ruledlabelfont=normalfont,labelsep=colon,strut=off

# Unveiling and Mitigating Generalized Biases of DNNs through the Intrinsic
Dimensions of Perceptual Manifolds

Yanbiao Ma , Licheng Jiao , Fang Liu , Lingling Li , Wenping Ma , Shuyuan Yang
, Xu Liu , Puhua Chen This work was supported in part by the Key Scientific
Technological Innovation Research Project of the Ministry of Education, the
Joint Funds of the National Natural Science Foundation of China (U22B2054),
the National Natural Science Foundation of China (62076192, 61902298,
61573267, 61906150, and 62276199), the 111 Project, the Program for Cheung
Kong Scholars and Innovative Research Team in University (IRT 15R53), the
Science and Technology Innovation Project from the Chinese Ministry of
Education, the Key Research and Development Program in Shaanxi Province of
China (2019ZDLGY03-06), and the China Postdoctoral Fund (2022T150506).
_(Corresponding author: Licheng Jiao.)_ The authors are with the Key
Laboratory of Intelligent Perception and Image Understanding of the Ministry
of Education of China, International Research Center of Intelligent Perception
and Computation, School of Artificial Intelligence, Xidian University, Xian
710071, China (e-mail<EMAIL_ADDRESS>lchjiao@mail.xidian.edu.cn). 0000-0002-8472-1475 0000-0003-3354-9617
0000-0002-5669-9354 0000-0002-6130-2518 0000-0001-8872-2195
0000-0002-4796-5737 0000-0002-8780-5455 0000-0001-5472-1426

###### Abstract

Building fair deep neural networks (DNNs) is a crucial step towards achieving
trustworthy artificial intelligence. Delving into deeper factors that affect
the fairness of DNNs is paramount and serves as the foundation for mitigating
model biases. However, current methods are limited in accurately predicting
DNN biases, relying solely on the number of training samples and lacking more
precise measurement tools. Here, we establish a geometric perspective for
analyzing the fairness of DNNs, comprehensively exploring how DNNs internally
shape the intrinsic geometric characteristics of datasets—the intrinsic
dimensions (IDs) of perceptual manifolds, and the impact of IDs on the
fairness of DNNs. Based on multiple findings, we propose Intrinsic Dimension
Regularization (IDR), which enhances the fairness and performance of models by
promoting the learning of concise and ID-balanced class perceptual manifolds.
In various image recognition benchmark tests, IDR significantly mitigates
model bias while improving its performance.

##  1 Introduction

Deep neural networks (DNNs) have demonstrated unprecedented capabilities in
solving complex problems and handling large-scale data sets [1, 2, 3], with
large models [4, 5, 6] based on these networks becoming the forefront of
artificial intelligence research. While striving for high performance,
fairness and bias in models have emerged as key factors affecting the
practical application of artificial intelligence [7, 8, 9, 10]. Although
previous studies have tended to attribute model bias to imbalances in sample
sizes [11, 12, 13, 14, 15], recent research [16, 17, 18] has revealed that
biases still exist in models even when sample sizes are balanced (see Fig.1a).
This indicates that deeper factors are affecting the fairness of models.
Therefore, exploring other intrinsic properties of data that affect model
fairness, and developing more universal and comprehensive methods for
predicting model biases, constitute an important and challenging task that is
crucial for enhancing the fairness of models.

Figure 1: a On a dataset containing the same number of samples per category,
the trained model still has a significant bias. b Geometric Perspectives and
Quantitative Results of Information Compression and Classification by DNN,
with d representing the depth of the DNN. The process of information
compression by DNN can be understood as gradually obtaining low-dimensional
perceptual manifolds along the internal layers of the DNN. Low-dimensional
perceptual manifolds benefit the performance of subsequent tasks. c Each
category corresponds to a data manifold, which is mapped to perceptual
manifolds in the DNN. As the depth of the DNN increases, the perceptual
manifolds gradually separate from each other and the dimensions decrease,
facilitating the correct identification of each category by the model. d Data
manifolds are constructed on a dataset basis rather than by category. As the
depth of the DNN increases, the intrinsic dimensions of perceptual manifolds
corresponding to four benchmark image datasets show a trend of initially
increasing and then decreasing.

Natural images usually follow the manifold distribution law: data of one
category typically cluster near a low-dimensional manifold embedded within a
high-dimensional space [19, 20, 21]. The data manifold formed by the image
embeddings generated within the internal layers of DNNs is called the
perceptual manifold [18]. If the collection of images for each category is
considered a data manifold, then the process of data classification can be
understood as effectively untangling, reducing dimensions, and separating
these perceptual manifolds corresponding to different categories through the
layers of the DNN [18, 22] (as shown in Fig.1b and 1c). From this geometric
perspective, we hypothesize that if the data manifolds corresponding to
certain categories are more complex geometrically, the DNN may encounter
greater challenges in simplifying and reducing the dimensions of these
perceptual manifolds, resulting in the perceptual manifolds produced by the
last layers of the DNN still having relatively large intrinsic dimension (ID)
[23, 24]. The differences in the IDs of perceptual manifolds across categories
could lead to inconsistencies in the recognition capabilities of the
classifiers at the end of the DNN for different categories, thereby affecting
the fairness of the model.

In this study, we first comprehensively explore the impact of the ID of the
perceptual manifold generated by DNN on model performance and fairness.
Specifically, our research unfolds in the following areas: (1) How the ID of
the dataset’s corresponding perceptual manifold affects the overall
performance of the model. (2) How the learning process of the model affects
the ID of the perceptual manifold. (3) Whether there is a correlation between
the ID differences of the perceptual manifolds corresponding to different
categories and model fairness. (4) How the model’s learning process affects
the imbalance of IDs among the perceptual manifolds corresponding to
categories. A significant finding is that the imbalance in ID among class
perceptual manifolds is significantly related to model fairness. Based on
multiple observations, we propose Intrinsic Dimension Regularization (IDR),
which enforces the model to learn more concise and ID-balanced perceptual
manifolds by adding additional constraints to the existing optimization
objective. This regularization aims to alleviate model bias and enhance
overall performance. Simultaneously, we devise a training strategy to
efficiently utilize IDR, addressing the challenge of estimating intrinsic
dimensions when embeddings of all samples are not accessible in each
iteration. Experiments on multiple benchmark image datasets, both balanced and
unbalanced, demonstrate that IDR can significantly mitigate model biases while
also enhancing the overall performance of the model. We hope this work will
facilitate a deeper understanding of the underlying mechanisms of deep
learning while offering practical solutions to mitigate potential biases in
artificial intelligence systems.

## Estimation of the Intrinsic Dimension

Given a set of embeddings $Z=[z_{1},\dots,z_{m}]\in\mathbb{R}^{p\times m}$
corresponding to an image dataset, $Z$ is typically distributed near a low-
dimensional perceptual manifold $M$ embedded in the $p$-dimensional space,
akin to a two-dimensional plane in three-dimensional space. The intrinsic
dimension $ID(M)$ of the perceptual manifold is such that $d<p$. A higher
intrinsic dimension indicates a more complex perceptual manifold. The
following describes how to use TLE to estimate the intrinsic dimension of the
perceptual manifold formed by $Z=[z_{1},\dots,z_{m}]\in\mathbb{R}^{p\times
m}$.

The primary method for estimating intrinsic dimension involves analyzing the
distribution of distances between each point in the dataset and its
neighboring points, and then estimating the dimensionality of the local space
based on the rate of growth of distances or other statistic. Assuming that the
distribution of samples is uniform within a small neighborhood, and then uses
a Poisson process to simulate the number of points discovered by random
sampling within neighborhoods of a given radius around each sample [24].
Subsequently, by constructing a likelihood function, the rate of growth in
quantity is associated with the surface area of a sphere. Given any embedding
$z_{i}$ in the dataset and its set of $k$ nearest neighbors $V$, the Maximum
Likelihood Estimator (MLE) of the local intrinsic dimensiona at $z_{i}$ is
given by:
$ID_{MLE}(z_{i})=-(\frac{1}{k}\sum_{j=1}^{k}ln\frac{r_{j}(z_{i})}{r_{k}(z_{i})})^{-1},$
where $r_{j}(z_{i})$ represents the distance between $z_{i}$ and its $j$-th
nearest neighbor. TLE [25] no longer assumes uniformity of sample distribution
in local neighborhoods, thus closer to the true data distribution. It assumes
the local intrinsic dimension to be continuous, thereby utilizing nearby
sample points to stabilize estimates at $z_{i}$. Specifically, the estimate of
the intrinsic dimension at $z_{i}$ using TLE is given by

$\begin{split}ID_{TLE}(z_{i})=-\Bigg{(}&\frac{1}{\left|V_{*}\right|^{2}}\sum_{\begin{subarray}{c}v,w\in
V_{*}\\\ v\neq
w\end{subarray}}\Bigg{[}\ln\frac{d_{z_{i}}(v,w)}{r_{k}(z_{i})}\\\
&+\ln\frac{d_{z_{i}}(2z_{i}-v,w)}{r_{k}(z_{i})}\Bigg{]}\Bigg{)}^{-1},\end{split}$

where $V_{*}=V\cup\\{z_{i}\\}$, and $d_{z_{i}}(v,w)$ is defined as
$(r_{k}(z_{i})(w-v)\cdot(w-v))/(2(z_{i}-v)\cdot(w-v))$. Furthermore, the
global intrinsic dimensionality of the perceptual manifold is estimated as the
average of local intrinsic dimensionalities:

$ID_{TLE}(Z)=\frac{1}{m}\sum_{i=1}^{m}ID_{TLE}(z_{i}).$

In this study, apart from using
$ID(Z)=\frac{(tr(ZZ^{T}))^{2}}{tr((ZZ^{T})^{2})}$ [26] to estimate the
intrinsic dimension of the perceptual manifold in intrinsic dimension
regularization, TLE was employed in all other experiments to estimate the
intrinsic dimensiona of the perceptual manifold. We will provide the code for
implementing intrinsic dimension estimation at
https://github.com/mayanbiao1234/Geometric-metrics-for-perceptual-manifolds.

