# CADGL: Context-Aware Deep Graph Learning for Predicting Drug-Drug
Interactions

Azmine Toushik Wasi, Taki Hasan Rafi, Raima Islam, Šerbetar Karlo, and Dong-
Kyu Chae† This paper was produced by the IEEE Publication Technology Group.
They are in Piscataway, NJ.Manuscript received April 19, 2021; revised August
16, 2021.T. H. Rafi and D.K. Chae are with the Department Computer Science,
Hanyang University, South Korea. E-mail:{takihr<EMAIL_ADDRESS>A. T.
Wasi is with Shahjalal University of Science and Technology, Bangladesh.
E-mail<EMAIL_ADDRESS>R. Islam was with BRAC University,
Bangladesh. E-mail<EMAIL_ADDRESS>S. Karlo was with University of
Cambridge, UK. E-mail<EMAIL_ADDRESS>Author.

###### Abstract

Examining Drug-Drug Interactions (DDIs) is a pivotal element in the process of
drug development. DDIs occur when one drug’s properties are affected by the
inclusion of other drugs. Detecting favorable DDIs has the potential to pave
the way for creating and advancing innovative medications applicable in
practical settings. However, existing DDI prediction models continue to face
challenges related to generalization in extreme cases, robust feature
extraction, and real-life application possibilities. We aim to address these
challenges by leveraging the effectiveness of context-aware deep graph
learning by introducing a novel framework named CADGL. Based on a customized
variational graph autoencoder (VGAE), we capture critical structural and
physio-chemical information using two context pre-processors for feature
extraction from two different perspectives- local neighborhood and molecular
context, in a heterogeneous graphical structure. Our customized VGAE consists
of a graph encoder, a latent information encoder, and an MLP decoder. CADGL
surpasses other state-of-the-art DDI prediction models, excelling in
predicting clinically valuable novel DDIs, supported by rigorous case studies.

###### Index Terms:

Graph neural networks, computational drug discovery, drug-drug interactions,
molecular interaction networks, pharmaceutical informatics, context-aware
learning

##  I Introduction

Drug-Drug Interactions (DDIs) study is a key component of the drug development
process. When multiple medications are taken at the same time, complicated
interactions between them can occur (illustrated in Figure 1). These
interactions can provide important information about potential side effects
and safety issues that can have a big impact on the creation and refinement of
novel pharmacological agents. DDIs signify that when more than one or several
drugs are taken simultaneously by patients, the potency of such medications
may produce therapeutic effects or cause adverse consequences to arise.
Unfortunately, identifying and predicting DDIs is challenging due to the vast
number of potential drug interactions - making clinical examination of all
combinations unfeasible, and resource-intensive.

Figure 1: Drug-Drug Interactions Prediction Scenario

However, recent advancements in Graph Neural Networks (GNNs) suggest promising
potential for DDI prediction. Originally, GNNs are renowned for their
capability to represent nodes (like drugs in a drug-drug interaction network)
and edges (like relationships between drugs) in a low-dimensional latent space
that captures the subtleties of their interactions due to their topological
and semantic characterization properties [1, 2, 3]. Notable GNNs include Graph
Convolutional Networks (GCN) [4], Graph Attention Networks (GAT) [5], and
Graph Isomorphism Networks (GIN) [6]. Owing to the success of GNNs, there have
been remarkable studies on DDI prediction based on GNNs: MSAN [7] uses a
Transformer-based GNN architecture; MHCA-DDI [8] focuses on polypharmacy side
effects; GMPNN [9] utilizes message-passing neural networks and a co-attention
mechanism for predicting pairwise substructure interactions; SSI-DDI [10]
employs substructure-substructure interaction modeling using Transformers; Ngo
et al. [11] utilize variational graph autoencoder (VGAE) to capture
polypharmacy side effects.

Despite the recent advances in DDI prediction, certain issues still remain
unsolved and overlooked in this domain. Firstly, lack of generalization is a
common weakness among the models discussed above. Most models face challenges
when it comes to adapting to new graph structures or changes in node feature
distribution. Relying heavily on molecular structure, MHCA-DDI [8] lacks
flexibility for generalization. SSI-DDI [10] faces limitations in terms of
performance being affected by changes in drug order, while GMPNN [9] may
struggle to capture complex mechanisms beyond substructures. Secondly, feature
extraction plays a crucial role in tackling complex tasks like DDI prediction.
In this scenario, there are more than 1700 nodes representing drugs and 86
different types of interactions, resulting in millions of possible
combinations. Furthermore, the challenge is intensified by the fact that any
two drugs can have multiple interactions, and specific molecular structures of
drug pairs contribute to distinct interactions. Take Ticlopidine (id DB00169)
and Doxorubicin (id DB00997), for instance; their combination may both
increase bioavailability (interaction type 5) and decrease serum concentration
(interaction type 9) [12]. To effectively capture these complex interaction, a
robust feature extraction process is vital for any ML model to learn and
predict novel DDIs, avoiding problems like overfitting, underfitting, biases,
noises, etc. In terms of robust feature extraction, MSAN [7] attempts to
overcome overfitting through similarity-based modules and substructure-
dropping augmentation but struggles with feature extraction compared to graph
autoencoders. SSI-DDI utilizes molecular graphs for segmentation and
recognizes pairwise interactions, but can leak noisy information. GMPNN
utilizes co-attention to learn drug substructures from molecular graphs, but
struggles to capture intricate mechanisms beyond substructures, whereas GCN
[4] falls short in distinguishing graphs with distinct edges and uniform
nodes. Finally, they all lack real-world representation and use capabilities.
We argue that developing DDI models must be accompanied by proper real-world
case studies analysis to check if previous clinical studies approve their
applicability initially.

In order to address all aforementioned limitations, we propose CADGL, a novel
Context-Aware Deep Graph Learning framework. It is based on the VGAE
architecture, but unlike the traditional VGAEs, our framework consists of two
encoders: a context-aware graph encoder and a latent information encoder, and
an MLP-based decoder. Our graph encoder aggregates node and edge values and
positions across multiple layers to capture and process graph features from
different perspectives, enabling robust feature extraction. Our approach has
demonstrated greater effectiveness compared to these models in feature
extraction and prediction, as evidenced by the experimental studies presented.

Our contributions are summarized into four folds:

  * •

To the best of our knowledge, our work is the first attempt to exploit a
context-aware framework based on our customized VGAE, incorporating both graph
structural and latent information encoding to effectively capture the graph
data for enhanced DDI prediction.

  * •

Our novel context-aware deep graph encoder involves two contextual pre-
processors, namely Local and Molecular Context Pre-processors, to collect
relations from different contexts and perspectives. It then encodes the
acquired features via self-supervision, resulting in improved performance on
downstream tasks and invariant representations that solve different issues
posed by existing models.

  * •

Experimental results show that CADGL outperforms all other state-of-the-art
DDI prediction systems, with a detailed analysis of the strengths and
necessity of different strategies adopted by our CADGL.

  * •

Our model is able to predict valuable novel DDIs with strong clinical
potential, supported by rigorous case studies confirming their applicability
in drug development. We also analyze the broader societal implications of the
drugs, which can be developed by newly identified DDIs, after comprehensive
clinical experiments and studies.

##  II Related Works

###  II-A Drug-Drug Interactions

While DDIs are prevalent in real-world scenarios, there remains a scarcity of
research addressing this issue. The proposed custom CNN [13] operates on
graphs, enabling prediction streams using graphs of various sizes and shapes,
extending molecular feature extraction beyond circular fingerprints. GraphDTA
[14] aims to predict drug-target affinities, employing GNNs to create
representations by constructing graphs for each molecule type. Lim et al. [15]
incorporate 3D structural data of protein-ligand binding poses into an
adjacency matrix, differentiating inter-molecular relationships via gating and
distance-aware graph attention. To predict DDIs, Torng and Altman [16]
suggests a two-staged graph-convolutional network where an unsupervised graph-
autoencoder is utilized in the initial step to learn fixed-size approximations
of protein pockets.

Figure 2: The overall framework of our CADGL. After taking input, structural
data is processed with local and molecular context processors. These are
passed into self-supervised graph attention for graph encoding. The output
passes through two FC blocks with properties data for developing latent
distribution and generate encodings. An MLP decoder extracts information,
while a classifier gauges potential link likelihoods.

###  II-B Context-Aware Learning in Drug-Interaction

Context-Aware Learning is a research topic that explores how to use additional
information (such as location, time, user preferences, etc.) to improve the
performance of machine learning models. Different contexts are usually used in
Drug-Interaction111Alongside DDI, Drug-Interaction also covers Drug-Target
Interaction, and Drug-Protein Interaction. These GNN tasks offer motivation in
DDI research due to similarity in approaches. related works, like molecular
structure [7], relations [17], etc. MSAN [7] proposes a novel model to
effectively predict potential Drug-Drug Interactions (DDIs) using context of
molecular structures of drug pairs. RaSECo [17] aims to enable all drugs,
including new ones, to distill effective relation-aware embeddings (RaEs),
thereby facilitating drug pair (DP) embedding to improve DDI prediction
performance.

###  II-C Autoencoders in Drug-Interaction

Variational Graph Autoencoder (VGAE) [18] is a framework for unsupervised
graph learning where interpretable latent representations are developed using
latent attributes for undirected graphs. To transform continuous code space
vectors into tiny graphs by employing neural networks, GraphVAE [19], a
molecule generation framework that consists of a decoder that outputs a
probabilistic fully-connected graph which discards difficulty in chemical
property representation.

##  III Method and Algorithm

Figure 2 illustrates the overview of our proposed CADGL framework. CADGL uses
two encoders (graph and latent information) and an MLP-based decoder. Given
the structural inputs (shown in lower left corner of Figure 2), we derive
feature vectors using the context-aware deep graph encoder; subsequently, the
latent information encoder refines these vectors to produce a latent
information encoded feature vector, which is then combined with other inputs
of physical and chemical properties. Finally, we feed it into the decoder,
which generates a final prediction.

###  III-A Problem Formulation

GNNs utilize various message passing mechanisms to aggregate information from
a center node’s neighborhood and update its representation, using different
techniques focusing on capturing different sets of features. Let $\mathcal{G}$
be a graph represented as $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{X})$,
where $\mathcal{V}$ represents the nodes, $\mathcal{E}$ denotes the edges, and
$\mathcal{X}$ is a matrix expressing the node features. Each node
$v\in\mathcal{V}$ and each edge $e\in\mathcal{E}$ are associated with their
mapping function $\phi(v):\mathcal{V}\rightarrow\mathcal{A}$ and
$\varphi(e):\mathcal{E}\rightarrow\mathcal{R}.\mathcal{A}$ and $\mathcal{R}$
denote the node type set and edge type set, respectively, where
$|\mathcal{A}|+|\mathcal{R}|>2$. The DDI prediction task involves binary
classification using two tuples:
$\left(\mathcal{D}_{1},\mathcal{D}_{2},t\right)$ and
$\left({\mathcal{F}_{1}},{\mathcal{F}_{2}},t\right)$, where $\mathcal{D}$
denotes a drug structural data (chemical molecular graphs) obtained from
$\mathcal{G}$ and ${\mathcal{F}}$ stands for physio-chemical properties of the
drugs that come from $\mathcal{X}$, which is originated from RDKit [20] using
the DrugBank [12] dataset. The goal is to predict if two drugs
$\mathcal{D}_{1},\mathcal{D}_{2}$ will interact under a specific interaction
type $t$.

###  III-B Context-Aware Deep Graph Encoder

Context-aware deep graph encoder encodes different contextual information
effectively. Our context-aware deep graph encoder (the lower area containing
the three green blocks in Figure 2), which is responsible for robust
extraction of molecular structural information and generating feature vectors,
comprises of two auxiliary information pre-processing units: local context
processor (LCP) and molecular context processor (MCP), and a self-supervised
graph attention network. LCP and MCP extract local structural and molecular
characteristics and help the self-supervised network with relevant information
to work effectively.

