# Fingerprint Pore Detection: A Survey

Azim Ibragimov  Mauricio Pamplona Segundo
Computer Science & Engineering  University of South Florida
<EMAIL_ADDRESS>
###### Abstract

This work presents the first survey on fingerprint pore detection. The survey
provides a general overview of the field and discusses methods, datasets, and
evaluation protocols. We also present a baseline method inspired on the state-
of-the-art that implements a customizable Fully Convolutional Network, whose
hyperparameters were tuned to achieve optimal pore detection rates. Finally,
we also reimplementated three other approaches proposed in the literature for
evaluation purposes. We have made the source code of (1) the baseline method,
(2) the reimplemented approaches, and (3) the training and evaluation
processes for two different datasets available to the public to attract more
researchers to the field and to facilitate future comparisons under the same
conditions. The code is available in the following repository:
https://github.com/azimIbragimov/Fingerprint-Pore-Detection-A-Survey

##  I INTRODUCTION

Due to fingerprint recognition’s adherence to the fundamental biometrics
concepts of permanence and distinctiveness, it has been extensively researched
during the past decade. This led to the development of numerous Automatic
Fingerprint Identification Systems (AFIS). To distinguish between genuine and
imposter individuals, the majority of AFIS use Level 1 (i.e., global
characteristics - ridge flow and pattern type) and Level 2 (i.e., local
features - minutiae and ridge skeleton) features [1]. These techniques achieve
high recognition accuracy but rely on high-quality, low resolution images to
do so. [2] Due to this, many researchers have become interested in using Level
3 features. They include all dimensional attributes of a ridge, such as ridge
path deviation, width, shape, pores, edge contour, incipient ridges, breaks,
creases, scars, and other permanent details. This wide range of fingerprint
attributes made many researchers interested in using L3 features as
supplemental data for matching [1]. This statement is also supported by [3]
who claims that latent print examiners constantly consider L3 features while
performing their duties since level 3 fingerprint features were included in
the FBI standard. [1]

Fingerprint sweat pores were proven to be perpetual, immutable, and unique
[4]. The perpetuity, immutability, and uniqueness of fingerprint sweat pores
make them a perfect fit for recognition purposes. [3]

To capture L3 features, one needs high-resolution, high-quality fingerprint
images. The Scientific Working Group on Friction Ridge Analysis, Study, and
Technology (SWGFAST) proposed a minimum scanning resolution of 1 000 dpi,
which is considered to be high-resolution, for latent images [5]. These high-
resolution images can capture locations and shapes of fingerprint pores, which
can be later used for recognition purposes.

Sizes of fingerprint sweat pores range from 88 to 220 microns and they come in
a variety of forms, including round, elliptical, oval, square, rhomboid, and
triangular. Additionally, the location of the sweat pores can vary. They are
typically located in the middle of a fingerprint ridge (closed pores), but
they can sometimes occasionally be open to the side (open pores). [4] Figure 1
illustrates examples of sweat pores that are present in a high-resolution
fingerprint image.

Figure 1: Sweat pores that are present in a high-resolution fingerprint image.
Blue circles represent open pores, and yellow circles represents closed pores.

This phenomena led to development of innovative AFIS systems that rely on pore
detection. Since pores’ shape and size can vary greatly, detecting them can be
difficult [6, 7, 8]. This difficulty inspired researchers to enhance pore
detection results, and significant progress has been done so far [9, 10, 11].

However, there are numerous issues associated with the publications that have
been released in the past 25 years. We believe that addressing these problems
would positively affect the quality of future publications. These issues
include, but are not limited to, varying evaluation procedures, various
training and testing protocols, difficulties in duplicating the results of
other studies, and a lack of publications that make their source code publicly
available. These factors lead to publications that are hard to analyze and
compare. This publication aims to analyze these issues, offer solutions for
some of them, and hopefully support future publications in this research
field.

To our knowledge, no thorough survey on pore detection has ever been carried
out. This paper aims to bring clarity to the subject, compare existing works,
and figure out the best practices for fingerprint pore extraction

This study divides pore detection publications into two major categories:
handcrafted approaches and machine learning approaches. Techniques developed
by experts using various functions, transformations and filters to locate
pores are referred to as handcrafted approaches. Techniques that use neural
networks to locate pores are referred to as machine learning methods. For a
while, handcrafted methods were preferred, but in recent years, there has been
a surge in the number of publications using methods based on machine learning.
Figure 2 displays the number of publications that have employed either
handcrafted or machine learning approaches to locate pores.

’94’05’07’08’10’12’17’18’19’21$4$$3$$2$$1$$1$$1$$1$$1$$2$$1$$2$HCML Figure 2:
Number of publications that have employed either handcrafted or machine
learning approaches to locate pores. Blue bars represent number of
publications that have employed handcrafted approaches and red bars represent
number of publications that have employed machine learning approaches.

As a general rule, machine learning approaches tend to display better results
when compared to handcrafted approaches. This can be seen when comparing
various works that are either implement machine learning [12, 13, 14, 15] or
hand-crafted approaches [16, 10]

Our contributions are:

  * •

The first survey on fingerprint pore detection, which covers existing
materials and methods and summarizes the current state-of-the-art performance.

  * •

A baseline approach inspired by state-of-the-art works, whose final
configuration was decided based on extensive experiments.

  * •

The reimplementation of three works from the literature [17, 18, 11].

  * •

A public repository containing the source code of reimplementations, baseline,
and experiments to help future comparisons in this research field.

We organize this survey as follows. Section II introduces the different
existing approaches divided into three main categories: handcrafted, machine
learning, and hybrid approaches. Section III presents the datasets used in the
literature to report pore detection performance. Section IV introduces our
proposed baseline and the layout definition process. Section V presents
results reported in the literature and results obtained in this work for the
proposed baseline and three reimplemented works. Finally, Section V-D presents
our final remarks and discusses future directions.

##  II Pore detection methods

This paper covers several publications on pore detection. Ideally, they should
follow these typical steps:

  1. 1.

Use a publicly available dataset of fingerprint images with annotated pore
locations;

  2. 2.

Propose a new method or modifications to an existing one seeking to advance
the literature on this topic;

  3. 3.

Employ a reproducible and fair protocol to validate the results of the new
method;

  4. 4.

Compare the reported results with other publications, and discuss the
advantages and shortcomings of the proposed method.

In this section, we focus on the design aspects of each work. While most
publications have unique approaches, it is possible to organize them into the
following categories: handcrafted (skeleton-based and filtering-based),
machine learning, and hybrid approaches. The following subsections present
details about each of these categories.

###  II-A Handcrafted skeleton-based methods

The early stages of pore detection research relied on skeleton tracking, a
popular technique for fingerprint-related applications (e.g., ridge
reconstruction [19] and minutia extraction [20]). With it, researchers showed
that pore analysis could significantly improve fingerprint recognition results
and sparked interest in the field [6, 21]

The core idea in this category of approaches is to binarize the fingerprint so
that pores become white-pixel blobs and then to skeletonize the white areas so
that pores show up as short-line segments. For instance, Stosz and Alyea [6]
navigate through these lines starting from endpoints (white pixels with a
single white neighbor) and then stop if they reach another endpoint or a
bifurcation within a maximum distance. Getting another endpoint means we have
a closed pore, while a bifurcation indicates an open pore. Figure 3 presents a
skeletonized fingerprint image with closed and open pores highlighted in
yellow and red circles, respectively.

(a)

(b)

(c)

Figure 3: An example of (a) fingerprint image along its (b) binarized and (c)
skeletonized versions with open (red circles) and closed pores (yellow
circles).

Kryszczuk  _et al._ [21] exploited image preprocessing to improve the quality
of the obtained skeleton. In this context, they used a Gabor wavelet filtering
to enhance the ridges before skeletonization. Figure 4(a) illustrates the
shape of the Gabor wavelet. As observed, this wavelet will produce strong
responses for image locations containing bright and dark areas side-by-side, a
common characteristic in fingerprint images. Figure 8(b) shows one example of
the Gabor wavelet filtering outcome. This work spotlights one of the main
weaknesses of this type of approach, which is its dependence on the skeleton
quality to detect pores correctly. The next categories ease this problem by
not relying on complicated leading stages.

###  II-B Handcrafted filtering-based methods

Before the deep learning era, filtering-based strategies were the most common
approach to detect pores [1, 10, 22, 23, 24, 25, 26]. These works rely on
human experts’ ability to devise a sequence of filtering steps capable of
highlighting pores in fingerprint images to a point where their detection
becomes effortless. In the literature, popular strategies exploit the fact
that fingerprints have evenly distributed pores along their friction ridges
and that pores are small white blobs of pixels in fingerprint images.

Jain  _et al._ [1] used Gabor and Mexican Hat wavelets to highlight ridges and
pores, respectively. Figure 4(b) illustrates the shape of the Mexican Hat
wavelet. As observed, it will produce strong responses for small bright areas
surrounded by dark areas. This result, combined with the strong response in
ridge areas produced by Gabor wavelets, will doubly reinforce the intensity of
pixels in pore areas making the detection simply a matter of binary
thresholding and blob localization. Other works also adopted isotropic kernels
as pore models: Ray  _et al._ [10] used a Gaussian kernel, Parsons  _et al._
[27] used the Difference of Gaussians, and Mngenge  _et al._ [23] the
Laplacian of Gaussian.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: Illustrations of (a) a Gabor wavelet, commonly used to segment
fingerprint ridges, and of (b) a Mexican Hat wavelet [1], (c) a Gaussian
kernel [10], (d) the Difference of Gaussians [27], (e) the Laplace of Gaussian
[23], and (f) an Anisotropic filter [22], all employed to highlight pores.
Figure 5: Diagram of the CNN architecture proposed by Shen  _et al._ [14].

Zhao  _et al._ [22] presented a dynamic anisotropic pore model to handle open
pores better since ridges do not surround them entirely (Figure 4(f)). This
model includes angle and scale parameters to adapt to common pore variations.
They estimate these parameters from local ridge orientation and frequency.
Rodrigues and Borges [25] built upon this work by using anisotropic filters
instead of Gaussians for a DoG-based detection.

Finally, some works invested in an image processing style of solution. Malathi
_et al._ [24] used watershed transform to locate “water-filled” pores. Lemes
_et al._ [11] counted the number of transitions between white and black pixels
in the border of a size-adjusted circle around a pixel to decide whether it is
a pore or not.

However, researchers found this group of techniques computationally expensive
and hard to adapt to unseen imaging conditions (_e.g._ , image resolution,
quality, and finger pressure). For this reason, they started exploring machine
learning methods to overcome these issues.

###  II-C Machine Learning approaches

Unlike handcrafted methods that rely on experts’ knowledge to devise
techniques to locate pores, machine learning leverages annotated data to model
pores automatically. This approach type received lots of attention after deep
neural networks became popular. A neural network is a learning system loosely
inspired by the brain. It consists of layers of interconnected neurons that
find relevant patterns within the training data. Within the pore detection
field, a specific type of neural network - Convolutional Neural Network (CNN)
[28] \- is a unanimous choice. One reason is CNN’s ability to take advantage
of data with spatial contexts, such as images. The convolutional layers can
learn several filters to extract the most significant visual features for a
particular problem.