## Results

We trained classical deep neural networks [27, 28, 29] on multiple benchmark
image datasets [30, 31, 32] for experimental exploration. Each model underwent
$10$ training sessions with different random seeds. All experimental results
represent the average of models trained with different initialization
parameters.

### Perceptual Manifold in Deep Neural Network

In the neural system, when neurons receive stimuli from the same category with
different physical features, a perceptual manifold is formed [33, 18]. The
formation of perceptual manifolds helps the neural system to perceive and
process objects of the same category with different features distinctly.
Recent studies [34, 35] have shown that the response of deep neural networks
to images is similar to human vision and follows the manifold distribution
law. Specifically, embeddings of natural images are distributed near a low-
dimensional manifold embedded in a high-dimensional space. Given a set of data
$X=[x_{1},\dots,x_{m}]$, and a trained deep neural network,
$Model=\\{f(x,\theta_{1}),g(z,\theta_{2})\\}$, where $f(x,\theta_{1})$ and
$g(z,\theta_{2})$ represent the representation network and classifier of the
model, respectively. The representation network extracts $p$-dimensional
embeddings $Z=[z_{1},\dots,z_{m}]\in\mathbb{R}^{p\times m}$ for $X$, where
$z_{i}=f(x_{i},\theta_{1})\in\mathbb{R}^{p}$. The point cloud manifold formed
by the set of embeddings $Z$ is referred to as the perceptual manifold in deep
neural networks. In the following sections, the intrinsic dimension of the
perceptual manifold are all estimated using TLE [25] if not otherwise
specified. We describe the TLE in the Methods section.

Figure 2: The Relationship between Intrinsic Dimensions of Perceptual
Manifolds and Model Performance, and the Learning Process. a The intrinsic
dimensions of perceptual manifolds generated by the last hidden layer of DNN
exhibit a negative correlation with the overall performance of the DNN across
four benchmark image datasets. b The correlation between the intrinsic
dimensions of perceptual manifolds generated by different layers of DNN and
the overall performance of the DNN. Across the four benchmark datasets, there
is an increasing negative correlation between the intrinsic dimensions of
perceptual manifolds and the overall performance of the DNN as the depth
increases. c As training progresses, the intrinsic dimensions of perceptual
manifolds corresponding to datasets generated by multiple benchmark networks
gradually decrease.

### The Intrinsic Dimension of Perceptual Manifold Exhibits Negative
Correlation with DNN Performance

DNNs compress information through hierarchical feature extraction [22, 36,
37], thereby forming lower-dimensional image representations in the last
hidden layer. As shown in Fig.1b, we utilized a well-trained DNN to extract
image embeddings from the entire dataset at different layers and quantified
the variation of the IDs of the perceptual manifolds with increasing depth of
layers (Fig.1d). The results reveal a pattern where the IDs of the perceptual
manifolds initially increase and then decrease with the deepening of layers,
rather than exhibiting a monotonic decrease. This phenomenon can be explained
as DNNs, in the shallow layers, untangling the coupling between perceptual
manifolds corresponding to different categories by increasing dimensions [20].
Please note that the perceptual manifold referred to in this section is
constructed from the entire dataset.

The compressive ability of DNNs may be associated with their performance [37,
38]. We conducted a more comprehensive and in-depth study from a geometric
perspective. Initially, we evaluated the overall performance of all well-
trained models on the test set. Then, we extracted embeddings of image sets
from the last hidden layer of DNNs and estimated the corresponding IDs of the
perceptual manifolds. Fig.2a illustrates the distribution of model performance
against the IDs of the perceptual manifolds generated. We found that: (1)
Lower ID of the perceptual manifold corresponds to higher accuracy of model on
the test set. (2) With the increasing complexity of network structures, the ID
of the generated perceptual manifold tends to be lower. These findings suggest
that DNNs capable of generating perceptual manifolds with lower intrinsic
dimensions exhibit stronger feature extraction and integration capabilities.
This provides a novel geometric constraint or objective for model training and
optimization, namely, enhancing model performance by reducing the dimension of
the perceptual manifold.

We further explored whether the IDs of perceptual manifolds from other layers
of DNNs could predict the models’ performance. Assuming the depth of the model
is $d$, we extracted image embeddings from the $\lceil d/5\rceil$, $\lceil
2d/5\rceil$, $\lceil 3d/5\rceil$, $\lceil 4d/5\rceil$ and $\lceil d\rceil$
layers of model. Fig.2b illustrates the Pearson correlation coefficients
(PCCs) between the IDs of perceptual manifolds generated from different layers
of the model and the model accuracy. The results indicate that with the
deepening of network layers, there is an increasingly strong correlation
between the IDs of perceptual manifolds and the overall performance of the
model. This phenomenon provides more specific guidance on the potential
application of intrinsic dimension as a geometric constraint, particularly
applied to the last hidden layer.

Figure 3: The negative correlation between the intrinsic dimensions of class
perceptual manifolds generated by the last hidden layer of the DNN and the
class accuracy. a Extracting image embeddings for each class from the last
hidden layer of the DNN and calculating the intrinsic dimensions of class
perceptual manifolds for each embedding set. b The scatter plot visualizes the
distribution of class accuracy against the intrinsic dimensions of class
perceptual manifolds, while the bar chart further illustrates their Pearson
correlation coefficients (PCCs). Figure 4: The impact of different layers of
the DNN and the learning process on the correlation between the intrinsic
dimensions of class perceptual manifolds and class accuracy. a Extracting
image embeddings for each class from different layers of well-trained DNNs and
calculating the intrinsic dimensions of class perceptual manifolds. b The
correlation between the intrinsic dimensions of class perceptual manifolds
generated by different layers and class accuracy. c The existing optimization
objectives are almost unable to reduce the imbalance of intrinsic dimensions
among perceptual manifolds.

### Learning Reduces the Intrinsic Dimension

To use ID as a training objective or an additional constraint, it is necessary
to explore how ID changes during the learning process, to enhance optimization
objectives rather than conflicting with them. During the training of DNNs, we
saved the model parameters every $10$ epochs and used these models to extract
embeddings of the entire image dataset at the last hidden layer. Fig.2c shows
the IDs of perceptual manifolds at different training stages. We observed a
clear trend: as training progresses, the IDs of perceptual manifolds gradually
decreases. This result suggests that deep learning models tend to learn more
concise data representations. Perceptual manifold with lower ID may imply that
data points are more tightly clustered and concentrated in the embedding
space, which helps the model learn more discriminative features. While this
phenomenon aligns with our intuition, we have quantitatively revealed its
existence for the first time through analysis, providing a solid empirical
foundation for future research, rather than relying solely on qualitative
analysis.

### ID Imbalance between Class Perceptual Manifolds Significantly Correlates
with Model Fairness

We have demonstrated that when considering the entire dataset as a single data
manifold, there exists a negative correlation between the IDs of perceptual
manifolds generated by the last hidden layer and the overall performance of
the DNNs. Furthermore, we are curious whether the model’s preferences for
different categories are related to the IDs of the corresponding perceptual
manifolds. We first define a measure for the imbalance of ID. As illustrated
in Fig.3a, for a $C$-classification task, let the IDs of the perceptual
manifolds corresponding to the $C$ categories be $ID_{1},ID_{2},\dots,ID_{C}$.
The imbalance of ID is defined as
$IF(ID)=\max\\{ID_{1},ID_{2},\dots,ID_{C}\\}/\min\\{ID_{1},ID_{2},\dots,ID_{C}\\}$
[21, 39]. A higher $IF(ID)$ indicates a greater imbalance. Similarly, the
model bias, $IF(ACC)$, is defined as the ratio of the highest class accuracy
to the lowest class accuracy.

As shown in Fig.3a, we extracted embeddings of each class of images generated
by the last hidden layer of the DNNs and saved them separately. Subsequently,
we estimated the IDs of the perceptual manifolds corresponding to each class
embedding and calculated the Pearson correlation coefficients between them and
the class accuracy. Fig.3b illustrates the distribution of IDs of each
perceptual manifold against its corresponding class accuracy. We reached two
key conclusions: (1) For the same model, the IDs of perceptual manifolds
corresponding to classes are negatively correlated with their accuracies. (2)
For classes, as the complexity of the model structure increases, the IDs of
the generated perceptual manifolds tend to decrease. These two conclusions
provide a novel geometric perspective for explaining model fairness. Moreover,
they offer a potential geometric constraint for mitigating model bias by
balancing and reducing the IDs of perceptual manifolds corresponding to all
classes.

