# Estimation of non-uniform blur using a patch-based regression convolutional
neural network (CNN)

Luis G. Varela, Laura E. Boucheron, Steven Sandoval, David Voelz, and Abu
Bucker Siddik
Klipsch School of Electrical and Computer Engineering
New Mexico State University
Las Cruces, NM 88003
{varelal<EMAIL_ADDRESS>

###### Abstract

The non-uniform blur of atmospheric turbulence can be modeled as a
superposition of linear motion blur kernels at a patch level. We propose a
regression convolutional neural network (CNN) to predict angle and length of a
linear motion blur kernel for varying sized patches. We analyze the robustness
of the network for different patch sizes and the performance of the network in
regions where the characteristics of the blur are transitioning. Alternating
patch sizes per epoch in training, we find coefficient of determination scores
across a range of patch sizes of $R^{2}>0.78$ for length and $R^{2}>0.94$ for
angle prediction. We find that blur predictions in regions overlapping two
blur characteristics transition between the two characteristics as overlap
changes. These results validate the use of such a network for prediction of
non-uniform blur characteristics at a patch level.

##  1 Introduction

The effects of atmospheric turbulence in an image are commonly modeled as a
spatially-varying blur [1, 2] and spatially-varying blurred images can be
modeled as a superposition of locally linear motion blurs [3]. Linear motion
blur, i.e., motion blur that can be described by motion in a line, is commonly
used as a model for camera shake, camera platform movement, and moving objects
in images [4, 5, 6]. In this work, we parametrize linear motion blur by length
and angle and estimate those parameters in image patches using a regression
convolutional neural network (CNN). In previous work it was demonstrated that
a regression CNN can accurately predict length and angle of motion blur in
uniformly blurred images and that those predictions can be used in non-blind
image deblurring methods to recover the latent sharp image [7]. The work in
[7] was extended in [8] to a patch-based approach to estimate non-uniform
blur. This paper extends the work in [8] with a comprehensive study of the
effect of patch size on the accuracy of parameter estimation and estimation
accuracy for patches that overlap different motion blur characteristics.

There are two main contributions of this work. First, the network is trained
to be robust to a range of patch sizes through a training approach that
alternates patch size per epoch and by removing the last two convolution
layers from the network in [8] to allow predictions on patches smaller than
size 32. This results in a network that can be used for prediction of non-
uniform blur at a wide range of patch sizes, depending on the expected spatial
scale of the blur. This is in contrast to most patch-based methods which are
designed to operate on a fixed-size patch, usually $30$ pixels, e.g., [9, 3].
Second, we study blur characteristic predictions in patches that overlap
regions of two blur characteristics which, to the best of our knowledge, has
not been studied before. This provides important insights for the prediction
of blur characteristics in non-uniformly blurred images for methods that
operate on image patches. Image patches can be expected to span a range of
blur characteristics and blur transitions in non-uniformly blurred images and
the specific performance of blur prediction networks in such scenarios is
important to understand.

The remainder of this paper is organized as follows. Section 2 presents an
overview of models for uniform and non-uniform image blurring and a discussion
of related work for image blur estimation and removal. Section 3 presents the
patch-based regression CNN architecture, details of training, and an overview
of the dataset used for training and testing. Section 4 discusses results,
including the effects of patch size on length and angle estimation and effects
of overlapping blur characteristics within patches. Section 5 provides
conclusions and a brief discussion of future work.

##  2 Background

###  2.1 Image Blur

A uniformly blurred image can be modeled as

$I_{\text{b}}=K*I_{\text{s}}+\eta,$ (1)

where $I_{\text{b}}$ is the blurred image, $K$ is the blur kernel,
$I_{\text{s}}$ is the sharp latent image, $\eta$ is additive noise, and $*$ is
the convolution operator (or correlation operator as is commonly assumed in
image filtering). This formulation assumes a uniform blur because the same
kernel $K$ is applied across the entire image. A non-uniform blur can be
represented as [10]

$I_{\text{b}}[m,n]=\sum_{s}\sum_{t}K_{m,n}[s,t]I_{\text{s}}[m+s,n+t]+\eta,$
(2)

where $K_{m,n}$ is a blur kernel centered at pixel $I_{\text{s}}[m,n]$, the
summation over $s$ and $t$ are defined over the extent of the kernel, and the
indices for sharp image $I_{\text{s}}$ implement a correlation (rather than a
convolution). Thus, in the most general case for Eq. (2), each pixel
$I_{\text{s}}$ has a different blur kernel operating on its neighborhood,
defined by the extent of the blur kernel. Note that if $K_{m,n}=K$, i.e., the
same blur kernel is applied to all pixels, then the non-uniform blur in Eq.
(2) reduces to the uniform blur of Eq. (1).

###  2.2 Estimating Image Blur Parameters and Deblurring

Many approaches to deblurring and blur kernel estimation prior to deep
learning used a maximum a posteriori (MAP) framework with implicit
regularization using image priors [4, 6, 11, 12, 13] or explicit edge
prediction [14, 15]. The MAP approach is an iterative optimization with
alternate iterations predicting the blur kernel and the latent sharp image.
Levin et al. [16, 17], however, posit that such an approach tends to favor a
solution in which the kernel is an impulse and the “deblurred” image is the
blurry image instead of converging to the true blur kernel and latent sharp
image. As such, Levin et al. [16, 17] advocate for approaches that only
estimate the blur kernel (they use an expectation-maximization (EM) framework
with either a Gaussian or sparse prior) and use of non-blind deblurring
methods to estimate the latent sharp image. Having demonstrated the utility of
using estimated blur kernels in non-blind deblurring in our previous work on
uniformly blurred images [7], here we focus on estimation of the blur kernel
in non-uniformly blurred images.

Recent methods have considered deep learning approaches to estimate the blur
kernel or blur field [18, 9, 19, 10], estimate the latent sharp image [20], or
both [21]. These methods use CNNs [19, 9, 21, 18], fully convolutional
networks (FCNs) [10], or generative adversarial networks (GANs) [20]. CNNs
have been used to enhance image edges as input to a classical MAP approach for
uniformly-blurred images [19], to classify whether an image is blurry or sharp
for use as regularization in a MAP framework [21], to predict blur kernels via
classification for uniformly-blurred image patches [9], and to classify the
blur type followed by a general regression network to estimate the blur
parameters [18]. In contrast, we use a regression CNN to directly estimate the
blur kernel or blur field by estimating motion blur parameters in patches.

In this work we extend the regression CNN of [7, 8] which can be trained as a
regressor on image patches to estimate length and angle as a parametrization
of linear motion blur kernels. The use of regression in this formulation
allows us to explore a continuous range of length and angle parameters for
linear motion blur. This work improves the results from [8] by introducing an
alternate patch size training. Furthermore, we create non-uniformly blurred
images by blurring two regions of the image with different characteristics and
use the network to predict blur in patches that overlap the two blur
characteristics. This gives important insights into the behavior of the
network when analyzing patches that include multiple blur characteristics.

##  3 Patch-Based Regression Convolutional Network

###  3.1 Motion Blur Parametrization

We parametrize linear motion blur kernels by the magnitude of the blur length,
$r$, and the orientation angle of the blur, $\phi$. In previous work [7],
exploration of a suite of length and angle pairs $(r,\phi)$, $r\in[1,100]$,
$\phi\in[-90,90)$ demonstrated that not all $(r,\phi)$ combinations result in
a unique blur kernel with shorter length kernels tending to have fewer unique
angles. The COCO dataset [22] was used to create a blurred dataset with the
collection of unique linear motion blur kernels with $(r,\phi)=(1,0)$ used to
designate a non-blurred image. A modified version of VGG16 [23] was trained,
where the last fully connected layer was modified to have two outputs (for
length and angle) and to predict values in the range $[0,1]$ using a sigmoid
activation for normalized values of $r$ and $\phi$. Here, we consider a
modified range of length $r$ to be commensurate with the maximum patch size
used in training, i.e., $r\in[1,\min(N_{\textrm{max}},100)]$, where
$N_{\textrm{max}}$ is the maximum patch size used in training and 100 is the
maximum blur length considered in [7]. The need for this constraint is further
discussed in Section 3.3.

###  3.2 Network Architecture

As in [8], in order to input dimensions other than a fixed $224\times 224$
pixels, we replace the flattening layer in the VGG16 network with a global
average pooling (GAP) layer. To mitigate issues associated with the fewer
outputs of a GAP layer compared to a fully connected layer, the number of
filters in later convolutional layers is increased and the number of nodes in
the fully connected layers is decreased. This results in the network
architecture shown in Table 1, the same architecture as used in [8]. While
this network can accept an arbitrary size input patch of $N\times N$ pixels,
$N$ is lower bounded by 32. This is due to Block 5 having an output size of
$N/32$ since each block in the model is followed by a max pooling layer with
stride $(2,2)$. Here, we remove Block 5 to lower the bound of minimum patch
size to 16. The same simulation from [8] trained on patch size 112 was run
with the removal of Block 5 and we noted no substantial changes in results
(see Table 2 for results).

