J

Lolla Liang Kusne Takeuchi<EMAIL_ADDRESS>
[a]Poolesville High School, Poolesville, MD 20837, USA [b]NIST Center for
Neutron Research, NIST, Gaithersburg, MD, 20899, USA [c]Department of
Materials Science and Engineering, University of Maryland, College Park, MD
20742, USA [d]Materials Measurement Laboratory, NIST, Gaithersburg, MD, 20899,
USA [e]Maryland Quantum Materials Center, College Park, MD, 20742 USA

# A Semi-Supervised Approach for Automatic Crystal Structure Classification

Satvik  Haotong  A. Gilad  Ichiro

###### Abstract

The structural solution problem can be a daunting and time consuming task.
Especially in the presence of impurity phases, current methods such as
indexing become more unstable. In this work, we apply the novel approach of
semi-supervised learning towards the problem of identifying the Bravais
lattice and the space group of inorganic crystals. Our semi-supervised
generative deep learning model can train on both labeled data- diffraction
patterns with the associated crystal structure, and unlabeled data,
diffraction patterns that lack this information. This approach allows our
models to take advantage of the troves of unlabeled data that current
supervised learning approaches cannot, which should result in models that can
more accurately generalize to real data. In this work, we classify powder
diffraction patterns into all 14 Bravais lattices and 144 space groups (we
limit the number due to sparse coverage in crystal structure databases), which
covers more crystal classes than other studies. Our models also drastically
outperform current deep learning approaches for both space group and Bravais
Lattice classification[billengePDF, garcia2019learning, ryu2019deep,
ziletti2018insightful, tiong2020identification] using less training data.

Semi-supervised model to predict crystal structure from powder neutron
diffraction patterns.

##  1 Introduction

The first step to understanding the properties of a crystalline material at a
microscopic level starts with identifying the crystal structure. However,
identifying the crystal structure is nontrivial. The first part of crystal
structure determination is indexing. There are several programs which can be
used, such as Dicvol06[dicvol], Topas[Topas], GSAS II[gsas] or N-TREOR[treor,
treor90]. These programs output a set of space groups and lattice parameters
that could represent the crystal. Using Le Bail[lebail] and Pawley[pawley]
refinements, the space group that fits the diffraction pattern the best can be
identified. Rietveld [rietveld1967line, rietveld1969profile] refinement can
then be applied to profile the lattice parameters and check the space group.
In the presence of impurity phases, this approach becomes more expensive as
peaks must be selected manually or tolerance levels must be tuned to discard a
certain number of peaks.

One of the approaches to identifying the position of atoms in a crystal is the
charge flipping algorithm (CFA) [CFA_first, Palatinus_summary,
baerlocher2007charge]. CFA is an iterative approach that relies on Fast
Fourier Transforms to determine the crystal structure of a material[fft,
superflip]. For CFA, the unit cell and Bravais Lattice have to be known–that
is, we must have already have succeeded with some degree of indexing. CFA also
cannot handle impurity phases, which are prevalent in many real-world samples.

To automate this task, recent research looks to supervised neural networks
(NN)s to analyze diffraction patterns. Supervised learning is an approach that
seeks to learn a functional mapping between data and their labels. The benefit
of NNs is that they, unlike CFA, do not require additional parameters such as
the Bravais Lattice or lattice parameters. Although some approaches use NNs to
aid in the Rietveld refinement[ozaki2020automated, chang2020deep, reviewpaper,
reviewpaper2], others use NNs to classify diffraction patterns based on the
crystal structure. These classifiers can be trained to identify impurity
phases and can be tailored towards specific detectors or parameters. For
example, Ryu et al.[ryu2019deep] trained a NN to classify diffraction patterns
of crystals that had defects. Liu et al.[billengePDF] used the pair density
function with powder neutron diffraction data for space group classification.
Ziletti et al.[ziletti2018insightful] used a convolutional neural network to
classify simulated single crystal diffraction x-ray image data into 8 space
groups.

A number of studies represent powder diffraction patterns as 2D images.
However, the information is inherently one dimensional. Previous groups likely
used the image approach to take easy advantage of trained models developed by
the machine learning community. Unfortunately, this one to many mapping could
introduce more complexities to the model. Garcia-Cardona et
al.[garcia2019learning] was one of the only studies to examine neutron
scattering data and used a 1-dimensional (1D) approach with simulated powder
diffraction data to both differentiate perovskites into 5 crystal systems and
tune the lattice parameters using regression. This study only looked at a
small subset of crystals

A significant challenge with NNs is that they struggle to generalize to new
datasets. Most models that predict the space group of a material use less than
100 space groups in their training dataset, which limits their application to
new diffraction patterns. However, large labeled diffraction datasets are
often rare as labeling them is an expensive task. For this reason, we use a
semi-supervised model, which takes advantage of both labeled and unlabeled
data during training[SGAN, zhu2009introduction, kingma2014semi, kipf2016semi].
We employ a generative network that can extract features from the unlabeled
data distribution and match these features with the corresponding crystal
structure. This allows semi-supervised learning to be used on more datasets,
especially ones where labels are not available.

In this study, we propose a 1D semi-supervised model for Bravais Lattice and
space group classification using powder neutron diffraction data. Our NNs are
trained with data spanning 144 space groups, making our network more
applicable than previous studies. The models used in this study are freely
available and can be downloaded.[github]

##  2 Methods

###  2.1 Data

To test our approach under conditions where we know the correct answer, we
work with simulated datasets. Our data was taken from the Inorganic Crystal
Structure Database (ICSD), which contains structural information about more
than 210,000 crystals [ICSD, disclaimer]. A total of 138,362 diffraction
patterns were simulated using Topas[Topas]. For the Bravais Lattices, we
combine the rhombohedral and tetragonal classes for a total of 14 classes with
”F”, ”I”, ”P” and ”C” representing the face centered, body centered, primitive
and base centered lattices respectively. We note that there is an inherent
class imbalance in the ICSD, as shown in Figure 1. The most prevalent classes
in this dataset were the primitive hexagonal, the face centered cubic, and the
primitive orthorhombic lattices. The least represented lattices were the face
centered orthorhombic and body centered orthorhombic lattices.

(a) (b)

Figure 1: Shows the (a) Bravais Lattice and (b) space group class distribution
of the simulated data.

For the space groups, we use 136,454 of the simulated diffraction patterns. We
used only the space groups that had more than 50 diffraction patterns, leaving
us with 144 out of the 225 space groups present in the ICSD.The most frequent
space group is 62 (Pnma) which is orthorhombic, and accounts for the
disproportionately large orthorhombic (P) diffraction patterns in the ICSD
dataset. A complete list of the 144 space groups used is shown in Table 1.

Crystal System | Space Groups
---|---
Triclinic | 1, 2
Monoclinic | 4-15
Orthorhombic | 18-20, 26, 29, 31, 33, 34,36, 38, 40-47, 51-53, 55-67, 69-74
Tetragonal |  | 82, 85-88, 92, 96, 99, 100, 107, 109, 113, 114, 119
---
121, 122, 123, 125-131, 135-137, 139-142
Trigonal | 143, 144, 146-148, 150, 152, 154-157, 159-167
Hexagonal | 173-176, 180, 182, 185-187, 189-194
Cubic | 197-201, 203-206, 212, 213, 215-218, 220, 221, 223-227, 229, 230
Table 1: List of space groups sorted by crystal system.

In this study, we use a one-dimensional approach rather than the traditional
two-dimensional image one. Our dataset consists of diffraction patterns of
powders. Examples of the 1D diffraction patterns are shown in Figure 2. To
normalize these diffraction patterns, we divide all intensities in each
diffraction pattern by the maximum intensity. This ensures that the new
maximum intensity is equal to 1 and the minimum is equal to 0.

(a) (b)

Figure 2: Example 1D diffraction patterns for (a) Hexagonal (P) crystals and
(b) Orthorhombic (P) crystals

###  2.2 Models

We use two approaches to classify the diffraction patterns: a supervised
approach using Convolutional Neural Networks (CNNs) and a semi-supervised
approach using a Semi-Supervised Generative Adversarial Network (SGAN).

###  2.3 Supervised Model

We use a 1D ResNet-18, a Residual Network[ResNet], model to identify the
crystal structure of the diffraction patterns. ResNets are examples of
Convolutional Neural Networks (CNNs), which are commonly used for image
classification network algorithms. CNNs consist of convolutional layers, which
are responsible for extracting high-level features, such as edges and colors,
from images. These layers are used to create a feature map consisting of the
most relevant characteristics of the image. To create this map using 1D data,
a filter of size $n$ is applied to a larger sequence with size $m$, and the
dot product of every $n$ consecutive values and the filter is computed. This
generates a smaller matrix that only includes the relevant features[cnn].