Furthermore, we explored the correlation between the IDs of class perceptual
manifolds generated by other layers of the DNN and class accuracy (as shown in
Fig.4a), demonstrated in Fig.4b. Considering the classification model as a
combination of representation network and classifier, we found that: (1) The
correlation between the IDs of class perceptual manifolds in the last hidden
layer and class accuracy is consistently higher than other layers. (2) There
is no fixed pattern of correlation variation in the shallow layers, but at the
end of the representation network, the correlation is always at a “low point”.
The first phenomenon determines the optimal application position for geometric
constraints used to mitigate model bias, namely, the last hidden layer. The
second phenomenon aligns with the motivation behind decoupled training
paradigms [14, 40, 41, 15] in imbalance learning, where the model’s preference
is primarily caused by the classifier, while the representation network can
obtain relatively unbiased image embeddings. These conclusions derived from a
geometric perspective provide deeper insights into understanding the
widespread issue of model unfairness.

### How Learning Affects Intrinsic Dimensional Imbalance

The bias of a trained DNN is closely related to the ID of the class perceptual
manifold produced by the last hidden layer. This led us to wonder about how
much the existing optimization objective affects the degree of imbalance in
the IDs during the learning process. During the training of each
classification model, we saved the model parameters every 10 epochs. Upon
training completion, we extracted embeddings of each class image from the last
hidden layer of all models. Then, we estimated the IDs of perceptual manifolds
corresponding to each class, along with their imbalance levels. Fig.4c
illustrates the trend of the degree of imbalance of IDs with training epochs.
We observed that all models attempt to reduce the ID imbalance during the
learning process, but they lack strong constraints, resulting in weak effects.
The high negative correlation between ID imbalance and model bias further
substantiates this point.

### Balancing the Intrinsic Dimensions of Class Perceptual Manifolds

Figure 5: a&b The computation process of Intrinsic Dimension Regularization
(IDR) and its corresponding training scheme. a We present two examples
($ID_{1}=10,ID_{2}=8,ID_{C}=2$ and $ID_{1}=ID_{2}=ID_{C}=8$) to verify whether
$L_{ID}$ satisfies the first and second design principles. It can be observed
that when $ID_{1}>ID_{2}>\dots>ID_{C}$, then
$L_{ID_{1}}>L_{ID_{2}}>\dots>L_{ID_{C}}$. $L_{ID}$ is a decreasing function
with respect to $ID$, thus fulfilling principle (1). When
$ID_{1}=ID_{2}=\dots=ID_{C}$, then $L_{ID_{1}}=L_{ID_{2}}=\dots=L_{ID_{C}}=0$.
Clearly, $L_{ID}$ conforms to principle (2). When
$ID_{i}=\min\\{ID_{1},\dots,ID_{C}\\}$, then $L_{ID_{i}}=0$, thus the purpose
of $L_{ID}$ is to make the intrinsic dimensions of all class perceptual
manifolds close to $min\\{ID_{1},\dots,ID_{C}\\}$. Obviously,
${\textstyle\sum_{i=1}^{C}}ID_{i}\geq C\cdot\min\\{ID_{1},\dots,ID_{C}\\}$,
therefore $L_{ID}$ satisfies principle (3). b End-to-end training scheme. We
construct a queue to store the covariance matrices generated at each
iteration, which is capable of storing at most the covariance matrix generated
in a complete epoch. As training progresses, the latest generated covariance
matrix is continuously used to update the oldest covariance matrix in the
queue. When applying $L_{ID}$, the stored covariance matrices in the queue can
be used for its calculation. c The bias of multiple DNN models is
significantly reduced by measuring the standard deviation of class accuracy,
demonstrating the efficacy of intrinsic dimension regularization.d Using
ResNet-50 as the backbone network, model performance under different values of
$\alpha$ when employing $L_{ID}$. e IDR improves the overall performance of
the model on multiple datasets.

In this study, we propose Intrinsic Dimension Regularization (IDR), which aims
to improve the performance and fairness of DNNs by reducing the imbalance of
IDs among class perceptual manifolds. In order to learn ID balanced and
concise perceptual manifolds, IDR needs to adhere to the following three
principles:

  * (1)

The larger the ID of a class perceptual manifold, the stronger the penalty
imposed on it. This principle is based on our experimental observation:
reducing ID is often associated with an improvement in model performance.

  * (2)

When IDs are balanced, the penalty intensity for each perceptual manifold is
the same.

  * (3)

The sum of IDs for all perceptual manifolds tends to decrease.

Given a $C$-classification task and a DNN, suppose the $C$-dimensional
embedding of each class’s images in the last hidden layer is represented as
$Z_{i}=[z_{i}^{1},\dots,z_{i}^{m_{i}}]\in\mathbb{R}^{C\times m_{i}}$, where
$i=1,\dots,C$, and $m_{i}$ denotes the number of samples in class $i$. The
intrinsic dimensionality of the perceptual manifold corresponding to class $i$
is estimated as
$ID_{i}=\frac{(tr(Z_{i}Z_{i}^{T}))^{2}}{tr((Z_{i}Z_{i}^{T})^{2})}$ [26].
Intrinsic Dimension Regularization term is formally defined as
$L_{ID}=\sum_{i=1}^{C}-\log(\frac{ID_{i}^{-1}}{\max\\{ID_{1}^{-1},\dots,ID_{C}^{-1}\\}})$.

As shown in Fig.5a, $L_{ID}$ conforms to three design principles. Considering
the differentiability of the loss function, we use a smoothed form of the
$\min$ function and its transformation to approximate the $\max$ function,
resulting in the final form of $L_{ID}$:

$\begin{split}L_{ID}^{smooth}&=\sum_{i=1}^{C}-\log\left(\frac{ID_{i}^{-1}}{-\log\left(\sum_{i=1}^{C}e^{-ID_{i}}\right)^{-1}}\right)\\\
&=\sum_{i=1}^{C}-\log\left(\frac{-\log\left(\sum_{i=1}^{C}e^{-ID_{i}}\right)}{ID_{i}}\right).\end{split}$

During model training, $L_{ID}$ can serve as a supplementary term to the
existing loss function, typically formulated as
$L=L_{original}+\frac{\log_{\alpha}epoch}{(\frac{L_{ID}^{smooth}}{L_{original}}).detach()}\times
L_{ID}^{smooth}$, where $\alpha>1$. The purpose of the
$(\frac{L_{ID}^{smooth}}{L_{original}}).detach()$-term is to keep the values
of $L_{ID}$ and $L_{original}$ at the same scale. By adjusting
$\log_{\alpha}epoch$, the trade-off between $L_{ID}$ and $L_{original}$ can be
controlled, with the influence of $L_{ID}$ gradually increasing with the
increase in epochs. The parameter $\alpha$ determines the rate of growth of
$L_{ID}$. When $\alpha$ equals the epoch, $\log_{\alpha}epoch$ equals $1$.
Therefore, a smaller $\alpha$ can cause the weight $\log_{\alpha}epoch$ to
reach $1$ earlier. However, applying $L_{ID}$ in practical applications faces
a challenge: the IDs of class perceptual manifolds keeps changing continuously
during training, thus requiring real-time updates. Re-extracting embeddings of
the entire dataset at each iteration not only interrupts training but also
greatly increases time costs. Feature slow shift phenomenon [42, 17] indicates
that as training progresses, the shift of sample embeddings becomes
increasingly small, to the extent that historical versions of embeddings can
approximate the current version. Inspired by this observation, we devise an
end-to-end training scheme to apply IDR, as visualized in Fig.5b. The key
steps of this scheme are summarized as follows:

  * (1)

In the first epoch of training, compute and store the covariance matrix of
embeddings from each class at each iteration. Use $Z_{i}^{j}{Z_{i}^{j}}^{T}$
to denote the covariance matrix belonging to class $i$ at the $j$-th
iteration. Thus, each class possesses $N/BS$ covariance matrices, where $N$
represents the total sample numbers of the dataset, and $BS$ denotes the batch
size.

  * (2)

From the second epoch to the $K$-th epoch, continuously update the oldest
batch of covariance matrices using the covariance matrices newly computed at
each iteration. The purpose of this step is to wait until the shift of
embeddings becomes sufficiently small.

  * (3)

Starting from the ($K+1$)-th epoch, we not only continue to update the
covariance matrices but also utilize the latest $N/BS$ covariance matrices to
approximate the overall covariance matrix
$Z_{i}{Z_{i}}^{T}=\frac{1}{N}\sum_{j=1}^{N/BS}Z_{i}^{j}{Z_{i}^{j}}^{T}$.
Through this process, we are able to estimate the intrinsic dimension of each
class perceptual manifold and avoid interrupting the training. Subsequently,
we compute and apply $L_{ID}$.