Shen  _et al._ [14] used the Fully Convolutional Network (FCN) [29]
architecture displayed in Figure 5 for pore detection. Their architecture has
five blocks with two convolutional and one max-pooling layer each, and a
residual connection between the third and fifth blocks. Convolutional layers
are followed by ReLU activation [30] (except the last one) and max-pooling
layers by batch normalization [31]. A final convolutional layer projects the
output into a 1-dimensional space with sigmoid activation. This network takes
17x17 image patches as input, and the outcome indicates whether the input
patch has a pore on its center or not. Figure 6(a) displays samples of patches
centered on pores, and Figure 6(b) patches not centered on pores. They train
the network with focal loss to overcome data unbalancing issues since the
number of non-pore patches is much higher than the number of pore patches. For
inference, the FCN generates a pore intensity map with the probability of each
pixel being the center of a pore, as shown in Figure 7(a). The non-maximum
suppression (NMS) post-processing algorithm [32] merges neighboring pixels
with high probability values into a single detection to obtain unique
coordinates for each pore. The final result is a binary pore map, as shown in
Figure 7(b), with discrete points representing the detected pores.

(a) Positive patch samples

(b) Negative patch samples

Figure 6: Patches with (a) positive cases and (b) negative cases that are fed
into CNN

(a)

(b)

Figure 7: (a) Pore Intensity Map and (b) Binary Pore Map

Other works presented similar approaches [16, 13, 12, 18, 33, 34, 17, 9] with
minor modifications when compared to Shen  _et al._ ’s work. Jang  _et al._
[16] employed soft labels to the training patches to ease the transition
between pore and non-pore labels using a Gaussian distribution. Anand  _et
al._ [13] exploited residual networks [35] to facilitate the training of deep
models. Labati  _et al._ [9] included a second CNN to refine pore detection
results and remove possible errors from the pore map. Finally, Wang  _et al._
[33] and Ding  _et al._ [34] used a U-Net [36] architecture instead of relying
on a patch-based classification.

###  II-D Hybrid Approaches

Hybrid approaches combine handcrafted and machine learning techniques in a way
that overcomes the limitations of each strategy. This way, they can rely on
human ingenuity to overcome training data limitations and ease the learning
task.

Zhao  _et al._ [12] devised a filter-based image enhancement to facilitate
pore detection using a CNN. Like in other filtering-based approaches, they
binarize the fingerprint (Figure 8(a)) and segment ridges using Gabor filters
(Figure 8(b)) and locate white pixels in both images with an XOR operation to
be used as pore candidates (Figure 8(c)). The result becomes the input of a
CNN that finds actual pores in the input image (Figure 8(d)). The post-
processing stage also uses an XOR operation between the CNN output and the
ridges map to remove false detections (Figure 8(e)).

(a)

(b)

(c)

(d)

(e)

Figure 8:  (a) Binary pore map, (b) ridge map, (c) Results of applying XOR
operation on binary and ridge maps, (d) Pore Intensity Map, (e) Binary Pore
Map.

##  III Datasets

For publications on pore detection, datasets are essential. They must include
high-resolution fingerprint images and annotated details about pores and their
locations in this field of study. To our knowledge, eight datasets have been
used in the past by pore detection publications. Table I provides detailed
information about these datasets.

Ref. |  Name |  dpi |  Size |  Public? |  Year
---|---|---|---|---|---
[37] |  NIST SD 4 |  500 |  10 |  No* |  2005
[22] |  PolyU-HRF |  1200 |  30 (DBI), 120 (DBII) |  No* |  2010
[38] |  GEN |  3800 |  66 |  No |  2016
[12] |  DB Touch-less, Latent |  1000 |  80 |  Yes |  2018
[39] |  IITI HRF |  1000 |  20 |  No |  2021
[34] |  Sub-cutan-eous |  2000 |  60 |  No |  2021
[11] |  NB_ID |  1000 |  30 |  No |  2014
[40] |  L3-SF |  1200 |  740 |  Yes |  2020
TABLE I: Summary of datasets that are used by publications in pore extraction.
No* refers to datasets that were publicly available in the past, but not
anymore

Researchers must follow an evaluation protocol for each dataset. This protocol
outlines how to use the dataset for experimentation. Earlier publications that
proposed handcrafted methods used datasets for testing purposes only. They
developed their algorithms and evaluated their performance on the entire
dataset. More recent papers that use machine learning techniques divide the
datasets into training, validation, and testing subsets. These subsets serve
as a means for model learning, hyperparameter tuning, and performance
evaluation. This section will refer to the datasets from Table I and outline
how pore detection publications used them.

###  III-A NIST SD 4 (withdrawn)

One of the earliest works on pore identification used the NIST Special Dataset
4 (NIST SD 4) [10]. To our knowledge, this is the only publication to use this
dataset, which is no longer distributed due to the lack of documentation
required by NIST [37]. The dataset contains thousands of 500 dpi fingerprints,
and Ray  _et al._ [10] annotated pore locations of 10 cherry-picked images for
their pore detection experiments.

###  III-B PolyU HREF (withdrawn)

The following dataset, the Hong-Kong Polytechnic University High-Resolution
Fingerprint Database (PolyU-HRF) [22], was used by numerous publications [22,
10, 41, 25, 9, 18, 33, 16, 17, 14, 42]. It consists of 3170 images with 1200
dpi and has two pore-annotated subsets. The first subset, DBI, is composed of
partial fingerprints of size $320\times 240$. This set contains 30 images with
annotated pore locations. The second subset, DBII, has full-size fingerprint
images with $640\times 480$ pixels. This subset has 120 images with pores
annotated by Teixeira and Leite [43]. Unfortunately, this dataset is no longer
distributed by the authors.

###  III-C IITI-HRF (public)

The Indian Institute of Technology Indore High-Resolution Fingerprint Database
(IITI-HRF) [39] is a more recent dataset and consists of two subsets. The
first one, IITI-HRFC, contains 6400 images with 1000x1000 pixels at 1000 dpi.
The second, IITI-HRFP, is a 320x240 center crop of the first subset. The
authors provide pore annotations for 20 images of the IITI-HRFP subset [39].
This is currently the only publicly available dataset of high-resolution
fingerprints.

###  III-D L3-SF (public)

L3 synthetic fingerprint (L3-SF) database [40] is a publicly available dataset
that contains synthetic high-resolution fingerprint images with the same
structure as the PolyU-HRF DBI subset. The dataset also contains a subset of
740 images with 1200 dpi and annotations of pore locations. As it is composed
of synthetic fingerprints, L3-SF avoids privacy-related issues for dataset
distribution.

###  III-E Private datasets

Different works created their own private datasets for pore detection
analysis, but these were not used by other works later on. The private
datasets are:

####  III-E1 GEN (private)

Genovese  _et al._ [38] created a dataset to develop a pore extraction
technique for touchless fingerprint images. Touchless fingerprint images do
not require fingers to be placed on sensors, thus allowing for the prevention
of hygiene problems and distortions [38]. Their dataset consists of
$1500\times 1000$ that are centered on fingerprint core. A dataset comprised
66 touchless fingerprint pictures.

####  III-E2 DB Touch-less, DB Latent (private)

Labati  _et al._ [9] created two datasets, DB Touch-less and DB Latent, with
80 annotated images in total. These images have resolution of at least 1000
dpi and were used to evaluate pore detection in heterogeneous conditions.

####  III-E3 Subcutaneous (private)

The datasets discussed in this publication so far have only utilized 2D
images. Optical coherence tomography was used by Ding  _et al._ [34] to gather
data about 0–3 mm beneath the skin’s surface. It contains 60 pore-annotated
images with $1500\times 1200$ pixels at 2000 dpi.

####  III-E4 NB_ID (private)

They created the NB_ID dataset to address the problem of newborn
identification. It contains three subsets, NB_ID_A, NB_ID_B, and NB_ID_II,
with thousands of newborn palmprints and footprints. For their experiments,
Lemes  _et al._ [11] randomly selected ten images per subset and annotated
pores in a 100x100 patch from each of them.

##  IV CNN Baseline specification

Figure 9: Baseline CNN Architecture

Reproducing the results of other papers is difficult, as none of them released
their source code. To address this problem, we release every pore detection
method implemented in our publication with instructions on how to replicate
the results of our experiments. Our first implemented approach is a CNN
baseline inspired by existing machine learning approaches (Section II-C). We
designed a customizable FCN, and Figure 9 illustrates its architecture. It
consists of applying $3\times 3$ convolutions with valid padding to the input
patch until the outcome is a $1\times 1$ value representing the probability of
the patch being centered on a pore. Thus, the number of layers depends on the
input patch size. This network can be modified in the following ways:

Patch size:

Input patches must fully enclose pores. A small size would not capture the
entire pore, while a large size would bring too much unnecessary information.
Recent publications used patch size of $17\times 17$ [14, 18, 17, 33] and
$19x19$ [39] pixels. In our experiments, we considered patch sizes of
$13\times 13$, $15\times 15$, $17\times 17$, and $19\times 19$ pixels.

Pore radius:

In the labelling process, all pixels inside a circle with radius $r$ centered
on the pore are labeled with $1$, and other pixels receive the label $0$. This
radius must be be able to represent shape of a pore. Recent publications used
$r$ values of $3$ [18, 17] and $5$ [39]. In our experiments, we considered $r$
values of $4$, $5$, and $6$.

Number of trainable layers:

The depth of the network determines its learning capacity. But the more layers
we use, the more trainable parameters we have. Some publications used max-
pooling layers [14, 33] to reduce the dimensionality of intermediary tensors
without requiring any trainable parameter. We considered architectures with
and without max-pooling layers in our experiments. When using max-pooling, we
replace every second convolutional layer with a max-pooling with the same
kernel size, thus obtaining an architecture with half of the trainable layers.

Residual connections:

Residual connections add skip connections between layers or blocks of layers
to speed up the training and avoid the vanishing gradients problem in deep
networks. Recent publications used residual networks for pore detection [39,
14]. We evaluated architectures with and without residual connections for
internal blocks (all except the first and last blocks).

Soft labels:

In hard labeling, pixels receive label $0$ or $1$. Meanwhile, soft labeling
assigns a value in the range $[0,1]$ proportional to the distance between the
pixel and the center of the closest pore. Su  _et al._ [18] utilized soft
labels in their approach. In our approach, we labeled pixels inside a circle
with radius $r$ around a pore with the value $(r-d)/r$, and other pixels with
$0$. Our experiments evaluate training with hard and soft labels.

