# Multi-input segmentation of damaged brain in acute ischemic stroke patients
using slow fusion with skip connection

Luca Tomasetti Corresponding Author<EMAIL_ADDRESS>Department of
Electrical Engineering and Computer Science, University of Stavanger, 4021
Stavanger, Norway Mahdieh Khanmohammadi  Department of Electrical Engineering
and Computer Science, University of Stavanger, 4021 Stavanger, Norway Kjersti
Engan  Department of Electrical Engineering and Computer Science, University
of Stavanger, 4021 Stavanger, Norway Liv Jorunn Høllesli  Department of
Electrical Engineering and Computer Science, University of Stavanger, 4021
Stavanger, Norway Stavanger Medical Imaging Laboratory (SMIL), Department of
Radiology, Stavanger University Hospital, 4019 Stavanger, Norway  Kathinka
Dæhli Kurz  Department of Electrical Engineering and Computer Science,
University of Stavanger, 4021 Stavanger, Norway Stavanger Medical Imaging
Laboratory (SMIL), Department of Radiology, Stavanger University Hospital,
4019 Stavanger, Norway

###### Abstract

Time is a fundamental factor during stroke treatments. A fast, automatic
approach that segments the ischemic regions helps treatment decisions. In
clinical use today, a set of color-coded parametric maps generated from
computed tomography perfusion (CTP) images are investigated manually to decide
a treatment plan. We propose an automatic method based on a neural network
using a set of parametric maps to segment the two ischemic regions (core and
penumbra) in patients affected by acute ischemic stroke. Our model is based on
a convolution-deconvolution bottleneck structure with multi-input and slow
fusion. A loss function based on the focal Tversky index addresses the data
imbalance issue. The proposed architecture demonstrates effective performance
and results comparable to the ground truth annotated by neuroradiologists. A
Dice coefficient of 0.81 for penumbra and 0.52 for core over the large vessel
occlusion test set is achieved. The full implementation is available at:
https://git.io/JtFGb.

##  1 Introduction

A cerebral stroke is the second most common cause of death among adults
worldwide [21]. Cerebral stroke can be divided into two general categories:
ischemic and hemorrhagic stroke. Ischemic stroke approximately represents 80%
of the totality of the strokes [16]. The ischemic brain tissue is divided into
two distinct regions during an ischemic stroke: the ischemic core (infarcted
tissue) and the ischemic penumbra, a hypoperfused but viable tissue region.
Fast and correct visualization of the salvageable penumbra and the
irreversibly damaged core tissues can benefit medical doctors for treatment
planning in acute stroke patients (AIS).

Computed Tomography (CT) or Magnetic Resonance Imaging (MRI) are the two of
the modalities used to diagnose acute stroke patients [6]. CT is preferred in
many centers due to its high sensitivity for detecting hemorrhage, rapid scan
times, and widespread availability. Information about clinical severity are
calculated, at hospital admission, using the National Institutes of Health
Stroke Scale (NIHSS), and color-coded parametric maps (2D PMs) are generated
using the 4D CT Perfusion (CTP) imaging usually performed immediately after
hospital admission. PMs are estimated to evaluate the changes in the tissue
density over the injection of a contrast agent over time. Time-to-peak (TTP),
time-to-maximum ($\text{T}_{\text{Max}}$), cerebral blood flow (CBF), and
cerebral blood volume (CBV), are all examples of PMs, derived from pixel
information of a time density curve, generated from a CTP study [11]. Also,
the maximum intensity projection (MIP) is found as the maximum Hounsfield unit
value over the time sequence of the CTP providing a 3D volume. In addition to
diagnosing acute stroke, CT is also necessary for treatment decisions, with
CTP being an essential modality with the ability to assess the penumbra and
core.

Deep neural network (DNN) models have been proven to be an effective and
beneficial tool for classification and segmentation tasks in many medical
image analysis applications. Various research groups have focused their effort
on the study of ischemic strokes, and some methods have been developed for
classifying and segmenting the infarct core [2, 5, 9, 13]. These methods rely
on a set of PMs as input and ground truth images generated through follow-up
images acquired hours after the stroke onset. Nevertheless, the methods
mentioned above only segment _core_ regions. However, it could be more
beneficial to acquire knowledge also about the penumbra regions since it is
crucial for the treatment decisions during the first stages of the ischemic
stroke [15, 20]. To the best of our knowledge, Tomasetti et al. was the first
research group to segment both _core and penumbra_ using machine learning and
deep learning approaches [19, 20].