Fig.5d illustrates the impact of different values of $\alpha$ on model
performance. It can be observed that increasing $L_{ID}$ in the mid-to-late
stages of training can more effectively enhance the overall performance of the
model. We validated the effectiveness of IDR on four benchmark image datasets
[30, 31, 32] using DNN models based on both convolutional and transformer
architectures [27, 28, 29]. Fig.5c demonstrates the standard deviation of
class accuracy on the test set before and after employing IDR. It can be
observed that the introduction of $L_{ID}$ significantly improves the fairness
of deep neural networks. Additionally, $L_{ID}$ comprehensively enhances the
performance of multiple baseline models, particularly achieving good results
on datasets with balanced sample sizes such as CIFAR-10, CIFAR-100, and Mini-
ImageNet (see Fig.5e). This performance is exciting because in the past,
researchers have not paid much attention to model fairness on datasets with
balanced sample sizes. In conclusion, this work provides a novel geometric
perspective and tool for studying the fairness of deep neural networks,
prompting AI researchers to pay attention to a broader range of imbalance
phenomena.

## Conclusion and Future Works

In this study, we first explored the association between the intrinsic
dimension of perceptual manifold and the overall performance and fairness of
the model from the geometric perspective of data classification. We then
proposed intrinsic dimension regularization (IDR) to reduce and balance the
intrinsic dimensions of perceptual manifolds corresponding to each class,
thereby mitigating model bias. Most previous studies have assumed that classes
with fewer samples are weak classes. Following this assumption, DNN models
trained on datasets with perfect class balance should not exhibit bias, or in
other words, DNNs should be entirely fair. However, recent research indicates
that classes with fewer samples are not always challenging to learn, and
models trained on class-balanced datasets may even exhibit significant bias. A
significant departure from them is that measuring model fairness from a
geometric perspective is no longer exclusive to long-tailed datasets but may
also be applicable to datasets with perfectly balanced samples. Our study has
opened up a geometric perspective on researching the fairness of deep neural
networks. From this perspective, all geometric characteristics concerning the
complexity of perceptual manifolds might be predictive of model fairness.
Therefore, extending a range of geometric measures for evaluating model
fairness is a direction for our future research.

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# Low Energy Nuclear Reactions Through Weak Interactions

K. Ramkumar<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology, Kanpur  Harishyam Kumar<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology, Kanpur  Pankaj Jain<EMAIL_ADDRESS>Department of Space, Planetary & Astronomical Sciences & Engineering, Indian
Institute of Technology, Kanpur

(June 2024)

###### Abstract

We consider the possibility that low energy nuclear reactions (LENR) arise due
to the conversion of proton to neutron through weak interactions. The
resulting neutron forms a short-lived virtual state, which then gets captured
by another nucleus through photon emission. This whole process happens under
the framework of second order perturbation theory. We find that the rate of
this process is negligibly small in most cases. However, in the presence of a
resonance the rate can be substantial and observable.

##  1 Introduction

There is considerable experimental evidence that nuclear fusion reactions may
take place at observable rate even at very low incident energy [1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11]. Theoretically, despite several attempts, it has proved to
be difficult to find a mechanism which can explain this phenomenon. The
proposed mechanisms include, screening contributions [12, 13, 14], correlated
states [15, 16], electroweak interactions [17], formation of nuclear clusters
[18], electronic deep orbit solution to Dirac equation [19] and phonon
contributions [20]. Besides these, it has been suggested that at second order
in perturbation theory, new fusion processes may open up which may not be as
suppressed as the standard first order process [21, 22, 23, 24, 25]. In this
case the process takes place by two interactions. The first interaction, which
may take place due to the presence of another particle in medium [21, 25] or
by photon emission [22, 23, 24], creates a virtual short-lived state. This is
not an eigenstate of energy. It is formed by superposition of eigenstates of
all energies and hence energy conservation is not applicable at individual
vertices, as is governed by the rules of the second order perturbation theory.
Of course, there is overall conservation of energy. Due to the contribution of
eigenstates of all energies, the Coulomb barrier may not be prohibitive and
hence it is possible that the fusion process can proceed at observable rates.
However, it has been shown that even in this case the rates may be quite small
unless some special conditions are met [22, 23, 24, 25].

In the present paper we consider a specific process which proceeds at second
order in perturbation theory. We consider fusion of a proton with a nucleus X
which has $Z$ protons and $A$ nucleons. The initial state particles are
assumed to have low energy of order eV. The Coulomb repulsion between these
two particles is prohibitive and the standard fusion process would lead to
extremely small reaction rates which are unobservable in laboratory. We assume
that the first interaction happens due to a weak process in which the incident
proton is converted to a virtual neutron with emission of a neutrino and a
positron. Subsequently, the neutron gets captured by the target nucleus $X$
through emission of a photon. As mentioned earlier, the intermediate nuclear
state is not an eigenstate of the Hamiltonian and we need to sum over all
eigenstates. In all such second order processes, it has been observed that
while the amplitude is large for some of these eigenstates, the sum has a
tendency to cancel and become very small [22, 23, 24, 25]. The amplitude turns
out to be large for eigenstates whose wave vector is approximately opposite to
that of the particles emitted at the first vertex. Several mechanisms have
been suggested to evade this cancellation. These include, presence of
resonance in the nuclear spectrum [22, 23, 24] and medium effects [22, 23, 24,
25]. As we shall see in this paper, the weak interaction induced process also
suffers from the same problem, i.e. the sum over all states tends to cancel.
We then examine the contribution of a resonance. We find that if a resonance
is present with suitable properties, the rate can be substantial. The
mechanism appears to be such that it may be valid in general and not just for
the specific weak interaction process being considered in this paper.

We consider the reaction of proton and a nucleus $X$ with atomic mass number
$A$ to produce a nucleus with mass number $A+1$. As nuclear by products,
positron, neutrino and high energy gamma photon are emitted. The process can
be expressed as,

${}^{1}{\rm
H}\>+\>{}^{A}X\>\longrightarrow\>{}^{A+1}X\>+\>e^{+}\>+\>\nu\>+\>\gamma(w)$
(1)

For simplicity, we assume that initially the proton and ${}^{A}X$ form a
molecular bound state. The two perturbations involved are:

  * 1.

Conversion of proton to neutron through weak interactions

  * 2.

Capture of neutron by ${}^{A}X$ with emission of a photon through
electromagnetic interactions.

The process is schematically illustrated in Fig. 1.

Figure 1:  Schematic illustration of the process given in Eq. 1.

##  2 The H$-$X reaction at second order in perturbation theory

We take the initial H$-$X molecular state to be s-wave ($l=0$). Here we are
interested in displaying the basic mechanism which is likely to have a much
wider application and not confined to the particular process being considered
here. In particular, it may also be applicable to cases in which the first
interaction is not a weak process, as considered in Ref. [22, 23, 24, 25]. As
an example, we may consider X to be the nickel nucleus with $A=58$. In this
case, the final state nucleus is nickel with $A=59$. As we shall argue, the
same mechanism is applicable to a wide range of nuclei that might have a sharp
resonance at relatively low energy. We may treat the nuclear system within the
framework of the shell model, in which ground state of the nickel nucleus
corresponds to the $2p_{3/2}$ level with a binding energy of approximately 9
MeV. The intermediate state neutron in this case would have $l=0$.

Let $\vec{r}_{1}$ and $\vec{r}_{2}$ be the position vectors of the AX and
proton respectively. Furthermore
$\vec{R}=(m_{1}\vec{r}_{1}+m_{2}\vec{r}_{2})/M$ and
$\vec{r}=\vec{r}_{2}-\vec{r}_{1}$ denote the center of mass and relative
coordinate respectively. Here $m_{1}$ and $m_{2}$ are the masses of particles
AX and the nucleon respectively and $M=m_{1}+m_{2}$. Here we shall treat
proton and neutron as an isospin doublets, with proton as isospin up and
neutron as isospin down. Hence the nucleon wave function is represented as

$\Psi=\begin{bmatrix}\psi_{p}\\\ \psi_{n}\end{bmatrix}$ (2)

in the isospin space. The parameter $m_{2}=m_{N}$ represents the mean mass of
these two particles. We denote the neutron and proton mass difference as,
$m_{n}-m_{p}=\Delta m$. We split the Hamiltonian of the system, such that,

$H=H_{0}+H_{I}$ (3)

where $H_{0}$ is the unperturbed Hamiltonian and $H_{I}$ is the time dependent
perturbation, which gets contributions both from weak and electromagnetic
interactions. The unperturbed Hamiltonian is given by,

$H_{0}\>=\>\frac{p_{1}^{2}}{2m_{1}}+{p_{2}^{2}\over 2m_{2}}-{1\over
2}\tau_{3}\>\Delta m\>c^{2}+V$ (4)

where $V$ represents the potential energy term, including both the nuclear and
molecular potential and $\tau$ are the Pauli matrices, acting as generators in
the isospin space. Here the kinetic energy part is taken to be isospin
symmetric, with the isospin breaking terms considered as part of the
potential. The term in Eq. 4 proportional to $\tau_{3}$ arises due to the mass
difference between neutron and proton. This term can be justified by
considering the non-relativistic limit of the Dirac equation, as shown in the
Appendix.