A ResNet was used in this study to overcome the degradation problem, which
occurs when neural networks are too dense that the accuracy saturates and then
quickly degrades[ResNet]. ResNets are characterized by their residual blocks,
which contain convolutional layers with an identity function. Figure 3 shows
the model architecture used for the ResNet-18, and includes an example of a
ResNet block used in this model. During training, we randomly selected 90% of
the data to use as the training set and the remaining 10% of the data was used
to test the model. This testing dataset was distinct from the training one, so
the model did not learn from the testing data.

(a) (b)

Figure 3: (a) shows ResNet block used in this study and (b) shows the
ResNet-18 architecture. In both figures, the orange, purple, green, red, gray,
blue, and yellow layers represent 1D Convolutional Layers, Dropout layers,
Layer Normalization layers, GeLU activation functions, 1D Max Pooling layers,
1D Adaptive Max Pooling Layers, and Fully Connected layers respectively. The
white circles with ”+” signs in them represent the addition of two layers. In
(b), each ResNet block was repeated twice, shown by the ”2x” next to each
block. The ”nc” and ”ks” represent the number of channels and the kernel
stride for each convolutional layer.

###  2.4 Semi-Supervised Model

We also use a Semi-Supervised Generative Adversarial Network (SGAN)[SGAN, GAN,
improvedgans]. The SGAN consists of two models: a Generator and a
Discriminator. The Generator tries to fool the Discriminator with fake
diffraction patterns while the Discriminator aims to differentiate between
real and fake diffraction patterns. The Discriminator also classifies the real
labeled data into the corresponding crystal structure class.

####  2.4.1 Generator

The purpose of the Generator is to sample the latent space, a high dimensional
feature space, to generate realistic diffraction patterns. The inputs to the
Generator were sampled from a random normal distribution with a mean of 0 and
standard deviation of 1. The Generator consists of 1D Convolutional Transpose
layers, 1D Batch Normalization Layers, and a LeakyReLU activation function.
The Convolutional Transpose layers are used to upsample the data[dcgan,
cnntranspose]. The Batch Normalization layers standardize the output of each
layer, which reduces error when the model tries to generalize to new
inputs[batchnorm] and has also shown to reduce mode collapse, a major problem
in GANs[dcgan]. Mode collapse occurs when the generator only produces a few
distinct diffraction patterns despite the latent space input. The Leaky
ReLU[leakyrelu] activation with $\alpha=0.2$ is used rather than ReLU to
reduce the vanishing gradients problem[dcgan]. Graphs of the ReLU and the
Leaky ReLU activation functions are shown in Figure 4. For negative values,
the derivative for the Leaky ReLU function is equal to $\alpha$, but for the
ReLU function, it is equal to 0. By having a non-zero derivative for all
values, the Leaky ReLU is used to combat the sparse gradient problem that
occurs while training GANs. Due to our normalization method, which was
dividing all values in a diffraction pattern by the maximum intensity, the
Discriminator’s inputs were in the range from 0 to 1. For this reason, a
Sigmoid activation was applied to the last layer of our Generator, rather than
the hyperbolic tangent function recommended by Salisman et al.[improvedgans]
Figure 5 shows the model architecture of the Generator.

(a) (b)

(c) (d)

Figure 4: Shows (a) ReLU, (b) Leaky ReLU with $\alpha=0.2$, (c) Derivative of
ReLU, (d) Derivative of Leaky ReLU with $\alpha=0.2$ Figure 5: Shows the
architecture for the Generator. The blue, greed, orange and red boxes
represent 1D Convolutional Transpose Layers, LeakyReLU activations, Batch
Normalization layers, and Sigmoid activations respectively. The ”nc” and ”ks”
in the blue boxes represent the number of channels and the kernel size
respectively for each 1D Convolutional Transpose Layer.

####  2.4.2 Discriminator

The Discriminator has two objectives: differentiate between real and generated
data, and classify the real data into the correct class. To do this, we use
the same 1D ResNet-18 model described in Section 2.3, but an activation
function to the last fully connected layer, as proposed by Salisman et
al[improvedgans]. This activation function is shown in Equation 1, and is a
version of the softmax activation. In this equation, $l_{k}(x)$ represents the
logit for class $k$ with data $x$. By doing this, we eliminate the need for a
second output layer, and instead use only the logits from the classification
layer. By applying this activation function, diffraction patterns with larger
logits, which signify more confident predictions, will be classified as ”real”
whereas diffraction patterns with smaller logits will be classified as ”fake.”
This encourages the Discriminator to be more confident in its predictions,
which sharpens the decision boundary between classes. The Discriminator’s
architecture is shown in Figure 6.

$\begin{split}D(x)&=\frac{Z(x)}{Z(x)+1}\\\
Z(x)&=\sum_{k=1}^{K}\exp(l_{k}(x))\\\ \end{split}$ (1)

Figure 6: Model architecture for ResNet-18 discriminator. The orange, purple,
green, red, gray, blue, and yellow layers represent 1D Convolutional Layers,
Dropout layers, Layer Normalization layers, GeLU activation functions, 1D Max
Pooling layers, 1D Adaptive Max Pooling Layers, and Fully Connected layers
respectively. The white circles with ”+” signs in them represent the addition
of two layers. In (b), each ResNet block was repeated twice. The ”nc” and ”ks”
represent the number of channels and the kernel stride for each convolutional
layer.

While training the Discriminator, there are two modes: supervised and
unsupervised. During unsupervised training, the Discriminator acts the same
way it would in a regular GAN as it tries to determine that the generated
diffraction patterns are fake and the data drawn from the unsupervised set is
real. In the supervised mode, the Discriminator is trained to predict the
class label for real samples. Training in the unsupervised mode can help the
Discriminator extract features from the data and training on the supervised
data will allow the Discriminator to use those extracted features for
classification.

####  2.4.3 Loss Functions and Objective Functions

The modified min-max loss proposed by Goodfellow et al.[GAN] was used for the
adversarial loss between the networks. The Generator’s objective function that
it tries to maximize is shown in Equation 2. $\theta_{g}$ represents the
parameters in the Generator, $z^{(i)}$ represents the random values in the
latent space. $G(z^{(i)})$ is the generated diffraction pattern from the
Generator and $-\log(D(G(z^{(i)})))$ is the probability that the Discriminator
predicts that the generated pattern is real.

$\nabla\theta_{g}\frac{1}{m}\sum_{i=1}^{m}-\log(D(G(z^{(i)})))$ (2)

For the discriminator, the objective function to be maximized is shown in
Equation 3. $\theta_{d}$ represents the parameters in the Discriminator,
$x^{(i)}$ is unsupervised real data, and maximizing $D(x^{(i)})$ implies that
the model can identify real data. Like the Generator’s objective function,
$z^{(i)}$ represents the random values in the latent space and $G(z^{(i)})$ is
the generated diffraction pattern from the latent space. Increasing the value
of $1-\log(D(G(z^{(i)})))$ shows that the Discriminator can determine that the
generated patterns are fake. Equation 3 also includes the categorical cross
entropy loss, which is shown in the term $\sum_{i}^{C}t_{i}\log(s_{i})$. Here,
$C$ represents the number of classes, $t_{i}$ shows whether the $i$th class is
the label of the diffraction pattern and $s_{i}$ is the discriminator’s
prediction.

$\nabla\theta_{d}\frac{1}{m}\sum_{i=1}^{m}[\log(D(x^{(i)}))+\log(1-D(G(z^{(i)})))]+\sum_{i}^{C}t_{i}\log(s_{i})$
(3)

####  2.4.4 Training Details

Figure 7 shows the training pipeline used in the SGAN. We train our SGAN using
four different amounts of labeled training data. In all scenarios, we randomly
select 10% of the data as testing data, which is distinct from the labeled
training data and the unlabeled training data. In the first scenario, we use
5% of the data as labeled training data and 85% as unlabeled training data. In
the second, we use 10% of the data as labeled training data and 80% as
unlabeled training data. In the third, we use 25% of the data as labeled
training data and 65% of the data as unlabeled training data and finally we
use 50% of the data as labeled training data with 40% of the data as unlabeled
training data. We also train our supervised ResNet with the same 5%, 10%, 25%
and 50% of the data to compare the accuracy of the SGAN to the purely
supervised approach.

Figure 7: Shows the architecture for the Semi-Supervised GAN.