####  III-B1 Local context processor (LCP).

This module is responsible for processing local neighbourhood information
using a graph convolutional neural network that utilizes a weight-sharing
mechanism to enable efficient learning of node embeddings from the input
graphs. It transforms each node’s representation by aggregating information
from its neighboring nodes and itself.

To begin with, we make a structural input $x$ (see lower left corner of Figure
2) by using a function
$f_{graph}:\left(\mathcal{D}_{1},\mathcal{D}_{2},t\right)\rightarrow x$ that
maps the given tuple to the input vectors, based on the concatenation and
union operations on node features. Then the formulation of this layer is:

${h}_{i}^{L}={W}{x}_{i}+{W}\cdot\operatorname{MEAN}({x}_{j},\forall
j\in\mathcal{N}(i))$ (1)

where ${W}$ is a learnable parameter matrix for this layer and the MEAN
operation aggregates the neighbor nodes ($\mathcal{N}(i)$) of $i$. The output
${h}_{i}^{L}$ is sent to the self supervised learning block as input.

####  III-B2 Molecular context processor (MCP).

This processor leverages standard molecular feature extraction methods based
on circular fingerprints. Drug molecules often contains circular structures
like benzene and benzene-based substances, which play a vital role forming
chemical bond and connections. MCP collects and processes these information
for better prediction. Its formulation is:

${h}_{i}^{M}={W}_{1}^{(deg(i))}{x}_{i}+{W}_{2}^{(deg(i))}\sum_{j\in\mathcal{N}(i)}\mathbf{x}_{j}$
(2)

where ${W}_{1,2}$ are model parameters for this layer, and ${h}_{i}^{M}$ is
the output. $deg$ denotes degree of a node, which is the count of its
neighboring nodes to which it is directly connected by a edge. This process
trains a unique weight matrix for each potential vertex degree in the
molecular graph.

The outputs from LCP and MCP are concatenated to ${h}$:

${h}=concat({h}_{i}^{L},{h}_{i}^{M})$ (3)

####  III-B3 Self-supervised graph attention network (SSGAttn).

Self-supervised learning approaches for training GNNs have effective learning
capabilities [21]. This network enhances model performance by incorporating a
self-supervised learning objective, which encourages the encoder to learn
representations that are invariant to graph transformations. This layer
augments the models’ ability to generalize to unseen data, making it more
robust and effective. The formulation is:

${x}_{o}=\alpha_{ii}W_{s}{{h}_{i}}+\sum_{j\in\mathcal{N}(i)}\alpha_{ij}W_{s}{{h}_{j}}$
(4)

Here, ${h}_{.}$ is the input achieved from the pre-processors after
concatenation, ${W}_{s}$ is a learnable parameter matrix, and ${x}_{o}$ is the
output of the entire graph encoder network. $\alpha$ is related to the
attention type. In this layer, we specially use the max attention, namely
$\alpha^{\text{MX}}$, computed by:

$\displaystyle\alpha^{\mathrm{MX}}_{ij}$
$\displaystyle=\frac{\exp\left(\operatorname{Leaky}\operatorname{ReLU}\left(e_{ij}^{\mathrm{MX}}\right)\right)}{\sum_{k\in\mathcal{N}(i)\cup\\{i\\}}\exp\left(\operatorname{LeakyReLU}\left(e_{ik}^{\mathrm{MX}}\right)\right)}$
(5) $\displaystyle e^{\mathrm{MX}}_{ij}$
$\displaystyle={a}^{\top}\left[W_{s}{{h}_{i}}\|W_{s}{{h}_{j}}\right]\cdot\sigma\left(\left(W_{s}{{h}_{i}}\right)^{\top}\cdot
W_{s}{{h}_{j}}\right)$

The self-supervision loss is applied to this layer, which uses positive and
negative edge sampling. The loss term is described in Section III-D3.

####  III-B4 Graph normalization layer.

Graph normalization layers are used to normalize output values from
convolutional layers using GraphNorm [22] before proceeding to the next layer.
Its general equation is:

${x}_{norm}=\frac{{x}-\zeta\odot\mathbb{E}[{x}]}{\sqrt{\operatorname{Var}[{x}-\zeta\odot\mathbb{E}[{x}]]+\epsilon}}\odot\gamma+\beta$
(6)

where $\zeta,\beta,\text{ and }\gamma$ are learnable parameters, $\epsilon$ is
a hyperparameter to control stability, and $\mathbb{E}[]$ and
$\operatorname{Var}$ denote the expectation and variation. For our graph
normalization after the LCP, it takes ${h}_{i}^{L}$ as input $x$ and the
output ${x}_{norm}$ is renamed to ${h}_{i}^{L}$ and sent to the concat layer
(Eq.(3)). Respectively, after the MCP, it takes ${h}_{i}^{M}$ as input $x$ and
the output ${x}_{norm}$ is renamed to ${h}_{i}^{M}$ and sent to the concat
layer.

###  III-C Latent Information Encoder and MLP Decoder

Traditional Graph AEs lack the ability to generate predictions for links on
new and unknown high-level graphs [11] due to the complexity and high-
dimensional nature of data. But our model employs a latent information encoder
to extract and combine features (substructure identification, atom centrality,
ECFP6, aromaticity modeling, atom classification, physiochemical parameters,
chemical feature categories, and other core chemistry attributes) from the
RDKit library [20].

To begin with, we use another mapping function
$f_{feature}:\left({\mathcal{F}_{1}},{\mathcal{F}_{2}},t\right)\rightarrow
X_{f}$ that maps the property information to the input vectors. Then, along
with the molecular structural encoded data ${x}_{o}$, the property information
$X_{f}$ can be represented into lower-dimensional latent space. This helps in
reducing the complexity and sparsity of data. In addition, this enables us to
extract relevant latent variables through two fully connected layers that
provide mean $\mu$ and log of variance $\log\sigma$ to create a low-
dimensional multivariate normal distribution. We calculate $\mu$ and
l$\log\sigma$ by:

$\displaystyle\mu=W_{\mu}[\Psi(x_{o})\mathbin{\|}\Psi(X_{f})],\log\sigma=W_{\sigma}[\Psi(x_{o})\mathbin{\|}\Psi(X_{f})]$
(7)

where $W_{\mu}$ and $W_{\sigma}$ are the model parameter matrices, $\Psi$
signifies the normalization, and $\left[\hskip 8.5359pt\parallel\hskip
8.5359pt\right]$ indicates concatenation. $\mu$ and $\log\sigma$ are the mean
and log of variance vectors (length of 64), respectively.

Now, our model is able to produce new node links and interaction possibilities
on intricate graphs at a high level of complexity by condensing all features
into a low dimensional latent space using $\mu$ and $\log\sigma$ developing a
posterior distribution:

$\displaystyle e=\mu+\kappa(\log\sigma)\cdot\exp(0.5\cdot\log\sigma)$ (8)

where $e$ is the output, reparameterized (generating non-uniform random
numbers by transforming a base distribution) from $\mu$ and $\log\sigma$.
$\kappa$ is a function that generates a normal distribution (
$\mathcal{N}(0,1)$) arbitrarily.

Finally, our decoder incorporates a series of MLP layers to compute possible
edges between corresponding nodes using the output generated by the encoders:

$\displaystyle Z=\text{MLP}({e})$ (9)

where ${e}$ is the input and $Z$ is the output vectors. Then, the final
prediction (i.e., an edge value) on whether an interaction will occur or not
is performed by:

$\small p=\sigma(z),z=\text{FCL}(Z)$ (10)

where $z$ is a scalar value produced by a fully connected layer (FCL) and $p$
is the final output probability, ranging from 0 to 1. $\sigma(.)$ denotes the
sigmoid function.

Algorithm 1 CADGL

1: Input: $(\mathcal{D}_{1},\mathcal{D}_{2},t)$ for each combination

2: Output: $p$

3: Map structural information
$f_{graph}:\left(\mathcal{D}_{1},\mathcal{D}_{2},t\right)\rightarrow x$

4: Map physio-chemical features $f_{features}\rightarrow X_{f}$

5: for DDI combinations in $x,X_{f}$ do

6: Local Context Processor: $x\rightarrow{h}_{i}^{L}$

7: Molecular Context Processor: $x\rightarrow{h}_{i}^{M}$

8: Concatenate both: $({h}_{i}^{L},{h}_{i}^{M})\rightarrow{h}$

9: Self-Supervised Graph Attention Net: ${h}\rightarrow x_{o}$

10: Latent Encoding: $X_{f},x_{o}\rightarrow\mu,\log\sigma$

11: Reparameterization: $\mu,\log\sigma\rightarrow e$

12: Decoding: $e\rightarrow Z$

13: Probability calculation: $Z\rightarrow p$

14: end for

###  III-D Training Objectives

For training our CADGL, we employ three loss functions, one is based on the
cross-entropy (CE), one is for the self-supervision layer (SS), and the other
is based on Kullback-Leibler Divergence (KL). The final loss function is a
linear combination of the three loss terms without using any hyper-parameter,
in order to avoid the complex hyper-parameter tuning.

$l_{CADGL}=l_{CE}+l_{SS}+l_{KL}$ (11)

Note that the impact of each loss function is examined and reported in
supplementary materials.

####  III-D1 Cross-entropy loss (CE Loss)

CE loss is designed to minimize the difference between the predicted
probabilities of the positive edges and 1, and the predicted probabilities of
the negative edges and 0, as follows:

$\small
l_{CE}=-\frac{1}{n}\sum_{i=1}^{n}\left(p_{i}^{\prime}\log\sigma\left(p_{i}\right)+\left(1-p_{i}^{\prime}\right)\log\left(1-\sigma\left(p_{i}\right)\right)\right)$
(12)

Here, $p_{i}$ is the output of the $i$-th drug pair (in equation 9), where $i$
ranges from 1 to $n$. The $\sigma(\cdot)$ denotes sigmoid function and label
of the $i$-th drug pair is denoted by $p_{i}^{\prime}$.

####  III-D2 KL-divergence loss (KL Loss)

This loss is for our latent information encoder (Eq. (8)). The KL-divergence
loss [23] is a measure of the difference between two probability
distributions. Here, it measures the difference between the learned
distribution (with parameters $\mu$ and log $\sigma$) and a unit Gaussian. It
is used to stabilize the distribution by keeping a stable Gaussian shape, as
follows:

$l_{KL}={KL}(q(z|x)\|p(z))=\frac{1}{2}\sum_{j=1}^{D}(1+2\log\sigma_{j}-\mu_{j}^{2}-\sigma_{j}^{2})$
(13)

here $q(z|x)$ is the learned distribution with $\mu$ and $\log\sigma$, $D$
denotes dimension and $p(z)$ is the unit Gaussian distribution with mean 0 and
variance 1.

####  III-D3 Self-Supervision loss (SS Loss)

It is for the self-supervised graph encoder. We define the self-supervised
optimization objective as a binary cross-entropy loss, following [24]:

$l_{SS}=-\frac{1}{\left|\mathbb{E}\cup\mathbb{E}^{-}\right|}\sum_{(j,i)\in\mathbb{E}\cup\mathbb{E}^{-}}I(i,j)\cdot\log\phi_{ij}^{l}+I(i,j)\cdot\log\left(1-\phi_{ij}^{l}\right)$
(14)

where $I(i,j)$ is an indicator function that outputs 1 if $i==j$ and 0
otherwise. We use a subset of $\mathbb{E}\cup\mathbb{E}^{-}$ (positive and
negative sampling of edges) sampled by probability $p_{e}\in(0,1]$ at each
training iteration for a regularization effect from randomness.
$\phi_{ij}^{l}$ denotes the probability that an edge exists between $i$ and
$j$ nodes.