The initial state wavefunction can be expressed as

$\Psi_{i}(\vec{r},\vec{R})=\psi_{i}(\vec{R})\chi_{i}(\vec{r})|I_{3}=1/2>$ (5)

where $I_{3}$ refers to the third component of the isospin. Here we indicate
only the isospin of the nucleon state. We assume the center of mass
wavefunction $\psi(\vec{R})$ to be a plane wave. This will not give any
essential contribution and here we focus on the relative coordinate. We assume
that the system is in the $l=0$ ground state. This state can be obtained by
using an appropriate molecular potential. In application to LENR, we are
interested in getting an order of magnitude estimate to determine if the rate
may be anywhere close to being observable in laboratory. For this purpose, we
simply use a Gaussian form of the wave function peaked at the typical bond
length for nickel hydrogen system, which is expected to be about 3.7 atomic
units. Hence we assume that $\chi_{i}(\vec{r})$ takes the form

$\chi_{i}(\vec{r})={N_{i}\over\sqrt{4\pi}}\ e^{-(r-r_{0})^{2}/\Delta^{2}}$ (6)

Here $N$ is the normalization factor and we take $r_{0}=3.5$ atomic units, and
$\Delta\approx 0.3$. The value of $\Delta$ is dictated by the requirement that
the wave function is heavily suppressed as $r\rightarrow 0$ due to Coulomb
repulsion. With the choice of parameters, it is clear that this region would
contribute neglibly to the reaction rate.

In order to compute the final state nuclear wave function we may assume the
shell model potential,

$V_{X}(r)={-V_{0}\over 1+\exp[(r-R_{X})/a_{X}]}$ (7)

where, $V_{0}=50$ MeV, $a_{X}=0.524$ fm and $R_{X}=1.25A^{1/3}$ [26]. With
these parameters we obtain the energy eigenvalue for the 2p state of nickel to
be $-9.2$ MeV. This corresponds to the ground state of the 59Ni and is in good
agreement with the observed value of $-9$ MeV. We express the corresponding
$l=1$ wave function as,

$\phi_{X,m}(\vec{r})={u_{X}(r)\over r}Y_{1}^{m}$ (8)

Here we are interested in determining whether the rate may be anywhere close
to being observable. For this purpose we may simplify our computation by
choosing a particular final state. Hence in our calculation we set $m=1$.
Furthermore we take the spin state of the intermediate neutron and the initial
proton to be such that $m_{S}=1/2$. This would correspond to a Fermi
transition in which the nucleon spin does not flip. The total spin quantum
numbers of the final state nucleus 59Ni in this case are $J=3/2$ and
$m_{J}=3/2$.

The center of mass wave function of the final state nucleus is taken to be a
plane wave. The intermediate state wave function of the neutron-X system with
$E>0$ and $l=0$ may be computed by using the nuclear potential given in Eq. 7.
It can be expressed as

$\Psi_{n}(\vec{r},\vec{R})=\psi_{n}(\vec{R})\chi_{n}(\vec{r})|I_{3}=-1/2>$ (9)

As we shall see later, a non-negligible contribution is obtained only when the
intermediate state is close to a resonance. In order to study this, it is
convenient to directly model the phase shifts rather than use a potential
model. The standard low energy expansion of the phase shift $\delta$ for
$l=0$, corresponding to the intermediate state, is given by

$k\cot\delta=-a_{1}+a_{2}k^{2}+...$ (10)

where we have not displayed higher order terms in $k^{2}$. Here $a_{1}$ and
$a_{2}$ are parameters related to the scattering length and effective range
respectively. These can be obtained from the potential model. Using this
expansion we can directly model the required wave function, as discussed in
the next subsection.

The transition amplitude at second order in perturbation theory is given by,

$\bra{f}T(t_{0},t)\ket{i}=\left(\frac{-i}{\hbar}\right)^{2}\sum_{n}\int_{t_{0}}^{t}dt^{\prime}\>\bra{f}e^{i\>H_{0}t^{\prime}/\hbar}H_{I}(t^{\prime})e^{-i\>H_{0}t^{\prime}/\hbar}\ket{n}\\\
\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\>\bra{n}e^{i\>H_{0}t^{\prime\prime}/\hbar}H_{I}(t^{\prime\prime})e^{-i\>H_{0}t^{\prime\prime}/\hbar}\ket{i}$
(11)

where the interaction Hamiltonian HI1 gets contribution both from weak and
electromagnetic interactions. We express this as

$H_{I}=H_{I_{1}}+H_{I_{2}}$ (12)

where $H_{I_{1}}$ and $H_{I_{2}}$ are respectively the weak and
electromagnetic contributions. The weak interaction part, corresponding to the
Fermi transition being considered here, is proportional to [27],

$H_{I_{1}}\propto G_{V}\tau_{-}$ (13)

where $\tau_{-}$ is the isospin lowering operator and $G_{V}$ is the weak
interaction coupling constant [27]. It is given by

$G_{V}/(\hbar c)^{3}=1.149\times 10^{-11}{\rm MeV}^{-2}$

The weak interaction Hamiltonian also contains terms which lead to creation of
a neutrino and a positron. These are not explicitly shown in Eq. 13.

The electromagnetic perturbation can be expressed as,

$H_{I_{2}}=\sum_{i}{Z_{i}e\over
cm_{i}}\vec{A}(\vec{r}_{i},t)\cdot\vec{p}_{i}+\sum_{i}{eg_{i}\over
2m_{i}c}\vec{S}_{i}\cdot\vec{B}(\vec{r}_{i},t)$ (14)

where $Z_{i}$, $m_{i}$ and $\vec{p}_{i}$ are respectively the charge, mass and
momentum vector of the particle $i$. Here we have also included a magnetic
moment coupling with $g_{i}$ and $\vec{S}_{i}$ representing the $g$ factors
and the spin operators respectively. In Eq. 14, $\vec{A}$ is the
electromagnetic field, given by,

$\vec{A}(\vec{r},t)={1\over\sqrt{V}}\sum_{\vec{k}}\sum_{\beta}c\sqrt{\hbar\over
2\omega}\left[a_{\vec{k},\beta}(t)\vec{\epsilon}_{\beta}e^{i\vec{k}\cdot\vec{r}}+a^{\dagger}_{\vec{k},\beta}(t)\vec{\epsilon}^{\,*}_{\beta}e^{-i\vec{k}\cdot\vec{r}}\right]$
(15)

and $\vec{B}=\vec{\nabla}\times\vec{A}$.

As explained earlier, the proton converts into a neutron through a Fermi
transition. Next, the neutron gets absorbed by the nucleus through an
electromagnetic interaction. We assume that this is an electric transition
where the charged particles in the nucleus AX emits a photon while the neutron
gets absorbed. A complete calculation is complicated since it would require us
to use the full multiparticle wave function of the nucleus $X$. Here we assume
that AX is a point particle of charge $Ze$, which reduces the problem into an
effective one body problem.

###  2.1 Resonant Contribution

In this subsection we make a model for the case in which the intermediate
state gets contribution from a resonance. We are interested in relatively low
energies. At such energies, the neutron absorption cross section shows a $1/v$
behaviour where $v$ is the velocity of the incident neutron [26]. This is
expected since the transition matrix element is expected to behave as a
constant and we obtain a $1/v$ from the flux factor. Besides that we see very
sharp resonances for many nuclei even at relatively low energies or order
1-10000 eV [28, 29, 30]. Using date available from BNL National Nuclear Data
Center, one finds that 58Ni also displays a resonance at roughly 7 keV with a
width of roughly 7 eV. As we shall see this can contribute substantially to
the reaction being considered. Here we are interested in modelling the phase
shifts in the vicinity of a resonance.

It is convenient to consider the standard square well potential with
$V=-V_{0}$ for $r<r_{0}$ and $V=0$ for $r>r_{0}$. We consider the intermediate
state wave function for $l=0$. We can express it as
$\chi_{n}=N_{n}u_{n}(r)/r$, where $N_{n}$ is the normalization factor. The
wave function $u_{n}(r)$ for the square well for wave number $k_{n}$ is given
by,

$\displaystyle u_{n}$ $\displaystyle=$ $\displaystyle{A\over
k_{n}}\sin(k_{n1}r)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm for}\
r<r_{0}$ $\displaystyle u_{n}$ $\displaystyle=$ $\displaystyle{B\over
k_{n}}\sin(k_{n}r)+{C\over k_{n}}\cos(k_{n}r)\ \ \ \ \ {\rm for}\ r\geq r_{0}$
(16)

where $k_{n1}^{2}=k_{n}^{2}+2mV_{0}/\hbar^{2}$ and

$\displaystyle B$ $\displaystyle=$ $\displaystyle
A\left[\sin(k_{n1}r_{0})\sin(k_{n}r_{0})+{k_{n1}\over
k_{n}}\cos(k_{n1}r_{0})\cos(k_{n}r_{0})\right]$ $\displaystyle C$
$\displaystyle=$ $\displaystyle
A\left[\sin(k_{n1}r_{0})\cos(k_{n}r_{0})-{k_{n1}\over
k_{n}}\cos(k_{n1}r_{0})\sin(k_{n}r_{0})\right]$ (17)

The coefficient $A$ is fixed by setting $B^{2}+C^{2}=1$. We use plane wave
normalization and at large $r$,
$u_{n}(r)\rightarrow\sin(k_{n}r+\delta(k_{n}))/k_{n}$ where $\delta(k_{n})$ is
the phase shift. As $k_{n}\rightarrow 0$, $A\rightarrow k_{n}$ and hence the
nuclear matrix element becomes independent of $k_{n}$, as expected.
Furthermore $B$ and $C$ are proportional to a constant and $k_{n}$
respectively.