Table 2 shows the hyperparameters used in the ResNet and the SGAN.

Hyperparameter | Supervised | Generator | Discriminator
---|---|---|---
Optimizer | Adam | Adam | Adam
Learning Rate | 1e-4, | 1e-6 | 5e-6
Dropout Rate | 0.1 | None | 0.1
Batch Size | 64 | 32 | 32
Nonlinear Activations | GELU | Leaky ReLU& Sigmoid | GELU
Table 2: Hyperparameters used in the supervised ResNet, Generator, and
Discriminator

We used Pytorch[torch] as a deep learning framework. To accelerate training,
each model was trained on 8 NVIDIA Tesla V100 Tensor Cores.

##  3 Results and Discussion

###  3.1 Supervised Model

Our supervised ResNet trained on 90% of the dataset had an accuracy of 88%.
The confusion matrix for the Bravais Lattice model is shown in Figure 8. By
plotting the predicted Bravais Lattice against the actual Bravais Lattice, the
confusion matrix provides more information about the sets of classes that the
network misclassified. If the model had a perfect testing accuracy, the values
along the principal diagonal would sum to 100% as the network would have
classified every diffraction pattern correctly. Again, there is some clear
imbalance in the sampled ICSD dataset, with orthorhombic (F) and orthorhombic
(I) having the least samples. From the confusion matrix, we can see that
despite the fact that orthorhombic (P) is the most prevalent class, the model
misclassifies some of these as monoclinic (P) crystals. The network also has
trouble differentiating between triclinic (P) and monoclinic (P) diffraction
patterns as both of these classes have low symmetries, agreeing with previous
studies[garcia2019learning].

Figure 8: Shows the confusion matrix for the supervised ResNet-18

For the space group identification, our model had a top 1 accuracy of 80.6%
and a top-5 accuracy of 90.27% across all 144 space groups. Accuracy was
measured by dividing the number of correctly classified diffraction patterns
in the testing set by the total number of patterns in the testing set. Top-5
accuracy is the percentage of samples for which the actual space group was one
of the model’s top 5 predictions. This outperforms most current models that we
are aware of: Liu et al. [billengePDF] used machine learning with a pair-wise
distribution function with a top-1 accuracy of 71% and a top-5 accuracy of 90%
across 45 space groups. Tiong et al.[tiong2020identification] classified x-ray
diffraction data into 8, 20, 49, and 72 space groups. Their accuracy decreased
from 99% to 80% for 8 and 72 space groups respectively, implying that this
accuracy would decrease further if their model was trained on more space
groups. Aguilar et al.[sg_1d] had a top-2 accuracy greater than 80% across all
space groups but used a dataset consisting of 650,000 diffraction patterns,
more than 5 times the size of the dataset used in this study. However, Aguilar
et al. used a 1D network, suggesting that a 1D approach can lead to more
accurate predictions. We note that we did not take advantage of data-
augmentation.

Study |  | Number of
---
Space Groups
Type of Data | Accuracy
Our Model | 144 | Powder Diffraction | 80%
Liu et al.[billengePDF] | 45 | PDF | 71%
Ziletti et al.[ziletti2018insightful] | 8 | 2D XRD | 96.3%
Tiong et al.[tiong2020identification] | 72 | 2D XRD | 80.2%
Table 3: Comparison to other supervised space group classifiers

###  3.2 Semi-Supervised Model

We compare the accuracy of the SGAN to the accuracy of the supervised model in
Table 4. The SGAN consistently outperforms the purely supervised model,
showing that the semi-supervised approach has the potential to be more
applicable in the real world. A graph comparing the accuracy of the supervised
and semi supervised models is shown in Figure 9. This graph shows that
although the accuracy of the SGAN is impacted by a lack of data, the
difference between the SGAN’s accuracy and the supervised model’s accuracy is
greatest when only 5% of the data is used.

Percentage of data |  | Supervised
---
Bravais Lattice
| SGAN
---
Bravais Lattice
| Supervised
---
Space Group
| SGAN
---
Space Group
5% | 61% | 70% | 54% | 60%
10% | 68% | 74% | 61% | 65%
25% | 76% | 80% | 68% | 72%
50% | 82% | 85% | 75% | 78%
Table 4: Comparing the accuracy of the SGAN to the purely supervised approach
with different amounts of labeled training data for both Bravais Lattice and
Space Group classification Figure 9: Graph comparing the accuracies of the
SGAN and supervised ResNet for Bravais Lattice and Space Group identification.

##  4 Conclusion

In this study, we use both CNNs and a semi-supervised GAN to investigate
supervised and semi-supervised approaches for crystal structure
classification. We demonstrate that semi-supervised GANs can prove to be more
accurate with limited amounts of labeled data for Bravais Lattice and space
group classification. Further, we explore a 1D approach rather than a
traditional 2D one. Our 1D model is more accurate than 2D image models, which
agrees with previous results in literature. Our semi-supervised model is also
more applicable to real datasets which will lack large amounts of labeled
data.

In the future, we would like to train the SGAN to identify impurity phases and
to test the method on real data sets. Acknowledgements

The authors are grateful to Austin McDannald and Hui Wu from NIST and Brian
Toby from Argonne National Laboratory. We acknowledge Stephan Rühl for the
ICSD. We acknowledge support from the Center for High Resolution Neutron
Scattering (CHRNS) a national user facility jointly funded by the NCNR and the
NSF under Agreement No. DMR-2010792. The work at the University of Maryland
was supported by NIST grant No. 60NANB19D027.


$f^{\top}=(1,2)$ is a semiflow: the Kirchhoff’s law is easily verified for the
nodes $t_{1}$ and $t_{2}$. Moreover, it is obvious that $f$ is a minimal
semiflow and its support is minimal as well.

It models a formal machine semi-deciding whether a given natural number $n$ is
even or null. Here, we just mean that it will always be possible to find a
sequence of transitions such that the Petri Net stops ($t_{1}$ and $t_{2}$ are
not enabled anymore, i.e. there exists a sink) given $q_{0}(A)=2k$ where
$k\in\mathbb{N}$, regardless of $q_{0}(B)$, the initial marking of $B$.
Actually, we can do slightly better and also prove that this Petri Net is live
if and only if $f^{\top}q_{0}=2i+1$ where $i\in\mathbb{N^{+}}$.

Figure 9: This tiny Petri Net is live if and only if $f^{\top}q_{0}$ is an odd
number greater than one whatever is the initial marking of $B$.

The scalar product $f^{\top}q$ does not vary, we have the following invariant:
for any marking $q$ reachable from an initial marking $q_{0}$,
$f^{\top}q=q(A)+2q(B)=f^{\top}q_{0}$. This can be rewritten as
$q(A)=f^{\top}q_{0}-2q(B)$ which means that $q(A)$ and $f^{\top}q_{0}$ have
always the same parity and this will never change 282828Let’s point out that
we just used the phrases always and never will as mentioned in Section 4.1: we
expressed an invariant about the parity of the marking of the place $A$. This
is not the usual formulation for an invariant generated by a semiflow. (if we
admit that zero is even). Therefore, if $f^{\top}q_{0}>2$ and is odd, then
$q(A)$ remains always odd. If $q(A)=1$ and $f^{\top}q_{0}>2$ then $q(B)>1$ and
$t_{2}$ is enable from $q$; it can be deduced that the Petri Net is live since
$q(A)$ remains odd; if $q(A)>1$ then $t_{1}$ is enable from $q$; again, the
liveness of the Petri Net can be easily deduced. Reversely, if the Petri Net
is live then $f^{\top}q_{0}>2$ since the enabling threshold of $t_{2}$ is
$f^{\top}Pre(\cdot,t_{2})=3$. Moreover, if $f^{\top}q_{0}$ is even, then
$q(A)$ is even and the Petri Net is not live ($t_{1}$ can be enabled until $A$
has no token which means that it is not possible to enable $t_{1}$ or $t_{2}$
anymore) $\square$

Since $f>0$, we have $A$ and $B$ bounded by $f^{\top}q_{0}$ for any initial
marking $q_{0}$. $\\{A,B\\}$ is therefore structurally bounded.

What is remarkable about the analysis of this tiny example is that it was not
necessary to develop the reachability graph in order to decide whether or not
the Petri Net is live or bounded. We could analyze the Petri Net even
partially ignoring the initial marking (i.e. considering $q_{0}(A)$ as a
parameter and without even considering the value taken by $q_{0}(B)$ ).