Patch size |  Pore Radius |  Max Pooling |  Res. connec-tions |  DBI I F-score |  DBII II F-score
---|---|---|---|---|---
13 |  4 |  ✗ |  ✗ |  89.45 |  90.61
13 |  5 |  ✗ |  ✗ |  88.67 |  90.40
13 |  6 |  ✗ |  ✗ |  89.03 |  90.50
13 |  4 |  ✓ |  ✗ |  82.85 |  84.91
13 |  5 |  ✓ |  ✗ |  85.20 |  85.74
13 |  6 |  ✓ |  ✗ |  85.41 |  87.15
13 |  4 |  ✗ |  ✓ |  87.77 |  89.78
13 |  5 |  ✗ |  ✓ |  87.77 |  89.82
13 |  6 |  ✗ |  ✓ |  88.00 |  89.87
13 |  4 |  ✓ |  ✓ |  82.67 |  84.23
13 |  5 |  ✓ |  ✓ |  85.62 |  87.87
13 |  6 |  ✓ |  ✓ |  86.24 |  86.80
15 |  4 |  ✗ |  ✗ |  90.62 |  91.11
15 |  5 |  ✗ |  ✗ |  89.69 |  91.44
15 |  6 |  ✗ |  ✗ |  89.70 |  91.26
15 |  4 |  ✓ |  ✗ |  88.22 |  87.74
15 |  5 |  ✓ |  ✗ |  88.01 |  89.58
15 |  6 |  ✓ |  ✗ |  87.87 |  89.69
15 |  4 |  ✗ |  ✓ |  88.88 |  90.28
15 |  5 |  ✗ |  ✓ |  87.65 |  90.00
15 |  6 |  ✗ |  ✓ |  88.00 |  90.25
15 |  4 |  ✓ |  ✓ |  87.65 |  89.22
15 |  5 |  ✓ |  ✓ |  88.09 |  89.76
15 |  6 |  ✓ |  ✓ |  87.79 |  89.44
17 |  4 |  ✗ |  ✗ |  89.79 |  90.64
17 |  5 |  ✗ |  ✗ |  90.90 |  92.24
17 |  6 |  ✗ |  ✗ |  90.45 |  91.40
17 |  4 |  ✓ |  ✗ |  87.66 |  86.56
17 |  5 |  ✓ |  ✗ |  88.41 |  86.67
17 |  6 |  ✓ |  ✗ |  88.41 |  84.33
17 |  4 |  ✗ |  ✓ |  87.91 |  89.77
17 |  5 |  ✗ |  ✓ |  87.49 |  89.90
17 |  6 |  ✗ |  ✓ |  87.91 |  90.05
17 |  4 |  ✓ |  ✓ |  83.19 |  86.33
17 |  5 |  ✓ |  ✓ |  85.39 |  87.73
17 |  6 |  ✓ |  ✓ |  82.70 |  84.26
19 |  4 |  ✗ |  ✗ |  91.11 |  92.09
19 |  5 |  ✗ |  ✗ |  90.91 |  91.92
19 |  6 |  ✗ |  ✗ |  90.01 |  91.39
19 |  4 |  ✓ |  ✗ |  89.67 |  90.47
19 |  5 |  ✓ |  ✗ |  89.09 |  90.81
19 |  6 |  ✓ |  ✗ |  88.64 |  90.45
19 |  4 |  ✓ |  ✓ |  88.82 |  90.16
19 |  5 |  ✓ |  ✓ |  88.61 |  90.39
19 |  6 |  ✓ |  ✓ |  87.49 |  90.11
19 |  4 |  ✗ |  ✓ |  87.12 |  89.43
19 |  5 |  ✗ |  ✓ |  87.88 |  89.49
19 |  6 |  ✗ |  ✓ |  88.63 |  90.70
TABLE II: Experiment results on PolyU-HRF dataset for different patch sizes and pore radii, with or without max-pooling, and with or without residual connections. All these experiments use soft labels (see Table III). Soft labels |  DBI I F-score |  DBII II F-score
---|---|---
✗ |  88.32 |  90.5
✓ |  90.90 |  92.24
TABLE III: Effects of using hard or soft labels when using $17\times 17$ input
patches, pore radius equal to $5$, and not including max-pooling layers or
residual connections.

###  IV-A Training details

Aside from the previously mentioned parameters, all other parameters were
identical for all experiments to prevent other factors from affecting the
performance results. We use a batch size of $128$ patches, the Adam optimizer
with a learning rate of $0.001$, early stopping when the training does not
improve on validation for $10$ consecutive epochs. We investigate the best
combination of parameters using the PolyU-HRF dataset. We use 120 annotated
images from DBII [43], which are split into training (90), validation (5) and
testing (25), and we also use 30 images from DBI for testing.

###  IV-B Suggested configuration

We have conducted a grid search over the customizable parameters of our
baseline to assess their impact on pore detection performance. Each
combination of parameters was repeated three times to reduce the effects of
the random initialization, and we report the average results. The results for
these experiments are shown in Tables II and III, and a graphical comparison
is shown in Figure 10. After analyzing the results, we ended up with the
following model configuration:

  * •

Patch sizes of $17\times 17$ and $19\times 19$ produce the best results. These
patch sizes are large enough to capture various shapes and sizes of pores. We
used $17\times 17$ patches since their models are more resource efficient and
produce results comparable to $19\times 19$.

  * •

Pore radius of $4$ and $5$ produce the best results for the PolyU-HRP images
with 1200 dpi. We used a pore radius of $5$ since it provided better results
for a $17\times 17$ patch.

  * •

Not using max-pooling layers produces better results, meaning that using more
layers produce more accurate results. It is known that the expressive power of
neural networks grow exponentially with the number of layers [44], so having
more convolutional layers increased the ability of our baseline distinguishing
between pore and non-pore patches.

  * •

The use of residual connections decreased the performance. We attribute this
result to the fact that residual networks perform better with very deep
networks, while our CNN models are shallow.

  * •

Soft labels produced better results than hard labels, as we do not have
neighboring patches with opposite labels.

13151719$89.69$$90.63$$90.9$$91.11$$90.61$$91.44$$92.24$$92.09$Test ITest II

456$91.11$$90.91$$90.45$$92.04$$92.24$$91.39$Test ITest II

PresentAbsent$89.28$$90.7$$91.11$$92.24$Test ITest II

PresentAbsent$90.63$$91.11$$91.44$$92.24$Test ITest II

Figure 10:  Affect of (a) patch sizes, (b) pore radius, (c) max pooling, and
(d) residual connections on detection results.

##  V Performance evaluation

###  V-A Correct detection criteria

One major issue in pore detection publications and their respective evaluation
protocols is the lack of a standard criterion to determine if a pore detection
is correct. Nowadays, these publications use different criteria to report
their performance, making the comparison of different works a complicated
task. Table IV summarizes the evaluation criteria of existing publications. In
this work, we adopt the bidirectional correspondence (_i.e._ , a pore
detection is correct if it is the closest detection to its closest ground
truth pore), as this is the only criterion that does not accepts multiple
detections for a single ground truth pore and also does not require any
distance threshold.

Ref |  Criteria
---|---
[38] [42][41] [15] |  Euclidean distance with empirical threshold
[9, 14, 17, 34] |  Bidirectional correspondence
[9][13] [12] |  Euclidean distance with threshold proportional to the average ridge width
[25] |  Manhattan distance with empirical threshold
TABLE IV: Criteria used by existing publication to determine if a pore detection is correct. Protocol |  Ref |  Training |  Validation |  Testing
---|---|---|---|---
I |  [22, 10, 1] |  0 |  0 |  24 (DBI)
II |  [41, 25, 1, 10, 22, 18] |  0 |  0 |  30 (DBI)
III |  [18] |  20 (DBI) |  0 |  10 (DBI)
IV |  [16, 12] |  18(DBI) |  6(DBI) |  6(DBI)
V |  [42] |  24 (DBI) |  0 |  6 (DBI)
VI |  [17] |  15 (DBI) |  5 (DBI) |  10 (DBI)
VII |  [33] |  70 (DBII) |  0 |  30 (DBI)
VIII |  [14, 13, 16, 17] |  90 (DBII) |  0 |  30 (DBI), 30 (DBII)
TABLE V: Data split protocols employed in publications using PolyU-HRF.

###  V-B Literature evaluation using PolyU-HRF

Although the PolyU-HRF dataset is no longer publicly available, it was heavily
used in the literature of pore analysis. But even when using the same dataset,
different works use different evaluation protocols (_e.g._ , data splits and
correct detection criteria) that hamper a direct comparison between them.
Table V lists all evaluation protocols used for PolyU-HRF and all publications
that used the same protocol. We can observe that early handcrafted approaches
used protocols that require no training or validation data. Meanwhile, machine
learning publications used many different protocols, the most popular being
Protocol VIII. Protocol VIII benefits from having an extensive training set
and two testing subsets with slightly different conditions (partial (DBI) vs.
full-size fingerprints (DBII)).

Table VI shows the results of publications that have used the PolyU-HRF
dataset. We have included an F-score column reporting the harmonic mean of the
true detection and false detection rates reported by the listed works. Even
though so many different protocols were used, we can observe that machine
learning approaches (Protocols III to VIII) clearly outperform handcrafted
approaches (Protocols I and II). This is an expected outcome, as a similar
behavior was observed in many different computer vision research fields.
However, picking the best machine learning approach is not as obvious. If we
group the protocols by the subset of data used for training, we get Protocols
III to VI for DBI and Protocols VII and VIII for DBII. In the first group, the
best performance was obtained by Ali  _et al._ [42], and, in the second group,
by Shen  _et al._ [14]. Both works use FCNs with some residual connections,
with Ali  _et al._ [42] using a shallower network combined with a handcrafted
post-processing step. Although there is no clear winner between the two,
Protocol VIII is more well founded than than Protocol V, making the results
reported by Shen  _et al._ [14] more reliable. Finally, we can see that the
performance of our baseline is comparable to the top-performing works for the
two most popular protocols (VI and VIII).

Protocol |  Ref |  True Detection |  False Detection |  F-score
---|---|---|---|---
I |  [22] |  84.8 |  17.6 |  83.58
|  [10] |  60.6 |  30.5 |  64.74
|  [1] |  75.9 |  23.0 |  76.44
II |  [41] |  86.1 |  8.6 |  88.67
|  [1] |  75.9 |  23.0 |  76.44
|  [22] |  84.8 |  17.6 |  83.58
|  [25] |  83.1 |  14.9 |  84.08
|  [10] |  63.4 |  20.4 |  70.58
|  [11] |  90.8 |  11.1 |  89.83
III |  [18] |  88.6 |  0.4 |  93.77
IV |  [16] |  93.09 |  8.64 |  92.21
|  [12] |  93.14 |  4.39 |  94.35
V |  [42] |  96.69 |  4.18 |  96.24
VI |  [17] |  91.95 |  8.88 |  91.53
|  [18] |  90.31 |  23.99 |  82.54
|  Baseline |  92.87 |  8.43 |  92.21
VII |  [33] |  83.65 |  13.89 |  84.86
VIII |  [14] |  95.56 (DBII), 92.81 (DBI) |  8.1 (DBII), 7.3 (DBI) |  93.69 (DBII), 92.75 (DBI)
|  [13] |  94.49 (DBII), 93.78 (DBI) |  8.5 (DBII), 8.5 (DBI) |  92.97 (DBII), 92.62 (DBI)
|  [16] |  91.32 (DBII), 89.54 (DBI) |  8.5 (DBII), 8.5 (DBI) |  91.40 (DBII), 89.50 (DBI)
|  [17] |  95.14 (DBII), 91.13 (DBI) |  17.82 (DBII), 16.52 (DBII) |  88.18 (DBI) 87.13
|  [18] |  88.79 (DBII), 82.56 (DBI) |  14.49 (DBII), 12.61 (DBII) |  87.12 (DBII) 84.90 (DBI)
|  Baseline |  93.74 (DBII), 91.79 (DBI) |  9.21 (DBII) 9.96 (DBI) |  92.24 (DBII) 90.90 (DBI)
TABLE VI: Protocol table of publications that utilized PolyU Dataset

###  V-C Literature evaluation using L3-SF

Since the PolyU-HRF dataset is no longer distributed by the authors, we ran
experiments using the L3-SF dataset so that researchers without access to the
PolyU dataset can compare their results to our baseline approach. We also
included our reimplementation of three literature approaches: Lemes  _et al._
[11], Dahia  _et al._ [17], and Su  _et al._ [18]. These reimplementations
obtained results on PolyU-HRF that are comparable to the original works when
using the same protocol, and we made their code publicly available as well.
Since L3-SF contains 740 images, we divided them into five subsets and ran
5-fold cross validation. Table VII shows the obtained results for the four
considered approaches. As observed, machine learning approaches obtained a
higher performance, although Dahia  _et al._ ’s did not obtain consistent
results for this dataset.