It is highly time-consuming to collect and label medical data, and transfer
learning is a popular approach to solve problems related to medical images [3,
22]. Over the past years, there have been numerous examples of transfer
learning architectures used for various tasks in disparate domains pre-trained
with the ImageNet dataset [4, 22]. Additionally, early and slow fusion
approaches, with or without inflation, have been proven to improve accuracy in
video classification [4, 8], and medical diagnosis [12, 14]. An early fusion
approach combines input information at the beginning of the process, allowing
a network to increase the performances of the system using cross-correlation
between data [7]. The slow fusion approach slowly merges input information
throughout the model permitting higher layers to access more global
information [8]. An inflating technique has been proposed to use pre-trained
weights from image classification networks in video classification models,
expanding the filters from 2D (image-based) to 3D (video-based) [4].

A proper understanding of _both_ ischemic regions is a major requisite for
initial treatment decisions; however, it has not been fully explored in the
previous researches; thus, in this work, we propose a DNN architecture to
simultaneously segment both core and penumbra regions in AIS patients. We
implemented a structure that was inspired by a multi-scale model proposed by
Wetteland et al. [22]; however, while Wetteland’s model used the same image
but with different magnifications as input, we propose a different approach.
Our model uses a multi-input CNN with slow fusion, based on transfer learning
from VGG-16 models pre-trained on ImageNet. We want to investigate if the
usage of the PMs in combination with MIP volume and NIHSS as input can produce
meaningful results in the segmentation of ischemic regions, as this is already
calculated and in use in clinical settings. The paper contributes with:

  * •

A fully-automatic DNN method (Fig. 1) to segment both _core and penumbra_ ,
using color-coded PMs acquired shortly after hospital admission when an AIS is
expected.

  * •

A slow fusion multi-input approach is tested combining the PMs with other
images and/or patient information.

  * •

The model is trained and tested using a dataset of patients affected by
different levels of vessel occlusion for generalizing the input.

  * •

Manual annotations based on experts’ assessment of the CTP with PMs and MIP
are used as ground truth.

##  2 Dataset

Figure 1: Proposed architecture with multi-input and slow fusion. The network
has a convolutional part composed of four to five VGG-16 models. Skip
connections combined with feature concatenation from VGG-16 models ensure slow
data fusion. On the left, an overview of the feature extraction part of the
VGG-16 architecture.

The study was approved by the Regional ethic committee project 2012/1499. The
dataset included CTP scans from 152 patients collected at Stavanger University
Hospital (SUS) between January 2014 and August 2020. NIHSS score was available
for all patients. Based on the level of vessel occlusion using CT angiography,
large vessel occlusion (LVO) was defined as occlusion of a large, proximal
artery. Non-LVO was defined as patients with perfusion deficits with occlusion
of a smaller, more distal artery or with perfusion deficits without visible
artery occlusion. Patients were divided into three groups based on the vessel
occlusion severity: 77 patients with LVO, 60 Non-LVO patients, and 15 without
ischemic stroke (WIS).

Two neuroradiologists with 16 and 3.5 years of clinical experience delineated
ground truth core and penumbra regions in an in-house developed software tool.
These delineations were based on visual information in the CBF, CBV, TTP,
$\text{T}_{\text{Max}}$ and MIP images acquired directly after the CTP at
hospital admission. PMs and MIP have a $512\times 512$ pixel resolution,
displaying only the brain region. Each patient study contains a number of
brain slices between 13 and 27 for each scan. Furthermore, the MRI examination
performed within one to three days after the CT examination was studied and
used in assistance to generate the ground truth images since follow-up MRI
exhibits the final infarct core. The dataset was randomly split into training
(58%), validation (20%), and testing sets (22%). The training set resulted in
89 patients (42 LVO, 38 Non-LVO, 9 WIS); the validation set resulted in 30
patients (16 LVO, 11 Non-LVO, 3 WIS), and the remaining 33 patients for
testing (19 LVO, 11 Non-LVO, 3 WIS).