Here we directly directly model $\delta(k_{n})$ assuming presence of a
resonance. Let us assume that the phase shift can be modelled exactly by Eq.
10, ignoring higher order terms. We obtain,

$\displaystyle\sin{\delta}$ $\displaystyle=$
$\displaystyle{-k_{n}\over\sqrt{k_{n}^{2}+(a_{1}-a_{2}k_{n}^{2})^{2}}}$
$\displaystyle\cos{\delta}$ $\displaystyle=$
$\displaystyle{a_{1}-a_{2}k_{n}^{2}\over\sqrt{k_{n}^{2}+(a_{1}-a_{2}k_{n}^{2})^{2}}}$
(18)

In terms of the phase shift, $u(r)=\sin(k_{n}r+\delta)/k_{n}$ for $r\geq
r_{0}$. This leads to the model, $C=-A/b$ and
$B=A(a_{1}-a_{2}k_{n}^{2})/(bk_{n})$ close to resonance with

$A={bk_{n}\over\sqrt{k_{n}^{2}+(a_{1}-a_{2}k_{n}^{2})^{2}}}$ (19)

where $b$ is a constant which is chosen so that this leads to nuclear matrix
element identical to the one obtained by the shell model potential (Eq. 7) in
the limit $k_{n}\rightarrow 0$. These have exactly the same behaviour in the
limit $k_{n}\rightarrow 0$ as the corresponding coefficients in the potential
model, given in Eq. 17. Hence in this case also the nuclear matrix element
goes as constant as $k_{n}\rightarrow 0$. Our model for the phase shift
displays the correct behaviour at very low energies and in the vicinity of a
resonance. We will use this to obtain the resonance contributions for the
calculation at second order in perturbation theory.

###  2.2 Weak interaction perturbation

The initial proton state $I_{3}=1/2$ while the intermediate neutron state has
$I_{3}=-1/2$, where $I_{3}$ is the third component of the isospin operator.
Let the integral over $t^{\prime\prime}$ in eq 11, which corresponds to the
amplitude for the first vertex in Fig. 1, be represented by $I_{2}$. We
obtain,

$I_{2}=\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\>\bra{n}e^{i\>H_{0}t^{\prime\prime}/\hbar}H_{I_{1}}(t^{\prime\prime})e^{-i\>H_{0}t^{\prime\prime}/\hbar}\ket{i}$
(20)

The positron and the neutrino wave functions are given by
$\exp(i\vec{k}_{e}\cdot\vec{r})/\sqrt{V}$ and
$\exp(i\vec{k}_{\nu}\cdot\vec{r})/\sqrt{V}$ respectively. The operator
$\tau_{-}$ leads to the transition between the $T_{3}=1/2$ proton state to
$T_{3}=-1/2$ neutron state. Hence we obtain

$I_{2}={G_{V}\over
V}\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\>e^{i(E_{e}+E_{\nu}+E_{n}-E_{i})t^{\prime\prime}/\hbar}\int
d^{3}rd^{3}R\psi^{*}_{n}(\vec{R})\chi^{*}_{n}(\vec{r})e^{-i\vec{K}\cdot\vec{r}_{2}}\psi_{i}(\vec{R})\chi_{i}(\vec{r})$
(21)

where, $E_{i}$, $E_{n}$, $E_{e}$ and $E_{\nu}$ are the energies of initial
state, the intermediate nuclear state, the emitted positron and the emitted
neutrino respectively. Note that the rest mass difference in proton and
neutron is included in the difference $E_{n}-E_{i}$. Furthermore,

$\vec{K}=\vec{k}_{e}+\vec{k}_{\nu}$ (22)

is the sum of the wave vectors of positron and neutrino. Integrating over
time, we obtain

$I_{2}=-{i\hbar G_{V}\over
V}\>{e^{i(E_{e}+E_{\nu}+E_{n}-E_{i})t^{\prime}/\hbar}\over
E_{n}+E_{e}+E_{\nu}-E_{i}}\tilde{I}_{2}$ (23)

where,

$\tilde{I}_{2}=\int
d^{3}rd^{3}R\psi^{*}_{n}(\vec{R})\chi^{*}_{n}(\vec{r})e^{-i\vec{K}\cdot\vec{r}_{2}}\psi_{i}(\vec{R})\chi_{i}(\vec{r})$
(24)

and we have dropped the term depending on $t_{0}$ since it will vanish in the
limit $t_{0}\rightarrow-\infty$. Here $\psi_{n}(\vec{R})\chi_{n}(\vec{r})$
represents the intermediate state wave function. We assume that the center of
mass dependence of mass dependence is a plane wave, i.e.,

$\psi_{n}(\vec{R})={1\over\sqrt{V}}e^{i\vec{K}_{n}\cdot\vec{R}}$ (25)

where $\vec{K}_{n}$ is the wave vector corresponding to the center of mass.
Setting $\vec{r}_{2}=\vec{R}+m_{1}\vec{r}/M$ we obtain

$\tilde{I}_{2}={(2\pi)^{3}\over
V}\delta^{3}(\vec{K}_{n}+\vec{K})I^{\prime}_{2}$ (26)

where we have set $\psi_{i}(\vec{R})=1/\sqrt{V}$ and

$I^{\prime}_{2}=\int
d^{3}r\chi^{*}_{n}(\vec{r})e^{-i\vec{K}\cdot\vec{r}m_{1}/M}\chi_{i}(\vec{r})$
(27)

###  2.3 Transition Matrix Element

Besides the weak interaction, the transition matrix element also get
contribution from the electromagnetic interaction. We can express it as,

$\langle
f|T|i\rangle=-{1\over\hbar^{2}}\sum_{n}\int_{t_{0}}^{t}dt^{\prime}e^{i(E_{f}-E_{n})t^{\prime}/\hbar}\langle
f|H_{I_{2}}(t^{\prime})|n\rangle I_{2}$ (28)

Here we get contribution from only the first term on the right hand side of
Eq. 14 if we ignore the magnetic moment of the neutron. The first term gets
contribution from the nucleus AX with charge $Ze$ and mass $Am_{N}$. The final
state wave function can be written as,

$\Psi_{f}(\vec{r},\vec{R})=\psi_{f}(\vec{R})\chi_{f}(\vec{r})$ (29)

The center of mass dependence is taken to be a plane wave

$\psi_{f}(\vec{R})={1\over\sqrt{V}}e^{i\vec{K}_{f}\cdot\vec{R}}$ (30)

Let the wave vector of the emitted photon be

$\vec{k}_{\gamma}=k_{\gamma}[\cos\theta_{2}\hat{z}^{\prime}+\sin\theta_{2}(\cos\phi_{2}\hat{x}^{\prime}+\sin\phi_{2}\hat{y}^{\prime})]$
(31)

The photon field operator in Eq. 14 has the exponential factor
$e^{-i\vec{k}_{\gamma}\cdot\vec{r}_{1}}$. This can be expressed as

$e^{i\vec{k}_{\gamma}\cdot\vec{r}_{1}}=e^{i\vec{k}_{\gamma}\cdot\vec{R}}e^{-i\vec{k}_{\gamma}\cdot\vec{r}m_{2}/M}$
(32)

Hence we obtain,

$\displaystyle\langle f|H_{I_{2}}(t^{\prime})|n\rangle$ $\displaystyle=$
$\displaystyle-i{Zem_{2}\over\sqrt{V}M}\sqrt{1\over
2\hbar\omega}(E_{f}-E_{n})e^{i\omega t^{\prime}}\int
d^{3}r^{\prime}\chi^{*}_{f}(\vec{r}\,^{\prime})\vec{\epsilon}_{\beta}\cdot\vec{r}{\,{}^{\prime}}\chi_{n}$
(33) $\displaystyle\times$ $\displaystyle{(2\pi)^{3}\over
V}\delta^{3}(\vec{K}_{f}+\vec{k}_{\gamma}-\vec{K}_{n})$

Here we have set the exponential factor
$\exp(-i\vec{k}\,^{\prime}_{f}\cdot\vec{r}\,^{\prime}m_{2}/M)\approx 1$ since
the integral over $r^{\prime}$ gets contributions only from very small values
of $r^{\prime}$.

As mentioned earlier, for an order of magnitude estimate we can choose a
particular final state. Hence we may consider only one of the two final state
photon polarization vectors. We take this to be,

$\vec{\epsilon}_{\beta}=-\sin\phi_{2}\hat{x}^{\prime}+\cos\phi_{2}\hat{y}^{\prime}$
(34)

The final state wave function $\chi_{f}$ is given in Eq. 8 with $m=1$.

$\chi_{f}(\vec{r})={u_{X}(r)\over r}Y_{1}^{1}$ (35)

In order to proceed further we need to specify the intermediate state wave
function $\chi_{m}$. Since the final state wave function has $l=1$ we expect
contributions both from $l=0$ and $l=2$. However, at low energies, $l=2$ is
expected to be much suppressed in comparison to $l=0$ [26]. Hence we keep the
contribution only from $l=0$ which leads to,

$\chi_{n}={R_{n}\over\sqrt{V}}={1\over\sqrt{V}}{u_{n}\over r}$ (36)

where at large distances, $R_{n}(r)\rightarrow j_{0}(k_{n}r)$, up to a phase
shift. We can project different harmonic components of the lepton final state
by expanding the exponential factor in $I^{\prime}_{2}$ (Eq. 27),

$e^{i\vec{k}\cdot\vec{r}}=4\pi\sum_{l=0}^{\infty}\sum_{m=-l}^{l}i^{l}j_{k}(kr)\
Y_{l}^{m*}(\theta_{k},\phi_{k})Y_{l}^{m}(\theta_{r},\phi_{r})$ (37)

Only the $l=0$ component of the leptonic final state contributes.

