11institutetext: National Information Processing Institute, Warsaw, Poland
11email<EMAIL_ADDRESS>
# Assessing generalization capability of text ranking models in Polish

Sławomir Dadas  Małgorzata Grębowiec

###### Abstract

Retrieval-augmented generation (RAG) is becoming an increasingly popular
technique for integrating internal knowledge bases with large language models.
In a typical RAG pipeline, three models are used, responsible for the
retrieval, reranking, and generation stages. In this article, we focus on the
reranking problem for the Polish language, examining the performance of
rerankers and comparing their results with available retrieval models. We
conduct a comprehensive evaluation of existing models and those trained by us,
utilizing a benchmark of 41 diverse information retrieval tasks for the Polish
language. The results of our experiments show that most models struggle with
out-of-domain generalization. However, a combination of effective optimization
method and a large training dataset allows for building rerankers that are
both compact in size and capable of generalization. The best of our models
establishes a new state-of-the-art for reranking in the Polish language,
outperforming existing models with up to 30 times more parameters.

###### Keywords:

Information Retrieval Text Ranking

##  1 Introduction

Text information retrieval is one of the most active research areas in natural
language processing. The popularity of this field primarily stems from its
broad applications and the challenges posed by steadily increasing volumes of
unstructured data, whether publicly available on the Internet or processed
internally by companies. Information retrieval methods are applied to various
problems, including search, question answering, summarization, clustering, or
plagiarism detection. Text retrieval originally relied on lexical methods such
as BM25 [17], which, with the advancement of deep learning, began to be
replaced or supplemented by neural approaches, particularly those leveraging
neural language models [31, 32].

Nowadays, a lot of attention is being paid to building end-to-end systems that
are designed to answer a user’s question based on data in a local knowledge
base. This challenge is typically addressed by integrating multiple
cooperating models, each specialized for a specific step in the process. Such
approaches are referred to as retrieval-augmented generation (RAG) [11, 12].
Figure 1 illustrates the typical architecture of the RAG system, consisting of
three models: retriever, reranker, and reader. The retriever is responsible
for the initial extraction of a set of documents relevant to the user’s query.
At this stage, the focus is primarily on the model’s efficiency - it should be
able to quickly select matching documents from a large collection, potentially
numbering in the millions or billions. Therefore, lexical methods based on
full-text indexes or neural text encoders along with vector indexes are
commonly employed. This allows for precomputing representations for documents
and storing them in a data structure that enables efficient searching. In the
next step, the retrieved documents are sorted using the reranker model. Since
the input consists of a small set of pre-selected documents, models used at
this stage may offer higher prediction quality at the expense of efficiency.
Frequently, employed methods compute similarity for each query-document pair,
requiring $n\times m$ comparisons for $n$ queries and $m$ documents per query.
In the final step of the process, the query along with the content of the
documents is passed to the reader, which generates the final answer.
Currently, large language models (LLMs) are most commonly used for this
purpose.

Figure 1: A diagram showing the three-step RAG workflow. In the first step,
the retriever model extracts a list of $k0$ best-matching documents from the
document collection based on the user’s query. Subsequently, the documents are
sorted using the reranker, which returns the top $k$ documents ordered by
their relevance to the query, where $k\leq k0$. The query is then passed to
the reader model, along with the context that includes the text of the
retrieved documents. The model is responsible for generating the final answer
using the context in either an extractive or abstractive manner.

This publication focuses on the reranking stage in the RAG process. It is
important to note that reranking is an optional step, as it would be possible
to skip it and pass the retriever results directly to the reader. The idea of
employing additional ranking models is based on the assumption that these
models have the ability to better assess the relevance of a document to the
query. Therefore, an additional sorting stage should improve the results from
the retrieval stage. Research indicates that in the case of supervised
learning, in which models are trained and evaluated on a dataset from the same
domain, the results of rerankers indeed surpass those obtained by retrievers
[28]. However, in the case of zero-shot evaluation, in which we train a model
on a single dataset and evaluate it on multiple datasets with different
structures and coming from various domains, this may not necessarily be true.
Recent publications on rerankers pay more attention to their generalization
ability and zero-shot performance [33, 13, 22]. The results of these
comparisons do not always favor rerankers. For some out-of-domain datasets,
these models are outperformed even by baseline methods such as BM25. Rosa et
al. [18] suggests that generalization capability of text rerankers increases
with model size, and only large rerankers with several billion parameters
achieve good results in a zero-shot setting. Another aspect worth noting is
the significant progress made in the field of dense retrievers over the past
years. Methods that were popular a few years ago, such as dense passage
retrievers (DPR) [9], achieved results worse than BM25 for zero-shot tasks
[23, 30]. In contrast, modern dense retrievers like E5 [24] or FlagEmbeddings
[26] demonstrate strong performance, surpassing BM25 significantly, as
confirmed by benchmarks such as BEIR [23]. During the same period, we have not
observed such significant progress for text ranking models, which may suggest
that the gap between these two groups of methods has narrowed.

###  1.1 Contributions

The aim of this article is to investigate the performance of rerankers for the
Polish language and compare their results with available retrieval models. For
this purpose, we use PIRB [6], an extensive benchmark covering 41 information
retrieval tasks for the Polish language. In this publication, we answer the
question of whether the available Polish and multilingual rerankers outperform
standalone retrieval-based solutions and whether they demonstrate sufficient
generalization capabilities. We also highlight the properties of models and
datasets that influence the quality of text reranking, attempting to provide
guidelines for when it is worthwhile to include reranker step in the RAG
process. More specifically, we make the following contributions:

  * $\bullet$

We evaluate several Polish and multilingual reranking models on a collection
of information retrieval tasks for the Polish language. In the evaluation, we
consider publicly available models as well as reproduce several baseline
methods described in the literature. Our experiments involve rerankers of
various scales, ranging from tens of millions to 13 billion parameters.