An interesting behavioral lesson to draw from this tiny example has to do with
liveness: once live, adding more tokens to some places of the Petri Net does
not guarantee that liveness will persist. As a matter of fact, while a minimum
of tokens are required (refer for instance to the notion of enabling threshold
in 5.3.2) for a PN to be live, in this example, adding one token to A will
make the Petri Net not live while adding a second token will make it live
again.

Even if adding more resources to the system under study can bring confidence
that some actions will eventually be performed, this does not necessarily
result in solving a deadlock issue; this could even create a deadlock
situation! Adding more resources is a false “good idea” that is encountered
with many students and engineers certainly taking its origin with the notion
of monotonicity seen section 3.9.

Figure 10: This second tiny Petri Net is live if and only if $g^{\top}q_{0}>k$
and is not a multiple of $k$ whatever is the initial marking of $B$. For
$k=1$, the Petri Net has no live transition whatever is the initial marking.
For $k=2$, we find again the Petri Net of Figure 9.

We can expand the tiny example of Figure 9 by adding a parameter $k$ such that
$k>1$ as in Figure 10 and call $TNK$ this Petri Net. This time, we have the
semiflow: $g^{\top}=(1,k)$ and we can show that the Petri Net is not live if
and only if $g^{\top}q_{0}\leq k$ or $g^{\top}q_{0}=nk$ where
$n\in\mathbb{N}$, independently of $q_{0}(B)$.

If $g^{\top}q_{0}<k$ then the enabling threshold of $t_{1}$ can never be
reached and neither $t_{1}$ nor $t_{2}$ can be executed. The Euclidean
division of $g^{\top}q_{0}$ by $k$ gives: $g^{\top}q_{0}=nk+i$ where $i<k$
then, $g^{\top}q\equiv i\ (mod\ k)$ therefore $q(A)\equiv i\ \forall q\in
RS(TNK,q_{0})$. If $i=0$ then we have $q(A)=nk-kq(B)$ and $t_{1}$ can be
enabled $n-q(B)$ times to reach a marking with zero token in $A$. If $i\neq 0$
and $g^{\top}q_{0}>k$ then $q(A)\neq 0$ and either $q(A)>k$ or $q(B)\neq 0\
\forall q\in RS(TNK,q_{0})$. In the first case, $t_{1}$ is enabled in the
second case $t_{2}$ is enabled. It is easy to conclude that the Petri Net
$TNK$ is live. $\square$

The loop between $A$ and $t_{2}$ prevents to draw this kind inof result with
the sole usage of the state equation (4).

Curiously, we conjecture that to design another Petri Net semi- deciding
whether a given number is odd is impossible.

###  7.2 Revisiting the mutual exclusion example

One could argue that the Petri Nets of Figures 5, 9, or 10 are relatively
simple and that it would be easy to obtain similar results without reasoning
with semiflows. This may be true (for instance, using the LRG of the Petri Net
of Figure 5 is particularly trivial), however, it must be stressed out that
semiflows allow to elegantly develop reasoning in the presence of parameters.

In this regard, let’s revisit and generalize our mutual exclusion example of
Section 4.2.3 by introducing few parameters. This time, we allow $k$ instances
of Program 1 and $l$ instances of Program 2 to run concurrently. The two edges
from S to ${Semp_{1}}$ and from ${Semv_{1}}$ to S are labeled by $x$; the two
edges from S to ${Semp_{2}}$ and from ${Semv_{2}}$ to S are labeled by $y$ (by
construction, we have $x>0$ and $y>0$ ). The initial state $q_{0}$ is being
defined with three non- negative parameters: $k$ tokens in $A$, $l$ tokens in
$D$, and $z$ tokens in S; and, $q_{0}(B)=q_{0}(E)=0$.

In this example, we authorize several instances of the same program to be
simultaneously in their critical section however, we still want to verify
mutual exclusion between instances of Program 1 and instances of Program 2
that is to say that we must have either $q(B)>0$ or $q(E)>0$ for any reachable
marking. From the four versions of invariant representing mutual exclusion in
section 4.2.3 only (ME) and (ME)’ still make sense in this parameterized
context. Therefore, we want to determine under which conditions (i.e. which
values of the parameters) the Petri Net is live and:

(ME): $q(B)\times q(E)=0$ for any marking reachable from $q_{0}$ is an
invariant.

The Petri Net Figure 11 has still 3 invariants directly deduced from 3
semiflows. The semiflows $f_{1}$ and $f_{2}$ are unchanged from Figure 5; the
third semiflow $sem_{2}$ is different and such that: $sem_{2}(B)=x$,
$sem_{2}(E)=y$, $sem_{2}($S$)=1$, $sem_{2}(p)=0$ for any other place $p$. For
any reachable marking $q$ from $q_{0}$,we have:

$I:x\times q(B)+y\times q(E)+q($S$)=z$

$F_{1}:q(B)+q(A)=k$

$F_{2}:q(D)+q(E)=l$

We propose to mead this proof in three steps.

Figure 11: k instances of Program1 and l instances of Program 2 can be
executed concurrently. This time, unlike the Petri Net in Figure 5, the mutual
exclusion invariant depends on the various values of the three parameters
$x,y,z$ independently of the parameters $k$ and $l$. This cannot be
straightforwardly deduced from any semiflow.

First, it is easy to prove that $q_{0}$ is a home state: let’s assume
$u\in\mathbb{N}$, $v\in\mathbb{N}$ and $q\in RS$ such that $q(B)=u$ and
$q(E)=v$. Therefore, we can enable the sequence $Semv_{1}^{u}Semv_{2}^{v}$ for
any value of $u$ and $v$. We then reach a marking $q^{\prime}$ such that
$q^{\prime}(B)=q^{\prime}(E)=0$. From $I,F_{1},F_{2}$, we easily conclude that
$q^{\prime}=q_{0}$ without any assumption on the values of the parameters
$k,l,z,x,y$.

Second, let us see that the Petri Net is live if and only if:

$k>0,l>0$ and $z\geq max(x,y)$.

If the Petri Net is live then $F_{1}$ (saying that there is no more than $k$
tokens in $A$) involves $k=q_{0}(A)>0$, $F_{2}$ involves $l=q_{0}(D)>0$ and
$I$ involves that $q(\textbf{S})$ is bounded by $z$, therefore we must have
$z\geq max(x,y)$ to have a chance to reach the enabling threshold of
$Semp_{1}$ or $Semp_{2}$ and enable them. Reversely, if the 3 inequalities
hold, we use the fact that $q_{0}$ is a home state to easily deduce that
$Semp_{i}$ are live then $Semv_{i}$ are also live (since they need only $B$ or
$E$ to contain one token that $Semp_{i}$ would have produced) and the Petri
Net is live.

Third, we can now prove that (ME) holds if and only if $z<x+y$ or $k=0$ or
$l=0$. The condition is obviously necessary otherwise if $(z\geq x+y)\ and\
(k\neq 0)\ and\ (l\neq 0)$ then we could enable the sequence
$Semp_{1}Semp_{2}$ from $q_{0}$ and (ME) does not hold. Reversely, if $k=0$ or
$l=0$ (ME) is obviously verified since one of the two program would have no
instance. Now, from $z<x+y$ and $I$ we can write:

$q(\textbf{S})=z-x\times q(B)-y\times q(E)<x+y-x\times q(B)-y\times q(E)$

If $q(B)$ and $q(E)$ were both strictly positive then we would have:
$q(\textbf{S})<0$ which is impossible. Therefore, $q(B)$ and $q(E)$ cannot be
both strictly positive for the same marking and the invariant is satisfied
independently of the values of $k$ and $l$.

We can conclude that this PN coupled with $q_{0}$ is live and (ME) is an
invariant if and only if $k>0$, $l>0$, and $z\in[max(x,y),x+y-1]$. $\square$

Moreover, let us assume $z<x+y$ and $x<y$. We have just seen that:
$q($S$)<x\times(1-q(B))+y\times(1-q(E))$. If $x<y$ then $0\leq q($S$)<-x\times
q(B)+y\times(2-q(E))$ which can hold ony if $q(E)<2$ which means that: (ME2):
if $x<y$ and $z<x+y$ then there cannot be more than one token in $E$ for any
reachable marking $q$ from $q_{0}$ is a second invariant for this Petri Net.

We can prove similarly that if $x=y$ and $z<x+y$ then we have:
$0<x\times(2-q(E)-q(B))$. therefore, (ME3): if $x=y$ and $z<x+y$ then
$q(B)+q(E)\leq 1$ for any reachable marking $q$ from $q_{0}$ is an invariant.