##  3 Method

The baseline model takes as input four PMs (CBF, CBV, TTP,
$\text{T}_{\text{Max}}$) of each 2D brain slice, MIP images and, for some
experiments, we also use the patient’s NIHSS score as input. The slow fusion
is done both by concatenating the feature vectors from the different inputs
and introducing skip connections from the different levels of the
convolutional part.

The architecture presents a structure that resembles the U-Net model [17].
Fig. 1 displays a representation of the architecture. The convolutional part
of the network is based on four to five distinct VGG-16 networks. This part is
used to extract low-resolution features. A detailed overview of a single
VGG-16 architecture is displayed on the right of Fig. 1. At the end of each
VGG-16 network, two convolutional layers followed by a dropout layer are
inserted to regularize the model. The VGG-16 models were pre-trained with the
ImageNet dataset. The deconvolutional section is a series of transpose layers
followed by convolutional layers. The transpose layers receive in input a
concatenation of the previous layer and the skip connection of the last
convolutional layer for each block in the VGG-16 architectures. This step is
performed through a concatenation layer, where various inputs are concatenated
together along the channel axis, providing a slow fusion of the low-level
features and the features of the VGG-16 models. All the convolutional layers
use a ReLU activation function, except the last layer, which uses a softmax
activation function that generates a probability vector for the three classes
involved (i.e., core, penumbra, and healthy brain).

The model’s output is a single 2D brain slice image with the same height and
width dimensions as the input PMs. The 3D volume is generated by concatenating
all the 2D brain slice images sequentially. To evaluate the accuracy of the
predicted outputs, we use three distinct metrics: the Dice Coefficient, the
Hausdorff distance, and the difference in volume among the predictions
($V_{p}$) and the ground truth images ($V_{g}$): $\Delta V=|V_{g}-V_{p}|$.

###  3.1 Class imbalance & Loss function

The dataset has imbalanced classes: 93.1% of the pixels belong to the healthy
brain class, 6.2% penumbra, and the remaining 0.7% core. This issue is even
more pronounced in the Non-LVO group, where 0.2% is core, 2.2% penumbra, and
the remaining 97.6% belongs to healthy brain tissue. To overcome this issue,
we build our model to focus on two aspects: the _loss function_ and the _Non-
LVO group_.

A generalized focal loss, based on the Tversky index (TI) [1, 18] was adopted.
The TI is a generalization of the Dice similarity coefficient. The selected
loss function was developed to address the data imbalance problem in medical
image segmentation, improving the trade-off between precision and recall when
training on small structures. $\gamma,\alpha$, and $\beta$ are hyper-
parameters of the Focal Tversky loss (FTL) [1]. Furthermore, during training,
we emphasized the misclassification of penumbra and core class, in patients in
the Non-LVO group, with a higher penalty in the loss because of the evident
imbalance among classes in this sub-group. A post-processing step is performed
before generating the predicted outcome: a binary mask of the entire brain
slice is created based on the MIP image to force the segmentation inside a
valid area (the brain tissue). Subsequently, from the softmax activation
function, the highest probability value for each pixel was selected.

##  4 Experiments & Results

In the reminder of the paper we define the models as:
$\text{SF}_{w}$(_input_), where $w\in${Frozen, Unfrozen, Gradual fine-tuning}
and the input are combination of PMs, MIP images (M), and NIHSS score (N). We
use the FTL as the loss function for our network. We perform three different
experiments: Exp-1) Hyper-parameter search for the poposed method (Fig. 1).
Exp-2) combination of inputs and freeze/unfreeze variations of VGG-16; Exp-3)
comparison of different input-fusion methods (Fig. 2). For all experiments,
the same setting was used: Adam [10] was used as the optimizer function. The
batch size was set to 2, and each model was trained for 1000 epochs. The
validation FTL was monitored, and an early stopping was invoked if there was
no improvement after 25 consecutive epochs.

In Exp-1 a hyper-parameter search is done running a large number of hyper-
parameter combinations (over the same model) for finding the optimal values
for the given task. We ran experiments with distinct values for FTL hyper-
parameters $\gamma\in[1,3],\alpha\in[0,1]$, and $\beta\in[0,1]$ as shown in
Table 1, where each value represents the average of the patients in the
validation set for the different severity levels.

Table 1: Exp-1: hyper-parameters search for the FTL loss, model
$\text{SF}_{F}$(PMs). Each value represents the average of the patients in the
validation set for the different severity levels and their standard deviation
(SD).