##  IV Implementation and Experiments

###  IV-A Experimental Settings

We use the DrugBank [12] dataset for our experiments. This dataset includes
1,703 drugs and a total of 191,870 drug pairs spanning 86 DDI types. It also
contains structural and chemical information about the drugs. To conduct our
analysis, we segregated the dataset into three subsets: a training set
comprising 115,185 drug pairs, a validation set containing 38,348 drug pairs,
and a test set consisting of 38,337 drug pairs. We perform all the experiments
05 times to collect experimental results. We train our model for a total of
300 iterations with the learning rate of 0.001.

###  IV-B Comparisons with State-of-the-arts

In this experiment, we employ the following baselines and state-of-the-arts:
GCN [4], GAT [5], GIN [6], SSI-DDI [10], MSAN [7], MHCA-DDI [8], DSN-DDI[25],
SA-DDI [26], VGAE [11] and GMPNN [9], to compare performance against CADGL.
Table I reports the comparison results. CADGL performs the best among all
competitors. Some models discussed in related works are not designed for DDI
tasks, meanwhile some of the codebases are missing, so we do not manage to
compare with them. Within the context of medical research, even slight
improvement in metrics can wield the power to determine matters of life and
death. Assessing by performance, we believe that our model holds great
significance and possesses the potential to yield meaningful contributions.

Figure 3: Validation Loss, AUROC and AUPRC score visualization of context pre-processors for first 80 epochs. Figure 4: Case study of new DDIs: (1) Ziprasidone and Ribociclib, (2) Secobarbital and Arotinolol, (3) Ajmaline and Ergocalciferol, (4) Terlipressin and Amitriptyline, (5) Treprostinil and Methylphenobarbital, and (3) Canagliflozin and Carvedilol. TABLE I: Experimental results ($\uparrow$) (mean$\pm$std). Model | Accuracy (%) | AUROC (%) | F1 (x100)
---|---|---|---
GCN (ICLR’17) | 81.42 $\pm$ 0.19 | 89.67 $\pm$ 0.14 | 82.94 $\pm$ 0.23
GIN (ICLR’19) | 96.08 $\pm$ 0.20 | 98.99 $\pm$ 0.04 | 96.18 $\pm$ 0.19
GAT (ICLR’22) | 94.00 $\pm$ 0.12 | 98.13 $\pm$ 0.05 | 94.22 $\pm$ 0.12
MHCA (ICML-W’19) | 78.51 $\pm$ 0.15 | 86.33 $\pm$ 0.28 | 83.31 $\pm$ 0.21
SSI-DDI (BiB’19) | 96.33 $\pm$ 0.09 | 98.95 $\pm$ 0.08 | 96.38 $\pm$ 0.09
GMPNN (BiB’22) | 95.30 $\pm$ 0.05 | 98.46 $\pm$ 0.01 | 95.39 $\pm$ 0.05
VGAE (NIPS-W’22) | 92.12 $\pm$ 0.09 | 91.62 $\pm$ 0.14 | 89.12 $\pm$ 0.08
SA-DDI (Ch.Sci.’22) | 96.23 $\pm$ 0.13 | 98.80 $\pm$ 0.08 | 96.29 $\pm$ 0.12
MSAN (CIKM’22) | 97.00 $\pm$ 0.09 | 99.27 $\pm$ 0.03 | 97.04 $\pm$ 0.08
DSN-DDI (BiB’23) | 96.94 $\pm$ 0.12 | 99.47 $\pm$ 0.08 | 96.93 $\pm$ 0.11
Ours (CADGL) | 98.21 $\pm$ 0.17 | 99.49 $\pm$ 0.14 | 97.79 $\pm$ 0.16

###  IV-C Ablation Studies

####  IV-C1 Effectiveness of context-aware deep graph encoder

Here in Table II, the study is conducted on our graph encoder and several
baseline graph encoders. We change only the graph encoder by baselines. The
latent information encoder and the decoder stay the same as our model in this
ablation study. The study finds that our graph encoder outperforms all other
alternatives, owing to its ability to aggregate node features by capturing
local information and improve acquired features via self-supervision.

TABLE II: Comparison of different graph encoders. Encoder | Accuracy (%) | AUROC | F1 Score
---|---|---|---
GCN | 92.12 $\pm$ 0.09 | 91.62 $\pm$ 0.14 | 89.12 $\pm$ 0.08
GAT | 94.14 $\pm$ 0.12 | 94.53 $\pm$ 0.05 | 92.22 $\pm$ 0.14
SAGE | 94.83 $\pm$ 0.31 | 95.68 $\pm$ 0.24 | 93.78 $\pm$ 0.11
GIN | 94.51 $\pm$ 0.13 | 95.74 $\pm$ 0.32 | 94.01 $\pm$ 0.29
Ours | 98.21 $\pm$ 0.17 | 99.49 $\pm$ 0.14 | 97.79 $\pm$ 0.16

####  IV-C2 Impact of context pre-processors.

The ablation study conducted in Table III highlights the significance of both
Local and Molecular Context Pre-processors (Local CP and Molecular CP) in
achieving enhanced model performance. MCP has a more notable impact on the
scores. The omission of either pre-processor notably reduces the model’s
performance, whereas the use of both enhances the scores.

TABLE III: Effectiveness of each pre-processors. Local CP | Molecular CP | Accuracy (%)
---|---|---
✗ | ✗ | -
✓ | ✗ | 91.910 $\pm$ 0.236%
✗ | ✓ | 93.662 $\pm$ 0.152%
✓ | ✓ |  98.211 $\pm$ 0.171%

Figure 3 illustrates the validation loss, AUROC and AUPRC scores during the
initial 80 epochs across various configurations. The results indicate that the
model employing MCP shows a rapid initial learning phase, albeit reaching
stabilization swiftly. Conversely, the combination of both LCP and MCP
demonstrates a comparatively slower learning pace, accompanied by more
pronounced improvements over time.

####  IV-C3 Effectiveness of our VGAE

Table I shows baseline model evaluations without a VGAE, while Table II
presents results with VGAE. Adding a latent information encoder and decoder to
GCN, which uses semi-supervised feature extraction, significantly boosts
accuracy from 81.42% to 92.12%. However, GIN encoder’s generative feature is
ineffective due to fully isolating test and validation nodes and edges,
hindering precise mapping to the latent space. GATs prioritize expressive node
representations, but in a generative setting, our graph encoder with LCP, MCP,
and SSGAttn components achieves the most optimal performance.

##  V Experimental Case Studies

For real-world impact, a DDI prediction model should be able to predict novel
undiscovered DDI in a very effective and productive manner, so that the drug
development process can be accelerated.

In Table IV, we present some novel possible DDIs to illustrate our model’s
capability to accurately predict potential DDIs. The DrugBank (
https://go.drugbank.com and ECGWave website https://ecgwaves.com/ ) is used to
obtain evidence for the DDIs and verify. The table also displays a set of
potential DDIs that are very likely to occur but have not been clinically
identified yet. We also present a clinical research analysis of the
undiscovered DDIs, exploring their possibility of occurring clinically in real
life. As they are still undiscovered, we will explore related studies to
determine if our model’s predictions align with real-world situations. More
examples are included in the appendix.

###  V-A Clinical Study Analysis on Novel DDIs

Here, we analyze clinical studies of some newly predicted DDIs for drug
development (visualized in Figure 4) identified by our model for potential
clinical application in order to validate predictions and model performance:

1\. QTc-prolonging activities of Ribociclib increases by Ziprasidone. QTc-
prolonging happens when it takes longer than usual for the heart to recharge
between beats [27]. Ziprasidone can effectively reduce the rate and time of
relapses in schizophrenia and it prolongs QTc interval and may block potassium
channels in cardiac cells [12]. Combination of Ziprasidone with Ribociclib can
increase the risk of QTc prolongation [28] as predicted by our model with
90.109% confidence, implicating probability of creating a novel drug which can
be beneficial in this case.

2\. Hypotensive activities of Arotinolol increases by Secobarbital.
Hypotension is a decrease in systemic blood pressure below accepted low
values. While there is not an accepted standard hypotensive value, pressures
less than 90/60 are recognized as hypotensive [29]. Our model suggests (with
99.883% confidence) that combining Arotinolol, which is being studied for its
potential as antihypertensive therapy [30], with Secobarbital having
anaesthetic, anticonvulsant, sedative, and hypnotic properties [12] may result
in increased efficacy in hypotensive activities.

3\. Arrhythmogenic fixation activities of Ajmaline increases by
Ergocalciferol. Arrhythmogenic activities means the ability of a substance or
condition to cause arrhythmias or abnormal heart rhythms. Ajmaline, a class 1A
antiarrhythmic, enhances heart rhythm by modulating sodium channels [31],
fixing arrhythmogenic problems. Ergocalciferol, a form of vitamin D, has been
shown to have a positive impact on cardiovascular health. [32] So, the
likelihood of intensifying arrhythmogenic fixation effects is noteworthy, as
noted by our model.

4\. Antihypertensive activities of Terlipressin decreases by Amitriptyline.
Antihypertensive activities of drugs refer to the ability of drugs to help
lowering high blood pressure. Terlipressin’s pharmacokinetic and molecular
attributes confer some benefits like preventing rebound hypotension upon drug
cessation [33]. Amitriptyline with analgesic properties, can lead to ECG
changes, and increased glucose level [34]. Given contradictory outputs, our
model suggestion of reduced antihypertension can be effective.

5\. Hypotensive activities of Treprostinil increases by Methylphenobarbital.
Treprostinil exhibits hypotensive effects by inhibiting pulmonary vasculature
vasoconstriction [35]. Methylphenobarbital is usually utilized for tension
headaches and migraines, exerting sedative effects through CNS depression.
Combining both substances can result in increased hypotensive effects as per
our model.

6\. Hypoglycemic activities of Canagliflozin increases by Carvedilol.
Hypoglycemic activity refers to the ability of a drug to help lowering the
blood sugar, often used for treating diabetes. Canagliflozin has been shown to
improve glycemic control in patients with type 2 diabetes (T2D) [36]. In a
study on recurrently hypoglycemic rats, carvedilol was found to improve
hypoglycemia awareness [37]. So, there is a significant potential for
Carvedilol to amplify the hypoglycemic effects of Canagliflozin, as suggested
by our model.

TABLE IV: Potential new DDIs. Serial Number | Drug 1 ($\mathcal{D}_{1}$) | Drug 2 ($\mathcal{D}_{2}$) | Interaction Type (t) | Probability of Occuring
---|---|---|---|---
New Drug-Drug Interactions (Clinically Verified via DrugBank)
1 | Verapamil | Pravastatin | Drug 2 Serum concentration increases ($\uparrow$) | 99.919%
2 | Voriconazole | Apixaban | Drug 2 Metabolism decreases ($\downarrow$) | 99.624%
3 | Fentanyl | Vortioxetine | Risk of adverse effects increases ($\uparrow$) | 99.549%
4 | Vemurafenib | Rifaximin | Drug 2 Serum Concentration increases ($\uparrow$) | 99.849%
5 | Erythromycin | Fentanyl | Drug 2 Metabolism decreases ($\downarrow$) | 99.385%
Possible Novel Drug-Drug Interactions (Not Clinically Verified Yet)
1 | Secobarbital | Arotinolol | Drug 2 hypotensive activities increases ($\uparrow$) | 99.883%
2 | Mesalazine | Nafamostat | Drug 2 anticoagulant activities increases ($\uparrow$) | 99.833%
3 | Voriconazole | Etoposide | Drug 2 Serum Concentration increases ($\uparrow$) | 99.816%
4 | Risperidone | Propofol | Risk of adverse effects increases ($\uparrow$) | 99.660%
5 | Pipamperone | Bupivacaine | Risk of adverse effects increases ($\uparrow$) | 99.503%
6 | Moricizine | Hydralazine | Drug 2 hypotensive activities increases ($\uparrow$) | 99.491%
7 | Isoniazid | Tazarotene | Drug 2 Metabolism decreases ($\downarrow$) | 99.420%
8 | Methylphenobarbital | Treprostinil | Drug 2 hypotensive activities increases ($\uparrow$) | 99.366%
9 | Nevirapine | Vinorelbine | Drug 2 Serum Concentration decreases ($\downarrow$) | 99.337%
10 | Dabrafenib | Pentamidine | Drug 2 QTc-prolongation increases ($\uparrow$) | 98.813%

##  VI Discussion

Continuous new drug development is important for many reasons, such as
treating new diseases, preventing the spread of infections, overcoming drug
resistance, creating better drugs with less side-effects, and improving the
quality of life for patients. However, developing new drugs is a complex and
costly process that requires scientific innovation, rigorous testing, and
regulatory approval [38], which can be accelerated utilizing DDIs. Our
approach has also demonstrated encouraging results in advancing new drug
development using DDIs. Our model presents intriguing DDIs discussed through
case studies involving CVD, hyper/hypotension, and hypoglycemic activities,
showing significant potential for clinical testing and substantial impact.