###  7.3 An example from the telecommunication industry

The example described in Figure 12 is a reduced 292929See [6, 27] for Petri
Nets reduction rules. version of a Petri Net published in [51] representing
two subscribers, “a caller” and a “callee,” having a conversation (places
$CLA$ and $CA$ respectively). Initially, they are in an idle state with places
$LA$ and $A$ marked with one token. Signals $PU,\ R$ are sent from the caller
to the callee and signals $S,\ F$ from the callee to the caller. The overall
desired behavior is that caller and callee cannot go back to their idle state
as long they have not received all the signals sent to them despite the fact
that they both can hang up at any time making the order in which signals $F$
and $R$ are sent and received undetermined.

From their idle state (place $LA$), the caller can pick up their phone
(transitions $t_{1}$) sending the signal $PU$ to the callee. From their idle
state (place $A$), the callee, upon receiving the signal $PU$, can pick up
their phone (transition $t_{7}$), send the signal $S$ and go to conversation
(place $CA$) from where they can hang up (transition $t_{8}$) at any time
sending the signal $F$ to the caller. Receiving the signal $S$, the caller can
go (transition $t_{2}$) to the conversation (place $CLA$). They can also hang
up at any time (transitions $t_{3},t_{4},t_{5}$) sending the signal $R$ to the
callee. After hanging up via $t_{4}$ or $t_{5}$, the caller will have to wait
(place $W$) until they receive the signal $F$ from the callee before going
back (transition $t_{6}$) to their initial idle state $LA$. The callee can go
back to their idle state $A$ only upon receiving the signal $R$ (transition
$t_{9}$).

The initial marking $M_{0}$ is such that $M_{0}(LA)=M_{0}(A)=1,\ M_{0}(p)=0$
for any other place: the modeled system is in its idle state.

Figure 12: This Petri Net has exactly three minimal semiflows of minimal
support constituting a minimal generating set over $\mathbb{Q}^{+}$:
$\mathcal{GB}_{1}=\\{f_{1},\ f_{2},\ f_{3}\\}$ such that:
$f_{1}(LA)=f_{1}(CLA)=f_{1}(W)=f_{1}(PU)=f_{1}(S)=\ 1,\ f_{1}(p)=\ 0$ for any
other place,
$f_{2}(LA)=f_{2}(PU)=f_{2}(F)=f_{2}(CA)=\ 1,\ f_{2}(p)=\ 0$ for any other
place,
$f_{3}(CLA)=f_{3}(S)=f_{3}(R)=f_{3}(A)=\ 1,\ f_{3}(p)=\ 0$ for any other
place.

####  7.3.1 Optimizing Sperner’s bound

First, let us notice that inequality (11) gives us the following bound for
$m$, the number of minimal supports: $m\leq\binom{9}{\left\lfloor
9/2\right\rfloor}=126$. However, it is easy to notice that transitions
$t_{1},t_{2},t_{8},\text{and}\ t_{9}$ have only one input and one output,
which means that any support including such an input also includes the
corresponding output to satisfy the equations associated to
$t_{1},t_{2},t_{8},\text{and}\ t_{9}$. The bound can be optimized to:
$m\leq\binom{5}{\left\lfloor 5/2\right\rfloor}=10$. It can further be improved
by reasoning on transitions $t_{5}\ \text{and}\ t_{6}$ and reach $m\leq 3$.

####  7.3.2 A proof scheme using minimal semiflows of minimal support

We want to verify the following property (known as safeness [27] p.489):

$\mathcal{P}=\forall M$ reachable from $M_{0},\forall p\in P,M(p)\leq 1$.

We also want to prove that the initial marking $M_{0}$ is a home state (i.e.,
a marking such that whatever the evolution of the Petri Net, it is always
possible to reach it back) from which it is easy to deduce that the Petri Net
is live.

These two important properties are proven hereunder, starting from
$\mathcal{GB}_{1}=\\{f_{1},f_{2},f_{3}\\}$ the set of minimal semiflows of
minimal support defined Figure 12 without considering every reachable marking.

Similarly as in equation (8), three invariants can directly be drawn from
$\mathcal{GB}_{1}$: for any reachable marking $M$ from $M_{0}$,
$f_{1}^{\top}M=f_{1}^{\top}M_{0}=1$, $f_{2}^{\top}M=f_{2}^{\top}M_{0}=1$,
$f_{3}^{\top}M=f_{3}^{\top}M_{0}=1$. 303030In the following, we will omit the
phrase “for any reachable marking $M$ from $M_{0}$”.

In other words, there is exactly one token in the support of any semiflow of
$\mathcal{GB}_{1}$. Hence, $\mathcal{P}$ is true since
$P=\bigcup_{i=1}^{i=3}\left\|f_{i}\right\|$.

In order to prove that $M_{0}$ is a home state, we start to prove that
$\mathcal{HS}=\\{M\ |\ M(LA)=1\\}$ is a home space (i.e., a set of markings
such that whatever is the evolution of the Petri Net, it is always possible to
reach one element of the set). If we consider the invariant deduced from
$f_{2}$, we know that any reachable marking $M$ is necessarily in one of the 4
following cases.

i) $M(LA)=1$, then $M\in\mathcal{HS}$,

ii) $M(F)=1$, since $f_{2}^{\top}M=1$, we have $M(LA)=M(PU)=M(CA)=0$.
$f_{1}^{\top}M=f_{1}(CLA)M(CLA)+f_{1}(W)M(W)+f_{1}(S)M(S)=1$ and it then
remains only 3 sub-cases to explore:

ii-1) $M(CLA)=1$, $t_{3}$ can occur from $M$ and we can reach
$M^{\prime}\in\mathcal{HS}$

ii-2) $M(W)=1$, $t_{6}$ can occur from $M$ and we can reach
$M^{\prime}\in\mathcal{HS}$

ii-3) $M(S)=1$, $t_{2}$ can occur from $M$ then we are in the sub-case ii-1).

iii) $M(CA)=1$, $t_{8}$ can occur from $M$ then we are in the case ii).

iv) $M(PU)=1$, then considering the invariants deduced from $f_{1}$ and
$f_{2}$, we have: $M(LA)=M(CLA)=M(W)=M(F)=M(S)=M(CA)=0$

From $f_{3}^{\top}M=1$, we have two sub-cases:

iv-1) $M(A)=1$, $t_{7}$ can occur from $M$ then we are in the case iii).

iv-2) $M(R)=1$, $t_{9}$ can occur then we are in the case iv-1) again.

From these simple 4 cases and 5 sub-cases, we directly deduce that
$\mathcal{HS}$ is a home space from where it is easy to conclude that $M_{0}$
is a home state and the Petri Net is live.

####  7.3.3 Further remarks

We let the interested reader develop the same proof scheme from
$\mathcal{GB}_{2}=\\{f_{1},\ g=f_{2}+f_{3},\ h=f_{1}+f_{3}\\}$ or from
$\mathcal{GB}_{3}=\\{g,\ h,\ l=f_{1}+f_{2}\\}$ which are minimal generating
sets over $\mathbb{Q}$ and find out that the analysis becomes more complex
even with small additional changes. We conclude that the smaller the support,
the more effective the analysis will be since the number of cases and sub
cases of the proof scheme depends on the number of elements of each considered
support.

The same proof scheme can be used with a colored Petri Net or the same Petri
Net enriched with two parameters to model $x$ callers and $y$ callees. In the
latter, the initial marking $M_{0}$ becomes: $M_{0}(LA)=x,\ M_{0}(A)=y,\
M_{0}(p)=0$ for any other place $p$. $\mathcal{GB}_{1}$ is unchanged, but its
three associated invariants become: $f_{1}^{\top}M=f_{1}^{\top}M_{0}=x$,
$f_{2}^{\top}M=f_{2}^{\top}M_{0}=x$, $f_{3}^{\top}M=f_{3}^{\top}M_{0}=y$. To
prove that $\mathcal{HS}=\\{M|\ M(LA)=x\\}$ is a home space, we can start from
a marking $M$ that satisfies $f_{2}^{\top}M=x$ and build a sequence of
transitions $\sigma_{1}$ that will reach a marking $M^{\prime}$ such that
$M^{\prime}(PU)<M(PU)$. $\sigma_{1}$ can then be iterated until a marking of
$\mathcal{HS}$ is finally reached. From there, it becomes easy to prove that
the Petri Net is live. At last, with the same scheme, it can be proven that
$\mathcal{HS}_{c}=\\{M|\ M(CLA)=M(CA)=z\ \text{and}\ z\leq min(x,y)\\}$ is a
home space meaning that it is always possible to have $z$ subscribers in a
conversation.