Table 1: Modified VGG16 model used for patch-based estimation. Layer | Description | Output Dimensions
---|---|---
Input | Input | $N\times N\times 3$
Block 1 |  $2\times$Conv | $(N/2)\times(N/2)\times 64$
Block 2 |  $2\times$Conv | $(N/4)\times(N/4)\times 128$
Block 3 |  $1\times$Conv | $(N/8)\times(N/8)\times 256$
$2\times$Conv | $(N/8)\times(N/8)\times 512$
Block 4 |  $3\times$Conv | $(N/16)\times(N/16)\times 1024$
Block 5 |  $3\times$Conv | $(N/32)\times(N/32)\times 2048$
Block 6 | Global Avg Pool | $1\times 2048$
FC1 | Fully Connected | 2048
FC2 | Fully Connected | 2048
FC3 | Output | 2
Notes: Each block is followed by a max pooling layer with stride $(2,2)$. All
convolutional and fully connected layers have a ReLU activation except the
final fully connected layer FC3 which has a sigmoid activation. Output
dimensions for convolution layers are described as length $\times$ width
$\times$ channels and all remaining layers are one-dimensional.

###  3.3 Training

The training of the network is similar to [8] with a few modifications to
adjust the training for different patch sizes. First, the normalization of $r$
is modified to be consistent with the maximum trained blur length
$N_{\textrm{max}}$ to utilize the full range of the sigmoid activation.
Second, we implement training wherein we change the input patch size per
epoch. As described in Section 4, this allows the network to specialize on
specific patch sizes within an epoch and improve performance across a range of
patch sizes. Thus, changing the input patch size per epoch results in a model
that is more robust in predicting blur characteristics across a wider range of
patch sizes. We use the blurred COCO dataset described in [7, 24] to train and
test the results of the architecture; subsets of the dataset are used as
described below.

Training the network with different patch sizes per epoch requires
modification of the data generator used in training. Since the blurred COCO
dataset described in [7, 24] contains images blurred with kernels of length
$r\in[1,00]$, images blurred with blur parameters that will not fit within the
patch size $N$ must be removed in training. For example a $32\times 32$ pixel
patch can accommodate a length-32 horizontal blur, but could accommodate a
length-$32\sqrt{2}$ blur at an angle of 45 degrees. As in [8], we keep blurred
images with $(r,\phi)$ such that

$r\leq\begin{cases}\sqrt{N^{2}(1+\tan^{2}(|\phi|)},|\phi|\leq 45^{\circ}\\\
\sqrt{N^{2}(1+\cot^{2}(|\phi|)},|\phi|>45^{\circ}\end{cases},$ (3)

where $N$ is the patch size. In short, this equation keeps blurred images from
the training set only if the lengths are short enough for the patch size to
accommodate by recognizing whether the horizontal or vertical axis uses the
full $N$ pixels and using a simple application of the Pythagorean theorem.
Here, blurred images from the training dataset are dynamically removed in each
epoch since the patch size $N$ may change per epoch. Implementation of Eq. (3)
is added to the data generator in training, ensuring that only those data that
satisfy Eq. (3) are seen in training. The data generator creates batches of
samples consisting of matched pairs of image patches and normalized labels by
randomly selecting from the entire dataset. For each sample, the labels are
denormalized (restored to their original range) and if the denormalized length
$r$ satisfies Eq. (3), the sample is appended to the batch. This
implementation of appending samples which satisfy Eq. (3) is important, as
removal of samples from a standard data generator batch can result in empty
batches (which we found resulted in the loss function returning not-a-number
(NaN) and training not converging).

We desire a single network that is robust to different patch sizes for maximum
flexibility in prediction of motion blur. For example, this single network
could predict blur characteristics at different granularities or be used for
prediction of blur in different levels of turbulence which are expected to
manifest on different length scales. Furthermore, we prefer that the network
be most accurate for prediction of blur characteristics at smaller patch
sizes, as we expect that will prove most accurate in the restoration of a non-
uniformly blurred image. Some limited generalization to other patch sizes was
seen in [8], also summarized in Table 2, where both length and angle
predictions were accurate for larger patches and accuracy suffered for patches
smaller than half the trained size. To this end we modified the data generator
to accept an array of patch sizes $P$ to specify the patch size per epoch. The
array of patch sizes is indexed circularly, looping back to the first element
after the last element. For example, a patch size array of $P=[32,64,112,224]$
would result in epochs being trained until convergence (change in the loss
function is $<10^{-12}$ for $15$ epochs) on patch sizes of
$32,64,112,224,32\ldots$.

###  3.4 Data

(a)  (b)  (c)

(d)  (e)

Figure 1: Examples of non-uniformly blurred images. (a) Original unblurred
image. (b) Length Horizontal: $\phi=0$ and $r=(5,15)$ in the (left, right).
(c) Length Vertical: $\phi=90$ and $r=(5,15)$ in the (top, bottom). (d) Angle
Horizontal: $r=15$ and $\phi=(-45,45)$ in the (left, right). (e) Angle
Vertical: $r=15$ and $\phi=(-45,45)$ in the (top, bottom).

The blurred COCO dataset previously described in [7, 24] is used to test
robustness of the network to patch size (Section 4.2). To test the network in
cases where the patch overlaps regions of different blur characteristics
(Section 4.3), a new non-uniformly blurred dataset is created, again using the
COCO dataset [22]. Here, we consider four cases of nonuniform blur, also
illustrated in Fig. 1. Fig. 1(b) illustrates the case where length is varied
horizontally, i.e., $\phi=0$, with the length changing from $r=5$ in the left
half of the image to $r=15$ in the right; we subsequently refer to images
blurred with this blur field as Length Horizontal. Fig. 1(c) illustrates the
case where length is varied vertically, i.e., $\phi=90$, with the length
changing from $r=5$ in the top half to $r=15$ in the bottom (Length Vertical).
Fig. 1(d) illustrates the case where angle is varied horizontally, with length
$r=15$ and the angle changing from $\phi=-45$ in the left half to $\phi=45$ in
the right (Angle Horizontal). Fig. 1(e) illustrates the case where angle is
varied vertically, with length $r=15$ and the angle changing from $\phi=-45$
in the top half to $\phi=45$ in the bottom (Angle Vertical). These non-
uniformly blurred images allow us to study the effects of prediction of blur
characteristics as image patches overlap with different blur characteristics
(Section 4.3).

##  4 Results and Discussion

###  4.1 Performance Metric

As in [7, 8], we use the coefficient of determination $R^{2}$ score [25] to
measure the performance of the model in predicting length and angle of blur
from blurry images. A more accurate model will have an $R^{2}$ closer to $1$,
a naïve model that predicts the data average will have an $R^{2}$ of $0$, and
models with predictions worse than the naïve model will have an $R^{2}$ score
$<0$.

###  4.2 Effects of Patch Size on Length and Angle Estimation

Table 2 shows the results of the training simulations from [7, 8] as a
baseline and new simulations with alternating patch size training as described
above. First, we test the effects of removing Block 5 from VGG16 which is
needed in order to train and test on patch sizes less than 32 pixels. We train
VGG16 without Block 5 on a patch size of 112, allowing us to directly compare
the results to those of full VGG16 trained on patch size 112 as reported in
[8]. We note comparable performance for length prediction and slightly
improved performance on angle prediction when tested on patch sizes of 64,
112, and 224. We see a downward trend in the $R^{2}$ score when testing on
patch sizes smaller than the trained size of 112, similar to the results seen
in [8]. Since the removal of Block 5 has not detrimentally affected prediction
on larger patch sizes and additionally allows training and testing on patch
sizes down to $16$, all new results in Table 2 are for an architecture without
Block 5.

Table 2: Coefficient of determination $R^{2}$ results for prediction of length $r$ and angle $\phi$ for different training configurations. Length $r$ |  |  |
---|---|---|---
| Test Patch Size
Training Config. | 16 | 29 | 30 | 31 | 32 | 64 | 112 | 224
224 [7] | - | - | - | - | - | - | - | 0.987
224 [8] | - | - | - | - | -1.072 | 0.314 | 0.798 | 0.978
112 [8] | - | - | - | - | -0.427 | 0.748 | 0.946 | 0.986
112 | -29.651 | - | - | - | -1.275 | 0.762 | 0.947 | 0.980
32, 64, 112, 224 | -1.145 | - | - | - | 0.594 | 0.865 | 0.958 | 0.985
30, 32, 32, 48, 64 | -0.657 | 0.734 | 0.765 | 0.752 | 0.815 | 0.358 | 0.426 | 0.447
29, 30, 31, 32, 33 | 0.129 | 0.786 | 0.787 | 0.788 | 0.837 | 0.847 | 0.831 | 0.810
Angle $\phi$ |  |  |
| Test Patch Size
Training Config. | 16 | 29 | 30 | 31 | 32 | 64 | 112 | 224
224 [7] | - | - | - | - | - | - | - | 0.994
224 [8] | - | - | - | - | 0.586 | 0.886 | 0.962 | 0.992
112 [8] | - | - | - | - | 0.751 | 0.748 | 0.946 | 0.986
112 | 0.317 | - | - | - | 0.822 | 0.968 | 0.987 | 0.997
32, 64, 112, 224 | 0.588 | - | - | - | 0.955 | 0.993 | 0.997 | 0.999
30, 32, 32, 48, 64 | 0.706 | 0.941 | 0.948 | 0.950 | 0.964 | 0.983 | 0.988 | 0.991
29, 30, 31, 32, 33 | 0.731 | 0.942 | 0.942 | 0.942 | 0.962 | 0.980 | 0.977 | 0.972
Notes: All results are tested on the entire test subset (21,789 blurry images)
of the blurred COCO dataset [7, 24]. Training configurations are presented as
the order in which patch sizes are presented per epoch (see Section 3.3). A
dash (-) indicates that the network was not tested on that patch size. Results
from [7, 8] were trained and tested on a modified VGG16 [23] architecture (see
Table 1) and new results use the modified VGG16 architecture with Block 5
removed.