It is also very important to clarify that our work does not guarantee the
development of novel drugs, nor do we lay claim to the ability to create new
pharmaceuticals directly. Instead, our focus lies on expediting the drug
discovery process by producing effective drug-drug interaction combinations
and gauging their potential efficacy. The primary goal is to provide valuable
insights so that clinical researchers can streamline their efforts, avoiding
exhaustive testing that consumes significant time and resources. Drug
development is an intricate and multifaceted undertaking. Our contribution is
positioned as a facilitative effort aimed at easing the burden on researchers
by showing the possibilities of new drugs with the help of artificial
intelligence. By identifying promising combinations and assessing their
potential effectiveness, our research aims to empower clinical researchers by
offering a valuable roadmap, enabling researchers to navigate the complexities
of drug development more efficiently. Our goal is to contribute to the
scientific community by reducing the time and effort expended on extensive
pre-clinical testing, ultimately expediting the translation of discoveries
into clinical trials.

##  VII Conclusion

In this paper, we introduce CADGL, a novel context-aware deep graph learning
framework that endeavors to predict DDIs. Our approach includes a three-
component variational graph autoencoder architecture with a graph encoder, a
latent information encoder, and a decoder. The context-aware deep graph
encoder captures node embeddings, the latent information encoder transforms
embeddings and properties enabling generative modeling, and an MLP decoder
predicts links between drug nodes. Effectiveness of CADGL is supported by
compelling evidence of possible new drugs which can be developed by newly
identified DDIs, and extensive experimentation confirms its superiority over
various state-of-the-art models.

## Acknowledgments

This work was partly supported by (1) Institute of Information &
communications Technology Planning & Evaluation (IITP) grant funded by the
Korea government (MSIT) (No.2020-0-01373, Artificial Intelligence Graduate
School Program (Hanyang University)) and (2) the National Research Foundation
of Korea (NRF) grant funded by the Korea government (*MSIT)
(No.2018R1A5A7059549). *Ministry of Science and ICT.

## References

  * [1] M. Zitnik, M. Agrawal, and J. Leskovec, “Modeling polypharmacy side effects with graph convolutional networks,” _Bioinformatics_ , vol. 34, no. 13, pp. i457–i466, 2018.
  * [2] Y. Yu, K. Huang, C. Zhang, L. M. Glass, J. Sun, and C. Xiao, “Sumgnn: multi-typed drug interaction prediction via efficient knowledge graph summarization,” _Bioinformatics_ , vol. 37, no. 18, pp. 2988–2995, 2021.
  * [3] X. Lin, Z. Quan, Z.-J. Wang, T. Ma, and X. Zeng, “Kgnn: Knowledge graph neural network for drug-drug interaction prediction.” in _IJCAI_ , vol. 380, 2020, pp. 2739–2745.
  * [4] T. N. Kipf and M. Welling, “Semi-supervised classification with graph convolutional networks,” in _International Conference on Learning Representations_ , 2017. [Online]. Available: https://openreview.net/forum?id=SJU4ayYgl
  * [5] S. Brody, U. Alon, and E. Yahav, “How attentive are graph attention networks?” in _International Conference on Learning Representations_ , 2022. [Online]. Available: https://openreview.net/forum?id=F72ximsx7C1
  * [6] K. Xu, W. Hu, J. Leskovec, and S. Jegelka, “How powerful are graph neural networks?” in _International Conference on Learning Representations_ , 2019. [Online]. Available: https://openreview.net/forum?id=ryGs6iA5Km
  * [7] X. Zhu, Y. Shen, and W. Lu, “Molecular substructure-aware network for drug-drug interaction prediction,” in _Proceedings of the 31st ACM International Conference on Information and Knowledge Management_ , ser. CIKM ’22. New York, NY, USA: Association for Computing Machinery, 2022, p. 4757–4761. [Online]. Available: https://doi.org/10.1145/3511808.3557648
  * [8] A. Deac, Y.-H. Huang, P. Veličković, P. Liò, and J. Tang, “Drug-drug adverse effect prediction with graph co-attention,” 2019.
  * [9] A. K. Nyamabo, H. Yu, Z. Liu, and J.-Y. Shi, “Drug–drug interaction prediction with learnable size-adaptive molecular substructures,” _Briefings in Bioinformatics_ , vol. 23, no. 1, 10 2021, bbab441. [Online]. Available: https://doi.org/10.1093/bib/bbab441
  * [10] A. K. Nyamabo, H. Yu, and J.-Y. Shi, “SSI–DDI: substructure–substructure interactions for drug–drug interaction prediction,” _Briefings in Bioinformatics_ , vol. 22, no. 6, 05 2021, bbab133. [Online]. Available: https://doi.org/10.1093/bib/bbab133
  * [11] K. N. Ngo, T. S. Hy, and R. Kondor, “Predicting drug-drug interactions using deep generative models on graphs,” in _NeurIPS 2022 AI for Science: Progress and Promises_ , 2022. [Online]. Available: https://openreview.net/forum?id=Hnax-9OBNtH
  * [12] D. S. Wishart, Y. D. Feunang, A. C. Guo, E. J. Lo, A. Marcu, J. R. Grant, T. Sajed, D. Johnson, C. Li, Z. Sayeeda, N. Assempour, I. Iynkkaran, Y. Liu, A. Maciejewski, N. Gale, A. Wilson, L. Chin, R. Cummings, D. Le, A. Pon, C. Knox, and M. Wilson, “DrugBank 5.0: a major update to the DrugBank database for 2018,” _Nucleic Acids Res._ , vol. 46, no. D1, pp. D1074–D1082, Jan. 2018.
  * [13] D. Duvenaud, D. Maclaurin, J. Aguilera-Iparraguirre, R. Gómez-Bombarelli, T. Hirzel, A. Aspuru-Guzik, and R. P. Adams, “Convolutional networks on graphs for learning molecular fingerprints,” in _Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 2_ , ser. NIPS’15. Cambridge, MA, USA: MIT Press, 2015, p. 2224–2232.
  * [14] T. Nguyen, H. Le, T. P. Quinn, T. Nguyen, T. D. Le, and S. Venkatesh, “GraphDTA: predicting drug–target binding affinity with graph neural networks,” _Bioinformatics_ , vol. 37, no. 8, pp. 1140–1147, 10 2020. [Online]. Available: https://doi.org/10.1093/bioinformatics/btaa921
  * [15] J. Lim, S. Ryu, K. Park, Y. J. Choe, J. Ham, and W. Y. Kim, “Predicting drug-target interaction using a novel graph neural network with 3d structure-embedded graph representation,” _Journal of chemical information and modeling_ , 2019. [Online]. Available: https://api.semanticscholar.org/CorpusID:201631371
  * [16] W. Torng and R. B. Altman, “Graph convolutional neural networks for predicting drug-target interactions,” _bioRxiv_ , 2018. [Online]. Available: https://api.semanticscholar.org/CorpusID:92451162
  * [17] “Relation-aware graph structure embedding with co-contrastive learning for drug–drug interaction prediction,” _Neurocomputing_ , vol. 572, p. 127203, 2024.
  * [18] T. N. Kipf and M. Welling, “Variational graph auto-encoders,” 2016.
  * [19] M. Simonovsky and N. Komodakis, “Graphvae: Towards generation of small graphs using variational autoencoders,” in _International Conference on Artificial Neural Networks_ , 2018. [Online]. Available: https://api.semanticscholar.org/CorpusID:3637466
  * [20] G. Landrum, P. Tosco, B. Kelley, Ric, sriniker, D. Cosgrove, gedeck, R. Vianello, NadineSchneider, E. Kawashima, D. N, G. Jones, A. Dalke, B. Cole, M. Swain, S. Turk, AlexanderSavelyev, A. Vaucher, M. Wójcikowski, I. Take, D. Probst, K. Ujihara, V. F. Scalfani, guillaume godin, A. Pahl, F. Berenger, JLVarjo, R. Walker, jasondbiggs, and strets123, “rdkit/rdkit: 2023_03_1 (q1 2023) release,” Apr. 2023. [Online]. Available: https://doi.org/10.5281/zenodo.7880616
  * [21] Y. Xie, Z. Xu, J. Zhang, Z. Wang, and S. Ji, “Self-supervised learning of graph neural networks: A unified review,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 45, no. 2, pp. 2412–2429, 2023.
  * [22] T. Cai, S. Luo, K. Xu, D. He, T.-Y. Liu, and L. Wang, “Graphnorm: A principled approach to accelerating graph neural network training,” in _Proceedings of the 38th International Conference on Machine Learning_ , ser. Proceedings of Machine Learning Research, M. Meila and T. Zhang, Eds., vol. 139. PMLR, 18–24 Jul 2021, pp. 1204–1215. [Online]. Available: https://proceedings.mlr.press/v139/cai21e.html
  * [23] M. Günder, N. Piatkowski, and C. Bauckhage, “Full kullback-leibler-divergence loss for hyperparameter-free label distribution learning,” 2022.
  * [24] D. Kim and A. Oh, “How to find your friendly neighborhood: Graph attention design with self-supervision,” in _International Conference on Learning Representations_ , 2021. [Online]. Available: https://openreview.net/forum?id=Wi5KUNlqWty
  * [25] Z. Li, S. Zhu, B. Shao, X. Zeng, T. Wang, and T.-Y. Liu, “DSN-DDI: an accurate and generalized framework for drug–drug interaction prediction by dual-view representation learning,” _Briefings in Bioinformatics_ , vol. 24, no. 1, p. bbac597, 01 2023. [Online]. Available: https://doi.org/10.1093/bib/bbac597
  * [26] Z. Yang, W. Zhong, Q. Lv, and C. Yu-Chian Chen, “Learning size-adaptive molecular substructures for explainable drug–drug interaction prediction by substructure-aware graph neural network,” _Chemical Science_ , vol. 13, no. 29, p. 8693–8703, 2022.
  * [27] J.-P. Valentin, P. Hoffmann, C. Ortemann-Renon, J. Koerner, J. Pierson, G. Gintant, J. Willard, C. Garnett, M. Skinner, H. M. Vargas, T. Wisialowski, and M. K. Pugsley, “The Challenges of Predicting Drug-Induced QTc Prolongation in Humans,” _Toxicological Sciences_ , vol. 187, no. 1, pp. 3–24, 02 2022. [Online]. Available: https://doi.org/10.1093/toxsci/kfac013
  * [28] D. Taylor, “Ziprasidone in the management of schizophrenia : the QT interval issue in context,” _CNS Drugs_ , vol. 17, no. 6, pp. 423–430, 2003.
  * [29] S. Sharma, M. F. Hashmi, and P. T. Bhattacharya, “Hypotension - statpearls - ncbi bookshelf,” _NCBI Bookshelf_ , Feb 2023. [Online]. Available: https://www.ncbi.nlm.nih.gov/books/NBK499961/
  * [30] H. Wu, Y. Zhang, J. Huang, Y. Zhang, G. Liu, N. Sun, Z. Yu, and Y. Zhou, “Clinical trial of arotinolol in the treatment of hypertension: dippers vs. non-dippers,” _Hypertens. Res._ , vol. 24, no. 5, pp. 605–610, Sep. 2001.
  * [31] J. Brugada, “The ajmaline challenge in brugada syndrome a useful tool or misleading information?” _European Heart Journal_ , vol. 24, no. 12, pp. 1085–1086, Jun. 2003. [Online]. Available: https://doi.org/10.1016/s0195-668x(03)00232-x
  * [32] B. A. Ference, I. D. Graham, L. Tokgozoglu, and A. L. Catapano, “Impact of lipids on cardiovascular health: Jacc health promotion series,” 2018.
  * [33] A. B. Pesaturo, H. R. Jennings, and S. A. Voils, “Terlipressin: Vasopressin analog and novel drug for septic shock,” _Annals of Pharmacotherapy_ , vol. 40, no. 12, pp. 2170–2177, Dec. 2006. [Online]. Available: https://doi.org/10.1345/aph.1h373
  * [34] H. M. Bryson and M. I. Wilde, “Amitriptyline,” _Drugs and Aging_ , vol. 8, no. 6, pp. 459–476, Jun. 1996.
  * [35] R. W. Chapman, Z. Li, M. R. Corboz, H. Gauani, A. J. Plaunt, D. M. Konicek, F. Leifer, C. E. Laurent, H. Yin, D. Salvail, C. Dziak, W. Perkins, and V. Malinin, “Inhaled hexadecyl-treprostinil provides pulmonary vasodilator activity at significantly lower plasma concentrations than infused treprostinil.” 2018.
  * [36] T. Minami, A. Kameda, and Y. Terauchi, “An evaluation of canagliflozin for the treatment of type 2 diabetes: an update,” _Expert Opinion on Pharmacotherapy_ , vol. 22, no. 16, pp. 2087–2094, Jun. 2021. [Online]. Available: https://doi.org/10.1080/14656566.2021.1939675
  * [37] R. Farhat, G. Su, and O. Chan, “Carvedilol prevents counterregulatory failure in recurrently hypoglycemic rats,” 2018.
  * [38] X. Lin, L. Dai, Y. Zhou, Z.-G. Yu, W. Zhang, J.-Y. Shi, D.-S. Cao, L. Zeng, H. Chen, B. Song, P. S. Yu, and X. Zeng, “Comprehensive evaluation of deep and graph learning on drug–drug interactions prediction,” _Briefings in Bioinformatics_ , vol. 24, no. 4, p. bbad235, 07 2023.