Model | Parameters |  Dice Coefficient (Avg.) $\pm$ SD
---|---|---
LVO | Non-LVO
$\gamma$ | $\alpha$ | $\beta$ | Penumbra | Core | Penumbra | Core
$\text{SF}_{F}$(PMs) | 1 | 1 | 1 |  0.68$\pm$0.2 |  0.29$\pm$0.3 | 0.30$\pm$0.3 |  0.13$\pm$0.2
0.3 | 0.7 |  0.36$\pm$0.2 |  0.37$\pm$0.3 |  0.06$\pm$0.1 |  0.21$\pm$0.3
0.5 | 0.5 |  0.66$\pm$0.2 |  0.35$\pm$0.3 | 0.30$\pm$0.4 |  0.19$\pm$0.2
0.7 | 0.3 |  0.68$\pm$0.2 |  0.31$\pm$0.3 |  0.28$\pm$0.3 |  0.17$\pm$0.2
4/3 | 0.3 | 0.7 |  0.68$\pm$0.2 |  0.35$\pm$0.3 |  0.29$\pm$0.4 |  0.17$\pm$0.2
0.5 | 0.5 |  0.69$\pm$0.2 |  0.36$\pm$0.3 | 0.30$\pm$0.3 |  0.17$\pm$0.3
0.7 | 0.3 | 0.71$\pm$0.1 |  0.37$\pm$0.3 |  0.27$\pm$0.3 | 0.22$\pm$0.3
1.5 | 0.3 | 0.7 |  0.67$\pm$0.2 |  0.30$\pm$0.3 |  0.27$\pm$0.3 |  0.14$\pm$0.2
0.5 | 0.5 |  0.67$\pm$0.2 |  0.30$\pm$0.3 |  0.29$\pm$0.3 |  0.14$\pm$0.2
0.7 | 0.3 |  0.70$\pm$0.3 |  0.37$\pm$0.2 |  0.29$\pm$0.3 |  0.20$\pm$0.3
2 | 0.3 | 0.7 |  0.67$\pm$0.2 |  0.00$\pm$0.0 | 0.30$\pm$0.4 |  0.00$\pm$0.0
0.5 | 0.5 |  0.70$\pm$0.2 |  0.00$\pm$0.0 |  0.28$\pm$0.3 |  0.00$\pm$0.0
0.7 | 0.3 |  0.70$\pm$0.2 | 0.39$\pm$0.3 |  0.29$\pm$0.3 |  0.20$\pm$0.3
3 | 0.3 | 0.7 |  0.00$\pm$0.0 |  0.00$\pm$0.0 |  0.00$\pm$0.0 |  0.00$\pm$0.0
0.5 | 0.5 |  0.70$\pm$0.2 |  0.00$\pm$0.0 | 0.30$\pm$0.4 |  0.00$\pm$0.0
0.7 | 0.3 |  0.68$\pm$0.2 |  0.37$\pm$0.3 |  0.29$\pm$0.3 |  0.19$\pm$0.3

Figure 2: Overview of the three models used for comparison in Exp-3: the early
fusion model ($\text{EF}_{w}$(_input_)); the early fusion with inflation
($\text{EFI}_{w}$(_input_)); $\text{SF}_{w}$(_input_) is described in Fig. 1.
Table 2: Statistical results over the validation set for the models divided
for Exp-2 and Exp-3. The last two rows contain results for Exp-3. Dice
coefficient, Hausdorff distance, and the volume difference are the metrics
considered to select the best model. Each value exhibits the average of the
patients in the validation set and the standard deviation (SD) for the
distinct groups. For the metrics: $\Uparrow$ indicates that higher values are
better, while with $\Downarrow$ lower values are preferable. Highlighted
values represent the best result for a specific class and metric. The selected
model is highlighted inside a red rectangle.

