard literature values
Stojku:2020wkh ; Stojku:2021yow , allowing systematic comparison of data and
the predictions from the simulation obtained using the same formalism and
parameter set. We recently also extended the formalism to include finite
orders in opacity Stojku2023  but found out that including these effects has a
minor impact on high-$p_{\perp}$ predictions. Thus, a computationally more
efficient version with one scattering center is used in this study.

To generate the medium-modified distribution of high-$p_{\perp}$ particles, we
use the following pQCD convolution formula:

$\displaystyle\frac{E_{f}d^{3}\sigma_{q}(H_{Q})}{dp^{3}_{f}}=\frac{E_{i}d^{3}\sigma(Q)}{dp^{3}_{i}}\otimes
P(E_{i}\rightarrow E_{f})\otimes D(Q\rightarrow H_{Q}),$ (3)

where $i$ and $f$ denote the initial parton ($Q$) and the final hadron
($H_{Q}$) respectively. $\frac{E_{i}d^{3}\sigma(Q)}{dp^{3}_{i}}$ represents
initial parton spectrum that is calculated in accordance with Ref. Kang:2012kc
; Sharma:2009hn ; Cacciari:2012ny . As discussed above, the energy loss
probability $P(E_{i}\rightarrow E_{f})$ has been computed within finite
temperature field theory. $D(Q\rightarrow H_{Q})$ represents the fragmentation
function, where DSS, BCFY, and KLP fragmentation functions have been used for
light hadrons, D, and B mesons, respectively. We consider the gluon mass
$m_{g}=\mu_{E}/\sqrt{2}$ Djordjevic:2003be  where $\mu_{E}$ is the
temperature-dependent Debye mass that is calculated following the procedure in
Ref. Peshier:2006ah . Further, we assume the mass of the light quark, charm,
and bottom quark to be $M=\mu_{E}/6$, 1.2 GeV, and 4.75 GeV, respectively. The
magnetic to electric mass ratio is $\mu_{M}/\mu_{E}=0.6$ Borsanyi:2015yka .
Also, we assume $\Lambda_{QCD}=0.2$ GeV and effective number of light quark
flavors $n_{f}=3$ in our study.

##  III Results

Figure 1: The averaged initial ($\tau=1$ fm) temperature distributions for
$p\in\\{1/3,0,-1/3,-2/3,-1\\}$ at 20-30% centrality. When averaging, the
participant angles of each event were aligned. The distributions in the upper
row were aligned with respect to the $\Psi_{2,2}$ angle, and in the lower row
w.r.t. $\Psi_{2,3}$, making the average ellipticity and triangularity visible.

Figure 2: The anisotropy parameters $\epsilon_{2,2}$ and $\epsilon_{2,3}$ as
function of $p$ at various centralities.

We characterise the initial state anisotropies with the coefficients
$\epsilon_{m,n}$ Niemi:2012aj

$\epsilon_{m,n}=-\frac{\int
dxdy\,r^{m}\cos\left[n(\phi-\Psi_{m,n})\right]e(x,y)}{\int dxdy\,r^{m}e(x,y)}$
(4)

where $(r,\phi)$ are the radial coordinates in the transverse plane, $e(x,y)$
is the energy density and $\Psi_{m,n}$ is the event plane angle:

$\Psi_{m,n}=\frac{1}{n}\arctan\frac{\int dxdy\,r^{m}\sin(n\phi)e(x,y)}{\int
dxdy\,r^{m}\cos(n\phi)e(x,y)}+\pi/n.$ (5)

To illustrate how the shape of the initial state depends on the parameter $p$,
we show in Fig. 1 the averaged initial temperature distributions at 20-30%
centrality for various values of $p$, and the anisotropy parameters
$\epsilon_{2,2}$ and $\epsilon_{2,3}$ as function of $p$ at various
centralities in Fig. 2. When averaging, the individual events were rotated to
align their participant angles $\Psi_{2,2}$ (upper row of Fig. 1) or
$\Psi_{2,3}$ (lower row of Fig. 1) to make the average ellipticity (upper row)
or triangularity (lower row) visible. As seen, the system becomes more
elliptically elongated and more triangular when $p$ is reduced.