Name |  F-score
---|---
Baseline |  87.20
DPF [11] |  71.91
CNN1 [18] |  86.86
CNN2 [17] |  58.18
TABLE VII: Baseline results on L3-SF

###  V-D Discussion and conclusion

This survey reviewed methods, datasets, and evaluation protocols used in pore
detection works. It provides an overview of the existing techniques and how
they compare to each other in terms of accuracy. Additionally, we presented a
baseline approach that reflects the core ideas of state-of-the-art
publications and made its source code publicly available. We also
reimplemented three other works from the literature and added their code to
our public repository. We hope to help future comparisons in this research
field by releasing our code for pore detection methods and performance
evaluation.

Although different approaches in the literature present high accuracy, the
performance in recent years seems to have stagnated. Since the performance in
existing datasets is not saturated, it would be interesting to investigate the
performance of human annotators in this task to verify the precision of the
ground truth annotations and their impact on the performance analysis of pore
detectors.

For future directions, many handcrafted and machine learning techniques could
be adapted to the problem of detecting pores and have never been tried before.
For instance, researchers could use multitask learning to estimate local
orientation and frequency, segment ridges, and detect pores and minutia
simultaneously. By including additional fingerprint-related tasks and forcing
the network to learn weights that are useful for different purposes, the
performance of pore detection itself could increase. Furthermore, graph neural
networks could be employed to explore pixel connectivity patterns for pore
detection or as a post-processing technique that learns the distribution of
pores in a fingerprint. Finally, all existing points deal with pores as 2D
points, mainly due to the limited annotations available. But pores have
different shapes and sizes. Researchers could explore this information not
only to obtain more accurate pore detectors but also to provide additional
information for pore-based fingerprint matching.

## References

  * [1] A. K. Jain, Y. Chen, and M. Demirkus, “Pores and ridges: High-resolution fingerprint matching using level 3 features,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 29, no. 1, pp. 15–27, 2007. [Online]. Available: https://ieeexplore.ieee.org/document/4016547?arnumber=4016547
  * [2] M. Pamplona Segundo and R. de Paula Lemes, “Pore-based ridge reconstruction for fingerprint recognition,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Workshops_ , June 2015.
  * [3] E. German, “Level 1, 2, and 3 details.” [Online]. Available: https://onin.com/fp/level123.html
  * [4] Locard, “Les pores et i’identification des criminals, biologica, vol.2, pp. 357-365,” 1912.
  * [5] “Swgfast scientific working group on friction ridge analysis study and technology,” 2006.
  * [6] J. D. Stosz and L. A. Alyea, “”automated system for fingerprint authentication using pores and ridge structure”, proc. spie conf. automatic systems for the identification and inspection of humans, vol. 2277, pp. 210-223,” 1994.
  * [7] A. R. Roddy and J. D. Stosz, “”fingerprint features-statistical analysis and system performance estimates,” in proceedings of the ieee, vol. 85, no. 9, pp. 1390-1421,” 1997.
  * [8] B. M. S. Swain and R. Dash, “Extended feature extraction technique from fingerprint,” 2014 annual ieee india conference (indicon),” 2014.
  * [9] R. Donida Labati, A. Genovese, E. Muñoz, V. Piuri, and F. Scotti, “A novel pore extraction method for heterogeneous fingerprint images using convolutional neural networks,” _Pattern Recognition Letters_ , vol. 113, pp. 58–66, 2018, integrating Biometrics and Forensics.
  * [10] M. Ray, P. Meenen, and R. Adhami, “A novel approach to fingerprint pore extraction,” in _Proceedings of the Thirty-Seventh Southeastern Symposium on System Theory, 2005. SSST ’05._ , 2005, pp. 282–286.
  * [11] R. de Paula Lemes, M. P. Segundo, O. R. Bellon, and L. Silva, “Dynamic pore filtering for keypoint detection applied to newborn authentication,” in _2014 22nd International Conference on Pattern Recognition_ , 2014, pp. 1698–1703.
  * [12] Y. Zhao, F. Liu, and L. Shen, “Fingerprint pore extraction using convolutional neural networks and logical operation,” in _Biometric Recognition_ , J. Zhou, Y. Wang, Z. Sun, Z. Jia, J. Feng, S. Shan, K. Ubul, and Z. Guo, Eds. Cham: Springer International Publishing, 2018, pp. 38–47.
  * [13] V. Anand and V. Kanhangad, “Pore detection in high-resolution fingerprint images using deep residual network,” _Journal of Electronic Imaging_ , vol. 28, no. 02, p. 1, apr 2019.
  * [14] Z. Shen, Y. Xu, J. Li, and G. Lu, “Stable pore detection for high-resolution fingerprint based on a cnn detector,” in _2019 IEEE International Conference on Image Processing (ICIP)_ , 2019, pp. 2581–2585.
  * [15] V. Anand and V. Kanhangad, “Porenet: Cnn-based pore descriptor for high-resolution fingerprint recognition,” _IEEE Sensors Journal_ , vol. 20, no. 16, pp. 9305–9313, 2020.
  * [16] H. U. Jang, D. Kim, S. M. Mun, S. Choi, and H. K. Lee, “Deeppore: Fingerprint pore extraction using deep convolutional neural networks,” in _IEEE Signal Processing Letters, Volume: 24, Issue: 12_ , December 2017. [Online]. Available: https://ieeexplore.ieee.org/document/8063406
  * [17] G. Dahia and M. P. Segundo, “Improving fingerprint pore detection with a small FCN,” _CoRR_ , vol. abs/1811.06846, 2018. [Online]. Available: http://arxiv.org/abs/1811.06846
  * [18] H.-R. Su, K.-Y. Chen, W. J. Wong, and S.-H. Lai, “A deep learning approach towards pore extraction for high-resolution fingerprint recognition,” in _2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , 2017, pp. 2057–2061.
  * [19] R. Malik and A. Masood, “Fingerprint enhancement and reconstruction,” in _Emerging Intelligent Computing Technology and Applications. With Aspects of Artificial Intelligence_ , D.-S. Huang, K.-H. Jo, H.-H. Lee, H.-J. Kang, and V. Bevilacqua, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009, pp. 660–669.
  * [20] A. Farina, Z. M. Kovács-Vajna, and A. Leone, “Fingerprint minutiae extraction from skeletonized binary images,” _Pattern Recognition_ , vol. 32, no. 5, pp. 877–889, 1999. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0031320398001071
  * [21] K. Kryszczuk, A. Drygajlo, and P. Morier, “Extraction of level 2 and level 3 features for fragmentary fingerprint comparison,” 2008.
  * [22] Q. Zhao, D. Zhang, L. Zhang, and N. Luo, “Adaptive fingerprint pore modeling and extraction,” _Pattern Recognition_ , vol. 43, no. 8, pp. 2833–2844, 2010\. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0031320310000944
  * [23] N. A. Mngenge, N. N. Nelufule, F. V. Nelwamondo, and N. Msimang, “Fingerprint pores extractor,” in _2012 NATIONAL CONFERENCE ON COMPUTING AND COMMUNICATION SYSTEMS_ , 2012, pp. 1–5.
  * [24] S. Malathi, S. Uma maheswari, and C. Meena, “Fingerprint pore extraction based on marker controlled watershed segmentation,” in _2010 The 2nd International Conference on Computer and Automation Engineering (ICCAE)_ , vol. 3, 2010, pp. 337–340.
  * [25] E. S. Rodrigues and V. R. P. Borges, “Pore detection in fingerprints based on image subtraction and anisotropic diffusion filtering,” in _2018 IEEE International Conference on Systems, Man, and Cybernetics (SMC)_ , 2018, pp. 2061–2066.
  * [26] S. Zaidi, S. K. Singh, and S. Tarar, “Isotropic pore detection algorithm for level 3 feature extraction,” in _Smart and Innovative Trends in Next Generation Computing Technologies_ , P. Bhattacharyya, H. G. Sastry, V. Marriboyina, and R. Sharma, Eds. Singapore: Springer Singapore, 2018, pp. 323–334.
  * [27] N. R. Parsons, J. Q. Smith, E. Thönnes, L. Wang, and R. Wilson, “Rotationally invariant statistics for examining the evidence from the pores in fingerprints,” _Law, Probability and Risk_ , vol. 7, no. 1, pp. 1–14, 08 2007. [Online]. Available: https://doi.org/10.1093/lpr/mgm018
  * [28] Y. LeCun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard, and L. D. Jackel, “Backpropagation applied to handwritten zip code recognition,” _Neural Computation_ , vol. 1, no. 4, pp. 541–551, 1989.
  * [29] J. Long, E. Shelhamer, and T. Darrell, “Fully convolutional networks for semantic segmentation,” in _2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2015, pp. 3431–3440.
  * [30] X. Glorot, A. Bordes, and Y. Bengio, “Deep sparse rectifier neural networks,” in _Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics_ , ser. Proceedings of Machine Learning Research, G. Gordon, D. Dunson, and M. Dudík, Eds., vol. 15. Fort Lauderdale, FL, USA: PMLR, 11–13 Apr 2011, pp. 315–323. [Online]. Available: https://proceedings.mlr.press/v15/glorot11a.html
  * [31] S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep network training by reducing internal covariate shift,” in _Proceedings of the 32nd International Conference on Machine Learning_ , ser. Proceedings of Machine Learning Research, F. Bach and D. Blei, Eds., vol. 37. Lille, France: PMLR, 07–09 Jul 2015, pp. 448–456. [Online]. Available: https://proceedings.mlr.press/v37/ioffe15.html
  * [32] J. Hosang, R. Benenson, and B. Schiele, “Learning non-maximum suppression,” in _2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2017, pp. 6469–6477.
  * [33] H. Wang, X. Yang, L. Ma, and R. Liang, “Fingerprint pore extraction using u-net based fully convolutional network,” in _Biometric Recognition_ , J. Zhou, Y. Wang, Z. Sun, Y. Xu, L. Shen, J. Feng, S. Shan, Y. Qiao, Z. Guo, and S. Yu, Eds. Cham: Springer International Publishing, 2017, pp. 279–287.
  * [34] B. Ding, H. Wang, P. Chen, Y. Zhang, R. Liang, and Y. Liu, “Subcutaneous sweat pore estimation from optical coherence tomography,” _IET Image Processing_ , vol. 15, no. 13, pp. 3267–3280, 2021.
  * [35] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in _2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2016, pp. 770–778.
  * [36] O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” in _Medical Image Computing and Computer-Assisted Intervention – MICCAI 2015_ , N. Navab, J. Hornegger, W. M. Wells, and A. F. Frangi, Eds. Cham: Springer International Publishing, 2015, pp. 234–241.
  * [37] “Nist special database 4, https://www.nist.gov/srd/nist-special-database-4.”
  * [38] A. Genovese, E. Muñoz, V. Piuri, F. Scotti, and G. Sforza, “Towards touchless pore fingerprint biometrics: A neural approach,” in _2016 IEEE Congress on Evolutionary Computation (CEC)_ , 2016, pp. 4265–4272.
  * [39] V. Anand and V. Kanhangad, “Cross-sensor pore detection in high-resolution fingerprint images,” _IEEE Sensors Journal_ , vol. 22, no. 1, pp. 555–564, 2022.
  * [40] A. B. Vieira Wyzykowski, M. P. Segundo, and R. de Paula Lemes, “Level three synthetic fingerprint generation,” in _2020 25th International Conference on Pattern Recognition (ICPR)_ , 2021, pp. 9250–9257.
  * [41] R. F. S. Teixeira and N. J. Leite, “Improving pore extraction in high resolution fingerprint images using spatial analysis,” in _2014 IEEE International Conference on Image Processing (ICIP)_ , 2014, pp. 4962–4966.
  * [42] M. Ali, C. Wang, and M. O. Ahmad, “An efficient convolutional neural network for fingerprint pore detection,” _IEEE Transactions on Biometrics, Behavior, and Identity Science_ , vol. 3, no. 3, pp. 332–346, 2021.
  * [43] R. F. S. Teixeira and N. J. Leite, “A new framework for quality assessment of high-resolution fingerprint images,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 39, no. 10, pp. 1905–1917, 2017.
  * [44] M. Raghu, B. Poole, J. Kleinberg, S. Ganguli, and J. S. Dickstein, “On the expressive power of deep neural networks,” in _Proceedings of the 34th International Conference on Machine Learning - Volume 70_ , ser. ICML’17. JMLR.org, 2017, p. 2847–2854.