Evaluating the angular part of the integral in Eq. 33, we obtain,

$\displaystyle\langle f|H_{I_{2}}(t^{\prime})|n\rangle$ $\displaystyle=$
$\displaystyle{Zem_{2}\over\sqrt{V}M}{4\pi\over 3}\sqrt{3\over 16\pi
E_{\gamma}}e^{-i\phi_{2}}(E_{f}-E_{n})e^{i\omega t^{\prime}}\int
dr^{\prime}r^{\prime\,2}u_{X}\chi_{n}$ (38) $\displaystyle\times$
$\displaystyle{(2\pi)^{3}\over
V}\delta^{3}(\vec{K}_{f}+\vec{k}_{f}-\vec{K}_{n})$

We therefore obtain,

$\displaystyle\langle f|T|i\rangle$ $\displaystyle=$ $\displaystyle
i{G_{V}\over\hbar V}{Zem_{2}\over\sqrt{V}M}\sqrt{\pi\over
3E_{\gamma}}e^{-i\phi_{2}}\sum_{n}\int_{t_{0}}^{t}dt^{\prime}e^{i(E_{f}+E_{\gamma}+E_{\nu}+E_{e}-E_{i})t^{\prime}/\hbar}I_{f}{E_{f}-E_{n}\over
E_{\nu}+E_{e}+E_{n}-E_{i}}I^{\prime}_{2}$ (39) $\displaystyle\times$
$\displaystyle{(2\pi)^{6}\over
V^{2}}\delta^{3}(\vec{K}_{f}+\vec{k}_{\gamma}-\vec{K}_{n})\delta^{3}(\vec{K}_{n}+\vec{K})$

where

$I_{f}=\int dr^{\prime}r^{\prime 2}u_{X}\chi_{n}$ (40)

The center of mass dependence is trivial. We need to take modulus square of
the transition amplitude and integrate over the final nuclear momenta after
inserting the density of states factor $Vd^{3}K_{f}/(2\pi)^{3}$. This simply
leads to a factor of unity while imposing the momentum conservation,

$\vec{K}_{f}+\vec{k}_{\gamma}+\vec{K}=0$ (41)

Here we focus on the relative variable. The reaction rate is given by

${dP\over dt}={V^{3}\over\Delta
T}\int{d^{3}k_{\gamma}d^{3}k_{e}d^{3}k_{\nu}\over(2\pi)^{9}}|\langle
f|T|i\rangle|^{2}$ (42)

The integrand depends only on the magnitude, $K$, of the vector $\vec{K}$ (Eq.
22) and not on the directions of the positron or neutrino momentum vectors.
Hence for performing the integration over these momenta, we can rotate our
coordinate axis and choose the new $z$-axis to point along the positron
momentum vector. Let $\theta$ be the angle between the neutrino and positron
momentum vectors. Hence we have,
$K^{2}=k^{2}_{\nu}+k^{2}_{e}+2k_{\nu}k_{e}\cos\theta$. This leads to,

${dP\over dt}={V^{3}\over(2\pi)^{9}\Delta T}\int
dk_{\gamma}k^{2}_{\gamma}d\Omega_{\gamma}dk_{e}k^{2}_{e}d\Omega_{e}dk_{\nu}k^{2}_{\nu}2\pi
d\cos\theta\,|\langle f|T|i\rangle|^{2}$ (43)

The photon and positron angular integrals simply give a factor of
$(4\pi)^{2}$. We convert the integral over $\cos\theta$ into an integral over
$K$ and obtain,

${dP\over dt}={4V^{3}\over(2\pi)^{6}\Delta T}\int
dk_{\gamma}k^{2}_{\gamma}dk_{e}k^{2}_{e}dk_{\nu}k_{\nu}^{2}\,{KdK\over
k_{\nu}k_{e}}\,|\langle f|T|i\rangle|^{2}$ (44)

A straightforward calculation gives,

${dP\over dt}={c\alpha G_{V}^{2}\over 6\pi^{3}(\hbar c)^{3}}\left({Zm_{2}\over
M}\right)^{2}\int
dk_{e}dk_{\nu}dKk_{e}k_{\nu}KE_{\gamma}\Bigg{|}\sum_{n}I_{f}{E_{f}-E_{n}\over
E_{\nu}+E_{e}+E_{n}-E_{i}}I^{\prime}_{2}\Bigg{|}^{2}$ (45)

along with the energy conservation condition,

$E_{f}+E_{\gamma}+E_{\nu}+E_{e}-E_{i}=0$ (46)

The integral over $K$ ranges from $|k_{e}-k_{\nu}|$ to $|k_{e}+k_{\nu}|$ where
$k_{e}$ and $k_{\nu}$ are the magnitudes of the positron and neutrino wave
vectors respectively. Here we confine ourselves to the case where dominant
contribution is obtained from relatively low intermediate state energies. The
maximum value we will consider is about 10 keV. Hence the energy numerator,
$E_{f}-E_{n}\approx-B$, where $B$ is the binding energy of the outer most
neutron in the final state nucleus. This will be taken to be of order MeV. The
energy denominator can be approximated as,

$E_{\nu}+E_{e}+E_{n}-E_{i}\approx E_{\nu}+E_{e}+(m_{n}-m_{p})c^{2}$ (47)

Since both the energy denominator and numerator are approximately independent
of the intermediate state energy, we can take them out of the sum over $n$.
This leads to,

$\displaystyle{dP\over dt}$ $\displaystyle=$ $\displaystyle{c\alpha
G_{V}^{2}\over 6\pi^{3}(\hbar c)^{3}}\left({Zm_{2}\over M}\right)^{2}\int
dk_{e}dk_{\nu}k_{e}k_{\nu}\int_{|k_{e}-k_{\nu}|}^{|k_{e}+k_{\nu}|}dKKE_{\gamma}$
(48) $\displaystyle\times$ $\displaystyle\left[{B\over
E_{\nu}+E_{e}+(m_{n}-m_{p})c^{2}}\right]^{2}\Bigg{|}\sum_{n}I_{f}I^{\prime}_{2}\Bigg{|}^{2}$

The sum over $n$ in the above equation can be expressed as

$\sum_{n}I_{f}I^{\prime}_{2}={4\pi V\over(2\pi)^{3}}\int
dk_{n}k_{n}^{2}I_{f}I^{\prime}_{2}$ (49)

Using Eq. 36, we obtain

$\sum_{n}I_{f}I^{\prime}_{2}={2N_{i}\over\pi}\int dk_{n}k_{n}^{2}\left(\int
dr^{\prime}r^{\prime 2}u_{X}R_{n}(r^{\prime})\right)\left(\int
drr^{2}R_{n}(r)j_{0}(Kr)\tilde{\chi}_{i}(r)\right)$ (50)

where we have explicitly substituted the initial state wave function and

$\tilde{\chi}_{i}=e^{-(r-r_{0})^{2}/\Delta^{2}}$ (51)

For small $k_{n}$, the integral,

$I_{f}=\int dr^{\prime}r^{\prime 2}u_{X}R_{n}\approx-7.6\times 10^{-11}$ (52)

in atomic units using the potential model given Eq. 7. This is applicable for
the nickel nucleus within the framework of shell model potential. For other
nuclei we expect a result with same order of magnitude assuming that a state
exists with binding energy of same order as in the case of nickel. We take the
numerical factor out of the $k_{n}$ integral and obtain,

$\sum_{n}I_{f}I^{\prime}_{2}={2N_{i}\over\pi}(-7.6\times 10^{-11})J$ (53)

where $J$ is the integral

$J=\int{k_{n}\over
K}dk_{n}{a_{1}\over\sqrt{k_{n}^{2}+(a_{1}-a_{2}k_{n}^{2})^{2}}}\int
dr\sin(k_{n}r+\delta)\sin(Kr)\chi(r)$ (54)

and $\delta$ is the phase shift. The factor
$a_{1}/\sqrt{k_{n}^{2}+(a_{1}-a_{2}k_{n}^{2})^{2}}$ arises due to the
intermediate state wave function in the nuclear matrix element. This is
normalized to unity in the limit $k_{n}\rightarrow 0$. The amplitude for the
full process in governed by this integral over $k_{n}$.

##  3 Results

We first study the integral $J$ for different models of the phase shift
$\delta$. We consider a simple model for the phase shift $\delta\approx
ak_{n}$, where $a$ is a relatively small constant. Such a behaviour is
expected for small values of $k_{n}$. In this case we find that the integral
$J$ is very small. In Fig. 2 we show $J$ (Eq. 54) as a function of the upper
limit $k_{nc}$ on the integration variable $k_{n}$. Here we have set $a=0.1$
and $K=10$. We find that although the integral becomes substantial for some
values of $k_{nc}$, this integral approaches zero as $k_{nc}$ becomes large,
leading to an extremely small rate.