Second, we introduce an alternating patch size training configuration as
discussed in Section 3.3 with the hypothesis that this will allow the network
to focus on smaller patch sizes and result in a more uniform $R^{2}$ score
when tested across different patch sizes. With a training configuration of
$P=[32,64,112,224]$, we see improved performance on smaller patch sizes
compared to a network trained on one patch size of 112. The downward trend in
$R^{2}$ score is still noticeable as patch size decreases, but the overall
range from the best and worst $R^{2}$ score has diminished compared to
training with a single patch size. Noting the lower performance on smaller
patch sizes, we trained a model with a training configuration of
$P=[30,32,32,48,64]$ with the hypothesis that the network would be forced to
better predict smaller patch sizes. Results demonstrate improved performance
on smaller patch sizes (29, 30, 31, 32), but reduced performance on larger
patch size (64, 112, 224). Finally, we trained a network with a configuration
of $P=[29,30,31,32,33]$ to specialize on patch sizes around 31 in order to
test the effect of predicting blur characteristics in overlapping blur patches
(Section 4.3). We note a much more uniform performance for both length and
angle prediction across a wide range of patch sizes, even up to 224.
Interestingly, it appears that important characteristics of blur can be
learned within a relatively small image patch and generalized to larger image
patches. It is this final configuration of $P=[29,30,31,32,33]$ that we use as
the trained network for subsequent results.

###  4.3 Analyzing Prediction of Blur Characteristics in Overlapping Patches

We study the effect of the patch overlapping different blur characteristics
with the non-uniform blur patterns discussed in Section 3.4. By definition,
nonuniformly blurred images have blur characteristics that change spatially
which leads to the question of how the network will predict blur in patches
that contain contributions from different blur characteristics. We randomly
select 10 images from the COCO dataset and apply the four non-uniform blur
patterns to each image. Those 10 blurred images are input to the trained
network (configuration $P=[29,30,31,32,33]$ as summarized in Table 2) and
length $r$ and angle $\phi$ are predicted for patches with size $31$ and
advance of 1 pixel in a direction perpendicular to the blur discontinuity. As
an example, for the Length Horizontal and Angle Horizontal blurs, the network
begins predicting characteristics for all patches at the left of the image,
advancing the patch position to the right in steps of one pixel, resulting in
varying degrees of overlap across the different blur characteristics.
Similarly, for Length Vertical and Angle Vertical, the network predicts blur
characteristics for patches at the top of the image, advancing down in steps
of one pixel.

(a)

(b)

Figure 2: Predicted blur characteristics versus percentage overlap for (a)
changing angle and (b) changing length. The solid black line is the linear
interpolation between the two blur characteristics as a point of reference.
The solid red and blue lines indicate average predictions across all $31\times
31$ patches in the 10 non-uniformly blurred COCO images and the filled region
is the corresponding standard deviation. The x-axis is labeled from $0\%$ to
$48\%$ overlap with the left side denoting leftward or upward regions and the
right side denoting rightward and downward regions.

When cropping a patch from an image with overlap between two blur
characteristics, we expect the prediction to be between the two blur
characteristics. Results for predicted characteristics for all $31\times 31$
patches in the 10 non-uniformly blurred COCO images are shown in Fig. 2(a) for
Angle Horizontal and Angle Vertical and Fig. 2(b) for Length Horizontal and
Length Vertical. Additionally, in Fig. 2, we plot the linear interpolation
between the two blur characteristics as a point of reference. The results for
change in angle [Fig. 2(a)] show average predictions with small variation and
a symmetric transition relative to the overlap. The results for change in
length [Fig. 2(b)] show average predictions with larger variation and less
symmetry relative to the overlap, but display a trend from shorter to longer
lengths from left to right (the horizontal case) or top to bottom (the
vertical case).

(a)

(b)

(c)

(d)

Figure 3: Predictions of blur characteristics displayed as heatmaps for four
example images blurred non-uniformly with (a) Angle Horizontal, (b) Angle
Vertical, (c) Length Horizontal, and (d) Length Vertical. Left column: the
non-uniformly blurred image. Middle column: the prediction as a heat map. Note
that the heat map is of smaller dimensions than the blurred image since the
predictions are patch-based. Right column: the average (solid line) and
standard deviation (shaded area) taken perpendicularly to the blur
discontinuity. Since each image is a different size, the average results in
the right column have different scales on the x-axis.

The wider variation in the results in Fig. 2(b) are in line with the results
summarized by the $R^{2}$ score, with length prediction resulting in lower
$R^{2}$ scores than angle prediction (see Table 2). The asymmetry in the
prediction with respect to the overlap is likely due to the fact that
prediction of a larger blur length will require a commensurately larger
overlap into that blur characteristic. From the results in Table 2, the
network has poorer performance on patches less than $\sim 31$. The results in
Fig. 2(b) may be evidence toward the fact that it is prediction of the longer
lengths that reduce the $R^{2}$ score of the network in smaller patch sizes.
We also note some bias in the estimation of length in Fig. 2(b). We believe
this is due to specific image characteristics confounding the length
prediction, further discussed below.

In Fig. 3 we plot a heatmap of predictions for example images blurred with
each of the four non-uniform blur patterns discussed in Section 3.4.
Consistent with the average results in Fig. 2, we see that the angle
predictions [Fig. 3(a) and (b)] are more uniform within the two halves of the
image as compared to the length predictions [Fig. 3(c) and (d)]. The results
in Fig. 3 also give some insight into cases where the network may struggle
with correct length prediction. In particular, many instances of
overestimation of length appear to be related to linear structures within the
images. For example, note that the length predictions for the wood floor
toward the bottom left of the image in Fig. 3(c) are longer than the ground
truth value $r=5$. The structure of the planks in the wood floor, however, is
likely confounding the correct prediction of length, providing image
structures that look like characteristics of a much longer blur than is
applied.

##  5 Conclusion

This work expands the uniform blur prediction from [7] and improves and
expands upon the results for a patch-based approach in [8]. The training
process is modified, alternating different patch sizes per epoch, improving
the robustness of the network to varying patch size. Results demonstrate that
the network can generalize to a wide range of patch sizes, with better
generalization to larger patches than smaller, indicating that a patch size of
approximately 30 pixels is ideal for encompassing important blur cues.

The study of blur characteristic prediction in patches that overlap two
regions with different blur indicates that the network predicts blur
characteristics in between the overlapped characteristics depending on the
level of overlap. Length prediction displayed a wider variance in prediction
versus overlap, consistent with the smaller coefficient of determination
scores seen for length prediction here and in [7, 8]. Observation of predicted
blur characteristics in example images indicate that specific linear
structures within images are likely the source of errors in length prediction.

There are several avenues of potential future work. The removal of the last
two convolutional layers produced no significant change in results; future
work may study of the necessary depth of the network. Also, it would be
interesting to explore the deblurring of the overlapped regions (using
approaches similar to those in [7]) and analyze artifacts introduced in the
deblurring of those overlapped regions. Finally, as our results seem to
indicate that a patch size of approximately 30 pixels is sufficient for
encompassing important blur cues in images, it could provide important
insights to explore other patch sizes. As other work, e.g., [9, 3], also
operate at the same patch size, specifically training our network toward a
smaller patch size, e.g., 16 pixels, may provide additional evidence toward an
optimal patch size for non-uniform blur estimation. Similarly, it may be
beneficial to analyze prediction of overlapped regions with smaller and larger
patch sizes.

## Data availability

Data underlying the results presented in this paper for uniformly-blurred
images can be generated from the COCO dataset [22] via the code available at
[24]. Data underlying the results for non-uniformly blurred images can be
generated from the COCO dataset [22] via the code available at [26].

## Acknowledgments

The authors gratefully acknowledge Office of Naval Research grant
N00014-21-1-2430 which supported this work.