###  7.4 A brief illustration of fairness and starvation

The Petri Net Figure 11 has a fairness issue despite the fact that it is live
for $k>0$, $l>0$, and $z\in[max(x,y),x+y-1]$. As soon as $k>1$ and $z\geq 2x$,
Program1 can starve Program2 by having constantly at least one instance of
program1 in its critical section: the marking of $B$ remains strictly positive
and an infinite loop between $Semp_{1}$ and $Semv_{1}$ lets no instance of
program2 to ever have the opportunity to enter in its critical section.
Similarly, as soon as $l>1$ and $z\geq 2y$, Program2 can starve Program1:
there exists an infinite computation such that $Semp_{1}$ can never have the
opportunity to be enable from any marking of this sequence.

Figure 13: There does not exist an infinite progression such that a Program
$i$ can prevent the other one to have the opportunity to execute its critical
section, in other words, to never have the possibility for $Semp_{j}$ (where
$j=3-i$) of being enabled.

The Petri Net Figure 13 addresses this issue since we can prove that the
transition $Semv_{1}$ is enabled only when the marking of $B$ is null. This
solution can be compared to another example page 111 of [6] where a privilege
is given to one Program to starve the other one. This time, it can be proven
that any infinite sequence offers to $Semp_{i}$ an infinite number of
opportunities to be enabled.

##  8 Conclusion

As soon as we can associate the behavior of a system under study with a set of
state variables and follow its evolution through sequences of states forming a
reachability graph (or a structure) where each edge is associated with a
transition, we can assume the existence of an underlying Transition System.
Doing so, invariants and home spaces can be checked or proven along all
possible sequences and help understanding and verifying the functioning of
such a system.

About Petri Nets, it can never be sufficiently stressed how the combination of
their two key concepts puts Petri Nets apart from many other conceptual
models. First, a bipartite graph (a topology) captures and models so well the
relationship and dependencies between actions and resources. Second, the
“token game” allows for simulations and construction of a reachability graph
given an initial marking (or state). This combination is clearly able to
depict concurrency and is supporting somehow a notion of parallelism. It has
been seen how semiflows create a link from the static topology (the bipartite
graph) to the dynamic evolution (the variation of the number of tokens) of the
Petri Net. They support constraints over all possible markings which greatly
help analyzing and discovering behavioral properties (even some unspecified
ones). Semiflows infer a class of invariants with a constant evaluation
function that can be computed from the initial marking of the Petri Net under
consideration. Last but not least, they can be characterized by a generating
set that can be used in order to support some elegant level of behavioral
analysis even with the presence of some level of parameterization.

By considering $\mathbb{N}$, then $\mathbb{Q^{+}}$, then $\mathbb{Q}$, the
size of a minimal generating set decreases as expected since an increasing
amount of possibilities to combine semiflows are provided. More interestingly,
if $m$ is the number of minimal supports in a Petri Net then:

a minimal generating set over $\mathbb{N}$ is finite and has at least $m$
elements,

a minimal generating set over $\mathbb{Q^{+}}$ has exactly $m$ elements,

a minimal generating set over $\mathbb{Q}$ has, at most, $m$ elements.

Moreover, we were able to regroup and clarify some key algebraic results
scattered in the literature by considering only the two notions of minimality
in terms of semiflow and support respectively.

The example of Section 7.3 showed how loose the Sperner’s bound is, as well as
a first pathway to improve upon it by considering the connections imposed by
the homogeneous system of equations (9). The same example provides reasons to
consider non-negative semiflows of minimal support. With that said, a
comprehensive complexity analysis of the complete process of computing a
generating set along with analyzing a Petri Net properties is required in
order to quantitatively evaluate the trade-off between manipulating a
generating set over $\mathbb{Q^{+}}$ or over $\mathbb{Q}$. Let us also point
out that in the literature, there exist additional reasons to consider minimal
supports described, for instance in [11].

We claimed that results in [8] or in [10] p. 68 need to be rephrased in
considering the formulation by the same authors in one of their previous
publications [68].

We believe that these results may be enriched along two different alleys. From
a mathematical point of view, the relation with integer linear programming or
convex geometry has been investigated many times, particularly in [9],
however, we believe it could be fruitful to look at the notion of toric
varieties and saturated semigroups [55]. From a Petri Net and, even more
broadly, from a transition system theory point of view, applying these new
results to a variety of models (for example, colored Petri Nets or any
transition system that can be associated with a system of equations such as
(9)) remains to be done.

Finally, we would like to stress the possibility to automate the proof scheme
exhibited throughout the example of Section 7.3. Even though some proof steps
can be shortened, we chose to develop our proofs in a systematic way to
provide a clear road map on how an algorithm could proceed for solving such
properties.

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aainstitutetext: Bogoliubov Laboratory of Theoretical Physics JINR, 141980
Dubna, Russia

# Vacuum Integration: UV- and IR-divergencies

I. V. Anikin<EMAIL_ADDRESS>
###### Abstract

In this note we present the important details regarding the massless vacuum
integrations which are not outlined in the literature. In particular, it has
been shown how the delta-function represents either UV-regime or IR-regime. In
the case of vacuum integration, we advocate the use of sequential approach to
the singular generated functions (distributions). The sequential approach is
extremely useful for many practical applications, in particular, in the
effective potential method.

##  1 Introduction

In different QFT-models, at the classical level, the effects of spontaneous
symmetry breaking are very important in the context of the geometrical
analysis of the Goldstone theorem. In this connection, the study of a vacuum
state as the potential minimum plays an significant role Vasilev:2004yr .
Meanwhile, the quantum corrections, that tend usually to distort the
geometrical picture, computed within the effective potential (EP) methods
allow to return again to the classical geometrical analysis of the models with
spontaneous symmetry breaking. In the standard EP-approaches, the quantum
corrections are given by the the vacuum integrations with the massive
propagators. However, the special interest is related to the vacuum
integrations with the massless propagators. It is mostly dictated by the use
of conformal symmetry (see for example Anikin:2020dlh ; Anikin:2023wkk ;
Anikin:2023ogb ).

On the other hand, working with the vacuum massless integrations, it demands
some careful considerations. Indeed, the general dimensional analysis suggests
that all vacuum integrations with the massless propagators lead to zero
Grozin:2005yg ; Grozin:2007zz . It is true except a particular case of
dimensionless integrand where the ultraviolet (UV), or infrared (IR), momentum
region is only under consideration. In this case, the arguments of dimensional
analysis cannot be applied.

In Gorishnii:1984te , it has been shown that the vacuum integration of
dimensionless and massless integrand is proportional to $\delta(n-D/2)$ where
the space dimension is defined as $D=d-2\varepsilon$
($d=2,4,6\,\,\text{etc.}$) and $n$ implies the propagator index. The delta-
function as a singular generated function (distribution) is a well-defined
linear functional on the suitable finite function space. In the case of
dimensional regularization, this space should be realized with the integration
measure as $d\varepsilon\,\varphi(\varepsilon)$ where $\varphi$ has a
localized support. However, it is not always convenient, even possible, to
deal with the measure as $d\varepsilon$ Anikin:2023wkk ; Anikin:2023ogb .
Moreover, owing to the symmetry properties, the delta-function is usually
hiding the information on the UV(or IR)-divergency.

Following Gorishni-Isaev’s method Gorishnii:1984te , we present all necessary
details on the vacuum integration where the delta-function has been treated in
the frame of the sequential approach Antosik:1973 ; Gelfand:1964 . We also
demonstrate how the delta-function represents the UV(IR)-regimes.

##  2 $\Delta_{F}(0)$-singularity

Let us consider the simplest case of scalar massless propagator
$\Delta_{F}(0)$ giving the tad-pole diagram. Using the Fourier transform, the
propagator $\Delta_{F}(0)$ can be write as 111For the sake of shortness, here
and in what follows the momentum loop normalization is hidden in $(d^{D}k)$.
Moreover, the Euclidian measure of momentum integrations has been implies.