# Self-energy correction to energy levels of highly charged ions in a
path integral formalism

S. Banerjee<EMAIL_ADDRESS>Max Planck Institute for Nuclear Physics,
Heidelberg, Germany  Z. Harman<EMAIL_ADDRESS>Max Planck Institute for
Nuclear Physics, Heidelberg, Germany

###### Abstract

Self-energy corrections to the energy levels of bound electrons are calculated
in the framework of path integrals. We arrive at the full fermion propagator,
using methods of functional integrals, in the form of Schwinger-Dyson equation
(SDE). From the full fermion SDE, the self-energy corrected propagator is
identified and the energy shift is obtained from the poles of the spectral
function. The numerical calculations are performed using complex contour
integrals and the B-spline representation of basis functions. We identify ions
with Lamb shifts observable via modern mass spectrometric methods.

##  I Introduction

Radiative corrections to energy levels have been at the focal point of
theoretical and experimental studies with atoms, and, more recently, highly
charged ions (HCI). Experimental advances in the production of HCI and the
measurement of their properties with unprecedented accuracy (see e.g. Refs.
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]) call for
versatile theoretical frameworks for the study of such systems. The
calculations pertaining to radiative corrections in quantum electrodynamics
(QED), most importantly, the self-energy (SE) effect, have been rigorously
studied and developed by a multitude of authors [19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. Historically, one of
the first such studies was performed by Gell-Mann, Low and Sucher using the
adiabatic $S$-matrix method [40, 41]. Later, a plethora of theoretical
formalisms based on Green’s functions were developed [42, 43, 44, 45, 46, 47,
48, 49].

In this Letter, we arrive at the SE corrected fermion propagator in the Furry
picture using the method of functional integrals conceived by Feynman [50] and
developed further by several authors, both in non-relativistic and
relativistic quantum mechanics [51, 52, 53, 54, 55, 56, 57, 58, 59]. The SE
correction to energy levels is evaluated for the first time in the path
integral framework. We derive the dressed bound-fermion propagator using the
Schwinger-Dyson equations (SDE). The derivation of the energy shift induced by
the interaction of the bound fermion with its own electromagnetic field is
performed by separating the expression into the so-called zero-, one-, and
many-potential terms [25], each of which are defined using perturbative path
integrals. The finite contributions due to these individual terms are
calculated numerically using complex contour integrals, extensively worked out
in Refs. [25, 27, 34]. Computations are performed by the known B-spline
representation of bound-electron basis states with existing numerical methods
[60].

The introduction of functional methods in atomic physics is also motivated by
the prospects of including non-electromagnetic interactions into precision
theory. For example, hadronic vacuum polarization effects have been calculated
by means of a quantum chromodynamic (QCD) Schwinger-Dyson approach [61, 62].
The improvement of experimental accuracy may necessitate in future the
inclusion of such QCD corrections in atomic spectra [63, 64, 65]. Prospects of
new physics searches with low-energy atomic precision experiments (see e.g.
[66, 67, 68, 69, 70, 2]) also suggest to employ a versatile formalism enabling
the description of various types of exchange bosons.

##  II Schwinger-Dyson equation for the bound-fermion propagator

We begin by deriving the complete expression for the dressed bound-fermion
propagator. This is accomplished by defining the SDE for the fermionic
propagator using path integrals, in analogy to Ref. [71]. The QED Lagrangian
is given as

$\displaystyle\mathcal{L}_{\text{QED}}(x)=$
$\displaystyle-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2\xi}(\partial^{\mu}A_{\mu})^{2}$
$\displaystyle+\bar{\psi}(x)(i\not{D}-m)\psi(x)-\bar{\psi}(x)e\gamma^{\mu}A_{\mu}(x)\psi(x)\,,$

where $D_{\mu}(x)=\partial_{\mu}(x)+ie\mathcal{A}_{\mu}(x)$,
$\mathcal{A}_{\mu}$ is the field of the nucleus; $A_{\mu}$ is the gauge-field
operator for the photon field, $\psi(x)$ is the field of the electron, $m$ is
its bare mass, $e$ is the elementary charge, and
$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$ is the
electromagnetic field operator or the curvature of the field. The $\gamma$ are
the usual Dirac matrices and $\mu,\nu\in\\{0,1,2,3\\}$ represent the Lorentz
indices. We apply the background field method [72, 73, 74], where we consider
the external field to be the classical background field of the nuclear charge
that is gauge invariant, and the photon gauge field is treated as a
fluctuation whose gauge has been fixed as seen in the second term of Eq. (II),
where $\xi$ is the gauge-fixing coefficient. We do not need to concern
ourselves with fixing the gauge for the external field since the effective
action in the presence or absence of a classical background field is
equivalent, and hence the choice of gauge plays no role. We can also safely
ignore this gauge-fixing term since we are interested in deriving the electron
propagator in the presence of an external field, and it does not contribute to
the effective action of the theory.

The generating functional is constructed using the above Lagrangian, and
Grassmann-valued sources $\eta$ and $\bar{\eta}$ of the fermion fields
$\bar{\psi}$ and $\psi$, respectively, and the source $J_{\mu}$ for the gauge
field:

$\displaystyle Z[\eta,\bar{\eta},J_{\mu}]$
$\displaystyle=\int\mathcal{D}\bar{\psi}\mathcal{D}\psi\mathcal{D}A\exp\\{i\int
d^{4}x[\mathcal{L}_{\text{QED}}$
$\displaystyle+J_{\mu}(x)A^{\mu}(x)+\bar{\psi}(x)\eta(x)+\bar{\eta}(x)\psi(x)]\biggr{\\}}\,,$

where $\mathcal{D}$ represents the integral measure over all field
configurations. To arrive at the SDE, we consider that the functional integral
of a total derivative is zero, i.e.,

$\displaystyle\int\mathcal{D}[\phi]\functionalderivative{\phi}=0\,,$

where $\functionalderivative{\phi}$ is the functional derivative w.r.t.
$\phi$, which represents any arbitrary field variable. For the electron
propagator, the derivative is taken with respect to the fermion field
$\bar{\psi}(x)$,

$\displaystyle\int\mathcal{D}[\bar{\psi}\psi
A]\frac{\delta}{\delta\bar{\psi}(x)}\exp\\{i[\mathcal{S}(\bar{\psi},\psi,A)$
(3)
$\displaystyle\quad+\int\differential[4]{x}(J_{\mu}(x)A^{\mu}(x)+\bar{\psi}(x)\eta(x)+\bar{\eta}(x)\psi(x))]\\}=0\,,$

with $\mathcal{S}=\int\differential[4]{x}\mathcal{L}_{\text{QED}}$ being the
action. Eq. (3) can be written in terms of a differential equation in the
generating functional $Z$

$\displaystyle\Biggl{[}\frac{\delta\mathcal{S}}{\delta\bar{\psi}(x)}$
$\displaystyle\left(-i\frac{\delta}{\delta
J_{\mu}},i\frac{\delta}{\delta\eta},-i\frac{\delta}{\delta\bar{\eta}}\right)$
$\displaystyle\qquad\qquad\qquad+\eta(x)\Biggr{]}Z[\eta,\bar{\eta},J_{\mu}]=0\,.$

We take the functional derivative of the action $\mathcal{S}$ by implementing
the Gâteux derivative method [75, 76], and arrive at the differential equation

$\displaystyle\Biggl{[}$
$\displaystyle\Biggl{(}i\not{\partial}-m-e\gamma^{\mu}\mathcal{A}_{\mu}-e\gamma^{\mu}(-i)\functionalderivative{J^{\mu}(x)}\Biggr{)}$
$\displaystyle\times(-i)\functionalderivative{\bar{\eta}(x)}+\eta(x)\Biggr{]}Z[\eta,\bar{\eta},J_{\mu}]=0\,.$

To obtain the two-point Green’s function of the electron, we perform another
derivation with respect to the source field $\eta(y)$ and represent it in
terms of the functional for the connected Green’s functions $W$, ($Z=e^{W}$)

$\displaystyle
e^{W[\eta,\bar{\eta},J_{\mu}]}\biggl{[}\delta(x-y)-\biggl{(}i\not{\partial}-m-e\gamma^{\mu}\mathcal{A}_{\mu}$
(6) $\displaystyle-ie\gamma^{\mu}\frac{(-i)\delta W}{\delta
J^{\mu}(x)}-ie\gamma^{\mu}(-i)\functionalderivative{J^{\mu}(x)}\biggr{)}S(x,y)\biggr{]}=0\,.$

Here, $S(x,y)$ is the bound-fermion propagator. We rewrite Eq. (6) in terms of
classical fields, in analogy to [71]

$\displaystyle\delta(x-y)-\biggl{(}i\not{\partial}-m-e\gamma^{\mu}\mathcal{A}_{\mu}$
(7) $\displaystyle-
ie\gamma^{\mu}A_{\mu}-ie\gamma^{\mu}(-i)\functionalderivative{J^{\mu}(x)}\biggr{)}S(x,y)=0\,.$

We obtain the equation

$\displaystyle(-i)\frac{\delta S(x,y)}{\delta J^{\mu}(x)}$ (8)
$\displaystyle\qquad=-i\int\differential[4]{z}\frac{\delta A_{\nu}(z)}{\delta
J^{\mu}(x)}\functionalderivative{A_{\nu}(z)}\left(\frac{\delta^{2}\Gamma}{\delta\psi(x)\delta\bar{\psi}(y)}\right)^{-1}\,.$

Using the expressions for the complete bound-electron propagator, photon
propagator and the electron-photon vertex function $\Gamma^{\mu}$, one obtains

$\displaystyle(-i)\frac{\delta S(x,y)}{\delta J^{\mu}(x)}$ (9)
$\displaystyle=-e\int\differential[4]{z}\differential[4]{u}\differential[4]{w}D_{\mu\nu}(x-z)S(x,w)\Gamma^{\nu}(w,u;z)S(u,y)\,.$

Using Eqs. (8) and (9), setting the external source fields equal to zero, and
considering that the nuclear field is a static scalar field, Eq. (7) reduces
to

$\displaystyle\delta(x-y)$ $\displaystyle=$
$\displaystyle(i\not{\partial}-m-e\gamma^{0}\mathcal{A}_{0})S(x,y)$
$\displaystyle+\int\differential[4]{u}\Sigma(x,u)S(u,y)\,,$

where

$\Sigma(x-y)=-ie^{2}\gamma^{\mu}\int\differential[4]{z}\differential[4]{w}D_{\mu\nu}(z-x)S(x,w)\Gamma^{\nu}(w,y;z)\,.$
(11)

To obtain the SDE for the bound propagator in coordinate space, we multiply
Eq. (II) throughout with the inverse propagator, yielding

$\displaystyle S^{-1}$
$\displaystyle(x,y)=(i\not{\partial}-m-e\gamma^{\mu}\mathcal{A}^{0})\delta(x-y)$
$\displaystyle-
ie^{2}\gamma^{\mu}\int\differential[4]{z}\differential[4]{w}D_{\mu\nu}(z-x)S(x,w)\Gamma^{\nu}(w,y;z)\,.$

This equation is pictorially represented in Fig. (1).