Figure 3: The charged hadron $R_{AA}$ and high-$p_{\perp}$ flow harmonics
$v_{2}$, $v_{3}$ and $v_{4}$ as function of transverse momentum in Pb+Pb
collisions at $\sqrt{s_{\text{NN}}}=5.02$ TeV for different initializations at
20-30% centrality. CMS (blue squares) CMS:2017xgk ; CMS:2016xef , ALICE (red
circles) ALICE:2018vuu ; ALICE:2018rtz , and ATLAS (green triangles)
ATLAS:2018ezv ; ATLAS:2017rmz  experimental data are also shown for
comparison. Figure 4: The same as Fig. 3, but at 40-50% centrality.

At low $p_{\perp}$, the $v_{2}$ and $v_{3}$ are strongly correlated with their
corresponding spatial anisotropies Niemi:2012aj , but at high $p_{\perp}$
their behavior is more complicated. This can be seen in Figs. 3 and 4, where
we show the calculated $R_{AA}$ and high-$p_{\perp}$ $v_{2}$, $v_{3}$ and
$v_{4}$ at 20-30% and 40-50% centrality, respectively. These results are for
Pb+Pb collisions at $\sqrt{s_{\text{NN}}}=5.02$ TeV, and compared with the
data. When $p$ is non-negative, high-$p_{\perp}$ $v_{2}$ increases with
decreasing $p$, i.e., with increasing $\epsilon_{2,2}$, but once $p$ becomes
negative $v_{2}$ hardly changes even if $\epsilon_{2,2}$ keeps increasing. On
the other hand, high-$p_{\perp}$ $v_{3}$ is _anticorrelated_ with
$\epsilon_{2,3}$ and tends to decrease when $\epsilon_{2,3}$ increases (i.e.,
when $p$ decreases), although when $p$ is non-negative, $v_{3}$ hardly
changes. This is surprising, but one has to keep in mind that the
high-$p_{\perp}$ anisotropies are not as sensitive to the initial shape of the
system as they are to the time-average of the shape and the temperature of the
system. Thus, at high $p_{\perp}$, we should not expect $v_{2}$ nor $v_{3}$ to
be directly related to the eccentricities $\epsilon_{2,2}$ and
$\epsilon_{2,3}$. Furthermore, as we have noted earlier Stojku:2021yow , the
high-$p_{\perp}$ observable which reflects the shape of the system best is not
$v_{2}$, but the ratio $v_{2}/(1-R_{AA})$.

The $R_{AA}$ hardly depends on the shape of the system, but as we have
discussed previously Stojku:2021yow , the initial temperature where the energy
loss sets in dominates $R_{AA}$. The $p$ value does not directly affect the
initial temperature, but since we require our calculations to reproduce the
observed multiplicity and $v_{2}\\{4\\}$, we have to adjust the initial
temperature accordingly. As seen in Fig. 1, the smaller the $p$, the smaller
the system. Reaching the same final multiplicity necessitates a larger initial
entropy density and, thus, higher temperature. On the other hand, the high
initial eccentricity necessitates higher shear viscosity to dampen the
low-$p_{\perp}$ $v_{2}$. Higher viscosity leads to higher entropy production
during the evolution, and to obtain the same final multiplicity, the initial
entropy density and, thus, temperature should be smaller. These two opposing
trends do not quite cancel each other but lead to the overall tendency of
initial temperature increasing with increasing $p$. The calculated $R_{AA}$,
shown in the upper left panels of Figs. 3 and 4, generally follows the
behavior of initial temperature: the larger the $p$ the lower the $R_{AA}$.
Based on these observations and comparison with the data333Because of the
large error bars and low statistics we do not pay attention to the $v_{3}$
data above $p_{\perp}\approx 10$ GeV., we can say that the observed $R_{AA}$
and $v_{3}$ favors values $p\gtrsim 0$, whereas high-$p_{\perp}$ $v_{2}$
favors values $p\lesssim 0$. Thus, we conclude that the high-$p_{\perp}$ data
favors the value $p\approx 0$ in agreement with the low-$p_{\perp}$ data
Bernhard:2019bmu .