$\displaystyle\Delta_{F}(0)=\int\frac{(d^{D}k)}{k^{2}}=\int(d^{D}k)\Big{\\{}C^{-1}(D,1)\int
d^{D}z\,\frac{e^{-ikz}}{\big{(}z^{2}\big{)}^{D/2-1}}\Big{\\}}$
$\displaystyle=C^{-1}(D,1)\int
d^{D}z\,\,\frac{\delta(z)}{\big{(}z^{2}\big{)}^{D/2-1}}\equiv\Gamma(D/2-1)\int(d^{D}z)\,\,\frac{\delta(z)}{\big{(}z^{2}\big{)}^{D/2-1}},$
(1)

where the integration measure $(d^{D}z)$ absorbs the normalization constant
$i(-\pi)^{D/2}$ arising from

$\displaystyle C^{-1}(D,n)=i(-\pi)^{D/2}\frac{\Gamma(D/2-n)}{\Gamma(n)}.$ (2)

If we assume that $D/2-1=0$, then the propagator in Eqn. (2) takes a form of

$\displaystyle\Delta_{F}(0)=\Gamma(0)\int(d^{D}z)\,\delta(z)\Rightarrow\Gamma(0),$
(3)

where, as well-known, the singularity of $\Gamma$-function can be presented as

$\displaystyle\Gamma(0)=\lim_{\epsilon\to 0}\Gamma(\epsilon)=\lim_{\epsilon\to
0}\Big{\\{}\frac{1}{\epsilon}+....\Big{\\}}.$ (4)

It is worth to notice that the condition given by $D/2-1=0$ should be applied
before the integration over $(d^{D}k)$ in Eqn. (2) in order to avoid the
uncertainty, see also Sec. 3.

On the other hand, according to Gorishnii:1984te , the vacuum integration
method applied to the Feynman propagator results in the delta-function. Let us
remind a key moment of Gorishni-Isaev’s method. Using the spherical system (in
the momentum Euclidian space), $\Delta_{F}(0)$ can be represented as

$\displaystyle\Delta_{F}(0)=\int\frac{(d^{D}k)}{k^{2}}=\frac{1}{2}\int
d\Omega\int_{0}^{\infty}d\beta\,\beta^{D/2-2},$ (5)

where $d\Omega$ gives the finite angle measure of integration. The replacement
$\beta=e^{y}$ leads to the following expression

$\displaystyle\Delta_{F}(0)=\frac{1}{2}\int
d\Omega\int_{-\infty}^{\infty}(dy)\,e^{iy\big{[}(-i)(D/2-1)\big{]}}=\frac{1}{2\,|i|}\delta\big{(}D/2-1\big{)}\,\int
d\Omega$ (6)

or, restoring all coefficients, it reads

$\displaystyle\Delta_{F}(0)=-\,2i\,\pi^{1+D/2}\,\delta(1-D/2)\Big{|}_{D=2}=-\,2i\,\pi^{2}\,\delta(0).$
(7)

So, for the case of $D=2$, the matching of Eqns. (3) and (7) gives the
following representation

$\displaystyle(-i)\,\Delta_{F}(0)=\Gamma(0)=-\,2\,\pi^{2}\,\delta(0).$ (8)

With this, we may conclude that $\delta(0)$-singularity can be treated as the
singularity of $\Gamma(0)$, see Eqn. (4). The same inference has been reached
by the different method, see Anikin:2020dlh . Notice that the physical (UV or
IR) nature of the mentioned singularity has been somewhat hidden.

In the dimensional regularization, the UV- and IR-divergencies are associated
with the small positive ($\varepsilon>0$) and negative ($\varepsilon<0$)
regularized parameter $\varepsilon$, respectively. In this connection, using
the $\alpha$-parametrization, we rewrite Eqn. (2) as

$\displaystyle\Delta_{F}(0)=\int\frac{(d^{D}k)}{k^{2}}=\Gamma(D/2-1)\int(d^{D}z)\,\,\frac{\delta(z)}{\big{(}z^{2}\big{)}^{D/2-1}}$
$\displaystyle=\int(d^{D}z)\,\,\delta(z)\,\Big{\\{}\int_{0}^{\infty}d\alpha\,\alpha^{D/2-2}\,e^{-\alpha
z^{2}}\Big{\\}}=\int_{0}^{\infty}(d\alpha)\,\alpha^{D/2-2}.$ (9)

Hence, one gets (modulo the normalization factor which is now irrelevant)

$\displaystyle\Delta_{F}(0)=\int\frac{(d^{D}k)}{k^{2}}=\int_{0}^{\infty}(d\alpha)\,\alpha^{D/2-2}\Rightarrow\frac{1}{D/2-1}\Big{\\{}\lim_{\alpha\to\infty}\alpha^{D/2-1}-\lim_{\alpha\to
0}\alpha^{D/2-1}\Big{\\}}.$ (10)

From Eqn. (12), one can see that the first term corresponds to the UV-
divergency, while the second term – to the IR-divergency. That is, we have

$\displaystyle\lim_{\alpha\to\infty}\alpha^{D/2-1}=[\infty]_{\text{UV}}\quad\text{if}\,\,\,D>2,$
$\displaystyle\lim_{\alpha\to
0}\,\,\alpha^{D/2-1}=\,\,[\infty]_{\text{IR}}\quad\text{if}\,\,\,D<2.$ (11)

In other words, if the dimensional parameter $\epsilon$ in $D=d-2\epsilon$ is
small one, $|\epsilon|<1$, and it varies from the negative to positive
variables, we have the following representation for $\Delta_{F}(0)$

$\displaystyle\Delta_{F}(0)\Big{|}_{d=2}$
$\displaystyle\Rightarrow\frac{1}{D/2-1}\Big{\\{}\Theta(D>2\,|\,\epsilon<0)\lim_{\alpha\to\infty}\alpha^{D/2-1}-\Theta(D<2\,|\,\epsilon>0)\lim_{\alpha\to
0}\alpha^{D/2-1}\Big{\\}}=0$ (12)
$\displaystyle\Rightarrow\delta\big{(}1-D/2\big{)}\Big{|}_{D\not=2}=0,$

where $\epsilon$ should be considered as an external independent parameter.
From Eqns. (2) and (12), in the dimensional regularization, one can see that
the positive small $\epsilon$ is regularizing the UV-divergency but not IR-
divergency. Thus, every of the methods gives the same final conclusion.

To conclude this section, we remind the other useful representation given by

$\displaystyle\Delta_{F}(0)=\lim_{z^{2}\to 0}\Delta_{F}(z^{2})=\lim_{z^{2}\to
0}\frac{1}{4\pi}\delta_{+}(z^{2})=\delta(0),\quad z\in\mathbb{E}^{4}$ (13)

which is in agreement with Eqns. (7) and (8).

##  3 Vacuum integration as a limit of non-vacuum integration

We now address to the relation between vacuum and non-vacuum integrations. In
the dimensional regularization procedure, we begin with the consideration of
two-point 1PI massless Green function given by

$\displaystyle\mathcal{I}(p^{2})=\int\frac{(d^{D}k)}{k^{2}(k^{2}+p^{2})}=(c.c.)\,(p^{2})^{D/2-2}\,G(1,1),$
(14)

where $(c.c)$ implies the coefficient constant and

$\displaystyle G(1,1)=\frac{\Gamma(-D/2+2)\Gamma^{2}(D/2-1)}{\Gamma(D-2)}.$
(15)

Using $D=4-2\epsilon$, we get

$\displaystyle\mathcal{I}(p^{2})=\int\frac{(d^{D}k)}{k^{2}(k^{2}+p^{2})}=(c.c.)\,(p^{2})^{-\epsilon}\,\frac{\Gamma(\epsilon)\Gamma^{2}(1-\epsilon)}{\Gamma(2-2\epsilon)}.$
(16)

In Eqns. (14) and (16), the scale dependence of $\mu^{2}$ is hidden as
irrelevant one.

The vacuum integration can be obtained from Eqn. (14) with the help of the
corresponding limit as

$\displaystyle\mathcal{V}_{2}\equiv\int\frac{(d^{D}k)}{(k^{2})^{2}}=\lim_{p^{2}\to
0}\mathcal{I}(p^{2}).$ (17)

There are, however, some subtleties of this limit which are now under our
considerations. Indeed, having used the $\alpha$-representation, let us
calculate the integral of Eqn. (14). We have the following

$\displaystyle\mathcal{I}(p^{2})=(c.c.)\int_{0}^{\infty}d\alpha
d\beta\frac{e^{-p^{2}\frac{\alpha\beta}{\alpha+\beta}}}{[\alpha+\beta]^{D/2}}=(c.c.)\int_{0}^{\infty}\lambda\lambda^{1-D/2}\int_{0}^{1}dxe^{-p^{2}\lambda
x\bar{x}},$ (18)

where

$\displaystyle\alpha=\lambda x_{1},\quad\beta=\lambda
x_{2},\quad\lambda\in[0,\,\infty].$ (19)

The next stage of calculations is to make a replacement as

$\displaystyle\tilde{\lambda}=p^{2}\lambda x\bar{x},\quad
d\tilde{\lambda}=p^{2}x\bar{x}d\lambda$ (20)

in the exponential function. This replacement simplifies the integrals and it
leads to the corresponding combination of $\Gamma$-functions denoted as
$G(1,1)$ Grozin:2005yg ; Grozin:2007zz . Ultimately, we reproduce the result
presented by Eqns. (14) and (16).