  * $\bullet$

Using distillation techniques, we train new highly efficient reranking models
for the Polish language. Our method involves leveraging the largest available
reranker based on the MT5-XXL architecture [27] as a teacher model in a
knowledge distillation procedure. We employ two distillation methods, one
based on directly mimicking the predictions of a large reranker, and the other
based on permutation modeling. The best of our models outperforms the teacher
model on the PIRB benchmark while being 30 times smaller in terms of the
number of parameters.

###  1.2 Related work

With the advancement of deep learning methods, models based on neural networks
are finding increasingly more applications in the field of information
retrieval, spanning from first-stage retrieval, through reranking, to the
extraction of specific information from documents. Early neural reranking
methods utilized architectures such as convolutional or recurrent networks
[8], leveraging pre-trained word embeddings. However, it was only with the
emergence of the transformer architecture that the topic received a
significant increase in attention. Today, many high-quality reranker models
are available for popular languages such as English and Chinese. Most such
solutions are based on encoder architectures like BERT [15, 28], or encoder-
decoder architectures like T5 [16, 33].

Beyond the development of neural methods and architectures, the availability
of training data is equally crucial. The progress we observe would not have
been possible without the release of the MS MARCO dataset [1], comprising
approximately 1 million queries and 9 million passages with relevance
judgments generated by the users of the Bing search engine. This enabled the
training of general-purpose rerankers that handle a wide range of topics and
domains. Datasets of a similar scale for languages other than English have
only started to emerge recently. Notable among these is mMARCO [2], which
includes data from the original MS MARCO translated into 13 languages, as well
as other recently released large collections such as Mr.TyDi [29] and MIRACL
[30]. Using these datasets, several publicly available multilingual rerankers
have been trained.

In the case of the Polish language, until recently, the topic of deep learning
methods for information retrieval was absent from the scientific literature.
The situation began to change only in the last two years, primarily concerning
the availability of datasets for information retrieval and question answering.
A significant number of individual datasets as well as collections comprising
several datasets have been prepared and made available during this period.
These include PolQA [21], MAUPQA [19], BEIR-PL [25] (a machine translated
version of the English BEIR [23] benchmark), as well as datasets used in the
PolEval-2022 Passage Retrieval task [10]. Most of the mentioned datasets have
been combined into the PIRB benchmark [6], for which the authors additionally
collected 10 new datasets by crawling QA sections on Polish websites. In
total, the benchmark included 41 information retrieval tasks for the Polish
language. In contrast to data availability, there is still substantial room
for improvement when it comes to models. Previous studies have primarily
focused on dense retrievers [20, 6]. The topic of reranking has only been
addressed by Wojtasik et al. [25]. The authors have also provided four
reranking models for the Polish language.

##  2 Evaluation

In this chapter, we describe the evaluation results of Polish and multilingual
rerankers on the Polish Information Retrieval Benchmark (PIRB). In our
experiments, we considered publicly available models as well as models
specifically trained by us for this study using publicly available source
code. Since reranking is the second stage of the RAG process, to conduct the
evaluation, we also used a retrieval model to initially select documents to be
sorted by each of the rerankers. For the first-stage retrieval, we utilized
mmlw-retrieval-roberta-large model, which, at the time of the experiments, was
the top-performing retriever in the PIRB benchmark according to the NDCG@10
metric. The model was used to extract the top 100 documents for each query,
which were then passed as input to the rerankers. Using the same retriever for
all rerankers allowed for a reliable comparison of their results both among
themselves and in relation to the performance of the retriever itself. It
should be noted that the computational cost of reranking is high, and some
datasets in the benchmark contain up to several hundred thousand queries.
Considering this, to enable efficient evaluation of multiple models, we
limited the number of queries to the first 1000 for each dataset. In the
following sections, we describe the evaluated methods and then proceed to the
analysis of the results.

###  2.1 Evaluated models

In our experiments, we included the following publicly available models:

  * $\bullet$

Polish rerankers \- The first group of methods consists of models trained as a
part of experiments conducted by Wojtasik et al. [25]. The authors evaluated
several baseline retrieval and reranking methods. They trained four rerankers
on the training split of the Polish MS MARCO dataset. This includes two types
of models and two sizes (base and large) for each type. The first type
includes standard cross-encoders with classification head based on the HerBERT
language model [14]. The second type utilizes a sequence-to-sequence approach,
in which the model takes the query-document pair as input and outputs a
"positive" token if the document is relevant to the query or a "negative"
token otherwise. These rerankers were created by fine-tuning plT5 language
model [4].

  * $\bullet$

Multilingual rerankers trained on mMARCO \- mMARCO is a multilingual version
of the MS MARCO dataset translated to 13 languages by researchers from the
University of Campinas, Brazil. After the release of this dataset, the authors
also published a set of rerankers trained by
them111https://huggingface.co/unicamp-dl. Most of these models are fine-tuned
versions of MT5 [27] with varying sizes, ranging from a few hundred million to
13 billion parameters.

In addition to the previously provided models, we also trained our own
rerankers for Polish, utilizing three popular learning methods from the
literature. For each method, we fine-tuned the models in two sizes, leveraging
Polish RoBERTa language models [5]. To train the models, we used data from the
Polish translation of MS MARCO. We employed the following methods:

  * $\bullet$

Binary cross entropy (BCE) loss with hard negatives \- The simplest and most
popular method for training rerankers is to treat the problem as binary
classification, in which relevant query-document pairs are positive samples
and non-relevant pairs are negative samples. The effectiveness of this method
depends on the selection of appropriate negatives that are difficult to
distinguish from positive pairs. In our procedure, we used a set of negatives
available in the Sentence-Transformers
library222https://huggingface.co/datasets/sentence-transformers/msmarco-hard-
negatives. The models were trained for 10 epochs with a batch size of 32 and
peak learning rate of 1e-5 with linear decay.