## References

  * [1] Bobby R. Hunt, Amber L. Iler, Christopher A. Bailey, and Michael A. Rucci.  Synthesis of atmospheric turbulence point spread functions by sparse and redundant representations.  Optical Engineering, 57(2):024101, 2018.
  * [2] Éric Thiébaut, Loïc Dénis, Ferréol Soulez, and Rahul Mourya.  Spatially variant PSF modeling and image deblurring.  In Adaptive Optics Systems V, volume 9909, pages 2211–2220. SPIE, 2016.
  * [3] Yuval Bahat, Netalee Efrat, and Michal Irani.  Non-uniform blind deblurring by reblurring.  In 2017 IEEE International Conference on Computer Vision (ICCV), pages 3306–3314, 2017.
  * [4] Rob Fergus, Barun Singh, Aaron Hertzmann, Sam T. Roweis, and William T. Freeman.  Removing camera shake from a single photograph.  In ACM SIGGRAPH 2006 Papers, SIGGRAPH ’06, page 787–794, New York, NY, USA, 2006. Association for Computing Machinery.
  * [5] Michael Hirsch, Christian J. Schuler, Stefan Harmeling, and Bernhard Schölkopf.  Fast removal of non-uniform camera shake.  In 2011 International Conference on Computer Vision, pages 463–470, 2011.
  * [6] Sunghyun Cho and Seungyong Lee.  Fast motion deblurring.  In ACM SIGGRAPH Asia 2009 Papers, SIGGRAPH Asia ’09, New York, NY, USA, 2009. Association for Computing Machinery.
  * [7] Luis G Varela, Laura E Boucheron, Steven Sandoval, David Voelz, and Abu Bucker Siddik.  Estimation of motion blur kernel parameters using regression convolutional neural networks.  arXiv preprint arXiv:2308.01381, 2023.
  * [8] Luis G Varela, Laura E Boucheron, David Voelz, Abu Bucker Siddik, and Steven Sandoval.  A patch-based regression convolutional neural network for motion blur estimation.  In SPIE vol. 12693 Unconventional Imaging, Sensing, and Adaptive Optics 2023, pages 415–424, 2023.
  * [9] Jian Sun, Wenfei Cao, Zongben Xu, and Jean Ponce.  Learning a convolutional neural network for non-uniform motion blur removal.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2015.
  * [10] Dong Gong, Jie Yang, Lingqiao Liu, Yanning Zhang, Ian Reid, Chunhua Shen, Anton Van Den Hengel, and Qinfeng Shi.  From motion blur to motion flow: A deep learning solution for removing heterogeneous motion blur.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2319–2328, 2017.
  * [11] Taeg Sang Cho, Sylvain Paris, Berthold K. P. Horn, and William T. Freeman.  Blur kernel estimation using the radon transform.  In CVPR 2011, pages 241–248, 2011.
  * [12] James H. Money and Sung Ha Kang.  Total variation minimizing blind deconvolution with shock filter reference.  Image and Vision Computing, 26(2):302–314, 2008.
  * [13] Jiaya Jia.  Single image motion deblurring using transparency.  In 2007 IEEE Conference on Computer Vision and Pattern Recognition, pages 1–8, 2007.
  * [14] Li Xu, Shicheng Zheng, and Jiaya Jia.  Unnatural L0 sparse representation for natural image deblurring.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2013.
  * [15] Dilip Krishnan, Terence Tay, and Rob Fergus.  Blind deconvolution using a normalized sparsity measure.  In CVPR 2011, pages 233–240, 2011.
  * [16] Anat Levin, Yair Weiss, Fredo Durand, and William T. Freeman.  Efficient marginal likelihood optimization in blind deconvolution.  In CVPR 2011, pages 2657–2664, 2011.
  * [17] Anat Levin, Yair Weiss, Fredo Durand, and William T. Freeman.  Understanding and evaluating blind deconvolution algorithms.  In 2009 IEEE Conference on Computer Vision and Pattern Recognition, pages 1964–1971, 2009.
  * [18] Ruomei Yan and Ling Shao.  Blind image blur estimation via deep learning.  IEEE Transactions on Image Processing, 25(4):1910–1921, 2016.
  * [19] Xiangyu Xu, Jinshan Pan, Yu-Jin Zhang, and Ming-Hsuan Yang.  Motion blur kernel estimation via deep learning.  IEEE Transactions on Image Processing, 27(1):194–205, 2018.
  * [20] Yungang Zhang, Yu Xiang, and Lei Bai.  Generative adversarial network for deblurring of remote sensing image.  In 2018 26th International Conference on Geoinformatics, pages 1–4, 2018.
  * [21] Lerenhan Li, Jinshan Pan, Wei-Sheng Lai, Changxin Gao, Nong Sang, and Ming-Hsuan Yang.  Learning a discriminative prior for blind image deblurring.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2018.
  * [22] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick.  Microsoft coco: Common objects in context.  In Computer Vision–ECCV 2014: 13th European Conference, Zurich, Switzerland, September 6-12, 2014, Proceedings, Part V 13, pages 740–755. Springer, 2014.
  * [23] Karen Simonyan and Andrew Zisserman.  Very deep convolutional networks for large-scale image recognition.  arXiv preprint arXiv:1409.1556, 2014.
  * [24] L. G. Varela and L. E. Boucheron.  Regression_Blur.  Github: https://github.com/DuckDuckPig/Regression_Blur.
  * [25] Davide Chicco, Matthijs J. Warrens, and Giuseppe Jurman.  The coefficient of determination R-squared is more informative than SMAPE, MAE, MAPE, MSE and RMSE in regression analysis evaluation.  PeerJ Computer Science, 7, 2021.
  * [26] L. G. Varela and L. E. Boucheron.  Regression_Patch _Blur.  Github: https://github.com/DuckDuckPig/RegressionPatchBlur.


# Propagator zeros and lattice chiral gauge theories

Maarten Golterman  Department of Physics and Astronomy, San Francisco State
University, San Francisco, CA 94132, USA  Yigal Shamir  Raymond and Beverly
Sackler School of Physics and Astronomy, Tel Aviv University, 69978, Tel Aviv,
Israel

###### Abstract

Symmetric mass generation (SMG) has been advocated as a mechanism to render
mirror fermions massive without symmetry breaking, ultimately aiming for the
construction of lattice chiral gauge theories. It has been argued that in an
SMG phase, the poles in the mirror fermion propagators are replaced by zeros.
Using an effective lagrangian approach, we investigate the role of propagator
zeros when the gauge field is turned on, finding that they act as coupled
ghost states. In four dimensions, a propagator zero makes an opposite-sign
contribution to the one-loop beta function as compared to a normal fermion. In
two dimensional abelian theories, a propagator zero makes a negative
contribution to the photon mass squared. In addition, propagator zeros
generate the same anomaly as propagator poles. Thus, gauge invariance will
always be maintained in an SMG phase, in fact, even if the target chiral gauge
theory is anomalous, but unitarity of the gauge theory is lost.

Introduction.  Chiral gauge theories play an important role in particle
physics. The Standard Model is a chiral gauge theory, and many theoretical
models for physics beyond the Standard Model are based on chiral gauge
theories as well. At present, no nonperturbative gauge invariant definition of
(anomaly free) nonabelian chiral gauge theories exists. Unlike the case of
vector-like asymptotically free gauge theories such as QCD, for which the
lattice provides a nonperturbative definition, the physics of chiral gauge
theories is understood mostly in a perturbative framework, supplemented by
incomplete and rather qualitative arguments based on large-$N$ expansions,
anomaly matching, duality relations, and other phenomenological approaches.

The nonperturbative construction of chiral gauge theories on the lattice is a
long-standing challenge, because of the fermion species-doubling problem. The
deep reasons underlying species doubling were first discussed by Karsten and
Smit [2], tying the phenomenon to the chiral anomaly, and then generalized by
Nielsen and Ninomiya [3]. In its simplest form, the doubling problem arises
when one considers a local free lattice hamiltonian in one spatial dimension.
The dispersion relation near $p=0$ will be $E=+p$ for one chirality. Because
of the periodicity of the Brillouin zone, unless there is a non-analyticity in
the dispersion relation, there must exist another point $p_{c}$ such that, for
small $\delta p=p-p_{c}$, the dispersion relation is $E=-\delta p$, which
signifies a fermion of the opposite chirality, the doubler.

During the 80s and 90s there was much activity in this field, which lead to a
better understanding of the fundamental obstacles, as well as to some
successes. For reviews, see Refs. [4, 5]. Building on the Ginsparg–Wilson
relation [6] as well as on earlier work by Kaplan [7] and by Narayanan and
Neuberger [8, 9, 10, 11], Lüscher achieved the construction of (anomaly-free)
abelian chiral gauge theories [12]. He also constructed nonabelian chiral
gauge theories to all orders in lattice perturbation theory [13, 14]. An
alternative approach, where the chiral gauge invariance is explicitly broken
on the lattice, and is only restored in the continuum limit, is the gauge-
fixing approach. The inclusion of a suitable gauge-fixing lattice action
ensures the existence of a novel critical point, where the target chiral gauge
theory emerges in the continuum limit [15, 16, 17, 18, 19, 20]. For another
proposal, see Ref. [21]. While these proposals are based on nontrivial
insights into the nature of the problem, it is still an open question whether
any of them will lead to a complete nonperturbative definition of
asymptotically free chiral gauge theories on the lattice.