# Restoring the Bloch-Nordsieck theorem in the electroweak sector of the
standard model

Axel Maas<EMAIL_ADDRESS>Franziska Reiner<EMAIL_ADDRESS>graz.at Institute of Physics, NAWI Graz, University of Graz, Universitätsplatz
5, A-8010 Graz, Austria

###### Abstract

The electroweak gauge symmetry cannot be broken in a literal sense due to
Elitzur’s theorem. Thus, asymptotic states need to be manifestly and non-
perturbatively gauge-invariant with respect to the electroweak symmetry. To
take this suitably into account perturbation theory augmented by the Fröhlich-
Morchio-Strocchi mechanism can be used. We show that this restores the Bloch-
Nordsieck theorem in electroweak processes in the standard model. This has
potentially substantial impact at, e. g., future lepton colliders, but has
only negligible effects at lower energies. We also demonstrate an alternative
implementation using PDFs, which allows an approach with manifest electroweak
Bloch-Nordsieck theorem also at hadron colliders.

##  I Introduction

Gauge symmetries can never be broken spontaneously due to Elitzur’s theorem
Elitzur:1975im ; Frohlich:1980gj ; Frohlich:1981yi . As a consequence, the
electroweak gauge symmetry is actually intact, and its apparent breaking is
only a consequence of a gauge choice Lee:1974zg . In fact, it is possible to
define gauges in which it remains unbroken Lee:1974zg ; Maas:2012ct , at the
expense of the applicability of perturbation theory Lee:1974zg . Hence, as has
been realized early on Banks:1979fi ; Fradkin:1978dv ; Osterwalder:1977pc ;
Frohlich:1980gj , physical, observable states need to be manifestly non-
perturbatively gauge-invariant, rather than only perturbatively BRST
invariant111The reason for BRST invariance to be insufficient is the Gribov-
Singer ambiguity Gribov:1977wm ; Singer:1978dk , which explicitly invalidates
perturbative BRST Fujikawa:1982ss , see Maas:2017wzi  for a review., also with
respect to the electroweak symmetry Frohlich:1980gj ; Frohlich:1981yi . This
requires to consider composite states rather than elementary ones as
asymptotic in and out states, very much like hadrons in QCD.

At first sight this seems to be in contradiction to the highly successful
description of experiments in terms of the elementary states in perturbation
theory. This is explained due to the Fröhlich-Morchio-Strocchi mechanism
Frohlich:1980gj ; Frohlich:1981yi . It shows that in the gauges usually
employed in perturbation theory many quantities are dominated, up to
corrections suppressed by powers of the ratio of the relevant energy scale to
the Higgs vacuum expectation value, by their usual perturbative expression.
Augmenting perturbation theory with the FMS mechanism appears to allow to take
the remaining deviations manifestly into account analytically222Similar
considerations have been followed in Jegerlehner:1985ch ; Jegerlehner:1984ia .
Maas:2020kda ; Dudal:2020uwb . This approach has been supported by lattice
calculations, and is reviewed in Maas:2017wzi .

In section II we will briefly review the mechanism and then apply it to
fermionic 2-to-2 processes at lepton colliders. We will extend here
Egger:2017tkd  to formulate the full FMS expression of the relevant matrix
elements. We thereby show how systematically at low energies the ordinary
results will appear, while at the same time cancellations ensure at high
energies that the violation encountered in standard perturbation theory
Ciafaloni:2000df ; Ciafaloni:2000rp  of the Bloch-Nordsieck theorem (BNT)
Bloch:1937pw  is absent. We will briefly discuss the implications for future
lepton colliders in a sample processes.

The same is true, in a more involved way, for hadronic initial states
Egger:2017tkd ; Maas:2017wzi . The strongly interacting nature does not allow
a purely perturbative approach as in the lepton case. Rather, we expand the
usual PDF language developed for $pp$ collisions in a gauge-invariant way in
section III, making the BNT manifestly maintained. This yields a few
surprising features of its own as to what the actual PDFs represent at high
energies, and what kind of sum rules they need to obey.

We wrap up the discussion in section IV with a number of possible further
steps. Some preliminary results have been made available in Reiner:2021bol .
We note that similar considerations will also apply to violations of the
Kinoshita-Lee-Naunberg theorem Kinoshita:1962ur ; Lee:1964is  in the final
state, and the remedies will follow completely analogously by replacing the
out states with composite states.

##  II Lepton scattering

###  II.1 Weakly gauge-invariant leptons

We consider in the following the usual standard model Lagrangian, supplemented
for simplicity by right-handed neutrinos, following Frohlich:1980gj ;
Frohlich:1981yi ; Egger:2017tkd ; Afferrante:2020fhd . Their existence is not
essential for this work, but they simplify considerably the technicalities.
Also, they allow for a very straightforward transfer of results between quarks
and leptons.

As emphasized in the introduction, field-theoretical arguments highlight that
the weak gauge symmetry is never spontaneously broken, and what is usually
called the Brout-Englert-Higgs (BEH) effect is a particularly suited choice of
gauge, which allows for a technically advantegous treatment of the dynamics
Elitzur:1975im ; Banks:1979fi ; Fradkin:1978dv ; Osterwalder:1977pc ;
Frohlich:1980gj ; Frohlich:1981yi ; Lee:1974zg . A full discussion of the
details and background can be found in the review Maas:2017wzi .

The most important consequence is that weakly gauge-dependent states,
especially the Higgs, the $W$ and $Z$ bosons, and all left-handed fermions
cannot be physical. Rather, just like in QCD, only manifestly gauge-invariant
composite states can be asymptotic Banks:1979fi ; Frohlich:1980gj ;
Frohlich:1981yi . We will concentrate here on the fermions, and refer for the
bosons again to the review Maas:2017wzi . In this context, it is useful to
write the Higgs field $\phi$ before separating off the vacuum expectation
value (vev) as

$X=\begin{pmatrix}\phi_{2}^{*}&\phi_{1}\\\
-\phi_{1}^{*}&\phi_{2}\end{pmatrix}.$ (1)

and thus the standard Higgs fluctuation mode as well as the three would-be
Goldstone bosons are still contained in $X$.

Fermions come now in two varieties. One are the left-handed Weyl spinors
$\psi^{L}$, which are gauged under the weak interaction in the fundamental
representation. I. e., they form doublets

$\psi^{L}=\begin{pmatrix}\nu^{L}\cr e^{L}\end{pmatrix}$ (2)

of which there are six, three for the lepton generations and three for the
quark generations. In addition, there are twelve right-handed Weyl spinors,
representing the six ungauged leptons and quarks of which always two can be
paired, e. g. $\nu^{R}$ and $e^{R}$, as an ungauged set corresponding to the
left-handed Weyl doublet. Besides the strong interactions separating quarks
and leptons the other differences arise due to the Yukawa, hypercharge, and
CKM/PMNS values for the, in total, 18 fermion fields.

At vanishing hypercharge and Yukawa couplings, there are several symmetries in
the weak sector of the standard model. First, there is the local weak gauge
symmetry $\mathrm{SU(2)_{w}}$. Second, there is a global $\mathrm{SU(6)_{Rf}}$
flavor symmetry of the right-handed Weyl fermions and an SU(3) left-handed
generation symmetry. These are present for quarks and leptons separately.
Finally, there is a less obvious global $\mathrm{SU(2)_{c}}$ symmetry which
acts only on the scalar doublet as a right-multiplication on $X$. Switching on
Yukawa couplings breaks these symmetries to the familiar pattern of lepton and
quark number, and a U(1) subgroup of $\mathrm{SU(2)_{c}}$. Gauging the latter
adds hypercharge.

Ignoring for a moment the BEH effect, there are for each generation and for
quarks and leptons separately four physical fermionic states in the theory,
which can be grouped into two chiral doublets. The first two states are the
flavor doublet of right-handed Weyl fermions $\chi^{R}$. One of them is the
right-handed charged lepton, $\chi^{R}_{2}=e^{R}$ and the other the right-
handed neutrino $\chi^{R}_{1}=\nu^{R}$. The other physical doublet is a gauge-
invariant, left-handed composite Weyl field, $\Psi^{L}=X^{\dagger}\psi^{L}$
Frohlich:1980gj ; Frohlich:1981yi ; Egger:2017tkd , which is a singlet with
respect to the non-Abelian gauge group but carries a global
$\mathrm{SU(2)_{c}}$ charge. The two components of this doublet will be
identified with the left-handed electron and the left-handed neutrino below.

In case non-zero Yukawa couplings break the global $\mathrm{SU(2)_{c}}$
symmetry and the flavor symmetry $\mathrm{SU(2)_{Rf}}$ to the diagonal
subgroup $\mathrm{SU(2)_{f}}$, this bound state transforms in the same way as
the right-handed fermion doublet. In this way it appears as if in the physical
spectrum the diagonal subgroup acts as an effective flavor symmetry for both
the left-handed sector and right-handed sector simultaneously. Note that the
two gauge-dependent components of the elementary left-handed Weyl fermion
$\psi^{L}$ do not transform under $\mathrm{SU(2)_{f}}$, but can be transformed
into each other via a gauge transformation and can therefore not be associated
with physically observable particles or flavors.

To connect to the usual perturbative picture, switch on the BEH effect in a ’t
Hooft gauge. The Higgs field can then be split into its vacuum fluctuations
$\eta$ and its vev. Using the gauge freedom, we conventionally chose the
vacuum expectation value to be in the real 2 direction. Thus, we have
$\langle\phi_{i}\rangle=\frac{v}{\sqrt{2}}\delta_{i2}$. At tree-level, this
yields the customary result that the gauged and ungauged Weyl spinors can be
combined into two Dirac spinors, each with a mass given by
$m_{f}=y_{f}v/\sqrt{2}$, forming the usual leptons Bohm:2001yx .