Model | Input |  Dice Coeff. (Avg.) $\pm$ SD $\Uparrow$ |  Hausdorff Dist. (Avg.) $\pm$ SD $\Downarrow$ |  $\Delta V$ (Avg.) $\pm$ SD (ml) $\Downarrow$
---|---|---|---|---
LVO | Non-LVO | LVO | Non-LVO | LVO | Non-LVO | WIS
PMs | MIP | NIHSS | Penumbra |  | Core
Exp-2 \- Layer weights: Frozen
$\text{SF}_{F}$(PMs) | X |  |  |  0.71$\pm$0.1 0.37$\pm$0.3 |  0.27$\pm$0.3 0.22$\pm$0.3 |  5.9$\pm$1.0 2.7$\pm$1.8 |  3.2$\pm$1.5 0.9$\pm$0.8 |  27.0$\pm$28 9.4$\pm$20 |  10.0$\pm$15 0.8$\pm$1.3 |  9.8$\pm$8.0 0.5$\pm$0.5
$\text{SF}_{F}$(PMs,M) | X | X |  |  0.69$\pm$0.2 0.36$\pm$0.3 |  0.29$\pm$0.3 0.20$\pm$0.3 |  5.9$\pm$0.9 3.0$\pm$1.7 |  3.1$\pm$1.4 1.0$\pm$0.8 |  25.9$\pm$27 10.0$\pm$21 |  5.5$\pm$6.0 0.8$\pm$1.0 |  8.1$\pm$7.0 0.5$\pm$0.3
$\text{SF}_{F}$(PMs,N) | X |  | X |  0.70$\pm$0.2 0.36$\pm$0.3 |  0.29$\pm$0.3 0.16$\pm$0.2 |  5.6$\pm$1.2 2.3$\pm$1.9 |  2.3$\pm$1.3 0.7$\pm$0.6 |  27.0$\pm$37 5.0$\pm$9.0 |  4.1$\pm$4.8 0.6$\pm$1.2 |  4.0$\pm$3.4 0.1$\pm$0.1
$\text{SF}_{F}$(PMs,M,N) | X | X | X |  0.68$\pm$0.2 0.34$\pm$0.3 |  0.3$\pm$0.3 0.18$\pm$0.3 |  5.7$\pm$1.4 2.5$\pm$1.8 |  2.4$\pm$1.2 0.8$\pm$0.7 |  29.9$\pm$37 6.3$\pm$12 |  2.8$\pm$3.0 0.5$\pm$1.0 |  3.9$\pm$4.0 0.0$\pm$0.0
Exp-2 \- Layer weights: Unfrozen
$\text{SF}_{U}$(PMs) | X |  |  |  0.70$\pm$0.2 0.34$\pm$0.3 |  0.29$\pm$0.4 0.24$\pm$0.3 |  5.4$\pm$1.3 2.7$\pm$1.7 |  2.1$\pm$1.5 0.7$\pm$0.7 |  29.8$\pm$36 6.5$\pm$14 |  4.2$\pm$6.9 0.5$\pm$0.7 |  0.1$\pm$0.1 0.0$\pm$0.0
$\text{SF}_{U}$(PMs,M) | X | X |  |  0.70$\pm$0.2 0.36$\pm$0.3 |  0.34$\pm$0.3 0.24$\pm$0.3 |  5.6$\pm$1.4 2.5$\pm$1.8 |  2.3$\pm$1.6 0.6$\pm$0.6 |  28.1$\pm$40 6.5$\pm$8.1 |  4.9$\pm$9.8 0.4$\pm$0.8 |  0.9$\pm$1.5 0.0$\pm$0.0