The last decade saw renewed interest in the mirror fermion approach. One
starts from a vector-like fermion spectrum containing both LH (left-handed)
and RH (right-handed) fields. For each irreducible representation of the gauge
group, the fermions of one chirality are included in the target chiral gauge
theory, while the fermion fields of the opposite chirality are unwanted
“mirror” fermions. Originally, it was proposed to take the continuum limit in
the broken (Higgs) phase [22, 23]. However, this approach does not allow for
full decoupling on the mirrors. The reason is that full decoupling requires
the Higgs vacuum expectation value $v$ to diverge in physical units when
taking the continuum limit. But since the mass of the gauge bosons in the
Higgs phase is $\sim gv$, this would imply that the gauge bosons decouple as
well. Keeping $v$ physical instead would imply that mirrors are also physical.
An interesting question is whether such an approach is phenomenologically
viable, but the underlying gauge theory remains vector-like, and not chiral.

Since then, the focus has shifted to attempts to decouple the mirror fermions
using some strong non-gauge interaction while at the same time keeping the
chiral gauge symmetry unbroken. In principle, this would allow for a complete
decoupling, with the mirrors obtaining a mass of order the cutoff of the
theory. For a recent review, see Ref. [24]. In an asymptotically free theory,
the gauge interaction itself is controlled by the gaussian fixed point, and
turning it on or off is not expected to change the elementary fermion
spectrum. The elementary fermion spectrum is thus controlled by the reduced
model, obtained by turning off the gauge interaction. The reduced model
contains the fermion fields and, possibly, additional scalar fields. The
original (target) chiral gauge symmetry $G$ turns into a global symmetry of
the reduced model.

The dynamical question is what the phase diagram of the reduced model looks
like [4, 5, 24]. There is always a free-fermion limit containing both LH and
RH massless fermions in the same (in general, reducible) representation of the
symmetry group $G$ to be gauged, which constitutes a vector-like spectrum. By
turning on specially designed multi-fermion or Yukawa interactions, one hopes
to achieve symmetric mass generation (SMG). In this paper, we use the term SMG
for a strong-coupling phase in the reduced model where (a) the mirror fermions
develop a mass gap of the order of the lattice cutoff, (b) the target chiral
fermions remain massless, and (c) the symmetry $G$ is unbroken. The low-energy
limit then consists of a chiral fermion spectrum in the desired representation
of the (unbroken) symmetry $G$. If the SMG paradigm is successful, one would
hope to recover the target chiral gauge theory when the gauge field is turned
back on.

Starting from the Smit–Swift [25] and Eichten–Preskill [26] models, many
unsuccessful attempts were made over the years to find an SMG phase, and the
dynamical reasons underlying this failure were investigated [27, 28, 29]. The
last decade or so has seen renewed interest in the 3450 model. This is an
anomaly free, two-dimensional abelian chiral gauge theory containing fermions
with charges 3 and 4 of one handedness, and fermions with charges 5 and 0 of
the opposite handedness.111 The zero charge fermion does not couple to the
gauge field, but it participates in the multi-fermion interactions that gap
the mirrors in the reduced model.  The focus on two-dimensional models is
motivated by the relative simplicity of gauge theories in two dimensions, with
the hope that lessons learned generalize to the more interesting case of four-
dimensional chiral gauge theories.

An attempt to gap the mirrors in the reduced version of the 3450 model was
made in Refs. [30, 31]. The vacuum polarization was calculated numerically,
but the result indicated that the mirrors did not decouple. A possible reason
for this failure was discussed in Ref. [32]. Recently, building on
developments in condensed matter physics [33], an SMG phase was reported in a
different reduced version of the 3450 model, with the SMG phase induced by
especially designed multi-fermion interactions [34, 35, 36].

The availability of a concrete construction allows us to reexamine the
question of whether the desired chiral gauge theory will indeed be recovered
when the gauge interaction is turned back on. The question is nontrivial for
the following reason. As already noted, the desired low-energy limit of the
reduced model is a theory of free, undoubled, massless chiral fermions. This
is essentially the domain of the no-go theorems. Indeed, it was argued in Ref.
[37] that, under some very general assumptions, the Nielsen–Ninomiya theorem
will be applicable to an effective low-energy hamiltonian of an unspecified
underlying theory, thus excluding a chiral spectrum in any reduced model
unless some of these assumptions are not satisfied.222As explained in Ref.
[18], the gauge-fixing approach of Refs. [15, 16, 20] evades the generalized
no-go theorem [37]. See also Ref. [17].

A caveat, already noted in Ref. [37], is that the fermion propagator may
contain a zero in the mirror channel. Equivalently, the effective hamiltonian
associated with the inverse propagator has a mirror pole, and thus is
nonlocal. While to our knowledge the issue was not directly investigated in
the SMG phase of the 3450 model, there are strong general arguments that an
SMG phase will always be accompanied by the appearance of a zero in the
propagator [33, 38, 39, 40, 41, 42, 43]. This zero takes the place of the
original massless mirror pole, and its essential features are captured by the
phenomenological expression

$P_{R,L}\,\frac{i\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p}{p^{2}+m^{2}}\ ,$
(1)

for the mirror fermion propagator valid near $p=0$, where $P_{R,L}={1\over
2}(1\pm\gamma_{5})$ are the chiral projectors. The large mass scale $m$
characterizes the mass gap in the mirror sector after SMG has taken place. In
the generic case, it is presumably on the order of the lattice cutoff.

The goal of this letter is to examine the consequences of a propagator zero of
a massive charged fermion in the reduced model when the gauge field is turned
back on. We will first discuss the contribution of a propagator zero to the
vacuum polarization, and then its impact on anomalies, in both two and four
dimensions.

Propagator zero and the vacuum polarization.  Before discussing the physics of
a propagator zero, let us briefly recall the calculation of the vacuum
polarization for normal RH or LH fields. Starting from the gauge invariant
lagrangian (in euclidean space)

${\cal L}=\overline{\psi}({\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}+ig\hbox to0.0pt{\hbox{$\mskip
3.0mu/$}\hss}A)P_{R,L}\psi\ ,$ (2)

the vacuum polarization diagram is

$\displaystyle\Pi^{R,L}$ $\displaystyle=$ $\displaystyle{1\over
2}\left\langle\left(-ig\,\overline{\psi}\hbox to0.0pt{\hbox{$\mskip
3.0mu/$}\hss}AP_{R,L}\psi\right)^{2}\right\rangle=\frac{g^{2}}{2}\,{\rm
tr}\,\\!\left(\frac{1}{\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\partial}\hbox
to0.0pt{\hbox{$\mskip 3.0mu/$}\hss}A\frac{1}{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}\hbox to0.0pt{\hbox{$\mskip
3.0mu/$}\hss}AP_{R,L}\right)$ (3) $\displaystyle=$
$\displaystyle\Pi^{e}\pm\Pi^{o}\ ,$

where $\Pi^{e}$ and $\Pi^{o}$ are the parity-even and parity-odd parts. In
$d=4$ the parity odd part vanishes, whereas in $d=2$ it gives rise to the
anomaly, which we discuss later.

Moving on, we consider a RH fermion field $\psi_{R}=P_{R}\Psi_{R}$,
$\overline{\psi}_{R}=\overline{\psi}_{R}P_{L}$, for which the propagator pole
has been replaced by a propagator zero. The momentum space RH propagator takes
the form (compare Eq. (1))

$\int
d^{d}x\,e^{-ipx}\left\langle\psi_{R}(x)\overline{\psi}_{R}(0)\right\rangle=P_{R}\,\frac{i\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p}{m^{2}}\Big{(}1+{\cal
O}(p^{2}/m^{2})\Big{)}\ .$ (4)

The (even) dimension of spacetime is $d$. For simplicity, we assumed that the
propagator zero occurs at $p=0$. Our results are unchanged if the zero occurs
at a different location in the Brillouin zone. The corrections to the leading
behavior are suppressed by powers of $p^{2}/m^{2}$ near the zero, and do not
affect our results either.

Keeping only the leading behavior, in operator language the RH propagator is

$\left\langle\psi_{R}\,\overline{\psi}_{R}\right\rangle=P_{R}\,\frac{\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\partial}{m^{2}}\ ,$ (5)

and the corresponding free effective lagrangian has a RH pole,

${\cal L}_{0}=m^{2}\,\overline{\psi}_{R}\,\frac{1}{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}\,\psi_{R}\ .$ (6)

This effective lagrangian is nonlocal. As we will see, this has consequences
for the gauged theory at low energy, even though the mirror fermion has a mass
gap.