However, the left-handed leptons are thus defined in a gauge-fixed way. The
decisive step to relate them to the gauge-invariant $\Psi^{L}$ is to apply the
FMS mechanism Frohlich:1980gj ; Frohlich:1981yi . Consider the physical left-
handed fermion333This construction automatically ensures the correct
assignment of hypercharge, and by extension of electric charge, to all bound
states Maas:2017wzi . It will therefore not be considered further in the
following. Frohlich:1980gj ; Frohlich:1981yi ,

$\Psi^{L}=X^{\dagger}\psi^{L}=\left(\frac{v}{\sqrt{2}}\mathbbm{1}+\eta\right)\psi^{L}=\frac{v}{\sqrt{2}}\begin{pmatrix}\psi_{1}^{L}\\\
\psi_{2}^{L}\end{pmatrix}+\eta\psi^{L},$ (3)

where the matrix-valued $\eta$ contains the usual fluctuation field identified
with the elementary Higgs boson and the Goldstone fields in the same manner as
$X$ contains $\phi$ in (1). To leading order in $v$, the first term, the field
$\Psi^{L}$ reduces to the elementary left-handed fermions, which are thus the
FMS-dominant constituents of the bound state $\Psi^{L}$. Of course, only the
total sum in (3) is gauge-invariant, and the leading order alone is not.

When now forming a propagator it follows

$\left\langle\Psi_{f_{1}}^{L}(p)\bar{\Psi}_{f_{2}}^{L}(-p)\right\rangle=\frac{v^{2}}{2}\left\langle\psi_{f_{1}}^{L}(p)\bar{\psi}^{L}_{f_{2}}(-p)\right\rangle+{\cal
O}\left(\frac{|p|}{v}\right).$ (4)

Thus, to all orders in perturbation theory and to leading order in $|p|/v$ the
propagator of the physical, composite fermion state is given by the gauge-
dependent elementary ones, i.e., the propagators of the left-handed charged
lepton and neutrino. Especially, the poles and thus the masses and widths
coincide. This was shown to all orders in perturbation theory for the Higgs
bound-state–elementary-state duality Maas:2020kda ; Dudal:2020uwb  and appears
to generalize straightforwardly to all other standard model particles
Maas:2020kda ; Dudal:2020uwb . In particular, propagators of the type (4)
indeed are necessarily also off-shell gauge-invariant when including the
finite polynomial in $v$ in each order in perturbation theory. Only at $|p|\gg
v$ they start to appreciably deviate from the perturbative result at the same
order Maas:2020kda ; Dudal:2020uwb .

In this way, the gauge-invariant Dirac spinor
$(\Psi_{f}^{L}\;\chi_{f}^{R})^{\mathrm{T}}$ describes the physical neutrinos
($f=1$) and charged leptons ($f=2$) with the same properties as the usual
gauge-dependent ones of perturbation theory Frohlich:1980gj ; Frohlich:1981yi
; Maas:2017wzi . This can be extended to include the hypercharge sector
Maas:2017wzi ; Egger:2017tkd , and quarks work completely analogously apart
from complications due to the strong interactions Maas:2017wzi ; Egger:2017tkd
.

All lattice results Maas:2017wzi ; Maas:2018ska ; Maas:2020kda ;
Afferrante:2020fhd ; Jenny:2022atm  as well as the fact that perturbation
theory is a very good approximation so far to experiments imply that the
corrections to perturbation theory are indeed small, at least at energies at
or below the electroweak scale. But the important bottom line is that the
physical left-handed leptons in the standard model are actually bound states
of the elementary ones and the Higgs field Frohlich:1980gj ; Frohlich:1981yi .

###  II.2 Scattering

In fact, the subleading contribution in (4) are not zero, and are thus, in
principle, detectable. In vector-boson scattering and form factors these have
indeed been isolated in a reduced standard-model setup on the lattice
Jenny:2022atm ; Maas:2018ska  and for the off-shell properties of the Higgs
and the vector bosons analytically Maas:2020kda ; Dudal:2020uwb . But it is
far from straightforward to observe them experimentally, yet. However, the
kinematic suppression in (4) and Maas:2020kda ; Dudal:2020uwb ;
MPS:unpublished ; Fernbach:2020tpa ; Egger:2017tkd  suggests that this will
change if the energy scale is substantially above the electroweak scale.

We therefore extend here the ad-hoc approximations in exclusive fermion
scattering of Egger:2017tkd . Consider a polarized cross-section of
$\bar{f}f\to\bar{F}F$, where the initial fermions $f$ and final fermions $F$
are distinct. If the energy scales are large enough compared to the fermion
masses, the scattering can be considered polarized, i. e. each of the $f$ and
$F$ are either completely left-handed or right-handed. In the purely right-
handed case both initial state and final state are entirely made-up from weak
gauge-singlets, and nothing will change. Thus, concentrate on at least one of
them being left-handed.

The relevant matrix elements are444Note that also the LSZ formalism needs to
be suitably augmented for external bound states, which is straightforward
Bohm:2001yx ; Weinberg:1995mt . What is not straightforward is how this
interacts with the FMS expansion beyond the leading term (8) below. This will
be presented elsewhere MPS:unpublished . However, as long as the Higgs
contents in the bound state is small compared to the one of the fermionic
component, this will only be a minor effect Egger:2017tkd .

$\displaystyle\left\langle\bar{f}^{R}f^{L}\bar{F}^{R}F^{L}\right\rangle$ (5)
$\displaystyle\left\langle\bar{f}^{L}f^{L}\bar{F}^{L}F^{L}\right\rangle,$ (6)

and $L$-$R$ permutations in (5). Following (3), this will reduce to the usual
perturbative expression to all orders in the coupling and to leading-order in
the FMS expansion Egger:2017tkd . By dimensional analysis, the higher orders
in the FMS expansion will be suppressed by $s/v^{2}$, where $s$ is the
relevant largest energy scale, e. g. the center-of-mass energy at tree-level.
Thus, to have something interesting happening, $s\gtrsim v^{2}$ is needed.

While an explicit calculation of first-order corrections is possible along the
lines of Maas:2020kda ; Dudal:2020uwb  and is under way MPV:unpublished ,
there is a much more generic effect if $s\gg v^{2}$, and thus $s\gg
m_{W/Z}^{2}$. This would be the situation at a TeV-scale lepton collider. In
such a case in the standard perturbative approach the fact that the external
left-handed fermions are gauge-dependent make itself felt by a violation
Ciafaloni:2000rp ; Bauer:2018xag  of the Block-Nordsieck theorem (BNT)
Bloch:1937pw . Because the initial states are gauge-non-singlets, the usual
summation over the full gauge multiplet needed for cancellations of infrared
divergencies in the BNT does not occur, and they remain in form of double
Sudakov logarithms $\sim\ln^{2}(s/m_{W}^{2})$ Ciafaloni:2000rp  from emissions
of (electro)weak gauge bosons Fadin:1999bq ; Ciafaloni:1999ub , cut off only
by the weak mass scale $m_{W}$. At a TeV lepton collider the effect is of the
same order as strong corrections, i. e. a couple of percent Ciafaloni:2000rp ,
and would grow with larger energies.

By its very construction, it is clear that a scattering process only involving
(5-6) necessarily respects the BNT in general. At the investigated very large
energies, the BEH effects is irrelevant, and all masses can be neglected.
Since the gauge multiplets are now complete, the proof that the BNT holds is
completely analogous to the one in QCD Bohm:2001yx , except for the fact that
the lower cutoff is fixed by the weak mass scale: Just as in QCD555Neglecting
the vev, this can also be shown using a coherent state approach completely
analougsly to the QCD case, except for the presence of the Higgs self-
interactions. However, if the vev is neglected, there is only an interaction
of four Higgs possible, which is thus as irrelevant as the four-gauge boson
interaction. any quark color can contribute, and thus need to be summed over,
all weak charges of the doublet (2) in the bound state (3) can equally likely
contribute, and thus need to be summed over. Or stated otherwise, by having
the bound state (3) rather than the weak doublet (2) involved as external
states in (5-6), the process becomes inclusive with respect to the gauge
doublet (2), with the Higgs components of the bound state playing the same
spectator role as the remaining spectator quarks of a proton in QCD. It is
only exclusive at the level of the global SU(2)${}_{\text{c}}$ components of
the bound state (3).

While this statement is necessarily true, it is nevertheless very interesting
how this works out in practice. And especially why this process does not
deviate at low, i. e. LEP2, energies, appreciably from the ordinary result
involving elementary particles as asymptotic states. Since the violation of
the BNT has at leading-double-log order only implications for the scattering
for two-left-handed particles Ciafaloni:2000rp , we will concentrate on this
case. Consider for concreteness the process $\bar{l}l\to\bar{q}q$, as here the
violation has been explicitly worked out in Ciafaloni:2000rp , and thus here
only the changes need to be addressed.

The corresponding matrix element is then

$\left\langle\bar{\Psi}^{L}_{2}\Psi^{L}_{2}\bar{Q}^{L}_{2}Q^{L}_{2}\right\rangle=\left\langle\underbrace{\bar{\psi}^{L}_{i}X_{i2}}\underbrace{X^{\dagger}_{2j}\psi^{L}_{j}}\underbrace{\bar{q}^{L}_{k}X_{k2}}\underbrace{X^{\dagger}_{2l}q^{L}_{l}}\right\rangle$
(7)

where $Q$ is the composite state for the left-handed top-type quark and
bottom-type quark constructed analogously to (3) from
$q^{L}=(b^{L},t^{L})^{T}$. For concreteness, we consider the
$\bar{e}e\to\bar{c}c$ process, and select the custodial indices accordingly to
be 2. Underbraces have been used to identify combinations, which make up
composite operators. This implies that all composite operators in the initial
state and final state involve both members of the weak fermion doublet (3)

First, we investigate what happens when performing the FMS mechanism in
leading order in $s/v$. In this case, $X=v1$, and the expression collapses as
Egger:2017tkd

$\left\langle\bar{\Psi}^{L}_{2}\Psi^{L}_{2}\bar{Q}^{L}_{2}Q^{L}_{2}\right\rangle=v^{4}\left\langle\bar{e}^{L}e^{L}\bar{c}^{L}c^{L}\right\rangle+\mathcal{O}\left(\frac{s}{v}\right),$
(8)

where $s$ is again the dominating energy scale, at tree-level the center-of-
mass energy Egger:2017tkd . Thus, to this order, we recover the usual
expression for the matrix element. This shows how the FMS mechanism recovers
the usual result, if the finite number of other terms in the FMS expansion can
be neglected. Thus, at LEP2 energies this should be a fairly good
approximation. However, the BNT violation also plays no role at these energies
as $s\sim m_{W}^{2}$ Ciafaloni:2000rp ; Bauer:2018xag , so it is no
contradiction that the violation appears to be recovered.

This can therefore be no longer be the case if $s\gg m_{W}^{2}$, like at
future lepton colliders. Assuming a fully exclusive measurement of the final
state SU(2)${}_{\text{c}}$ quantum numbers, however, it is safe to assume that
the Higgs fields in (7) act primarily as spectators to the FMS-dominant
constituent. This is also supported by investigations of the substructure of
the bound state (3) Egger:2017tkd ; Afferrante:2020fhd . This is akin to the
situation in hadron scattering where the other partons are also spectators. Of
course, the Higgs are then not leaving as debris, but will be needed to
construct again weakly gauge-invariant final states, dressing again the
elementary states Egger:2017tkd . In that sense, the process is different than
in QCD, where the dressing is obtained in fragmentation from soft gluons and
light quarks. Due to the large masses and mass defects, this does not happen
at this level in an electroweak process. However, at sufficiently high
energies, electroweak fragmentation of elementary weak particles into bound
states like (3) in the final state can be expected to also happen very
similarly to QCD.

As the process is then inclusive with respect to the weak doublet (2), it is
thus necessary to sum the cross-sections of the possible pairings. Ignoring
for a moment the final state, this implies that the total cross section is
given by666In fact, the same result would be obtained when using a PDF
description for the bound state (3), by setting $f_{e}=f_{\nu}=\delta(x)$ and
all Higgs PDFs to zero in (12).