Figure 1: Diagrammatic representation of the Schwinger-Dyson equation for the
electron propagator. The double line represents the propagator in the nuclear
Coulomb field, the wave line represents a virtual photon, while the thick line
depicts the full, dressed electron propagator.

##  III Derivation of the self-energy shift

The second term on the r.h.s. of Eq. (II), given by Eq. (11) gives the SE
corrected bound propagator and is the well-known SE operator. Fourier
transforming this with respect to the time variable and considering the
electron-photon vertex operator $\Gamma^{\nu}$ to lowest order, i.e. replacing
it with the matrix $\gamma^{\nu}$, we obtain

$\displaystyle\Sigma(E)=-ie^{2}\int\frac{\differential{\omega}}{2\pi}\gamma^{\mu}S(\mathbf{x},\mathbf{w};E-\omega)D_{\mu\nu}(\mathbf{z}-\mathbf{x};\omega)\gamma^{\nu}\,.$
(13)

Using this SE operator and Feynman rules [77], we can construct the SE
corrected Green’s function between points $i$ and $f$:

$\displaystyle G(\mathbf{x}_{f},E_{f};\mathbf{x}_{i},E_{i})\sim
ie^{2}\bigg{(}\frac{i}{2\pi}\bigg{)}^{2}\int d^{3}\mathbf{z}_{1}\int
d^{3}\mathbf{z}_{2}\int d\omega$ $\displaystyle\qquad\times
S(\mathbf{x}_{f},\mathbf{z}_{2};E_{f})e\gamma^{\mu}S(\mathbf{z}_{2},\mathbf{z}_{1};\eta)e\gamma^{\nu}$
$\displaystyle\qquad\times
S(\mathbf{z}_{1},\mathbf{x}_{i};E_{i})D_{\mu\nu}(\mathbf{z}_{1}-\mathbf{z}_{2};\omega)\delta(E_{f}-E_{i})\,.$
(14)

A Green’s function, e.g. that in Eq. (14) can also be given in the spectral
representation

$\displaystyle
G(\mathbf{x}_{f},\mathbf{x}_{i},E)=\sum_{n}\frac{\phi_{n}(\mathbf{x}_{f})\bar{\phi}_{n}(\mathbf{x}_{i})}{E-E_{n}(1-i\varepsilon)}\,$
(15)

in terms of the perturbed states ${\phi}_{n}$, where we have made the
replacement $E=E_{i}=E_{f}$, and $\varepsilon$ is infinitesimally small. The
spectral function $G_{a}(E)\equiv\bra{a}G(E)\ket{a}$ for a given atomic
reference state $\ket{a}$, which contains the perturbed eigenenergies $E_{n}$
of the basis states $\phi_{n}$, has a pole around the perturbed eigenenergy of
state $\ket{a}$:

$\displaystyle G_{a}(E)\approx\frac{C_{a}}{E-E_{a}}\,,$ (16)

where the constant $C_{a}$ is the residue term. While in case of the path
integral treatment of the quantum mechanical H atom the poles can be simply
seen in the analytical expression of the Green’s function [55, 78], here the
energies can be determined from the poles using complex contour integration
[79, 80, 44, 43]. Considering a small contour $\Gamma$ which surrounds an
isolated pole at the bound-state energy $E_{a}$, one can easily obtain

$\displaystyle\frac{1}{2\pi
i}\oint_{\Gamma}dE\,E\,G_{a}(E)=E_{a}C_{a}\,,\quad\text{and}$ (17)
$\displaystyle\frac{1}{2\pi i}\oint_{\Gamma}dE\,G_{a}(E)=C_{a}\,.$ (18)

The ratio of the above equations gives us the level energy

$\displaystyle E_{a}=\frac{\frac{1}{2\pi
i}\oint_{\Gamma}dE\,E\,G_{a}(E)}{\frac{1}{2\pi
i}\oint_{\Gamma}dE\,G_{a}(E)}\,.$ (19)

The energy shift with respect to the unperturbed (Dirac) energy is

$\displaystyle\Delta E_{a}=\frac{\frac{1}{2\pi i}\oint_{\Gamma}dE\,\Delta
E\,\Delta G_{a}(E)}{1+\frac{1}{2\pi i}\oint_{\Gamma}dE\,\Delta G_{a}(E)}\,,$
(20)

where

$\displaystyle\Delta E_{a}\equiv\Delta E_{a}^{(1)}+\Delta
E_{a}^{(2)}+\dots\,,$ $\displaystyle\Delta G_{a}\equiv\Delta
G_{a}^{(1)}+\Delta G_{a}^{(2)}+\dots\,,$

have been expanded in terms of the fine-structure constant, $\alpha$, in a
perturbation series, and $\Delta G_{a}=G_{a}(E)-\Delta G_{a}^{(0)}$; $\Delta
G_{a}^{(0)}=\frac{1}{E-E_{a}^{(0)}}$ being the zeroth-order contribution. We
expand the Eq. (20) in a geometric series and obtain

$\displaystyle\Delta E^{(1)}_{a}=\frac{1}{2\pi i}\oint_{\Gamma}dE\,\Delta
E\,\Delta G_{a}^{(1)}(E)\,.$ (21)

We can now construct the function in Eq. (16) from the Green’s function in Eq.
(14) and consider a single state with eigenenergy $E_{a}$,

$\displaystyle
G_{a}(E)=\bigg{(}\frac{i}{2\pi}\bigg{)}^{2}\frac{1}{(E-E_{a})^{2}}$ (22)
$\displaystyle\qquad\times\int
d\omega\sum_{n}\frac{\bra{an}I(\omega)\ket{na}}{E-\omega-
E_{n}(1-i\varepsilon)}\,,$

where we have introduced the photon exchange operator [34]
$I(\mathbf{z}_{1}-\mathbf{z}_{2};\omega)=e^{2}\alpha^{\mu}\alpha^{\nu}D_{\mu\nu}(\mathbf{z}_{1}-\mathbf{z}_{2};\omega)$
with $\alpha^{\mu}=\gamma^{0}\gamma^{\mu}$. Thus, we have the SE correction to
the bound-state energy level, using Eqs. (17)-(21), as

$\displaystyle\Delta E_{a}^{(1)}=\bra{a}\Sigma(E_{a})-\delta
m\gamma^{0}\ket{a}\,,$ (23)

where $\bra{a}\Sigma(E_{a})\ket{b}=\frac{i}{2\pi}\int
d\omega\sum_{n}\frac{\bra{an}I(\omega)\ket{nb}}{E-\omega-
E_{n}(1-i\varepsilon)}$; $\ket{an}$ denotes a two-electron tensor product
state and the $\delta m$ is the mass counter-term.

Eq. (23) is however fraught with divergences and we follow [27, 34] and
separate the SE shift into zero-, one-, and many-potential terms

$\displaystyle\bra{a}\Sigma(E_{a})\ket{a}$ $\displaystyle=$
$\displaystyle\bra{a}\Sigma^{(0)}(E_{a})\ket{a}$ $\displaystyle+$
$\displaystyle\bra{a}\Sigma^{(1)}(E_{a})\ket{a}+\bra{a}\Sigma^{(2+)}(E_{a})\ket{a}\,.$

The individual terms in Eq. (III) can be written in terms of the Coulomb-Dirac
Green’s functions [78], and the photon propagator using perturbative path
integrals[52, 54, 81]. The zero-potential term is given as

$\displaystyle\bra{a}\Sigma^{(0)}(E_{a})\ket{a}$ (25)
$\displaystyle=2i\alpha\int\differential{\omega}\int\differential[3]{\mathbf{r}_{1}}\differential[3]{\mathbf{r}_{2}}\psi^{\dagger}_{a}(\mathbf{r}_{2})\alpha^{\mu}G^{(0)}(E_{a}-\omega)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\alpha^{\nu}D_{\mu\nu}(\omega)\psi_{a}(\mathbf{r}_{1})\,,$

where $G^{(0)}(E_{a}-\omega)$ is the free electron Green’s function.

Similarly, the one-potential term can be expressed as

$\displaystyle\bra{a}\Sigma^{(1)}(E_{a})\ket{a}$ (26)
$\displaystyle=2i\alpha\int\differential{\omega}\int\differential[3]{\mathbf{r}_{1}}\differential[3]{\mathbf{r}_{2}}\psi^{\dagger}_{a}(\mathbf{r}_{2})\alpha^{\mu}G^{(1)}(E_{a}-\omega)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\alpha^{\nu}D_{\mu\nu}(\omega)\psi_{a}(\mathbf{r}_{1})\,,$

where the Green’s function for a single interaction of the electron with the
nuclear potential $V$ in terms of the free Green’s function is [52, 54, 81]

$\displaystyle G^{(1)}(\mathbf{r}_{2},\mathbf{r}_{1};E_{a}-\omega)$ (27)
$\displaystyle=\left[\int_{0}^{\infty}\differential[3]{\mathbf{x_{1}}}V(\mathbf{x_{1}})\right]\left[\prod_{i=0}^{1}G^{(0)}(\mathbf{x}_{i+1},\mathbf{x}_{i};E_{a}-\omega)\right]$

with $\mathbf{r}_{2}=\mathbf{x}_{2}$ and $\mathbf{r}_{1}=\mathbf{x}_{0}$.
Following the same methodology, the many-potential term is given by

$\displaystyle\bra{a}\Sigma^{(2+)}(E_{a})\ket{a}$ (28)
$\displaystyle=2i\alpha\int\differential{\omega}\int\differential[3]{\mathbf{r}_{1}}\differential[3]{\mathbf{r}_{2}}\psi^{\dagger}_{a}(\mathbf{r}_{2})\alpha^{\mu}G^{(2+)}(E_{a}-\omega)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\alpha^{\nu}D_{\mu\nu}(\omega)\psi_{a}(\mathbf{r}_{1})\,.$

The Green’s function for the many-potential term between the coordinates
$\mathbf{r}_{3}$ and $\mathbf{r}_{4}$, as seen in Fig. (2), with $n$
insertions of the nuclear interaction where $n\rightarrow\infty$, can be given
in terms of the free Green’s function as

$\displaystyle G(\mathbf{r}_{4},\mathbf{r}_{3};E_{a}-\omega)$ (29)
$\displaystyle=\sum_{n=0}^{\infty}\Bigg{\\{}\prod_{k=1}^{n}\left[\int_{0}^{\infty}\differential[3]{\mathbf{x}_{k}}V(\mathbf{x}_{k})\right]$
$\displaystyle\qquad\qquad\qquad\qquad\times\left[\prod_{i=0}^{n}G^{(0)}(\mathbf{x}_{i+1},\mathbf{x}_{i};E_{a}-\omega)\right]\Bigg{\\}}\,.$

This effectively gives us the exact Dirac-Coulomb Green’s function [78].