$\text{SF}_{U}$(PMs,N) | X |  | X |  0.72$\pm$0.2 0.36$\pm$0.3 |  0.29$\pm$0.3 0.23$\pm$0.3 |  5.6$\pm$1.1 2.6$\pm$1.8 |  2.7$\pm$1.4 0.9$\pm$0.7 |  28.8$\pm$28 5.2$\pm$10 |  7.8$\pm$12 0.5$\pm$0.7 |  3.4$\pm$2.7 0.1$\pm$0.1
$\text{SF}_{U}$(PMs,M,N) | X | X | X |  0.71$\pm$0.2 0.36$\pm$0.3 |  0.32$\pm$0.3 0.22$\pm$0.3 |  5.7$\pm$1.0 2.8$\pm$1.7 |  2.4$\pm$1.6 1.0$\pm$0.8 |  29.0$\pm$25 7.0$\pm$14 |  9.0$\pm$18 0.8$\pm$0.9 |  5.5$\pm$4.5 0.1$\pm$0.1
Exp-2 \- Layer weights: Gradual Fine-tuning
$\text{SF}_{G}$(PMs) | X |  |  |  0.71$\pm$0.2 0.35$\pm$0.3 |  0.30$\pm$0.3 0.19$\pm$0.3 |  5.8$\pm$1.0 2.4$\pm$1.8 |  3.0$\pm$1.4 0.8$\pm$0.7 |  29.3$\pm$25 4.7$\pm$7.6 |  11.8$\pm$22 0.6$\pm$0.8 |  6.2$\pm$5.5 0.0$\pm$0.0
$\text{SF}_{G}$(PMs,M) | X | X |  |  0.69$\pm$0.2 0.35$\pm$0.3 |  0.34$\pm$0.6 0.22$\pm$0.4 |  5.4$\pm$1.4 2.4$\pm$1.8 |  1.9$\pm$1.2 0.6$\pm$0.5 |  27.8$\pm$38 5.2$\pm$9.4 |  3.3$\pm$3.7 0.3$\pm$0.3 |  2.4$\pm$3.8 0.0$\pm$0.0
$\text{SF}_{G}$(PMs,N) | X |  | X |  0.72$\pm$0.2 0.37$\pm$0.3 |  0.31$\pm$0.3 0.21$\pm$0.3 |  5.3$\pm$1.4 2.4$\pm$1.7 |  2.3$\pm$1.4 0.7$\pm$0.6 |  26.0$\pm$35 4.4$\pm$7.0 |  6.5$\pm$14 0.5$\pm$0.7 |  1.0$\pm$1.7 0.0$\pm$0.0
$\text{SF}_{G}$(PMs,M,N) | X | X | X |  0.68$\pm$0.3 0.34$\pm$0.3 |  0.31$\pm$0.3 0.18$\pm$0.3 |  5.5$\pm$1.3 2.5$\pm$1.8 |  2.3$\pm$1.3 0.7$\pm$0.6 |  26.9$\pm$35 5.1$\pm$9.6 |  5.5$\pm$9.2 0.5$\pm$0.8 |  0.9$\pm$1.1 0.0$\pm$0.0
Exp-3 \- Layer weights: Gradual Fine-tuning
$\text{EF}_{G}$(PMs,N) | X |  | X |  0.68$\pm$0.2 0.26$\pm$0.3 |  0.32$\pm$0.3 0.19$\pm$0.3 |  5.6$\pm$1.4 2.4$\pm$1.9 |  2.3$\pm$1.5 0.7$\pm$0.6 |  31.8$\pm$47 8.0$\pm$21 |  5.3$\pm$10 0.7$\pm$0.9 |  1.8$\pm$0.3 0.0$\pm$0.0
$\text{EFI}_{G}$(PMs,N) | X |  | X |  0.67$\pm$0.2 0.29$\pm$0.3 |  0.29$\pm$0.3 0.16$\pm$0.2 |  6.0$\pm$0.8 2.5$\pm$1.9 |  3.3$\pm$1.5 0.7$\pm$0.6 |  25.5$\pm$28 5.3$\pm$10 |  6.3$\pm$7.1 0.6$\pm$1.2 |  6.7$\pm$4.7 0.0$\pm$0.0