  * $\bullet$

Knowledge distillation with mean squared error (MSE) loss \- Another way of
training is to use relevancy scores from another high-quality reranker and
teach the new model to mimic those predictions. This is therefore a regression
task because we directly use numerical labels denoting the level of relevance
between the query and the document. In this procedure, we used publicly
available pre-computed set of scores for the MS MARCO
dataset333https://zenodo.org/record/4068216/files/bert˙cat˙ensemble˙msmarcopassage˙train˙scores˙ids.tsv.
The models were trained for 10 epochs with batch size of 32 and peak learning
rate of 1e-5 with linear decay.

  * $\bullet$

Permutation distillation with RankNet loss \- The above two methods are based
on simple loss functions that consider each query-document pair independently.
More sophisticated techniques for training rerankers propose listwise
approaches, in which the loss is calculated based on the query and the set of
all documents assigned to that query. One such method is RankNet [3], in which
loss is computed based on the relative order of documents sorted by their
relevance to the query. The technique has been applied recently by Sun et al.
[22], who proposed to generate sorted lists of relevant documents by prompting
large language models, and then train smaller neural rerankers on this data.
We applied the same idea to train our models. First, we generated 10,000
samples from the MS MARCO dataset, where each sample consisted of a query and
a sorted list of 20 relevant documents. Then we used the scripts provided by
the authors to fine-tune the models for Polish. For training, we employed the
hyperparameters suggested in their article [22] \- the models were trained for
two epochs with a batch size of 32 and a constant learning rate of 5e-5.

###  2.2 Evaluation results

The results of our experiments are shown in Table 1. The table is divided into
sections based on the size of the evaluated models. The first group consists
of models of small and base sizes, ranging from tens to a maximum of a few
hundred million parameters. The subsequent groups include large models (>300
million parameters) and the largest models (>3 billion). For comparison, the
table also includes the results of first-stage retrieval methods: BM25 and
mmlw-retrieval-roberta-large dense retriever. The latter model is particularly
important since it was used to generate candidate documents for reranking.
Therefore, we expect rerankers to achieve results higher than those obtained
by this model, since only improving on the first-stage results makes the
reranker useful. Scores higher than the retriever are shown in blue.

The most important observation is that the majority of rerankers were unable
to surpass the results of the retrieval stage. Only two models achieved a
higher average NDCG@10 metric value for the entire benchmark – the largest
from the evaluated models (mt5-13b-mmarco-100k) and the polish-roberta-
large-v2 model fine-tuned using the RankNet method. The situation looks better
when broken down into individual task groups. Some types of datasets appear to
be easier for rerankers (e.g., PolEval or MAUPQA), while others pose
significant challenges (e.g., Web Datasets or BEIR-PL). Since the data from
the PolEval and MAUPQA groups were prepared by the same authors, we can
speculate that some steps in the process of building these datasets, such as
the initial selection of documents for labeling, might favor reranking models.
On the other hand, Web Datasets and BEIR-PL are characterized by considerable
data diversity, as they represent compilations of datasets from multiple
sources or prepared by different authors, sharing few common features.

Table 1: Evaluation results on PIRB. We report the average NDCG@10 metric values for all datasets (41 tasks) and for the individual task groups. Rerankers are divided into groups based on the number of parameters. The highest NDCG@10 value in each group is underlined. Highlighted in blue are the results of rerankers that are higher than those achieved first-stage retrieval. $*$ indicates models trained by us. Model name |  Average |  PolEval |  WebDS |  BEIR |  MAUPQA |  Other
---|---|---|---|---|---|---
Retrieval only
BM25 | 41.85 | 45.51 | 47.27 | 33.26 | 38.64 | 71.09
sdadas/mmlw-retrieval-roberta-large | 58.53 | 62.72 | 67.43 | 53.19 | 50.08 | 83.85
Small & base models
unicamp-dl/mMiniLM-L6-v2-mmarco-v2 | 49.28 | 58.76 | 45.41 | 40.61 | 49.80 | 78.10
clarin-knext/herbert-base-reranker-msmarco | 49.77 | 59.87 | 46.52 | 41.47 | 49.22 | 78.12
nreimers/mmarco-mMiniLMv2-L12-H384-v1 | 53.38 | 64.69 | 51.02 | 44.37 | 52.37 | 80.12
unicamp-dl/mt5-base-mmarco-v2 | 53.98 | 63.29 | 56.55 | 44.31 | 50.84 | 81.72
clarin-knext/plt5-base-msmarco | 55.39 | 63.40 | 60.65 | 45.69 | 51.29 | 81.61
polish-roberta-base-v2 (BCE)* | 52.56 | 64.23 | 51.35 | 44.27 | 49.73 | 79.67
polish-roberta-base-v2 (RankNet)* | 52.75 | 65.36 | 53.78 | 42.38 | 49.27 | 81.90
polish-roberta-base-v2 (MSE)* | 54.12 | 64.57 | 53.51 | 46.34 | 50.99 | 81.81
Large models (> 300m params)
clarin-knext/herbert-large-msmarco | 52.96 | 66.22 | 51.19 | 42.23 | 51.94 | 79.64
clarin-knext/plt5-large-msmarco | 57.36 | 65.97 | 64.11 | 46.98 | 52.25 | 84.47
polish-roberta-large-v2 (BCE)* | 56.83 | 69.04 | 60.29 | 45.80 | 52.64 | 84.29
polish-roberta-large-v2 (MSE)* | 57.28 | 68.02 | 60.62 | 48.00 | 52.56 | 83.98
polish-roberta-large-v2 (RankNet)* | 58.57 | 71.70 | 63.91 | 47.54 | 52.36 | 86.42
XL & XXL models (> 3b params)
unicamp-dl/mt5-3b-mmarco-100k-kdd-alltrain | 56.97 | 65.96 | 64.34 | 46.37 | 51.10 | 85.96
unicamp-dl/mt5-13b-mmarco-100k | 61.63 | 70.54 | 72.13 | 51.62 | 53.62 | 86.25

In order to further investigate the underlying factors affecting the
performance of the rerankers, we compared their results against those achieved
by the retriever, broken down into individual datasets. Charts illustrating
this comparison for two best-performing rerankers are shown in Figure 2. Each
chart shows 41 datasets from PIRB sorted by the absolute improvement in
NDCG@10 metric values compared to the retrieval stage.