$\sigma_{\bar{\Psi}_{L}^{2}\Psi_{L}^{2}\to X}\sim\sigma_{\bar{l}_{L}l_{L}\to
X}+\sigma_{\bar{l}_{L}\nu_{L}\to X}+\sigma_{\bar{\nu}_{L}l_{L}\to
X}+\sigma_{\bar{\nu}_{L}\nu_{L}\to X}.$ (9)

This neglects the Higgs spectator interactions, and will therefore have
corrections at higher orders. The fact that the Higgs behave as spectators has
been suppressed in the notation, just as with hadron-hadron collisions.
Nonetheless, because the net zero weak charge of the bound states can be
arbitrarily split between the Higgs spectator and the elementary lepton field,
a summation is necessary over all possible weak charge states, as given in
(9). This is again in complete analogy to color in QCD.

However, by relegating the Higgs to spectators, only the fermionic components
of the bound state (3) can now interact, especially also the left-handed
neutrinos. This is, because the experimentally prepared initial states are
SU(2)${}_{\text{c}}$ eigenstates and not unphysical gauge eigenstates. Just
like all colors are contained for every valence quark in the proton, now all
weak charges, and thus left-handed electrons and left-handed neutrinos both,
are contained in the initial SU(2)${}_{\text{c}}$ eigenstate, and thus need to
be summed over777Note that the usual differences of electrons and neutrinos,
like different masses and electric charges, are carried by the bound state (3)
Maas:2017wzi . Only to leading order in $v$ in (3) these properties are
projected onto them, as shown in (4)..

Now, each of the individual cross sections violates the Bloch-Nordsieck
theorem, which leads to a double-logarithmic Sudakov enhancement
Ciafaloni:2000rp . If $s\gg v$, they become

$\displaystyle\sigma_{\bar{l}^{L}l^{L}\to X}$
$\displaystyle=\sigma_{\bar{\nu}^{L}\nu^{L}\to X}=$ $\displaystyle A+S$ (10)
$\displaystyle\sigma_{\bar{l}^{L}\nu^{L}\to X}$
$\displaystyle=\sigma_{\bar{\nu}^{L}l^{L}\to X}=$ $\displaystyle A-S,$ (11)

where $A$ contains the non-enhanced part, and $S$ the enhancement proportional
to the exponentiated Sudakov logarithm. Inserting this into (9) immediately
shows that the enhancement cancels. The decisive step here is that it is
necessary to do the summation, as mandated by gauge invariance. That they
would cancel if summed follows from the BNT, and is well known
Ciafaloni:2000rp . Thus, the FMS mechanism explains how the summation can take
place at high energies as mandated by gauge symmetry without altering the low-
energy behavior accessed by LEP(2). Of course, this is now the behavior at
small and large energies compared to the Higgs vacuum expectation value. What
happens in the intermediate energy ranges where both behaviors transition into
each other needs a full evaluation of (7) in FMS-mechanism augmented
perturbation theory, which is relegated to future work MPV:unpublished .

However, to the current order, this exactly cancels the BNT violations. These
have been estimated to be of the same order as the leading-order QCD effects
at a 1 TeV lepton collider Ciafaloni:2000rp , and substantially larger at a 3
TeV lepton collider Ciafaloni:2000rp ; Bauer:2018xag , depending on
polarization. Having them removed by the BNT is therefore not a negligible
effect.

Of course, eventually it will no longer be justified to treat the Higgs as
spectators. If there are no non-trivial bound state contributions, this can be
accounted for by augmented perturbation theory MPV:unpublished . If this would
not be the case, a PDF-type language Egger:2017tkd , similar to the hadron
case treated in section III, would be necessary888See also Calmet:2000th ;
Calmet:2001rp ; Calmet:2001yd  for a similar approach with a different
motivation..

##  III Hadron scattering and PDFs

In principle, the same considerations apply as well with quarks in the initial
states, i. e. for hadron colliders like the LHC or (future) hadron colliders
Ciafaloni:2000rp ; Fernbach:2020tpa . In particular, the BNT is violated in
the standard approach, and this has a substantial impact at sufficiently high
energies Ciafaloni:2000rp ; Bauer:2018xag . At first sight, it appears to be
sufficient to follow the same prescription as before to fix it. But there is a
subtlety, arising from the need to use PDFs to describe the hadron structure.

In the standard approach a cross section will be determined as Bohm:2001yx

$\sigma_{PP\to
X}=\sum_{ij}\int_{0}^{1}dx\int_{0}^{1}dyf_{i}(x)f_{j}(y)\sigma_{\bar{i}j\to
X}(xp_{1},yp_{2}),$ (12)

where $i$ runs over all constituents of both hadrons, the $f_{i}$ are the
corresponding PDFs, and $X$ is the final state. Consider for the moment just
the first generation of quarks. It is worthwhile to recall how BNT violations
in the strong interactions are avoided. This happens as in the hard cross
section an unbiased sum over all colors is performed Bohm:2001yx . Each color
appears equally as initial state in the hard cross section. This is due to the
implicit presence of three PDFs for each quark, one for each color, which are
identical999This is, of course, only possible in perturbation theory, as
colored quarks are physical states in the usual BRST construction Bohm:2001yx
. However, the PDFs are genuine non-perturbative quantities, and the BRST
construction is broken non-perturbatively Fujikawa:1982ss , and in fact quarks
as initial states cannot act as physical states Lavelle:1995ty , and the PDFs
need to be defined in a non-perturbatively manifest gauge-invariant way
Ji:2013dva . The reason why such a factorization nonetheless works is that in
QCD there is a one-to-one correspondence between each quark and a gauge-
invariant physical observable, its flavor. But it is also this feature, the
flavor, which will be needed to be resolved at the electroweak level..

Because of the isospin sum rules

$\displaystyle\int
dx\left(f_{u^{L}}(x)-f_{\bar{u}^{L}}(x)+f_{u^{R}}(x)-f_{\bar{u}^{R}}(x)\right)=2$
(13) $\displaystyle\int
dx\left(f_{d^{L}}(x)-f_{\bar{d}^{L}}(x)+f_{d^{R}}(x)-f_{\bar{d}^{R}}(x)\right)=1$
(14) $\displaystyle f_{i^{L}}(x)=f_{i^{R}}(x)$ (15)

the PDFs for the up quark and the down quark can never be the same. But since
the left-handed up quarks and down quarks are the members of a weak multiplet
similar to (2), this seems to imply that BNT violation cannot be avoided. Of
course, a possibility appears to be to relax the equality of the left-handed
quarks and right-handed quarks, and this is indeed anyhow necessary at high
energies Bauer:2018xag . But the proton is a parity eigenstate, and thus fully
shifting isospin to the right-handed quarks seems to be at least unlikely.

The reason for this impasse is, of course, that (12) is really just the
leading order of the FMS expansion. To understand this, it is helpful to start
with the manifestly gauge-invariant description of the proton. To this end, it
is useful to follow a two-step procedure Maas:2017wzi ; Egger:2017tkd . The
left-handed quark doublet $q^{L}=(u^{L},d^{L})$ can be dressed in the same way
as leptons in (3). This implies that the physical flavor of left-handed quarks
is also nothing but the global SU(2)${}_{\text{c}}$ symmetry of the Higgs
field. Thus, left-handed quark flavor is the same for left-handed quarks and
left-handed leptons. Their distinction is entirely by their other quantum
numbers, like hypercharge, baryon number, and lepton number.

Construct then weakly-gauge-invariant Dirac spinors

$\displaystyle U$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\left((X)^{\dagger}q^{L}\right)_{1}\cr
u^{R}\end{pmatrix}$ (16) $\displaystyle D$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\left((X)^{\dagger}q^{L}\right)_{2}\cr
d^{R}\end{pmatrix}$

which transform suitably under the diagonal $\mathrm{SU(2)_{f}}$ group. A
nucleon operator can then be constructed as Egger:2017tkd ; Maas:2017wzi ;
Gattringer:2010zz

$\displaystyle N$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1+\gamma_{0}\right)\epsilon^{IJK}U_{I}\left(U^{T}_{J}C\gamma_{5}D_{K}\right)$
$\displaystyle=$
$\displaystyle\frac{\epsilon^{IJK}}{2}\begin{pmatrix}u^{R}-\left((X)^{\dagger}q^{L}\right)_{1}\cr\left((X)^{\dagger}q^{L}\right)_{1}-u^{R}\end{pmatrix}_{I}\times$
$\displaystyle\times\left(\left(\left((X)^{\dagger}q^{L}\right)_{1}^{J}\right)^{T}\tau^{2}\left((X)^{\dagger}q^{L}\right)_{2}^{K}-(u^{R}_{J})^{T}\tau^{2}d^{R}_{K}\right)$
$\displaystyle=$
$\displaystyle\frac{\epsilon^{IJK}}{2}\left(\begin{pmatrix}u^{R}-\phi_{2}u^{L}\cr\phi_{2}u^{L}-u^{R}\end{pmatrix}^{I}+\phi_{1}\begin{pmatrix}d^{L}\cr-d^{L}\end{pmatrix}^{I}\right)\times$
$\displaystyle\times\left(|\phi_{2}|^{2}(u^{L}_{J})^{T}\tau_{2}d^{L}_{K}-|\phi_{1}|^{2}(d^{L}_{J})^{T}\tau^{2}u^{L}_{j}-(u^{R}_{J})^{T}\tau^{2}d^{R}_{K}\right.$
$\displaystyle\left.+\phi_{2}\phi_{1}^{*}(u^{L}_{J})^{T}\tau^{2}u^{L}_{K}-\phi_{1}\phi_{2}^{*}(d^{L}_{J})^{T}\tau_{2}d^{L}_{K}\right).$

where $C$ is the charge conjugation matrix, $\tau^{2}$ is the second Pauli
matrix, and the capital indices enumerate color. The projector in front is
needed to project out positive parity, as needed for the proton.

In leading order in the Higgs vacuum expectation value, $\phi_{2}=v$ and
$\phi_{1}=0$, (III) collapses to the usual expression in QCD Gattringer:2010zz
. The term in bracket acts as a diquark with zero net flavor. Spin and flavor
is therefore carried entirely by the leading $U$, which mixes, due to the
projector, left-handed components and right-handed components. It is thus
visible how in a manifestly gauge-invariant description there are valence
Higgs degrees of freedom in addition to valence quarks, and the former carry
the flavor of the proton, very much in the same way as for leptons.

This now shows how (12) can be seen as the leading order FMS expansion of the
physical cross section. Every of the would-be quark PDFs describes the
probability to encounter a flavor carrier in the proton, which are given by
(16). Since the color indices play the same role as before, and at low enough
energy the structure (16) is not resolved, the hard cross section is then
having Higgs-quark bound states as initial states. The cross section can then
be FMS expanded like in (8). Keeping only the leading term yields finally the
standard expression for hadronic cross sections (12).

If all internal weak structure of the proton (III) would be entirely part of
the weak structure bound-state (16), it would be sufficient to switch from the
leading-order FMS expansion to the full expression, like in (9), at high
energies. This would maintain the BNT, at the expense of reinterpreting the
quark PDFs as physical flavor PDFs.