Figure 2: Diagrammatic representation of the many-potential Green’s function.
Single lines represent a free propagator, while wave lines terminated by a
cross represent the nuclear Coulomb field.

The energy shift due to the many-potential contribution can thus be written as

$\displaystyle\bra{a}\Sigma^{(2+)}(E_{a})\ket{a}$ (30)
$\displaystyle=2i\alpha\int\differential{\omega}\int\differential[3]{\mathbf{r}_{1}}\differential[3]{\mathbf{r}_{2}}\differential[3]{\mathbf{r}_{3}}\differential[3]{\mathbf{r}_{4}}\psi^{\dagger}_{a}(\mathbf{r}_{2})\alpha^{\mu}$
$\displaystyle\quad\times
G^{(0)}(\mathbf{r}_{2},\mathbf{r}_{4};E_{a}-\omega)V(\mathbf{r}_{4})G(\mathbf{r}_{4},\mathbf{r}_{3};E_{a}-\omega)$
$\displaystyle\qquad\quad\times
V(\mathbf{r}_{3})G^{(0)}(\mathbf{r}_{3},\mathbf{r}_{1};E_{a}-\omega)\alpha^{\nu}D_{\mu\nu}(\omega)\psi_{a}(\mathbf{r}_{1})\,.$

Using the spectral representation for the free and bound-electron propagators
[27], the energy shift is given as

$\displaystyle\bra{a}\Sigma^{(2+)}(E_{a})\ket{a}=\frac{i}{2\pi}\int\differential{\omega}$
(31)
$\displaystyle\times\sum_{\alpha,\beta,i}\frac{\bra{i}V\ket{\beta}\bra{a\beta}I(\omega)\ket{\alpha
a}\bra{\alpha}V\ket{i}}{(E_{a}-\omega-\epsilon_{\alpha}{\varepsilon^{+}})(E_{a}-\omega-\epsilon_{i}{\varepsilon^{+}})(E_{a}-\omega-\epsilon_{\beta}{\varepsilon^{+}})}\,,$

where $\ket{\alpha}$ and $\ket{\beta}$ are free-electron states, the $\ket{i}$
represents bound-electron states, and we use the notation
$\varepsilon^{+}=(1-i\varepsilon)$.

Following [27, 34], we introduce frequency-dependent effective basis functions

$\displaystyle\ket{\phi^{(\pm)}_{i}(\omega)}=\sum_{\alpha}\frac{\bra{\alpha}V\ket{i}}{\omega-\epsilon_{\alpha}(1\mp
i\varepsilon)}\ket{\alpha}\,,$ (32)

which simplify Eq. (31) to the form

$\bra{a}\Sigma^{(2+)}(E_{a})\ket{a}=\frac{i}{2\pi}\int\differential{\omega}\sum_{i}\frac{\bra{a\phi_{i}^{(-)}}I(\omega)\ket{\phi_{i}^{(+)}a}}{E_{a}-\omega-\epsilon_{i}({1-i\varepsilon})}\,.$
(33)

One can further reduce Eq. (33) by expanding the numerator inside the integral
on the r.h.s. in partial waves [27] such that the angular integrations can be
performed analytically, yielding the reduced many-potential term expressed
with generalized Slater integrals [34]. The zero- and one-potential terms with
the well-known loop functions are regularized in momentum space following Ref.
[34], and the numerical calculations for the zero-, one-, and many-potential
terms are performed using a B-spline representation of basis states [82, 83,
84], implementing the dual kinetic balance approach [85].

Ion |  $R_{rms}$(fm) | $1s$ | $2s$ | $2p_{1/2}$ | $2p_{3/2}$ | Ref.
---|---|---|---|---|---|---
Hg79+ | 5.475 | 206.08550(5) | 35.3186(2) | 3.155007(6) | 4.530466(6) |
5.475 | 206.1710(1) | - | - | - | [86]
5.4648 | 206.1760(2) | 35.3919(3) | 3.23417(2) | 4.602445 | [87]
0 | 207.18170(6) | 35.56777(1) | 3.243747(2) | 4.604480(2) | [88]
0 | 207.181639 | 35.5677713(2) | 3.243742 | 4.604478 | [87]
Pb81+ | 5.505 | 226.2388(1) | 39.2102(3) | 3.82741(2) | 5.093(2) |
5.5012 | 226.33237(5) | 39.28791(4) | 3.91122(2) | 5.168749 | [87]
0 | 227.61127(5) | 39.515797(6) | 3.924620(4) | 5.171414(2) | [88]
0 | 227.611278 | 39.515805 | 3.924616 | 5.171412 | [87]
U91+ | 5.863 | 354.938(2) | 65.312(2) | 9.4392(2) | 8.853(3) |
5.863 | - | 65.39(6) | 9.52(6) | - | [27]
5.8571 | 355.052(2) | 65.4218(6) | 9.55075(6) | 8.89290(3) | [87]
| 0 | 359.50196(7) | 66.295010(9) | 9.62526(1) | 8.9010555(3) | [88]
| 0 | 359.501961 | 66.295016 | 9.625267 | 8.901058 | [87]
Table 1: SE correction to the energy levels of H-like heavy ions, in eV. We
use a homogeneously-charged sphere as our nuclear model and compare our
results to Refs. [27, 87, 86], and point-like nucleus results from Refs. [88,
87].

##  IV Numerical results

As a test of our method, numerical results for the SE shift in H-like HCI are
shown in Table. 1 and compared to Refs. [27, 88, 87]. Our results are
generally in good agreement with existing tabulations. We used the model of a
homogeneously charged spherical nucleus, with root-mean-square nuclear radii
from Ref. [86]. The differences to other works originate from differences of
the nuclear radii and charge distribution models used.

With the increase of the atomic number, QED effects are boosted to the regime
where they are well observable with novel mass spectrometric methods with
uncertainties on the 1-eV level or below [89, 90, 5, 4]. Such ions will allow,
for the first time, the test of QED via measuring the mass difference of the
H-like ion and the bare nucleus, directly yielding the electronic binding
energy by exploiting the energy-mass equivalence relation. It is not only the
$1s$ hydrogenic ground state which features well observable QED effects, but
also excited states possess sizeable radiative shifts. E.g. the Lamb shift of
the $2s$ state approximates the SE correction to the binding energy of a Li-
like ion, which can be spectrometrically determined by measuring the mass
difference of the Li- and the He-like ions in their ground state. In our
approach presented here, electron interaction effects are neglected, which,
for heavy ions, is a justified first approximation. The formalism however may
be extended in future to many-electron systems, by allowing for the exchange
of photons between electrons. (We note that the Li-like sequence has been
extensively studied in the framework of other formalisms [91, 92, 93].)
Similarly, the SE results given for the $2p_{1/2}$ and $2p_{3/2}$ states (with
the subscripts denoting the total angular momentum $j$) approximate the
radiative shift of the binding energy of the valence electron in the B- and
N-like sequences, respectively. Very heavy ions in these charge states still
feature QED corrections accessible via mass spectrometry.

##  V Summary

We develop an alternate formalism for the evaluation of the SE shift of atomic
energy levels. The Green’s function for the SE corrected bound electron is
extracted from the Schwinger-Dyson equation derived using the functional
integral technique. We avoid the operator formalism and present an elegant and
intuitive framework that preserves all the symmetries of the system, and also
treats the background nuclear field non-perturbatively. This formalism can be
extended to study higher-order radiative corrections, and energy shifts in
many-electron systems. The functional methods developed in this work can be
naturally generalized to various hypothetical gauge bosons, enabling the
detailed study of new physics-effects in atomic spectra.

## Acknowledgements

Supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) – Project-ID 273811115 – SFB 1225.