Figure 2: Comparison of NDCG@10 scores between first-stage retrieval and
reranking for the top two rerankers, broken down into individual datasets from
the PIRB benchmark.

Particularly noteworthy are the _arguana_ and _quora_ datasets, as both
rerankers perform poorly on them. For the 13B model, these are the only two
datasets where this model significantly lags behind the retriever. For the
rest, the differences are either small, or the reranker demonstrates a clear
advantage. This can be explained by the fact that these two datasets are not
typical examples of question answering tasks. In the case of the _arguana_ ,
the goal is to find the best matching counterarguments to a given argument,
while the _quora_ task involves finding duplicates for a given question. It is
understandable that models fine-tuned on QA-oriented data may not generalize
well to other types of information retrieval tasks. However, it is worth
noting that dense retrievers handle such problems better, even when not
trained on similar data.

The smaller of the two rerankers performs worse on a larger number of
datasets. We can observe that these are often specialized datasets related to
a narrow domain of knowledge. Examples include _zapytajfizyka_ (physics),
_abczdrowie_ (medicine), _eprawnik_ (law), _pwn_ (linguistics), _scifact_
(science), _fiqa_ (finance), _specprawnik_ (law). In the case of the larger
reranker, there is also a slight decrease in quality on some of these
datasets, but overall, this model performs significantly better on specialized
domains. This confirms the observation of Rosa et al. [18] that with an
increase in the number of model parameters, its out-of-domain performance also
improves. In summary, most of the existing Polish and multilingual rerankers
fall short in quality compared to modern dense retrievers, and there is no
reason to use them in a zero-shot setting. The solution is either fine-tuning
a smaller model to the specific domain or task, or using rerankers with a
larger number of parameters that possess better generalization capabilities.

##  3 Knowledge distillation

In the previous chapter, we demonstrated that of the available Polish and
multilingual rerankers, only the largest MT5-XXL model performed significantly
better than the dense retriever while demonstrating a reasonably good
generalization ability for out-of-distribution data. Unfortunately, such large
rerankers have limited practical applications due to their high computational
requirements and long query processing time. With this in mind, in a
subsequent experiment we verified whether it was possible to train smaller
models using knowledge distillation methods to achieve performance comparable
to the 13B model. In this section, we summarize our findings.

We applied MSE and RankNet methods described in the previous section,
utilizing the same language models, i.e. polish-roberta-base-v2 and polish-
roberta-large-v2. The main difference was using a much larger dataset and
employing the 13B model as a teacher instead of publicly available pre-
computed set of predictions. Previously, a subset of the MS MARCO training
data was used. In this experiment, we expanded the training dataset to include
the following parts: 1) All queries from the Polish MS MARCO training split
(800k queries); 2) The ELI5444https://huggingface.co/datasets/eli5 dataset
translated to Polish (over 500k queries); and 3) A collection of questions and
answers crawled from the Polish website _znanylekarz_ (approximately 100k
queries). In total, the compiled dataset consisted of 1.4 million queries and
10 million documents. Then, for each query, we retrieved a set of best-
matching documents using three retrieval methods with different
characteristics: BM25, dense retriever, and SPLADE [7]. Each method returned
the top 16 results, so after removing duplicates we obtained at most 48
documents per query. In the last step, we used the 13B model to score all
query-document pairs. For the MSE method, the scores were used directly for
training. In the case of the RankNet method, for each query, we created an
ordered list by sorting the documents according to their score. We used the
same training hyperparameters as in the previous experiment, except for linear
decay and a lower learning rate of 2e-5 for the RankNet method.

The results of the newly trained models are shown in Table 2. We can observe
that the MSE distillation yielded reasonably good results, with both models
performing better than previously evaluated methods with a similar number of
parameters. However, RankNet distillation proved to be even more effective.
The large model trained with this method achieved performance higher than the
teacher model, while the base model also demonstrated strong results,
outperforming both MSE rerankers. The best of our models achieved the average
NDCG@10 of 62.65 on the PIRB benchmark, over 1 point higher than the teacher.
It also has 30 times fewer parameters (435M vs. 13B) and is over 33 times
faster. The average number of queries per second during the evaluation on a
single Nvidia A6000 GPU was 0.15 for the teacher model, 5.0 for polish-
roberta-large-v2 and 8.4 for polish-roberta-base-v2.

Table 2: Results of the distilled models on the PIRB benchmark. We report the average NDCG@10 metric values for all datasets and for the individual task groups. Model name |  Average |  PolEval |  WebDS |  BEIR |  MAUPQA |  Other
---|---|---|---|---|---|---
Retrieval only
sdadas/mmlw-retrieval-roberta-large | 58.53 | 62.72 | 67.43 | 53.19 | 50.08 | 83.85
Rerankers
polish-roberta-base-v2 (MSE) | 57.50 | 64.78 | 66.95 | 48.08 | 50.68 | 82.15
polish-roberta-large-v2 (MSE) | 60.27 | 67.54 | 71.44 | 51.13 | 51.92 | 84.87
polish-roberta-base-v2 (RankNet) | 60.32 | 67.92 | 70.39 | 51.61 | 52.48 | 83.36
polish-roberta-large-v2 (RankNet) | 62.65 | 70.77 | 73.81 | 54.05 | 53.54 | 86.01
unicamp-dl/mt5-13b-mmarco-100k (teacher) | 61.63 | 70.54 | 72.13 | 51.62 | 53.62 | 86.25

The experiment shows that by using a sufficiently large training dataset, we
were able to match and even surpass the performance of the multi-billion
teacher model. It also demonstrates clear advantage of permutation
distillation over the traditional distillation method based on mean squared
error loss. In Sun et al. [22] study for English, they trained models using
this technique on datasets of up to 100,000 queries. We believe that combining
this method with larger and more diverse training datasets would improve
performance for English and multilingual models as well.