We introduce the gauge field via minimal coupling, as usual. The effective
lagrangian becomes

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle
m^{2}\,\overline{\psi}_{R}\,\frac{1}{{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}+ig\hbox to0.0pt{\hbox{$\mskip
3.0mu/$}\hss}A}\,\psi_{R}$ $\displaystyle=$ $\displaystyle
m^{2}\,\overline{\psi}_{R}\left(\frac{1}{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}-ig\frac{1}{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}\hbox to0.0pt{\hbox{$\mskip 3.0mu/$}\hss}A\frac{1}{\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\partial}-g^{2}\frac{1}{\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\partial}\hbox to0.0pt{\hbox{$\mskip
3.0mu/$}\hss}A\frac{1}{\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\partial}\hbox
to0.0pt{\hbox{$\mskip 3.0mu/$}\hss}A\frac{1}{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}+\cdots\right)\psi_{R}$ $\displaystyle\equiv$
$\displaystyle{\cal L}_{0}+{\cal L}_{\rm int}\ .$

The second line effectively defines the inverse of ${{\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\partial}+ig\hbox to0.0pt{\hbox{$\mskip
3.0mu/$}\hss}A}$ in terms of its expansion in the coupling constant $g$, or
equivalently, in the external gauge field.333 We treat the gauge field
perturbatively, and thus the operator ${\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}+ig\hbox to0.0pt{\hbox{$\mskip 3.0mu/$}\hss}A$ cannot
have zero modes. The vacuum polarization now consists of a bubble and a
tadpole diagram. We start with the bubble diagram. Expanding $\exp(-\int
d^{d}x{\cal L}_{\rm int})$ to second order and using Eq. (5) gives

${1\over 2}\left\langle\left(igm^{2}\,\overline{\psi}_{R}\,\frac{1}{\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\partial}\hbox to0.0pt{\hbox{$\mskip
3.0mu/$}\hss}A\frac{1}{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}\,\psi_{R}\right)^{2}\right\rangle=\frac{g^{2}}{2}\,{\rm
tr}\,\\!\left(P_{R}\hbox to0.0pt{\hbox{$\mskip 3.0mu/$}\hss}A\frac{1}{\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\partial}\hbox to0.0pt{\hbox{$\mskip
3.0mu/$}\hss}A\frac{1}{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}\right)=\Pi^{L}\ .$ (8)

The tadpole contribution arises from expanding $\exp(-\int d^{d}x{\cal L}_{\rm
int})$ to first order,

$g^{2}m^{2}\left\langle\overline{\psi}_{R}\,\frac{1}{\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\partial}\hbox to0.0pt{\hbox{$\mskip
3.0mu/$}\hss}A\frac{1}{\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\partial}\hbox
to0.0pt{\hbox{$\mskip 3.0mu/$}\hss}A\frac{1}{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}\,\psi_{R}\right\rangle=-g^{2}\,{\rm
tr}\,\\!\left(P_{R}\hbox to0.0pt{\hbox{$\mskip 3.0mu/$}\hss}A\frac{1}{\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\partial}\hbox to0.0pt{\hbox{$\mskip
3.0mu/$}\hss}A\frac{1}{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}\right)=-2\Pi^{L}\ .$ (9)

The vacuum polarization of a RH propagator zero is thus

$\Pi^{L}-2\Pi^{L}=-\Pi^{L}=-\Pi^{e}+\Pi^{o}\ .$ (10)

Comparing with Eq. (3) we see that the parity-even part flips its sign (the
parity-odd part remains the same). A similar result is obtained for a LH
propagator zero.

Equation (10) is our main result. To start, it demonstrates that a propagator
zero acts as a coupled state at low energy in the gauge theory through the
poles it generates in the vertices. Let us concentrate on the parity-even part
$\Pi^{e}$, postponing the parity-odd part to our discussion of anomalies
below. Differentiating it twice with respect to $A_{\mu}$ gives the familiar
result

$\Pi^{e}_{\mu\nu}(k)=-\frac{g^{2}}{2}\int\frac{d^{d}p}{(2\pi)^{d}}\,\frac{{\rm
tr}\,({\hbox to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p}\gamma_{\mu}({\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}p}+{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}k})\gamma_{\nu})}{p^{2}(p+k)^{2}}\ .$ (11)

In the abelian case, this result is valid for a unit-charge field. In the
nonabelian case, the result is multiplied by ${\rm
tr}\,(T_{a}T_{b})=T\delta_{ab}$, where $T$ is the group trace in the fermion
representation.

The integral (11) is UV divergent and may be computed for example using
dimensional regularization. In four dimensions, after renormalization, the
result is

$\Pi^{e}_{\mu\nu}=\frac{g^{2}}{24\pi^{2}}(k^{2}\delta_{\mu\nu}-k_{\mu}k_{\nu})\log\left(\frac{k^{2}}{4\pi\mu^{2}}\right)+\cdots\
,$ (12)

where $\mu$ is the renormalization scale. The dots stand for a finite term,
which is subleading compared to the logarithm. The logarithm in Eq. (12)
determines the fermion contribution to the one-loop beta function. Because
$\Pi^{e}$ flips sign between Eqs. (3) and (10), it follows that the
contribution of a propagator zero to the one-loop beta function has the same
absolute value as a normal fermion, but an opposite sign. Similarly, the
contribution of a propagator zero to the imaginary part of the vacuum
polarization will be negative, in violation of the optical theorem. The
propagator zero thus acts as a ghost state, which ruins the unitarity of the
gauge theory in the SMG phase.

A similar result was found in the 80s in Refs. [44, 45], where a specific
nonlocal lattice action [46] was studied in which the fermion doublers were
replaced by poles in the action, or equivalently, zeros in the fermion
propagator. We have employed an effective lagrangian approach to generalize
this result, finding that the effect occurs in complete generality.

Let us illustrate the consequences of these findings for the mirror fermion
approach. For definiteness, we start with a four-dimensional lattice gauge
theory of massless domain-wall fermions, in which the LH spectrum on one wall
constitutes the fermion spectrum of the target chiral gauge theory. The RH
spectrum on the other wall is the mirror spectrum. Before we turn on any
interactions to gap the mirrors, the theory is vector-like. Let the
contribution of these Dirac fermions to the coefficient of the one-loop beta
function be $b_{f}$. This contribution is split evenly between the LH and RH
fields, with each contributing $b_{f}/2$. The one-loop contribution of the
target chiral fermions is thus $b_{f}/2$.

Now let us assume that the RH fermions have been gapped in an SMG phase, with
each RH propagator pole replaced by a propagator zero. What we find is that
the total fermion contribution to the one-loop beta function will now vanish,
because the contribution of the LH target chiral fermions, which is $b_{f}/2$,
is cancelled by that of the RH propagator zeros, which is $-b_{f}/2$.

In two dimensions, restricting ourselves to the abelian case for simplicity,
the vacuum polarization gives rise to a photon mass squared $m_{\rm ph}^{2}$
which, in turn, is proportional to the sum of the U(1) charges squared [47]
(for a textbook discussion, see Ref. [48]). The target chiral gauge theory,
which consists of half of the massless fermion degrees of freedom of the
initial domain-wall fermion lattice theory (and contains chiral fields of both
chiralities), is expected to generate a mass squared $m_{\rm ph}^{2}/2$.
Again, we find that the contribution of the propagator zeros has the same
absolute value and an opposite sign, and is thus given by $-m_{\rm ph}^{2}/2$.
The result is a vanishing photon mass in the SMG phase. These effects, in both
four and two dimensions, reflect the role of the propagator zeros as coupled
ghost states.

Propagator zero and anomalies.  We first briefly recall the calculation of the
axial anomaly, starting with the two dimensional case. One way to perform the
calculation is to introduce both a vector gauge field $V_{\mu}$ and an axial
gauge field $A_{\mu}$, and consider the lagrangian

${\cal L}=\overline{\psi}\Big{(}{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}+\frac{ig}{2}({\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}V}+\hbox to0.0pt{\hbox{$\mskip
3.0mu/$}\hss}A\gamma_{5})\Big{)}\psi\ .$ (13)

If we set $V=A$, this reduces to

${\cal L}=\overline{\psi}({\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}+ig\hbox to0.0pt{\hbox{$\mskip 3.0mu/$}\hss}AP_{R})\psi\
.$ (14)

This lagrangian describes a RH fermion coupled to the gauge field together
with a free LH fermion. In dimensional regularization Eq. (13) is thus an
adequate starting point for the calculation of the anomaly of a RH fermion.
When calculating the anomaly using dimensional regularization, one must make
sure to use a consistent definition of $\gamma_{5}$, or of its two-dimensional
counterpart (see, for example, Refs. [48, 49]). Below, we will be careful to
only perform manipulations which are valid within dimensional regularization.
This means, for example, that when we calculate a diagram we are allowed to
anticommute $\gamma_{5}$ through $\hbox to0.0pt{\hbox{$\mskip 3.0mu/$}\hss}A$,
because the external gauge field (and momenta) are two- or four-dimensional by
assumption. However, we are not allowed to anticommute $\gamma_{5}$ through
the fermion propagator, because of its evanescent part.