Figure 2:  The integral $J$ (Eq. 54) as a function of $k_{nc}$, the upper
limit on the integration variable $k_{n}$ for the model $\delta=ak_{n}$, where
$\delta$ is the phase shift and $a=0.1$ is a constant.

We next consider the case when we are close to a resonance and use the model
given in Eq. 18. We first consider a resonance at small $k_{n}$. We find that
several nuclei show resonances in neutron absorption at energies as small as 1
eV, see for example [29]. We set $a_{1}=800$ and $a_{2}=8$ in Eq. 18, which
leads to a resonance, with a relatively narrow width, at $k=10$. In terms of
energy, the resonance lies at $E_{R}=0.74$ eV with a width of 0.02 eV. The
integral $J$ as a function of the upper cutoff on $k_{n}$ is shown in Fig. 3.
Here we have set $K=10.4$. We see that the integral saturates at a relatively
large value. We find that $J$ takes relatively large values for $K$ less than
roughly 11 and slowly decays for large $K$. In Fig. 4 we display results for
another case in which the resonance is present at a higher energy. In this
case we choose $a_{1}=2.17\times 10^{6}$ and $a_{2}=1.67$. Hence the resonance
is located at $k=1140.2$ or $E_{R}=9.6$ keV and has a relatively narrow width
of 10 eV. Here we have set $K=1140.6$. We again see a substantial
contribution.

Figure 3:  The integral $J$ (Eq. 54) as a function of $k_{nc}$, the upper
limit on the integration variable $k_{n}$ assuming a resonance at low energy.
Figure 4:  The integral $J$ (Eq. 54) as a function of $k_{nc}$, the upper
limit on the integration variable $k_{n}$ assuming a resonance at medium
energy.

In order to compute the rate we ignore the neutrino mass and use the
relativistic formula for the positron energy. For the parameters $a_{1}=800$,
$a_{2}=8$, the rate is found to be $4\times 10^{-26}$ per second. This is
found assuming that the nuclear matrix element is given by Eq. 52 which
corresponds to nickel nucleus and a binding energy of 9 MeV. In general, for
other nuclei this serves as an order of magnitude estimate, assuming that the
final state has sufficiently large binding energy so that the process is
exothermic. We find that the rate is sufficiently large to be observable. The
energy spectrum of photons shows a sharp peak at approximately 7 MeV. This is
essentially determined by energy conservation.

We next consider a case with resonance at higher energy, analogous to the
result shown in Fig. 4. We make small adjustment to the parameters so that the
resonance corresponds to the one seen in nickel with resonance energy of 7 keV
and width of 7 eV. This can be obtained by setting $a_{1}=1.9\times 10^{6}$
and $a_{2}=2.0$. In this case we find a much larger rate of about $10^{-16}$
per second. The peak in the photon spectrum is found to be around 3.5 MeV.
Hence we find a relatively large rate for the case of conversion of 58Ni to
59Ni which is observable in laboratory. For example, in an electrochemical
experiment we may be able to achieve of the order of, or larger than,
$10^{16}$ molecules composed of p-Ni. This would lead to a rate of one event
per second which can be observed by detecting the emitted photon at the energy
peaked about 3.5 MeV. Similar rates are expected for other nuclei. A detailed
comparison with experimental results is postponed to future research.

We point out that a substantial contribution is obtained only in cases where
the width of the resonance approximately matches the energy scale of
oscillations seen in the absence of resonance as a function of the
intermediate state energy eigenvalue, as seen in Fig. 2. If these two energy
scales are very different the rate is expected to be very small. Furthermore,
the resonance mechanism may be applicable to a wide range of processes and not
just the weak interaction process considered here. The rates in these cases
are expected to be much higher since they will not be subjected to the weak
interaction suppression.

##  4 Conclusion

In this paper we have considered a nuclear process in which an incident proton
reacts with a target heavy nucleus X to form an isotope of X with emission of
a positron, neutrino and a photon. The process gets dominant contribution at
second order in perturbation theory. At the first vertex the proton converts
into a virtual neutron through a weak process. At the second interaction the
neutron gets absorbed by the nucleus X with emission of a photon. We find that
the rate for such a process is negligible unless it is assisted by the
presence of a nuclear resonance. In the absence of resonance, we find
significant contributions to amplitude from many intermediate neutron energy
eigenstates. However this contributions oscillate about zero and the sum is
found to be negligible. In order to get a substantial contribution, the
resonance width has to match the energy scale of the oscillations. Assuming
the presence of such a resonance, we find that the rate is substantial and
observable. We argue that this condition can be met in many nuclei observed in
nature. We have directly applied our model to 58Ni which shows a resonance at
approximately 7 keV with a relatively narrow width of 7 eV. The rate in this
case is found to be approximately $10^{-16}$ per second which is
experimentally observable. Hence the proposed mechanism can be tested in
laboratory. The resonance mechanism is also applicable to many other processes
and not just confined to the weak process being considered here. A detailed
application to different processes and other nuclei as well as comparison with
experimental data is postponed to future research.

##  5 Appendix

In this equation we consider the Dirac equation for the isospin doublet,
proton-neutron system and perform its non-relativistic reduction. The wave
function in isospin space is given by Eq. 2. The Dirac equation can be written
as

$i\hbar{\partial\over\partial t}\begin{bmatrix}\psi_{p}\\\
\psi_{n}\end{bmatrix}=\left[c\vec{\alpha}\cdot\vec{p}+\beta m_{N}c^{2}-{1\over
2}\beta\Delta mc^{2}\tau_{3}+V\right]\begin{bmatrix}\psi_{p}\\\
\psi_{n}\end{bmatrix}$ (55)

Here the first two terms on the right hand side are isospin symmetric, and
proportional to the identity matrix in the isospin space, while the last two
terms break this symmetry. We point out that the potential $V$ is also a
$2\times 2$ matrix in isospin space. We denote its components by $V_{11}$,
$V_{12}$, $V_{21}=V^{*}_{12}$ and $V_{22}$. We denote the upper and lower
components in the Dirac space by the symbols $\phi$ and $\zeta$ respectively.
Hence for the proton, the equation becomes

$i\hbar{\partial\over\partial t}\begin{bmatrix}\phi_{p}\\\
\zeta_{p}\end{bmatrix}=\left[c\vec{\alpha}\cdot\vec{p}+\beta
m_{N}c^{2}-{1\over 2}\beta\Delta
mc^{2}+V_{11}\right]\begin{bmatrix}\phi_{p}\\\
\zeta_{p}\end{bmatrix}+V_{12}\begin{bmatrix}\phi_{n}\\\
\zeta_{n}\end{bmatrix}$ (56)

The corresponding equation for the neutron is,

$i\hbar{\partial\over\partial t}\begin{bmatrix}\phi_{n}\\\
\zeta_{n}\end{bmatrix}=\left[c\vec{\alpha}\cdot\vec{p}+\beta
m_{N}c^{2}+{1\over 2}\beta\Delta
mc^{2}+V_{22}\right]\begin{bmatrix}\phi_{n}\\\
\zeta_{n}\end{bmatrix}+V_{21}\begin{bmatrix}\phi_{p}\\\
\zeta_{p}\end{bmatrix}$ (57)

We next make the following transformation for the proton and neutron wave
functions [31],

$\phi_{p,n}=e^{-im_{N}c^{2}t/\hbar}\Phi_{p,n}\ \ \ \ ,\ \ \
\zeta_{p,n}=e^{-im_{N}c^{2}t/\hbar}Z_{p,n}$ (58)

The equations for the upper and lower components, $\phi_{p}$ and $\zeta_{p}$
can be written as

$\displaystyle i\hbar{\partial\over\partial t}\Phi_{p}$ $\displaystyle=$
$\displaystyle c\vec{\sigma}\cdot\vec{p}\,Z_{p}-{1\over 2}\Delta
mc^{2}\Phi_{p}+V_{11}\Phi_{p}+V_{12}\Phi_{n}$ (59) $\displaystyle
i\hbar{\partial\over\partial t}Z_{p}$ $\displaystyle=$ $\displaystyle
c\vec{\sigma}\cdot\vec{p}\,\Phi_{p}-2m_{N}c^{2}Z_{p}+{1\over 2}\Delta
mc^{2}Z_{p}+V_{11}Z_{p}+V_{12}Z_{n}$ (60)

We now make the standard leading order approximation. Using Eq. 60 we obtain,

$Z_{p}\approx{\sigma\cdot\vec{p}\over 2cm_{N}}\Phi_{p}$ (61)

Substituting this in Eq. 59, we obtain the following leading order equation
for $\Phi_{p}$ in the non-relativistic limit,

$i\hbar{\partial\over\partial t}\Phi_{p}={p^{2}\over 2m_{N}}\Phi_{p}-{1\over
2}\Delta mc^{2}\Phi_{p}+V_{11}\Phi_{p}+V_{12}\Phi_{n}$ (62)

Following similar steps, we obtain the corresponding equation for $\Phi_{n}$,

$i\hbar{\partial\over\partial t}\Phi_{n}={p^{2}\over 2m_{N}}\Phi_{n}+{1\over
2}\Delta mc^{2}\Phi_{n}+V_{22}\Phi_{n}+V_{21}\Phi_{p}$ (63)

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