##  4 Conclusions

In this publication, we have evaluated the performance of Polish and
multilingual text ranking models on PIRB, an exhaustive benchmark covering 41
information retrieval tasks for Polish. Most of the rerankers we have analyzed
perform worse than strong dense retriever for Polish, and their generalization
ability is low, except for models with a large number of parameters. In our
experiments, we also trained a set of new models, including baseline methods
as well as high-quality rerankers distilled from the largest available
multilingual model based on the MT5-XXL architecture. We showed that with a
diverse dataset and an effective training method, it is possible to build a
compact reranker that achieves results competitive with significantly larger
models. The best of our models sets new state-of-the-art for Polish reranking
tasks, outperforming existing solutions.

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11institutetext: Institut fÃr Theoretische Physik, Auf der Morgenstelle 14,
72076 TÃbingen (Germany)

# Gauge-fixed Lattice QCD and the dispersion relation of Wilson fermions

Giuseppe Burgio 11<EMAIL_ADDRESS>Hannes Vogt 11
<EMAIL_ADDRESS>
###### Abstract

We show that, when investigating Wilson-fermions correlation functions on the
lattice, one is bound to encounter major difficulties in defining their
dispersion relation, even at tree level. The problem is indeed quite general
and, although we stumbled upon it while studying Coulomb-gauge applications,
it also affects gauge fixed studies in covariant gauges, including their most
popular version, Landau gauge. In this paper we will discuss a solution to
this problems based on a redefinition of the kinematic momentum of the
fermion.

##  1 Introduction

The investigation of the IR-properties of non abelian gauge theories is still
one of the most interesting and fruitful chapters of particle physics. In
particular, lattice studies have contributed a great deal to our understanding
of the gauge invariant aspects of confinement, chiral symmetry breaking,
hadron spectrum etc. However, both from a theoretical and from a practical
point of view, there exists a broad interest in analysing whether and how such
properties are encoded in gauge-fixed Green’s functions, mainly since most
applicable continuum methods still heavily rely on them. The corresponding
first studies for QCD date back to the ’70s, see e.g. Casher:1974vf ;
Gribov:1977wm ; Mandelstam:1979xd ; Kugo:1979gm ; Cornwall:1981zr , and many
analytic and numerical investigations have since followed, both in covariant
Zwanziger:1991gz ; Zwanziger:1992qr ; Zwanziger:1993dh ; Roberts:1994dr ;
Cucchieri:1995pn ; Cucchieri:1997dx ; Cucchieri:1997ns ; Leinweber:1998uu ;
Skullerud:2000un ; Alkofer:2000wg ; Boucaud:2000ey ; Bonnet:2001uh ;
Bowman:2001xh ; Bonnet:2002ih ; Bowman:2002bm ; Zwanziger:2003cf ;
Bloch:2003sk ; Bowman:2004jm ; Gattnar:2004bf ; Cucchieri:2004sq ;
Bogolubsky:2005wf ; Schleifenbaum:2006bq ; Bogolubsky:2007bw ; Dudal:2009xh ;
Vandersickel:2012tz ; Schaden:2013ffa  and non-covariant gauges
Dell'Antonio:1991xt ; Zwanziger:1998ez ; Cucchieri:2000gu ; Cucchieri:2000hv ;
Cucchieri:2000kw ; Zwanziger:2002sh ; Feuchter:2004mk ; Greensite:2004ke ;
Epple:2006hv ; Nakagawa:2006fk ; Quandt:2007qd ; Epple:2007ut ; Quandt:2008zj
; Reinhardt:2008ek ; Burgio:2008jr ; Burgio:2009xp ; Campagnari:2010wc ;
Leder:2010ji ; Quandt:2010yq ; Reinhardt:2011fq ; Reinhardt:2011hq ;
Heffner:2012sx ; Burgio:2012bk ; Burgio:2012ph ; Pak:2013uba ; Vogt:2013jha ;
Greensite:2014bua ; Burgio:2015hsa ; Campagnari:2015zsa ; Heffner:2015zna ;
Vastag:2015qjd ; Burgio:2016nad ; Reinhardt:2017pyr . A consistent picture
emerges from this corpus, supporting Gribov’s original idea Gribov:1977wm :
restricting the functional integral to the first Gribov region indeed
generates a dynamical mass scale ${\cal{O}}\left(1\,\mathrm{GeV}\right)$.
Coulomb-gauge in particular, being the closest we have to a physical
description Christ:1980ku ; Adler:1984ri , offers many advantages; e.g., one
can show that the self energy of the gluon can in fact be described with
Gribov’s original formula
$\omega_{A}=\sqrt{|\vec{p}|^{2}+M^{4}/|\vec{p}|^{2}}$, again with
$M={\cal{O}}\left(1\,\mathrm{GeV}\right)$ Burgio:2008jr .

A lesson one constantly learns from all lattice investigations at fixed gauge
is that, to be able to extract meaningful informations, one must take
particular care in treating discretization artifacts, which often cloud the
sought results. For example, the gluon self-energy above could only be
obtained after devising a proper scheme to remove spurious contributions
caused by the discretized time Burgio:2008jr . Similarly, extracting the
Coulomb string tension directly from the Coulomb-kernel, which one naively
would consider the most straightforward prescription, turns out to be a quite
non trivial task; alternative, less intuitive definitions turn out to be more
efficient Burgio:2015hsa . This holds of course also for quark-quark
correlation functions, as the extraction of the chiral mass and of the self
energy for staggered fermions, necessary to detect any evidence of chiral
symmetry breaking and confinement, clearly exemplifies Burgio:2012ph .