If this is not the case, which may happen e. g. for multi-parton interactions,
or if the separation between strong and weak bound state effects is no longer
possible at high energies, this will require a different approach101010At such
energies the BEH effect becomes irrelevant, and thus factorization will work
as in QCD, supplemented by a scalar. Literally the same steps will just need
to be done, and the factorization scale is then given by the Higgs vev rather
than the hadronic scale.. In that case, it is necessary to introduce explicit
Higgs PDFs111111In Fernbach:2020tpa  this had been done only partially. With
hindsight, this needs to be reinterpreted as attempting to encode the
difference between the actual Higgs substructure and the conventional one only
effectively by a single additional PDF, which did not influence the sum rule.
Fernbach:2020tpa  at the valence level to maintain the sum rules (13-14), as
the flavor is carried by the custodial symmetry doublet eigenstate
$(-\phi_{1},\phi_{2}^{*})$, which mixes the weak doublet eigenstate
$(\phi_{1},\phi_{2})$. This can be accommodated by introducing four Higgs
PDFs, $f_{hij}$, in which $i$ and $j$ denote the weak and custodial
contributions, respectively. They therefore correspond directly to the entries
of the matrix $X$ in (1) in terms of the required parton to participate in the
hard interaction. This implies also

$f_{h\frac{1}{2}j}=f_{h-\frac{1}{2}j}\\\ $

to maintain gauge invariance and the BNT. Denoting for $j$ $u$ up-type flavor
and $d$ for down-type flavor, this ultimately replaces (13-14) by

$\displaystyle\int
dx\left(f_{u^{R}}(x)-f_{\bar{u}^{R}}(x)+f_{h\frac{1}{2}u}+f_{h-\frac{1}{2}u}\right)$
$\displaystyle=$ $\displaystyle 2$ $\displaystyle\int
dx\left(f_{d^{R}}(x)-f_{\bar{d}^{R}}(x)+f_{h\frac{1}{2}d}+f_{h-\frac{1}{2}d}\right)$
$\displaystyle=$ $\displaystyle 1.$

Of course, in $R_{\xi}$ gauges this appears to correspond to having unphysical
states as possible initial states. However the Goldstone-Boson equivalence
theorem relates them to longitudinally polarized vector boson matrix elements.

Note that in the baryon number sum rule still the left-handed quark PDFs
appear, but they obey

$f_{u^{L}}=f_{d^{L}}.$

In the sumrule for momentum and electromagnetic charge now all PDFs, including
the Higgs ones, appear.

This structure now automatically ensures that the BNT is also maintained in
calculations of hadronic collisions. Of course, the different PDFs will need
to be determined and evolved, including the other electroweak ones, along the
lines of Bauer:2018xag . However, if at sufficiently low energies the weak
substructure does not matter, it necessarily follows that

$\displaystyle\sum_{u_{L},d_{L},hkl}\int_{0}^{1}dx\int_{0}^{1}dyf_{i}(x)f_{j}(y)\sigma^{\text{elementary}}_{\bar{i}j\to
X}(xp_{1},yp_{2})$ $\displaystyle\approx$
$\displaystyle\sum_{u_{L},d_{L}}\int_{0}^{1}dx\int_{0}^{1}dy\tilde{f}_{i}(x)\tilde{f}_{j}(y)\sigma_{\bar{i}j\to
X}^{\text{elementary}}(xp_{1},yp_{2})$

need to hold, where the $\tilde{f}_{i}$ are now identical to the BNT-violating
standard PDFs and the superscript “elementary” implies that these are the
ordinary perturbative cross sections, i. e. the ones in leading order in the
Higgs vev of FMS-augmented perturbation theory. However, since the
factorization scale is crossed in this way, this is really not to be
understood as an exact statement anymore. In fact, from the high energy
perspective what happens is that either a valence left-handed quark from the
structure (16) or a left-handed quark radiated off from the valence Higgs
takes part in the hard interaction, and the Higgs becomes off-shell
suppressed. Neglecting the latter will then reinstantiate the BNT violation,
but sufficiently small to be irrelevant at low energies.

Alternatively, if the weak substructure can be treated entirely using
augmented perturbation theory, an approach like (9) would also be possible.
However, then a subtle problem arises, as this will not remove the strong
substructure. Thus, there will be still subprocesses in which the QCD cores of
the physical proton interact, which would need to revert again to a PDF
language. But, as is visible in expression (III), this process is inclusive
with respect to the left-handed weak doublets. Thus the QCD expression
involves at the QCD level not only proton-like states, but also other QCD-like
objects. For these, PDFs are not as readily available. This may require other
sources than experiments, e. g. using lattice calculations Ji:2013dva ;
Cichy:2018mum ; Constantinou:2020hdm , to follow such an approach instead of
introducing Higgs PDFs.

In addition, similar considerations also apply to further generations.
However, since off-diagonal intergeneration elements of the CKM matrix can be
recast as intergeneration Yukawa interactions of the Higgs, it may in these
cases probably be better to set up the PDF scheme in such a basis. However,
this is beyond the scope of this work, as the BNT restoration will work in the
same way also for the other generations.

##  IV Conclusion

Manifest gauge invariance enforces the use of composite operators to describe
physical states also for the weak interactions. However, the combination of
the Brout-Englert-Higgs effect and the FMS mechanism shows that at
sufficiently small energies in the standard model this does yield
quantitatively essentially the same results as if only perturbative gauge
invariance is respected. Serious problems only arise when kinematics are no
longer determined primarily by the FMS-dominant constituent, i. e. in most
cases at either extremely small Jenny:2022atm ; Maas:2018ska  or extremely
large energy transfers Maas:2020kda ; Egger:2017tkd ; Fernbach:2020tpa ;
Dudal:2020uwb . It then becomes necessary to revert to a full desciption of
the composite states.

We have investigated how this could restore the BNT by using manifestly weakly
gauge-invariant initial states, at least at leading order. This potentially
affects cross sections at very high energies considerably, especially at TeV-
scale future lepton colliders. The situation is the same at hadron colliders,
but here the PDF structure both suppresses the effect Ciafaloni:2000rp  and
also makes its resolution more involved. Likewise, we expect final states,
including fragmentation, to require a similar treatment to avoid violations of
the Kinoshita-Lee-Naunberg theorem, which we have not discussed in detail.

Augmenting perturbation theory with the FMS mechanism may allow for a
possibility to take these effects into account with only marginally more
effort than in usual perturbation theory. Given the potential impacts for
predictions for future colliders, this appears a small price to pay.

###### Acknowledgements.

We are grateful to S. Plätzer and R. Sondenheimer for helpful discussions and
to S. Plätzer for a critical reading of the manuscript, and to B. Webber for
prompting this research. This work was done within the scope of the FCC
Feasibility Study.

## References

  * (1) S. Elitzur,  Phys. Rev. D12, 3978 (1975).
  * (2) J. Fröhlich, G. Morchio, and F. Strocchi,  Phys.Lett. B97, 249 (1980).
  * (3) J. Fröhlich, G. Morchio, and F. Strocchi,  Nucl.Phys. B190, 553 (1981).
  * (4) B. Lee and J. Zinn-Justin,  Phys.Rev. D5, 3137 (1972).
  * (5) A. Maas,  Mod. Phys. Lett. A27, 1250222 (2012), 1205.0890.
  * (6) T. Banks and E. Rabinovici,  Nucl.Phys. B160, 349 (1979).
  * (7) E. H. Fradkin and S. H. Shenker,  Phys. Rev. D19, 3682 (1979).
  * (8) K. Osterwalder and E. Seiler,  Annals Phys. 110, 440 (1978).
  * (9) V. N. Gribov,  Nucl. Phys. B139, 1 (1978).
  * (10) I. M. Singer,  Commun. Math. Phys. 60, 7 (1978).
  * (11) K. Fujikawa,  Nucl. Phys. B223, 218 (1983).
  * (12) A. Maas,  Progress in Particle and Nuclear Physics 106, 132 (2019), 1712.04721.
  * (13) F. Jegerlehner and J. Fleischer,  Phys. Lett. B 151, 65 (1985).
  * (14) F. Jegerlehner and J. Fleischer,  Acta Phys. Polon. B 17, 709 (1986).
  * (15) A. Maas and R. Sondenheimer,  Phys. Rev. D 102, 113001 (2020), 2009.06671.
  * (16) D. Dudal et al.,  Eur. Phys. J. C 81, 222 (2020), 2008.07813.
  * (17) L. Egger, A. Maas, and R. Sondenheimer,  Mod. Phys. Lett. A32, 1750212 (2017), 1701.02881.
  * (18) M. Ciafaloni, P. Ciafaloni, and D. Comelli,  Phys. Rev. Lett. 84, 4810 (2000), hep-ph/0001142.
  * (19) M. Ciafaloni, P. Ciafaloni, and D. Comelli,  Nucl. Phys. B 589, 359 (2000), hep-ph/0004071.
  * (20) F. Bloch and A. Nordsieck,  Phys. Rev. 52, 54 (1937).
  * (21) F. Reiner and A. Maas,  PoS EPS-HEP2021, 449 (2022), 2110.07312.
  * (22) T. Kinoshita,  J. Math. Phys. 3, 650 (1962).
  * (23) T. D. Lee and M. Nauenberg,  Phys. Rev. 133, B1549 (1964).
  * (24) V. Afferrante, A. Maas, R. Sondenheimer, and P. Törek,  SciPost Phys. 10, 062 (2021), 2011.02301.
  * (25) M. Böhm, A. Denner, and H. Joos,  Gauge theories of the strong and electroweak interaction (Teubner, Stuttgart, 2001).
  * (26) A. Maas, S. Raubitzek, and P. Törek,  Phys. Rev. D99, 074509 (2019), 1811.03395.
  * (27) P. Jenny, A. Maas, and B. Riederer,  Phys. Rev. D 105, 114513 (2022), 2204.02756.
  * (28) A. Maas, S. Plätzer, and R. Sondenheimer,  2023\.
  * (29) S. Fernbach, L. Lechner, A. Maas, S. Plätzer, and R. Schöfbeck,  Phys. Rev. D 101, 114018 (2020), 2002.01688.
  * (30) S. Weinberg,  The Quantum theory of fields. Vol. 1: Foundations (Cambridge University Press, Cambridge, 1995).
  * (31) A. Maas, S. Plätzer, and F. Veider,  2023\.
  * (32) C. W. Bauer, D. Provasoli, and B. R. Webber,  JHEP 11, 030 (2018), 1806.10157.
  * (33) V. S. Fadin, L. N. Lipatov, A. D. Martin, and M. Melles,  Phys. Rev. D 61, 094002 (2000), hep-ph/9910338.
  * (34) P. Ciafaloni and D. Comelli,  Phys. Lett. B 476, 49 (2000), hep-ph/9910278.
  * (35) X. Calmet and H. Fritzsch,  Phys. Lett. B496, 161 (2000), hep-ph/0008243.
  * (36) X. Calmet and H. Fritzsch,  Phys. Lett. B525, 297 (2002), hep-ph/0107085.
  * (37) X. Calmet and H. Fritzsch,  Phys. Lett. B526, 90 (2002), hep-ph/0103333.
  * (38) M. Lavelle and D. McMullan,  Phys. Rept. 279, 1 (1997), hep-ph/9509344.
  * (39) X. Ji,  Phys. Rev. Lett. 110, 262002 (2013), 1305.1539.
  * (40) C. Gattringer and C. B. Lang,  Quantum chromodynamics on the lattice (Lect. Notes Phys., 2010).
  * (41) K. Cichy and M. Constantinou,  Adv. High Energy Phys. 2019, 3036904 (2019), 1811.07248.
  * (42) M. Constantinou et al.,  Prog. Part. Nucl. Phys. 121, 103908 (2021), 2006.08636.