## References

  * King  _et al._ [2022] S. A. King, L. J. Spieß, P. Micke, A. Wilzewski, T. Leopold, E. Benkler, R. Lange, N. Huntemann, A. Surzhykov, V. A. Yerokhin, J. R. J R Crespo López-Urrutia, and P. O. Schmidt, Nature 611, 43 (2022).
  * Sailer  _et al._ [2022] T. Sailer, V. Debierre, Z. Harman, F. Heiße, C. König, J. Morgner, B. Tu, A. V. Volotka, C. H. Keitel, K. Blaum, and S. Sturm, Nature 606, 479 (2022).
  * Micke  _et al._ [2020] P. Micke, T. Leopold, S. A. King, E. Benkler, L. J. Spieß, L. Schmöger, M. Schwarz, J. R. Crespo López-Urrutia, and P. O. Schmidt, Nature 578, 60 (2020).
  * Rischka  _et al._ [2020] A. Rischka, H. Cakir, M. Door, P. Filianin, Z. Harman, W. J. Huang, P. Indelicato, C. H. Keitel, C. M. König, K. Kromer, M. Müller, Y. N. Novikov, R. X. Schüssler, C. Schweiger, S. Eliseev, and K. Blaum, Phys. Rev. Lett. 124, 113001 (2020).
  * Schüssler  _et al._ [2020] R. X. Schüssler, H. Bekker, M. Braß, H. Cakir, J. R. Crespo López-Urrutia, M. Door, P. Filianin, Z. Harman, M. W. Haverkort, W. J. Huang, P. Indelicato, C. H. Keitel, C. M. König, K. Kromer, M. Müller, Y. N. Novikov, A. Rischka, C. Schweiger, S. Sturm, S. Ulmer, S. Eliseev, and K. Blaum, Nature 581, 42 (2020).
  * Kubiček  _et al._ [2014] K. Kubiček, P. H. Mokler, V. Mäckel, J. Ullrich, and J. R. Crespo López-Urrutia, Phys. Rev. A 90, 032508 (2014).
  * Sturm  _et al._ [2011] S. Sturm, A. Wagner, B. Schabinger, J. Zatorski, Z. Harman, W. Quint, G. Werth, C. H. Keitel, and K. Blaum, Phys. Rev. Lett. 107, 023002 (2011).
  * Gillaspy  _et al._ [2010] J. D. Gillaspy, C. T. Chantler, D. Paterson, L. T. Hudson, F. G. Serpa, and E. Takács, J. Phys. B 43, 074021 (2010).
  * Nakamura  _et al._ [2008] N. Nakamura, A. P. Kavanagh, H. Watanabe, H. A. Sakaue, Y. Li, D. Kato, F. J. Currell, and S. Ohtani, Phys. Rev. Lett. 100, 073203 (2008).
  * Gumberidze  _et al._ [2005] A. Gumberidze, T. Stöhlker, D. Banaś, K. Beckert, P. Beller, H. Beyer, F. Bosch, S. Hagmann, C. Kozhuharov, D. Liesen, F. Nolden, X. Ma, P. Mokler, A. Orsić-Muthig, M. Steck, D. Sierpowski, and S. T. amd Y. Zou, Phys. Rev. Lett. 94, 223001 (2005).
  * Brandau  _et al._ [2002] C. Brandau, T. Bartsch, A. Hoffknecht, H. Knopp, S. Schippers, W. Shi, A. Müller, N. Grün, W. Scheid, T. Steih, F. Bosch, B. Franzke, C. Kozhuharov, P. H. Mokler, F. Nolden, M. Steck, T. Stöhlker, and Z. Stachura, Phys. Rev. Lett. 89, 053201 (2002).
  * Gillaspy [1996] J. D. Gillaspy, Phys. Scr. T65, 168 (1996).
  * Stöhlker  _et al._ [2000] T. Stöhlker, P. H. Mokler, F. Bosch, R. W. Dunford, F. Franzke, O. Klepper, C. Kozhuharov, T. Ludziejewski, F. Nolden, H. Reich, P. Rymuza, Z. Stachura, M. Steck, P. Swiat, and A. Warczak, Phys. Rev. Lett. 85, 3109 (2000).
  * Marmar  _et al._ [1986] E. S. Marmar, J. E. Rice, K. Kallne, J. Kallne, and R. E. LaVilla, Phys. Rev. A 33, 774 (1986).
  * Deslattes  _et al._ [1984] R. D. Deslattes, H. F. Beyer, and F. Folkmann, J. Phys. B 17, L689 (1984).
  * Beiersdorfer  _et al._ [1989] P. Beiersdorfer, M. Bitter, S. von Goeler, and K. W. Hill, Phys. Rev. A 40, 150 (1989).
  * Decaux  _et al._ [1997] V. Decaux, P. Beiersdorfer, S. M. Kahn, and V. L. Jacobs, The Astrophysical Journal 482, 1076 (1997).
  * Tavernier  _et al._ [1985] M. Tavernier, J. P. Briand, P. Indelicato, D. Liesen, and P. Richard, J. Phys. B 18, L327 (1985).
  * Uehling [1935] E. A. Uehling, Phys. Rev. 48, 55 (1935).
  * Wichmann and Kroll [1956] E. H. Wichmann and N. M. Kroll, Phys. Rev. 101, 843 (1956).
  * Desiderio and Johnson [1971] A. M. Desiderio and W. R. Johnson, Phys. Rev. A 3, 1267 (1971).
  * Mohr [1974] P. J. Mohr, Ann. Phys. 88, 52 (1974).
  * Mohr [1975] P. J. Mohr, Phys. Rev. Lett. 34, 1050 (1975).
  * Gyulassy [1975] M. Gyulassy, Nuc. Phys. A 244, 497 (1975).
  * Snyderman [1991] N. J. Snyderman, Ann. Phys. 211, 43 (1991).
  * Indelicato and Mohr [1992] P. Indelicato and P. J. Mohr, Phys. Rev. A 46, 172 (1992).
  * Blundell [1992] S. A. Blundell, Phys. Rev. A 46, 3762 (1992).
  * Persson  _et al._ [1993] H. Persson, I. Lindgren, and S. Salomonson, Phys. Scr. 1993, 125 (1993).
  * Pachucki [1993] K. Pachucki, Ann. Phys. 226, 1 (1993).
  * Pachucki [1994] K. Pachucki, Phys. Rev. Lett. 72, 3154 (1994).
  * Mohr  _et al._ [1998] P. Mohr, G. Plunien, and G. Soff, Phys. Rep. 293, 227 (1998).
  * Jentschura  _et al._ [1999] U. D. Jentschura, P. J. Mohr, and G. Soff, Phys. Rev. Lett. 82, 53 (1999).
  * Fullerton and Rinker [1976] L. W. Fullerton and G. A. Rinker, Phys. Rev. A 13, 1283 (1976).
  * Yerokhin and Shabaev [1999] V. A. Yerokhin and V. M. Shabaev, Phys. Rev. A 60, 800 (1999).
  * Plunien and Soff [1999] G. Plunien and G. Soff, Quantum electrodynamics of highly charged ions, in  _Atomic Physics with Heavy Ions_, edited by H. F. Beyer and V. P. Shevelko (Springer Berlin Heidelberg, Berlin, Heidelberg, 1999) pp. 63–91.
  * Shabaev  _et al._ [2000] V. M. Shabaev, A. N. Artemyev, and V. A. Yerokhin, Phys. Scr. 2000, 7 (2000).
  * Yerokhin  _et al._ [2004] V. A. Yerokhin, P. Indelicato, and V. M. Shabaev, Phys. Rev. A 69, 052503 (2004).
  * Yerokhin  _et al._ [2021] V. A. Yerokhin, C. H. Keitel, and Z. Harman, Phys. Rev. A 104, 022814 (2021).
  * Sapirstein [1998] J. Sapirstein, Rev. Mod. Phys. 70, 55 (1998).
  * Gell-Mann and Low [1951] M. Gell-Mann and F. Low, Phys. Rev. 84, 350 (1951).
  * Sucher [1957] J. Sucher, Phys. Rev. 107, 1448 (1957).
  * Shabaev [1991] V. M. Shabaev, J. Phys. A Math. 24, 5665 (1991).
  * Shabaev [1990] V. M. Shabaev, Theor. Math. Phys. 82, 1573 (1990).
  * Shabaev [2002] V. M. Shabaev, Phys. Rep. 356, 119 (2002).
  * Braun  _et al._ [1984] M. Braun, A. Gurchumelia, and U. Safronova, Relativistic atom theory (1984).
  * Feldman and Fulton [1987] G. Feldman and T. Fulton, Ann. Phys. 179, 20 (1987).
  * Yennie [1989] D. R. Yennie, AIP Conf. Proc. 189, 393 (1989).
  * Feldman and Fulton [1988] G. Feldman and T. Fulton, Ann. Phys. 184, 231 (1988).
  * Adkins [1989] G. S. Adkins, AIP Conf. Proc. 189, 65 (1989).
  * Feynman [1949] R. P. Feynman, Phys. Rev. 76, 769 (1949).
  * Feynman and Hibbs [2010] R. P. Feynman and A. R. Hibbs,  _Quantum mechanics and path integrals_ (Dover Publications, Inc., 2010).
  * Bhagwat and Lawande [1989] K. Bhagwat and S. Lawande, Phys. Lett. A 135, 417 (1989).
  * Barut  _et al._ [1987] A. O. Barut, A. Inomata, and G. Junker, J. Phys. A 20, 6271 (1987).
  * Camblong and Ordóñez [2004] H. E. Camblong and C. R. Ordóñez, Int. J. Mod. Phys. A 19, 1413 (2004).
  * Duru and Kleinert [1979] I. Duru and H. Kleinert, Phys. Lett. B 84, 185 (1979).
  * Inomata [1982] A. Inomata, Phys. Lett. A 87, 387 (1982).
  * Ferraro [1992] R. Ferraro, Phys. Rev. D 45, 1198 (1992).
  * Inomata [1984] A. Inomata, Phys. Lett. A 101, 253 (1984).
  * Peak and Inomata [1969] D. Peak and A. Inomata, J. Math. Phys. 10, 1422 (1969).
  * Cakir [2020] H. Cakir,  _Quantum Electrodynamic Theory of Few-Electron Highly Charged Ions_ , Ph.D. thesis, Heidelberg University (2020).
  * Goecke  _et al._ [2011] T. Goecke, C. S. Fischer, and R. Williams, Phys. Lett. B 704, 211 (2011).
  * Bashir  _et al._ [2012] A. Bashir, L. Chang, I. C. Cloet, B. El-Bennich, Y.-X. Liu, C. D. Roberts, and P. C. Tandy, Commun. Theor. Phys. 58, 79 (2012).
  * Friar  _et al._ [1999] J. L. Friar, J. Martorell, and D. W. L. Sprung, Phys. Rev. A 59, 4061 (1999).
  * Karshenboim and Shelyuto [2021] S. G. Karshenboim and V. A. Shelyuto, Eur. Phys. J. D 75 (2021).
  * Breidenbach  _et al._ [2022] S. Breidenbach, E. Dizer, H. Cakir, and Z. Harman, Phys. Rev. A 106, 042805 (2022).
  * Jaeckel and Roy [2010] J. Jaeckel and S. Roy, Phys. Rev. D 82, 125020 (2010).
  * Flambaum  _et al._ [2018] V. V. Flambaum, A. J. Geddes, and A. V. Viatkina, Phys. Rev. A 97, 032510 (2018).
  * Berengut  _et al._ [2018] J. C. Berengut, D. Budker, C. Delaunay, V. V. Flambaum, C. Frugiuele, E. Fuchs, C. Grojean, R. Harnik, R. Ozeri, G. Perez, and Y. Soreq, Phys. Rev. Lett. 120, 091801 (2018).
  * Frugiuele  _et al._ [2017] C. Frugiuele, E. Fuchs, G. Perez, and M. Schlaffer, Phys. Rev. D 96, 015011 (2017).
  * Counts  _et al._ [2020] I. Counts, J. Hur, D. P. L. A. Craik, H. Jeon, C. Leung, J. C. Berengut, A. Geddes, A. Kawasaki, W. Jhe, and V. Vuletić, Phys. Rev. Lett. 125, 123002 (2020).
  * Bufalo  _et al._ [2011] R. Bufalo, B. M. Pimentel, and G. E. R. Zambrano, Phys. Rev. D 83, 045007 (2011).
  * DeWitt [1967] B. S. DeWitt, Phys. Rev. 162, 1195 (1967).
  * Abbott [1981] L. F. Abbott, Nucl. Phys. B 185, 189 (1981).
  * Abbott [1982] L. F. Abbott, Acta Phys. Polon. B 13, 33 (1982).
  * Gâteaux [1919] R. Gâteaux, Bull. Soc. Math. Fr. 47, 70 (1919).
  * Behmardi and Nayeri [2008] D. Behmardi and E. Nayeri, Appl. Math. Sci. (Ruse) 2 (2008).
  * Peskin and Schroeder [1995] M. E. Peskin and D. V. Schroeder,  _An Introduction to quantum field theory_ (Addison-Wesley, Reading, USA, 1995).
  * Kayed and Inomata [1984] M. A. Kayed and A. Inomata, Phys. Rev. Lett. 53, 107 (1984).
  * Messiah [1961] A. Messiah,  _Quantum Mechanics_, Quantum Mechanics No. Bd. 2 (Elsevier Science, 1961).
  * Lepage [1977] G. P. Lepage, Phys. Rev. A 16, 863 (1977).
  * Schulman [1981] L. S. Schulman,  _Techniques and Applications of Path Integration_ (Dover Publications, 1981).
  * Johnson  _et al._ [1988] W. R. Johnson, S. A. Blundell, and J. Sapirstein, Phys. Rev. A 37, 307 (1988).
  * Sapirstein and Johnson [1996] J. Sapirstein and W. R. Johnson, J. Phys. B 29, 5213 (1996).
  * Grant [2009] I. P. Grant, J. Phys. B 42, 055002 (2009).
  * Shabaev  _et al._ [2004] V. M. Shabaev, I. I. Tupitsyn, V. A. Yerokhin, G. Plunien, and G. Soff, Phys. Rev. Lett. 93, 130405 (2004).
  * Mohr and Soff [1993] P. J. Mohr and G. Soff, Phys. Rev. Lett. 70, 158 (1993).
  * Yerokhin and Shabaev [2015] V. A. Yerokhin and V. M. Shabaev, J. Phys. Chem. Ref. 44, 033103 (2015).
  * Mohr [1992] P. J. Mohr, Phys. Rev. A 46, 4421 (1992).
  * Kromer  _et al._ [2022] K. Kromer _et al._ , Eur. Phys. J. A 58, 202 (2022).
  * Filianin  _et al._ [2021] P. Filianin, C. Lyu, M. Door, K. Blaum, W. J. Huang, M. Haverkort, P. Indelicato, C. H. Keitel, K. Kromer, D. Lange, Y. N. Novikov, A. Rischka, R. X. Schüssler, C. Schweiger, S. Sturm, S. Ulmer, Z. Harman, and S. Eliseev, Phys. Rev. Lett. 127, 072502 (2021).
  * Sapirstein and Cheng [2011] J. Sapirstein and K. T. Cheng, Phys. Rev. A 83, 012504 (2011).
  * Kozhedub  _et al._ [2010] Y. S. Kozhedub, A. V. Volotka, A. N. Artemyev, D. A. Glazov, G. Plunien, V. M. Shabaev, I. I. Tupitsyn, and T. Stöhlker, Phys. Rev. A 81, 042513 (2010).
  * Yerokhin  _et al._ [2001] V. A. Yerokhin, A. N. Artemyev, V. M. Shabaev, M. M. Sysak, O. M. Zherebtsov, and G. Soff, Phys. Rev. A 64, 032109 (2001).