In contrast to the previous observables, the high-$p_{\perp}$ $v_{4}$ is
almost insensitive to the $p$ value (lower right panels of Figs. 3 and 4).
Furthermore, our generalized DREENA-A framework consistently underpredicts the
experimental data for charged hadron $v_{4}$. This discrepancy has been
previously highlighted in our earlier works Zigic:2022xks ; Karmakar:2023bxq ,
where variations in initial conditions, including the presence or absence of
initial transverse free streaming, and different shear viscosity-to-entropy
density ($\eta/s$) parametrizations led to similarly low predictions for
high-$p_{\perp}$ $v_{4}$. We have called this divergence between theory and
experiment the ”high-$p_{\perp}$ $v_{4}$ puzzle” Zigic:2022xks .

Our comprehensive analyses demonstrate that attempts to resolve this puzzle
through modifying parameters in fluid dynamical simulations—including
eliminating free streaming, delaying thermalization time, employing diverse
$\eta/s$ parametrizations, or varying the $p$ values for initial
conditions—have yielded negligible impact on high-$p_{\perp}$ $v_{4}$. If
upcoming high-luminosity experimental data corroborate these observations,
resolving the $v_{4}$ puzzle might necessitate alternative approaches to
initial state and/or energy loss models.

Figure 5: The D and B meson $R_{AA}$ (first row) and high-$p_{\perp}$ flow
harmonics $v_{2}$ (second row) and $v_{3}$ (third row) as function of
transverse momentum in Pb+Pb collisions at $\sqrt{s_{\text{NN}}}=5.02$ TeV for
different initializations. The left column depicts collisions at 20-30%
centrality and the right column at 50-60% centrality.

As known, the heavy quark energy loss is more sensitive to the details of the
evolution of the fireball than the light quark energy loss. Therefore, we have
also evaluated the $R_{AA}$ and high-$p_{\perp}$ anisotropies of D and B
mesons shown in Fig. 5. The overall behavior is the same as for charged
hadrons, but in particular $v_{3}$ becomes more sensitive to the initial
anisotropy, and is able to differentiate between all values of $p$ we tested.
This enhanced sensitivity to details of the evolution hints at the possibility
that once the high-luminosity data becomes available from sPHENIX and Run 3 at
the LHC, the study of D and B meson anisotropies may reveal structures so far
hidden, and which the T${}_{\text{R}}$ENTo model cannot describe.

Figure 6: Calculated charged hadron $R_{AA}$ (first row), high-$p_{\perp}$
$v_{2}$ (second row) and $v_{3}$ (third row) as function of transverse
momentum for $p=0$ and $-3$ and $\tau_{0}=1$ and 0.6 fm at 10-20%, 20-30%,
30-40% and 40-50% centrality classes.