In this paper we will show that for Wilson fermions extra care needs to be
taken when calculating fermionic Green’s functions. Indeed, the problem lies
in the very definition of a suitable kinematic momentum against which to plot
fermionic correlation functions, independently of the gauge, and thus affects
all gauge fixed investigation, e.g. also in Landau gauge Skullerud:2000un . Of
course, one could argue that other lattice fermions could be used for QCD
instead, since they should not suffer from such problems. However, the recent
surge in investigations of beyond the standard model scenarios, with many
lattice simulations performed for different models, has increased the interest
in studying the non perturbative properties of correlation functions in QCD-
like theories; depending on the model chosen, one will want to keep the
freedom of using the most convenient discretization for the fermionic fields.
Indeed, that’s how we noticed the issue in the first place - when trying to
apply the analysis of Burgio:2012ph  to SU(2) + (adjoint) Wilson fermions
configurations generated by the Edinburgh-Swansea-Odense collaboration as a
model for minimal walking Technicolor DelDebbio:2008zf ; DelDebbio:2010hu ;
DelDebbio:2010hx ; DelDebbio:2015byq .

##  2 The fermion self energy

The most general structure for the expectation value of the Dirac operator,
both in covariant and Coulomb gauges, reads Skullerud:2000un ; Burgio:2012ph :

$S^{-1}(\vec{p},p_{4})=i\,\vec{p}/A_{s}(\vec{p},p_{4})+i\,p/_{4}A_{t}(\vec{p},p_{4})+i\,p_{i}\gamma_{i}p_{4}\gamma_{4}A_{d}(\vec{p},p_{4})+B(\vec{p},p_{4})\mathbb{1}\,.$
(1)

$A_{d}$ can be shown to vanish in general, while of course $A_{s}(p)=A_{t}(p)$
for covariant gauges Skullerud:2000un ; Burgio:2012ph . For Coulomb gauge,
moreover, $A_{s}$, $A_{t}$ and $B$ can be shown to be $p_{4}$ independent
Burgio:2012ph . Factorizing $A_{s}=Z^{-1}$ out, Eq. (1) can be rewritten as
Skullerud:2000un ; Burgio:2012ph :

$S^{-1}(\vec{p},p_{4})=Z^{-1}(\vec{p})\left[i\,\vec{p}/+i\,p/_{4}\,\alpha(\vec{p})+M(\vec{p})\mathbb{1}\right]\quad\left(\mathrm{cov.}\;S^{-1}(p)=Z^{-1}({p})\left[i\,p/+M({p})\mathbb{1}\right]\right)\,,$
(2)

where $\alpha=A_{t}/A_{s}$ and $M=B/A_{s}$. The inverse Dirac propagator above
can be used to define the static quark propagator $S(\vec{p})=\int
dp_{4}\,S(\vec{p},p_{4})$, which, in analogy to the free case, corresponds up
to a factor $1/2$ to the Hamiltonian operator $H$. Contrary to covariant
gauges, this is straightforward to calculate in Coulomb gauge, yielding
Burgio:2012ph :

$H(\vec{p})=\frac{\displaystyle
Z(\vec{p})}{\displaystyle\alpha(\displaystyle\vec{p})}\frac{\displaystyle\sqrt{\vec{p}^{2}+M^{2}(|\vec{p}|)}}{\displaystyle
i\,\vec{p}/+M(\vec{p})\mathbb{1}}=\frac{\displaystyle
Z(\vec{p})}{\displaystyle\alpha(\displaystyle\vec{p})}\frac{\displaystyle-i\,\vec{p}/+M(\vec{p})\mathbb{1}}{\displaystyle\sqrt{\vec{p}^{2}+M^{2}(|\vec{p}|)}}\,.$
(3)

The dispersion relation of the fermion can therefore be directly read off from
Eq. (3), since the inverse coefficient of $-i\,\vec{p}/+M(\vec{p)}$
corresponds to the eigenvalue of $H$:

$\omega(|\vec{p}|)=\frac{\alpha(|\vec{p}|)}{Z(|\vec{p}|)}\sqrt{\vec{p}^{2}+M^{2}(|\vec{p}|)}\,.$
(4)

As shown in Burgio:2012ph , chiral symmetry breaking is encoded in
$M(|\vec{p}|)$, while for a confining theory both $\omega_{A}$ and $\omega$
should be IR-divergent. Non-perturbative Coulomb gauge therefore nicely
decouples the chiral symmetry and confinement properties of QCD-like theories,
making it an interesting tool in the analysis of beyond the standard model
scenarios Cossu:2008wh .

##  3 The kinematic momentum for Wilson fermions

If the self-energy in Eq. (4) had been obtained in the continuum, there would
be nothing one should be careful about. On the lattice, on the other hand, the
discrete momenta $k_{i}=\frac{2\pi}{aN_{i}}n_{i}$,
$k_{4}=\frac{2\pi}{aN_{t}}\left(n_{4}+\frac{1}{2}\right)$ do not define
suitable kinematic variables that can be related to the continuum ones. For
staggered and overlap fermions there is however an easy prescription:
calculate first, either analytically or numerically, the quark propagator in
the free case,
$S^{\text{(0)}}(k)=i\,\left(\sum_{\mu}C_{\mu}(k_{\mu})\gamma_{\mu}\right)+m\,\mathbb{1}$.
Isolating the coefficients of the (euclidean) $\gamma$-matrices defines now
the kinematic momenta: $p_{\mu}(k_{\mu})=C_{\mu}(k_{\mu})$. The reason why
such procedure works is of course that the $\mathbb{1}$-coefficient is $k$
independent. For Wilson fermions, on the other hand, the doubling problem is
solved by modifying exactly such coefficient, introducing an explicit
$k$-dependence:

$S^{\text{(0)}}(k)=\frac{i\,}{a}\sum_{\mu=1}^{4}\gamma_{\mu}\tilde{p}_{\mu}(k)+\left(m+\frac{a}{4}\sum_{\mu=1}^{4}p_{\mu}^{2}(k)\right)\,\mathbb{1}\,,$
(5)

with $\tilde{p}_{\mu}(k_{\mu})=\frac{1}{a}\sin(ak_{\mu})$ and
$p_{\mu}(k_{\mu})=\frac{2}{a}\sin(ak_{\mu}/2)$; a definition of the kinematic
momenta as for staggered/overlap fermions is therefore not possible. Previous
studies in Landau gauge Skullerud:2000un  have ignored the problem and used
$p_{\mu}$ instead; this however does not reproduce the correct dispersion
relation even in the free case, in contrast e.g. to $\omega_{A}$, which is a
"natural" function of $p_{\mu}(k_{\mu})$ Leinweber:1998uu .

To illustrate our proposal, start by considering the free static propagator:

$\int_{-\infty}^{\infty}\frac{dp_{4}}{2\pi}S_{0}(\vec{p},p_{4})=\frac{-i\,{\vec{p}/}+m\,\mathbb{1}}{2\sqrt{\vec{p}^{2}+m^{2}}}\equiv\frac{-i\,{\vec{p}/}+m\,\mathbb{1}}{2\omega_{0}}\,,$
(6)

where $\omega_{0}$ is the free dispersion relation, i.e. the eigenvalue of the
free Hamiltonian, the $p$ are all functions of $k$ and:

$S_{0}(\vec{p},p_{4})=\frac{-i\,p/_{4}-i\,{\vec{p}/}+m\,\mathbb{1}}{p_{4}^{2}+\vec{p}^{2}+m^{2}}\,.$
(7)

Notice now that the free dispersion relation can be directly obtained from the
free propagator:

$\omega_{0}^{-1}(k)=\frac{1}{\sqrt{\vec{p}^{2}(k)+m^{2}}}=\int_{-\infty}^{\infty}\frac{dp_{4}}{2\pi}\left|S_{0}(\vec{p}(k),p_{4})\right|^{2}\,,$
(8)

where

$\left|S_{0}(p)\right|^{2}=S_{0}^{\dagger}(p)S_{0}(p)=\frac{1}{p_{4}^{2}+\vec{p}^{2}+m^{2}}\,,$
(9)

The kinematic momentum can be thus "defined" via $S_{0}$, calculating
$\omega_{0}(k)$ through Eq. (8) and setting
$|\vec{p}(k)|=\sqrt{\omega_{0}^{2}(k)-m^{2}}$, very much in the spirit of the
gluonic "tree-level correction" of Leinweber:1998uu . Since in lattice
calculations, to avoid discretization errors arising from the breaking of
rotational invariance, one only considers diagonal momenta of the form
$\vec{k}=(k,k,k)$ Leinweber:1998uu , such prescription is sufficient for all
practical purposes. Explicitly (for the sake of readability the lattice
spacing will be set to $a=1$ in the following):

$\displaystyle S_{0}^{\text{lat}}(k)$
$\displaystyle=\frac{-i\,\gamma_{4}\sin(k_{4})-i\,\sum_{i}\gamma_{i}\sin(k_{i})+(m+\sum_{\mu}(1-\cos(k_{\mu})))\,\mathbb{1}}{\sin^{2}(k_{4})+\sum_{i}\sin^{2}(k_{i})+\left[m+\sum_{\mu}(1-\cos(k_{\mu}))\right]^{2}}\,,$
(10) $\displaystyle\left|S_{0}^{\text{lat}}(k)\right|^{2}$
$\displaystyle=\frac{1}{\sin^{2}(k_{4})+\sum_{i}\sin^{2}(k_{i})+\left[m+\sum_{\mu}(1-\cos(k_{\mu}))\right]^{2}}\,.$
(11)

Replacing ${\displaystyle{\int\frac{dp_{4}}{2\pi}}}$ with
${\displaystyle{\frac{1}{2\pi}\frac{4\pi}{N_{t}}\sum_{k_{4}}}}$ in Eq. (8) we
thus get:

$\displaystyle\omega_{0,m}^{-1}(k)$
$\displaystyle=\frac{1}{\sqrt{\vec{p}^{2}+m^{2}}}\equiv\frac{2}{N_{t}}\sum_{k_{4}}\left|S_{0}^{\text{lat}}(k)\right|^{2}\,,$
(12) $\displaystyle|\vec{p}({k})|$
$\displaystyle=\sqrt{\frac{1}{\left(\frac{2}{N_{t}}\sum_{k_{4}}\left|S_{0}^{\text{lat}}(k)\right|^{2}\right)^{2}}-m^{2}}\,.\
$ (13)

Setting now $m=0$ we get, as operative definition of the kinematic momentum:

$\displaystyle|\vec{p}(k)|^{-1}$
$\displaystyle=\frac{2}{N_{t}}\sum_{k_{4}}\left|S_{0}^{\text{lat}}(k)\right|^{2}$
$\displaystyle=\frac{2}{N_{t}}\sum_{n_{4}=-N_{t}/2+1}^{N_{t}/2}\frac{1}{\sin^{2}(k_{4})+\sum_{i}\sin^{2}(k_{i})+\left[\sum_{\mu}(1-\cos(k_{\mu}))\right]^{2}}\,.$
(14)

The values arising from Eq. (14) can be calculated once and for all for each
desired lattice size and stored to be used whenever needed. Figure 1 shows the
comparison between the "naive" fermionic kinematic momentum
$p(k)=2\sqrt{\sum_{i}\sin^{2}(k_{i}/2)}$, as used e.g. in Skullerud:2000un ,
and our definition Eq. (14) for a $128^{4}$ lattice. As one can see, the
differences are quite striking throughout the whole Brillouin-zone.