Now, the first mathematical subtlety is that if we suppose the limits
$p^{2}\to 0$ and $\epsilon\to 0$ are consequent ones, not simultaneous, it is
clear that these limits are not commutative operations, i.e.

$\displaystyle\big{[}\lim_{p^{2}\to 0},\,\lim_{\epsilon\to 0}\big{]}\not=0.$
(21)

On the other hand, if the limits are simultaneous ones we deal with the
uncertainty of $[0]^{0}$ which should be somehow resolved.

The second subtlety is related to the limit $p^{2}\to 0$ and the replacement
of Eqn. (20). Namely, in order to avoid the mentioned uncertainty, we have to
implement the limit $p^{2}\to 0$ before the possible replacement. In this
case, the limit of $p^{2}\to 0$ is well-defined operation and we finally
obtain that

$\displaystyle\lim_{p^{2}\to
0}\mathcal{I}(p^{2})=(c.c.)\int_{0}^{\infty}d\lambda\lambda^{1-D/2}=\frac{1}{2-D/2}\Big{\\{}\lim_{\lambda\to\infty}\lambda^{2-D/2}-\lim_{\lambda\to
0}\lambda^{2-D/2}\Big{\\}}$
$\displaystyle\equiv\int\frac{(d^{D}k)}{(k^{2})^{2}}=\mathcal{V}_{2}.$ (22)

##  4 $\delta(0)$-singularity

We are now in a position to discuss the treatment of $\delta(0)$-singularity
(or $\delta(0)$-uncertainty). To this aim, we follow to the sequential
approach to the singular generated functions (distributions).

From one hand, based on the dimensional analysis, we may conclude that all
massless vacuum integrations disappear, i.e.

$\displaystyle\mathcal{V}_{n}=\int\frac{(d^{D}k)}{[k^{2}]^{n}}=0\quad\text{for}\,\,n\not=D/2.$
(23)

However, the case of $n=D/2$ (or $n=2$ if $\varepsilon\to 0$) requires the
special consideration because the dimensional analysis argumentation does not
now work. Nevertheless, the nullification of $\mathcal{V}_{D/2}$ takes still
place but thanks to different reasons. It turns out, the ultraviolet and
infrared divergencies are cancelled each other. Hence, if only the ultraviolet
divergencies are under our consideration, $\mathcal{V}_{D/2}$ is not equal to
zero.

To demonstrate, we dwell on the vacuum integration which is externally the IR-
regularized one. It is necessary to remind that, in the space with
$D=d-2\epsilon$, the positive value of $\epsilon$ allows to avoid the UV-
divergency. In the spherical co-ordinate system, we write the following
representation

$\displaystyle\mathcal{V}_{2}=\int_{UV}\frac{(d^{D}k)}{[k^{2}]^{2}}\equiv\frac{\pi^{D/2}}{\Gamma(D/2)}\int_{\mu^{2}}^{\infty}d\beta\beta^{D/2-3}\quad\text{with}\,\,\,\beta=|k|^{2},$
(24)

where $\mu^{2}$ plays a role of IR-regularization and the angular integration
given by the measure $d\Omega$ is calculated explicitly. Next, calculating
$\beta$-integration, we reach the representation as

$\displaystyle\mathcal{V}_{2}=\frac{\pi^{2-\varepsilon}\mu^{-2\varepsilon}}{\Gamma(2-\varepsilon)}\,\frac{1}{\varepsilon}\Big{|}_{\varepsilon\to
0},$ (25)

where it is shown that the $\epsilon$-pole corresponds to the UV-divergency
only because the IR-divergency is absent by construction thanks for $\mu^{2}$.
This is a very-well known representation used, for example, in Grozin:2005yg ;
Grozin:2007zz .

On the other hand, we are able to calculate the vacuum integration by
Gorishni-Isaev’s method Gorishnii:1984te . In this case, $\mathcal{V}_{n}$
reads

$\displaystyle\mathcal{V}_{n}=\int\frac{(d^{D}k)}{[k^{2}]^{n}}=\frac{2i\,\pi^{1+D/2}}{(-1)^{D/2}\,\Gamma(D/2)}\delta(n-D/2).$
(26)

Supposing $D=4-2\varepsilon$, the only contribution is given by

$\displaystyle\mathcal{V}_{2}=\int\frac{(d^{D}k)}{[k^{2}]^{2}}=\frac{2i\,\pi^{3-\varepsilon}}{\Gamma(2-\varepsilon)}\delta(\varepsilon)\,\not=\,0.$
(27)

Hence, the delta-function of argument $\epsilon$ reflects the UV-divergency.
We specially stress that the representations of $\mathcal{V}_{2}$ given by
Eqns. (25) and (27) are equivalent.

The delta-function as a generated function (distribution) is a linear singular
functional (which cannot be generated by any locally-integrated functions)
defined on the suitable finite function space. Such a definition is absolutely
well but it is not unique one. Namely, the delta-function can be understood
with the help of the fundamental sequences of regular functionals provided the
corresponding weak limit, see for example Antosik:1973 ; Gelfand:1964 .
Besides, one of the delta-function representations is related to the following
realization

$\displaystyle\delta(t)=\lim_{\varepsilon\to
0}\delta_{\varepsilon}(t)\equiv\lim_{\varepsilon\to
0}\frac{St.F.(-\varepsilon\leq t\leq 0)}{\varepsilon},$ (28)

where $St.F.(-\varepsilon\leq t\leq 0)$ implies the well-known step-function
without any uncertainties.

Going back to Eqn. (27), one can see that the treatment of
$\delta(\varepsilon)$ as the linear (singular) functional on the finite
function space with $d\mu(\varepsilon)=d\varepsilon\phi(\varepsilon)$ meets
some difficulties within the dimensional regularization approach. Indeed, for
the practical use, $\varepsilon$ is not a convenient variable for the
construction of the finite function space because we finally need to be
focused on the limit as $\epsilon\to 0$.

Meanwhile, within the sequential approach Antosik:1973 ; Gelfand:1964 , the
delta-function might be considered as the usual singular (meromorphic)
function and the $\delta(0)$-singularity/uncertainty can be treated as a pole
of the first order Anikin:2020dlh ,

$\displaystyle\delta(0)=\lim_{\varepsilon\to
0}\delta_{\varepsilon}(0)\equiv\lim_{\varepsilon\to 0}\frac{1}{\varepsilon}.$
(29)

For the demanding mathematician, the representation of Eqn. (29) should be
understood merely as a symbol. That is, $\delta(0)$ denotes alternatively the
limit of $1/\epsilon$. This representation is also backed by the obvious fact
that Eqns. (25) and (27) are equivalent ones.

It is worth to notice that representation of $\delta(0)$ through the pole of
an arbitrary meromorphic function should be used very carefully. For example,
if we suppose that (here, $z\in\mathbb{E}^{4}$ and the delta-function is
assumed to be a functional on the finite function space)

$\displaystyle\big{[}\delta(z)\big{]}^{2}=\delta(0)\,\delta(z),$ (30)

the representation given by

$\displaystyle\delta(z)=\lim_{\epsilon\to
0}\delta_{\epsilon}(z),\quad\delta_{\epsilon}(z)=\frac{1}{\pi^{2}\epsilon^{4}}e^{-z^{2}/\epsilon^{2}}\Rightarrow\delta(0)\sim\delta_{\epsilon}(0)=\frac{1}{\pi^{2}\epsilon^{4}}$
(31)

does not satisfy the condition of Eqn. (30). Another informative example can
be found in Efimov:1973pjo .

##  5 Conclusion

To conclude, we have presented the important explanations regarding the
massless vacuum integrations. In the note, we have demonstrated the
preponderance of sequential approach where the singular generated functions
(distributions) are treated as a fundamental sequences of regular functionals.
Due to this treatment, the uncertainty as $\delta(0)$ can be resolved via the
meromorphic function of first order. Also, it has been shown in detail how the
delta-function represents either UV-regime or IR-regime.

## Acknowledgements

Our special thanks go to S.V. Mikhailov and L. Szymanowski for very useful and
stimulating discussions.

## References

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