The two dimensional axial anomaly arises from the bubble diagram with one $V$
vertex and one $A$ vertex. Starting from the lagrangian (13), this diagram is
given by

$\Pi^{AV}=\frac{g^{2}}{4}\,{\rm tr}\,\\!\left(\frac{1}{\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\partial}{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}V}\frac{1}{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}\hbox to0.0pt{\hbox{$\mskip
3.0mu/$}\hss}A\gamma_{5}\right)\ .$ (15)

In order to calculate the divergence of the axial current, one replaces $\hbox
to0.0pt{\hbox{$\mskip 3.0mu/$}\hss}A$ by ${\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}q}$, where $q$ is the external momentum. The well known result is
$\propto\epsilon_{\mu\nu}\partial_{\mu}V_{\nu}$. Alternatively, using the two-
dimensional relation
$\gamma_{\mu}\gamma_{5}\propto\epsilon_{\mu\nu}\gamma_{\nu}$ one can express
the parity-odd part of the vacuum polarization in terms of the parity-even
part, and then obtain the anomaly. For details see, e.g., Ref. [48].

Proceeding to the case of a propagator zero, our starting point is

${\cal L}=m^{2}\,\overline{\psi}\,\frac{1}{{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}+\frac{ig}{2}({\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}V}+\gamma_{5}\hbox to0.0pt{\hbox{$\mskip 3.0mu/$}\hss}A)}\,\psi\
.$ (16)

Note that this time $\gamma_{5}$ is to the left of $\hbox
to0.0pt{\hbox{$\mskip 3.0mu/$}\hss}A$. If we set $V=A$, we now obtain

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle
m^{2}\,\overline{\psi}\,\frac{1}{{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}+igP_{R}\hbox to0.0pt{\hbox{$\mskip
3.0mu/$}\hss}A}\,\psi$ $\displaystyle=$ $\displaystyle
m^{2}\left(\overline{\psi}_{R}\,\frac{1}{{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}+ig\hbox to0.0pt{\hbox{$\mskip
3.0mu/$}\hss}A}\,\psi_{R}+\overline{\psi}_{L}\,\frac{1}{\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\partial}\,\psi_{L}\right)\ .$

The second equality holds in two and four dimensions, where $\gamma_{5}$
anticommutes with the Dirac operator, showing the adequacy of this starting
point (compare Eq. (Propagator zeros and lattice chiral gauge theories)).
Again, for our perturbative calculations we define the inverses of all
differential operators that depend on external gauge fields through their
expansions in powers of $g$.

Like Eq. (13), also Eq. (16) allows for the use of dimensional regularization.
As in the case of the vacuum polarization, the two-dimensional anomaly of a
propagator zero involves tadpole and bubble diagrams. The details are similar,
and the final result is that the contribution of a propagator zero is
$\Pi^{AV}$, with a contribution $-\Pi^{AV}$ from the bubble diagram,444 The
minus sign comes from the location of $\gamma_{5}$ to the left of $\hbox
to0.0pt{\hbox{$\mskip 3.0mu/$}\hss}A$ in Eq. (16), instead of to the right in
Eq. (13).  and a contribution $+2\Pi^{AV}$ from the tadpole diagram. In other
words, a propagator pole and a propagator zero give rise to the same anomaly.
This is consistent with our previous result, that the parity-odd part of the
vacuum polarization is the same for a propagator pole and for a propagator
zero.

The four-dimensional case is similar. Starting from the lagrangian (13), the
usual triangle diagram is given by

$\Pi^{AVV}=-\frac{ig^{3}}{16}\,{\rm tr}\,\\!\left(\frac{1}{\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\partial}{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}V}\frac{1}{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}V}\frac{1}{\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}\hbox to0.0pt{\hbox{$\mskip
3.0mu/$}\hss}A\gamma_{5}\right)\ .$ (18)

For the contribution of a propagator zero, starting from the nonlocal
lagrangian (16) we have tadpole, bubble, and triangle diagrams. The triangle
diagram for a propagator zero turns out to be the same as for a propagator
pole,555 In this case the minus sign from the different placement of
$\gamma_{5}$ is canceled by the overall relative sign of the three vertices
contributing to the triangle diagram.  while the bubble and tadpole diagrams
cancel each other. The final result is again that a propagator pole and a
propagator zero give rise to the same anomaly.

A formal but quick way to keep track of the relative signs is to note that the
fermion partition function defined by the lagrangian (13) is ${\rm det}({\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\partial}+\frac{ig}{2}({\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}V}+\hbox to0.0pt{\hbox{$\mskip
3.0mu/$}\hss}A\gamma_{5}))$, whereas the partition function defined by the
nonlocal lagrangian (16) is ${\rm det}^{-1}({\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\partial}+\frac{ig}{2}({\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}V}+\gamma_{5}\hbox to0.0pt{\hbox{$\mskip 3.0mu/$}\hss}A))$. For
the purpose of perturbative calculations, the latter partition function may
alternatively be expressed as a path integral over a commuting complex field
$\phi$ carrying a spinor index, with lagrangian $\phi^{\dagger}({\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}\partial}+\frac{ig}{2}({\hbox
to0.0pt{\hbox{$\mskip 1.0mu/$}\hss}V}+\gamma_{5}\hbox to0.0pt{\hbox{$\mskip
3.0mu/$}\hss}A))\phi$. It is straightforward to see that the diagrammatic
expansion of these determinants reproduces the relative signs we found above.

Let us revisit the mirror fermion approach to the construction of lattice
chiral gauge theories in the light of our findings. As explained in the
introduction, this approach was based on two premises. The first premise is
that gapping the mirrors in an SMG phase will decouple them entirely from the
low-energy gauge theory. The second is that the target chiral fermion spectrum
must be anomaly free, in order to respect the gauge invariance of the lattice
theory after the mirrors have been gapped.

What we have found is that, if the mirror propagator poles are traded with
propagator zeros in the SMG phase, then they will remain coupled to the gauge
theory at low energy. Gauge invariance is in fact always maintained,
regardless of whether or not the fermion spectrum of the target chiral gauge
theory is anomaly free. The reason is that each propagator zero generates the
same anomaly as the corresponding propagator pole. Thus, the propagator zeros
will always cancel the anomaly of the target chiral fermions, as did the
original mirror propagator poles. The underlying reason for this result is
gauge invariance, which was exact before the SMG dynamics was turned on, and
remains so after. In two dimensions, an interesting relation was found between
SMG and anomaly cancelation expressed through the “boundary fully gapping
rules” of Ref. [34], but it turns out that this plays no role in arriving at
our conclusions about anomaly cancelation between the target and mirror
sectors of a model with symmetric mass generation.

At the same time, unitarity is always lost, because the propagator zeros
contribute to the one-loop beta function in four dimension, or to the photon
mass squared in two-dimensional abelian theories, as ghost states.

Acknowledgments.  This work was initiated at the Aspen Center for Physics,
which is supported by National Science Foundation grant PHY-2210452. YS wishes
to thank the organizers and the participants of the workshop “Emergent
Phenomena of Strongly-Interacting Conformal Field Theories and Beyond” for
stimulating discussions. In particular, YS thanks Cenke Xu and Yizhuang You
for discussions and correspondence on symmetric mass generation. YS also
thanks the Department of Physics and Astronomy at San Francisco State
University for hospitality. This material is based upon work supported by the
U.S. Department of Energy, Office of Science, Office of High Energy Physics,
Office of Basic Energy Sciences Energy Frontier Research Centers program under
Award Number DE-SC0013682 (MG). YS is supported by the Israel Science
Foundation under grant no. 1429/21.