Figure 1: Comparison between $p(k)=2\sqrt{\sum_{i}\sin^{2}(k_{i}/2)}$ and Eq.
(14) for a $128^{4}$ lattice ($\vec{k}=(k,k,k)$).

Figure 2 compares the (massive) free fermionic dispersion relations plotted
against the naive momentum and the one obtained from Eq. (14), with an
arbitrarily chosen bare mass. While the former shows a dependence which is
physically meaningless for all momenta and is bound to distort any IR results,
the latter gives (by definition!) exactly the expected physical behaviour
throughout the whole Brillouin-zone.

Figure 2: Massive fermionic dispersion relations vs. naive and improved
momentum from Eq. (14).

The above definition will suffice for most applications. Should one however
think that mass contributions in the definition of the kinematic momenta might
play a role in the application at hand, and therefore wish to extend the
prescription to $m\neq 0$, a bit of care is needed. As one can immediately
verify, for $m\neq 0$ Eq. (12) shifts the pole from $m$ to
$\frac{1}{2}m^{2}+m$, so that from Eq. (13) one gets $\vec{p}(k=0)\neq 0$,
moreover with the wrong slope at the origin,
$\left.\displaystyle{\frac{{d}{p}}{{d}{k}}}\right|_{k=0}\neq 1$. A "mass-
improved" prescription should be therefore defined as follows. Expand first
$\omega^{2}_{0,m}(k)$ from Eq. (12) around $k=0$ up to $2^{\mathrm{nd}}$ order
for diagonal momenta, obtaining:

$\omega_{0,m}^{2}(k)=\left[\frac{N_{t}}{2}\frac{1}{\sum_{k_{4}}\frac{1}{\sin^{2}\left(k_{4}\right)+3\sin^{2}\left(k\right)+\left[m+2\sin^{2}\left(\frac{k_{4}}{2}\right)+6\sin^{2}\left(\frac{k}{2}\right)\right]^{2}}}\right]^{2}=\omega^{2}_{0,m}(0)+\frac{1}{2}\left.\frac{\partial^{2}\omega_{0,m}^{2}(k)}{\partial
k^{2}}\right|_{k=0}k^{2}+\mathcal{O}(k^{4})\,.$ (15)

From this on can then directly define:

$|\vec{p}(k)|^{2}=\left[\omega_{0,m}^{2}(k)-\omega_{0,m}^{2}(0)\right]\left[\frac{1}{2}\left.\frac{\partial^{2}\omega_{0,m}^{2}(k)}{\partial
k^{2}}\right|_{k=0}\right]^{-1}\,,$ (16)

with:

$\displaystyle\omega^{2}_{0,m}(0)$
$\displaystyle=\left[\frac{N_{t}}{2\sum_{k_{4}}\frac{1}{S_{m}(k_{4})}}\right]^{2}\,,\quad\qquad\left.\frac{\partial^{2}\omega_{0,m}^{2}(k)}{\partial
k^{2}}\right|_{k=0}=\frac{N_{t}^{2}\sum_{k_{4}}\frac{6+6\left(m+2\sin^{2}\left(\frac{k_{4}}{2}\right)\right)}{S^{2}_{m}(k_{4})}}{2\left[\sum_{k_{4}}\frac{1}{S_{m}(k_{4})}\right]^{3}}\,,$
$\displaystyle S_{m}(k_{4})$
$\displaystyle=\sin^{2}(k_{4})+\left[m+2\sin^{2}\left(\frac{k_{4}}{2}\right)\right]^{2}=m^{2}+4(m+1)\sin^{2}\left(\frac{k_{4}}{2}\right)\,.$
(17)

##  4 Conclusions and outlook

In this paper we have given an improved definition of the kinematic momentum
for Wilson fermions, which we believe to be crucial if one wishes to extract
any information from the IR-behaviour of correlation functions. Such
definition, adapted accordingly, should be used in any gauge-fixed analysis
involving Wilson-fermions, e.g. also in Landau gauge. Concerning our original
motivation, i.e. the application of the analysis of Burgio:2012ph  to the
configurations used in DelDebbio:2008zf ; DelDebbio:2010hu ; DelDebbio:2010hx
; DelDebbio:2015byq , two further problem still need to be solved. First, for
Wilson fermions the coefficient $B$ in Eq. (1) is $k_{4}$ dependent, both at
tree level and in the interacting case. Normalizing $B$ to its tree level
expression, i.e. the $\mathbb{1}$ coefficient in Eq. (5), as proposed in
August:2013jia  for Landau gauge, could only work if the ratio for different
energies at fixed $\vec{k}$,
$\displaystyle{{B_{\text{lat}}(\vec{k},k^{1}_{4})}/{B_{\text{lat}}(\vec{k},k^{2}_{4})}}$
with $k^{1}_{4}\neq k^{2}_{4}$, would exactly match their equivalent ratios at
tree-level, which one can immediately verify not to be the case. Thus a
different prescription needs to be developed, a possible one being to average
the normalized ratios over $k_{4}$:

$B(\vec{k})=m\frac{1}{N_{t}}\sum_{k_{4}}\frac{B_{\text{lat}}(\vec{k},k_{4})}{B^{0}_{\text{lat}}(\vec{k},k_{4})}\,,$
(18)

where the factor $m$ is needed to guarantee $B(0)=m$. The second problem comes
from the fact that for adjoint Wilson fermions the massless limit at
$\beta=2.25$ is approached for a negative bare mass, $-am_{0}\approx 1.20$.
However this is neither a meaningful choice in Eq. (18) nor does it provide a
reasonable tree-level correction for $B$, as already pointed out for Landau
gauge August:2013jia . The proposal there was to use the PCAC mass instead,
which might be a good choice in the continuum limit, but can still be the
source of other uncertainty at finite lattice spacing. Further work will
definitely be needed to settle the issue.

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