As discussed in the Introduction, we had observed in our earlier works that
the reproduction of high-$p_{\perp}$ $v_{2}$ is possible only if the
transverse expansion of the fireball is delayed to $\tau\approx 1$ fm
Stojku:2020wkh . To study whether it would still be possible to start
transverse expansion earlier and fit the $v_{2}$ data if the initial
anisotropy was way larger than usually assumed, we cranked the $p$ parameter
of the T${}_{\text{R}}$ENTo model down to -3, and readjusted viscosity and
normalization to reproduce the low-$p_{\perp}$ data. The resulted
high-$p_{\perp}$ observables, $R_{AA}$, $v_{2}$ and $v_{3}$ of charged
hadrons, are shown in Fig. 6. A comparison of the solid blue ($p=0$ and
$\tau_{0}=1$ fm) and dashed orange ($p=-3$ and $\tau_{0}=1$ fm) curves reveals
behavior similar to previously discussed: Smaller $p$ reduces temperature and
increases both elliptic and triangular anisotropies. Consequently $R_{AA}$ and
$v_{2}$ increase whereas $v_{3}$ decrease. Interestingly, the increase in
$R_{AA}$ might allow an earlier start of energy loss without reducing $R_{AA}$
too much. The red curve ($p=-3$ and $\tau_{0}=0.6$ fm) reveals this to be the
case. $R_{AA}$ is almost identical to $R_{AA}$ obtained using $p=0$ and
$\tau_{0}=1$ fm, but the increase in $v_{2}$ is insufficient to compensate for
the reduction due to the earlier onset of transverse expansion. Furthermore,
both small $p$ and early onset of expansion reduce $v_{3}$, leading to smaller
$v_{3}$ than experimentally observed. To be sure of our conclusions, we
repeated the calculations using $p=-1$ and $-2$, and obtained results which
were between the ones obtained using $p=0$ and $p=-3$. One could ask what
would happen if we cranked $p$ down even further, say, down to minus eleven,
but as shown in Fig. 2, once $p\lesssim-2$, the anisotropies change very
little. Therefore, we dare to conclude that if there is an initial state which
is anisotropic enough to allow both early onset of transverse expansion and
reproduction of high-$p_{\perp}$ anisotropies, it cannot be modeled using
T${}_{\text{R}}$ENTo.

As mentioned in the Introduction and Section IIA, we repeated this study using
an even earlier start of transverse expansion and jet energy loss at
$\tau=0.2$ (results not shown), and it was even clearer that the increased
anisotropy was insufficient to compensate for the reduction of $R_{AA}$ and
high-$p_{\perp}$ $v_{2}$ due to early onset of expansion and energy loss 444We
remind the reader that in Stojku:2020wkh , we studied the early expansion
using both free streaming and fluid dynamics, and the results were the same:
Early transverse expansion strongly reduces high-$p_{\perp}$ $v_{2}$
irrespectively whether the expansion is fluid dynamical or not..

##  IV Summary

In the fluid-dynamical models of heavy-ion collisions, the initial state is
conventionally constrained by comparison to low-$p_{\perp}$ data. In this
study, we have provided a complementary approach by using high-$p_{\perp}$
data, i.e., QGP tomography, to constrain the initial state. In particular, we
carried out two distinct investigations:

i) We modulated the shape of the initial state by varying the parameter $p$ of
the T${}_{\text{R}}$ENTo model. We found out that while $R_{AA}$ and $v_{3}$
preferred values $p\gtrsim 0$, high-$p_{\perp}$ $v_{2}$ preferred $p\lesssim
0$. Thus, the best overall fit was obtained with $p\approx 0$, consistently
with the Bayesian analysis of low-$p_{\perp}$ data.

ii) We tested if larger elliptic anisotropy of the initial state, i.e., $p\ll
0$ would allow an earlier onset of transverse, in our case fluid-dynamical,
expansion. Our results show that, while lower $p$ values enhance
high-$p_{\perp}$ $R_{AA}$ and $v_{2}$, the enhancement is insufficient for
facilitating an earlier onset of transverse expansion in heavy-ion collisions.

We conclude by noting that by providing a complementary approach to constrain
the early stages of heavy-ion collisions, this research demonstrates the value
of employing jet tomography, in which both low- and high-$p_{\perp}$
theoretical and experimental data are collectively utilized to refine our
understanding of QGP properties.

Acknowledgements: This work is supported by the European Research Council,
grant ERC-2016-COG: 725741, and by the Ministry of Science and Technological
Development of the Republic of Serbia. PH and BK were also supported by the
program Excellence Initiative–Research University of the University of Wroclaw
of the Ministry of Education and Science. JA acknowledges the financial
support from the Academy of Finland Project No. 330448. JA’s research was also
funded as a part of the Center of Excellence in Quark Matter of the Academy of
Finland (Project No. 346325). This research is part of the European Research
Council Project No. ERC-2018-ADG-835105 YoctoLHC.

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