## References

  * [1]
  * [2] L. H. Karsten and J. Smit, Lattice Fermions: Species Doubling, Chiral Invariance, and the Triangle Anomaly, Nucl. Phys. B 183 (1981), 103.
  * [3] H. B. Nielsen and M. Ninomiya, Absence of Neutrinos on a Lattice. 1. Proof by Homotopy Theory, Nucl. Phys. B 185 (1981), 20 [erratum: Nucl. Phys. B 195 (1982), 541]; Absence of Neutrinos on a Lattice. 2. Intuitive Topological Proof, Nucl. Phys. B 193 (1981), 173-194.
  * [4] Y. Shamir, Lattice chiral fermions, Nucl. Phys. B Proc. Suppl. 47 (1996), 212-227 [arXiv:hep-lat/9509023 [hep-lat]].
  * [5] M. Golterman, Lattice chiral gauge theories, Nucl. Phys. B Proc. Suppl. 94 (2001), 189-203 [arXiv:hep-lat/0011027 [hep-lat]].
  * [6] P. H. Ginsparg and K. G. Wilson, A Remnant of Chiral Symmetry on the Lattice, Phys. Rev. D 25 (1982), 2649.
  * [7] D. B. Kaplan, A Method for simulating chiral fermions on the lattice, Phys. Lett. B 288 (1992), 342-347 [arXiv:hep-lat/9206013 [hep-lat]].
  * [8] R. Narayanan and H. Neuberger, Chiral fermions on the lattice, Phys. Rev. Lett. 71 (1993) no.20, 3251 [arXiv:hep-lat/9308011 [hep-lat]].
  * [9] R. Narayanan and H. Neuberger, A Construction of lattice chiral gauge theories, Nucl. Phys. B 443 (1995), 305-385 [arXiv:hep-th/9411108 [hep-th]].
  * [10] H. Neuberger, Exactly massless quarks on the lattice, Phys. Lett. B 417 (1998), 141-144 [arXiv:hep-lat/9707022 [hep-lat]];
  * [11] H. Neuberger, More about exactly massless quarks on the lattice, Phys. Lett. B 427 (1998), 353-355 [arXiv:hep-lat/9801031 [hep-lat]].
  * [12] M. Lüscher, Abelian chiral gauge theories on the lattice with exact gauge invariance, Nucl. Phys. B 549 (1999), 295-334 [arXiv:hep-lat/9811032 [hep-lat]].
  * [13] M. Lüscher, Weyl fermions on the lattice and the nonAbelian gauge anomaly, Nucl. Phys. B 568 (2000), 162-179 [arXiv:hep-lat/9904009 [hep-lat]].
  * [14] M. Lüscher, Lattice regularization of chiral gauge theories to all orders of perturbation theory, JHEP 06, 028 (2000) [arXiv:hep-lat/0006014 [hep-lat]].
  * [15] Y. Shamir, The Standard model from a new phase transition on the lattice, Phys. Rev. D 57 (1998), 132-146 [arXiv:hep-lat/9512019 [hep-lat]].
  * [16] M. F. L. Golterman and Y. Shamir, A Gauge fixing action for lattice gauge theories, Phys. Lett. B 399 (1997), 148-155 [arXiv:hep-lat/9608116 [hep-lat]].
  * [17] W. Bock, M. F. L. Golterman and Y. Shamir, Lattice chiral fermions through gauge fixing, Phys. Rev. Lett. 80 (1998), 3444-3447 [arXiv:hep-lat/9709154 [hep-lat]].
  * [18] W. Bock, M. Golterman and Y. Shamir, Gauge fixing approach to lattice chiral gauge theories: Part 2, Nucl. Phys. B Proc. Suppl. 63, 581-586 (1998) [arXiv:hep-lat/9709115 [hep-lat]].
  * [19] M. Golterman and Y. Shamir, Fermion number violation in regularizations that preserve fermion number symmetry, Phys. Rev. D 67 (2003), 014501 [arXiv:hep-th/0202162 [hep-th]].
  * [20] M. Golterman and Y. Shamir, SU(N) chiral gauge theories on the lattice, Phys. Rev. D 70 (2004), 094506 [arXiv:hep-lat/0404011 [hep-lat]].
  * [21] D. M. Grabowska and D. B. Kaplan, Nonperturbative Regulator for Chiral Gauge Theories?, Phys. Rev. Lett. 116 (2016) no.21, 211602 [arXiv:1511.03649 [hep-lat]].
  * [22] I. Montvay, A Chiral SU(2)-L x SU(2)-R Gauge Model on the Lattice, Phys. Lett. B 199 (1987), 89-94.
  * [23] I. Montvay, Higgs and Yukawa theories on the lattice, Nucl. Phys. B Proc. Suppl. 26 (1992), 57-77.
  * [24] E. Poppitz and Y. Shang, Chiral Lattice Gauge Theories Via Mirror-Fermion Decoupling: A Mission (im)Possible?, Int. J. Mod. Phys. A 25 (2010), 2761-2813 [arXiv:1003.5896 [hep-lat]].
  * [25] J. Smit, Fermions on a Lattice, Acta Phys. Polon. B 17 (1986), 531 ITFA-85-18. P. V. D. Swift, The Electroweak Theory on the Lattice, Phys. Lett. B 145 (1984), 256-260.
  * [26] E. Eichten and J. Preskill, Chiral Gauge Theories on the Lattice, Nucl. Phys. B 268 (1986), 179-208.
  * [27] M. F. L. Golterman, D. N. Petcher and J. Smit, Fermion interactions in models with strong Wilson-Yukawa couplings, Nucl. Phys. B 370, 51-68 (1992).
  * [28] W. Bock, A. K. De and J. Smit, Fermion masses at strong Wilson-Yukawa coupling in the symmetric phase, Nucl. Phys. B 388, 243-274 (1992).
  * [29] M. F. L. Golterman, D. N. Petcher and E. Rivas, Absence of chiral fermions in the Eichten-Preskill model, Nucl. Phys. B 395 (1993), 596-622 [arXiv:hep-lat/9206010 [hep-lat]].
  * [30] J. Giedt and E. Poppitz, Chiral Lattice Gauge Theories and The Strong Coupling Dynamics of a Yukawa-Higgs Model with Ginsparg-Wilson Fermions, JHEP 10, 076 (2007) [arXiv:hep-lat/0701004 [hep-lat]].
  * [31] C. Chen, J. Giedt and E. Poppitz, On the decoupling of mirror fermions, JHEP 04, 131 (2013) [arXiv:1211.6947 [hep-lat]].
  * [32] Y. Kikukawa, Why is the mission impossible? – Decoupling the mirror Ginsparg-Wilson fermions in the lattice models for two-dimensional abelian chiral gauge theories, PTEP 2019, no.7, 073B02 (2019) [arXiv:1710.11101 [hep-lat]].
  * [33] J. Wang and Y. Z. You, Symmetric Mass Generation, Symmetry 14, no.7, 1475 (2022) [arXiv:2204.14271 [cond-mat.str-el]].
  * [34] J. Wang and X. G. Wen, Nonperturbative regularization of (1+1)-dimensional anomaly-free chiral fermions and bosons: On the equivalence of anomaly matching conditions and boundary gapping rules, Phys. Rev. B 107, no.1, 014311 (2023) [arXiv:1307.7480 [hep-lat]].
  * [35] M. DeMarco and X. G. Wen, A Novel Non-Perturbative Lattice Regularization of an Anomaly-Free $1+1d$ Chiral $SU(2)$ Gauge Theory, [arXiv:1706.04648 [hep-lat]].
  * [36] M. Zeng, Z. Zhu, J. Wang and Y. Z. You, Symmetric Mass Generation in the 1+1 Dimensional Chiral Fermion 3-4-5-0 Model, Phys. Rev. Lett. 128, no.18, 185301 (2022) [arXiv:2202.12355 [cond-mat.str-el]].
  * [37] Y. Shamir, Constraints on the existence of chiral fermions in interacting lattice theories, Phys. Rev. Lett. 71, 2691-2694 (1993) [arXiv:hep-lat/9306023 [hep-lat]].
  * [38] Y. Z. You, Z. Wang, J. Oon and C. Xu, Topological number and fermion Green’s function for strongly interacting topological superconductors, Phys. Rev. B 90, no.6, 060502 (2014) [arXiv:1403.4938 [cond-mat.str-el]].
  * [39] K. Slagle, Y. Z. You and C. Xu, Exotic quantum phase transitions of strongly interacting topological insulators, Phys. Rev. B 91, no.11, 115121 (2015) [arXiv:1409.7401 [cond-mat.str-el]].
  * [40] Y. Xu and C. Xu, Green’s function Zero and Symmetric Mass Generation, [arXiv:2103.15865 [cond-mat.str-el]].
  * [41] S. Catterall and N. Butt, Topology and strong four fermion interactions in four dimensions, Phys. Rev. D 97, no.9, 094502 (2018) [arXiv:1708.06715 [hep-lat]].
  * [42] Y. Z. You, Y. C. He, C. Xu and A. Vishwanath, Symmetric Fermion Mass Generation as Deconfined Quantum Criticality, Phys. Rev. X 8, no.1, 011026 (2018) [arXiv:1705.09313 [cond-mat.str-el]].
  * [43] D. C. Lu, M. Zeng and Y. Z. You, Green’s Function Zeros in Fermi Surface Symmetric Mass Generation, [arXiv:2307.12223 [cond-mat.str-el]].
  * [44] M. Campostrini, G. Curci and A. Pelissetto, Can a Nonlocal Lattice Fermion Formulation Avoid the Doubling Problem?, Phys. Lett. B 193, 279-282 (1987).
  * [45] A. Pelissetto, Lattice nonlocal chiral fermions, Annals Phys. 182, 177 (1988).
  * [46] C. Rebbi, Chiral Invariant Regularization of Fermions on the Lattice, Phys. Lett. B 186, 200-204 (1987).
  * [47] J. S. Schwinger, Gauge Invariance and Mass. 2., Phys. Rev. 128, 2425-2429 (1962).
  * [48] M. E. Peskin and D. V. Schroeder, An Introduction to quantum field theory, Addison-Wesley, 1995.
  * [49] J. Collins, Renormalization, Cambridge University Press, 1984.