For Exp-2 we have combined our slow fusion multi-input baseline model with
different combinations of inputs to understand if diversity in the input can
improve training the model. The various combinations are shown in Table 2 with
the corresponding results. From the baseline input (PMs), we added MIP images
as input or the NIHSS score or combined both MIP images and NIHSS score.
Models for all these experiments were based on a multi-input with slow fusion.
VGG-16 models were trained with frozen weights, unfrozen weights, and a
gradual fine-tuning approach for each experiment. The latter setting was
developed in three steps: first, the model was trained with all the VGG-16
weights frozen; secondly, after monitoring the validation loss having no
improvements for 25 consecutive epochs, the bottom half of the weights were
unfrozen, and the training continued; at last, when no improvement in the
validation loss was detected again, all weights were unfrozen, the training
continued, and the validation loss was monitored. We have selected
$\text{SF}_{G}$(PMs,N) as the proposed model.

Table 3: Evaluation metrics for the selected method ($\text{SF}_{G}$(PMs,N))
over the 33 test patients (19 LVO, 11 Non-LVO, 3 WIS) manually annotated by
two experts neuroradiologists ($\text{NR}_{1}$ and $\text{NR}_{2}$).

Model |  Dice Coefficient (Avg.) $\pm$ SD $\Uparrow$ |  Hausdorff Distance (Avg.) $\pm$ SD $\Downarrow$ |  $\Delta V$ (Avg.) $\pm$ SD (ml) $\Downarrow$
---|---|---|---
LVO | Non-LVO | All | LVO | Non-LVO | All | LVO | Non-LVO | WIS | All
Penumbra |  | Core
$\text{SF}_{G}$(PMs,N) vs ($\text{NR}_{1}$ & $\text{NR}_{2}$) |  0.81$\pm$0.1 0.52$\pm$0.2 |  0.48$\pm$0.3 0.12$\pm$0.2 |  0.63$\pm$0.3 0.34$\pm$0.3 |  5.0$\pm$1.2 3.2$\pm$1.5 |  2.2$\pm$0.9 0.7$\pm$1.0 |  3.7$\pm$2.0 2.1$\pm$2.8 |  15.5$\pm$12 6.1$\pm$5.1 |  4.4$\pm$4.4 0.7$\pm$1.4 |  0.2$\pm$0.4 0.0$\pm$0.0 |  10.5$\pm$11 3.7$\pm$4.8
$\text{NR}_{1}$ vs. $\text{NR}_{2}$ |  0.78$\pm$0.1 0.44$\pm$0.2 |  0.65$\pm$0.2 0.15$\pm$0.2 |  0.67$\pm$0.2 0.30$\pm$0.3 |  5.1$\pm$$\pm$1.0 3.2$\pm$1.4 |  1.9$\pm$1.4 0.5$\pm$0.9 |  3.6$\pm$2.2 2.0$\pm$1.8 |  33.3$\pm$28 5.6$\pm$4.3 |  5.5$\pm$9.2 0.7$\pm$1.9 |  0.0$\pm$0.0 0.0$\pm$0.0 |  21.0$\pm$26 3.5$\pm$4.2
$\text{SF}_{G}$(PMs,N) vs. $\text{NR}_{1}$ |  0.84$\pm$0.1 0.57$\pm$0.2 |  0.48$\pm$0.3 0.12$\pm$0.2 |  0.64$\pm$0.3 0.37$\pm$0.3 |  4.9$\pm$1.1 3.0$\pm$1.4 |  2.2$\pm$1.0 0.7$\pm$1.0 |  3.6$\pm$1.9 1.9$\pm$1.7 |  11.2$\pm$11 6.2$\pm$5.7 |  4.4$\pm$4.4 0.7$\pm$1.4 |  0.3$\pm$0.4 0.0$\pm$0.0 |  7.9$\pm$9.4 3.8$\pm$5.2
$\text{SF}_{G}$(PMs,N) vs. $\text{NR}_{2}$ |  0.78$\pm$0.1 0.44$\pm$0.2 |  0.43$\pm$0.3 0.1$\pm$0.2 |  0.59$\pm$0.3 0.29$\pm$0.3 |  5.4$\pm$1.1 3.3$\pm$1.4 |  2.7$\pm$1.4 0.5$\pm$0.8 |  4.0$\pm$2.1 2.1$\pm$1.8 |  29.2$\pm$30 6.8$\pm$5.9 |  8.2$\pm$9.6 0.3$\pm$0.6 |  0.3$\pm$0.4 0.0$\pm$0.0 |  19.6$\pm$26 4.0$\pm$5.5

To understand if using multi-input and slow fusion is suitable for this task,
in Exp-3 we compared it with two models: an early fusion
($\text{EF}_{G}$(PMs,N)), and early fusion with inflation
($\text{EFI}_{G}$(PMs,N)), adopting the same multi-input idea but with
different fusion approacheas. The inflation approach converts 2D into 3D
layers, adding a temporal dimension. The inflation followed the idea of the
I3D network by Carreira et al. [4], where they introduced video classification
models with an inflated ImageNet pre-trained image classification
architecture. The setting and hyper-parameters of these two new architectures
are the same as the selected model to maintain a fair comparison. Fig. 2
presents an overview of the model architectures. For $\text{EF}_{G}$(PMs,N),
the four input parametric maps are concatenated together over the channel axis
to generate a single input volume passing through a 2D convolutional layer to
reduce the channel dimension to feed it to a single VGG-16 backbone.
Differently, $\text{EFI}_{G}$(PMs,N)’s inputs are concatenated over the time
dimension; the filters and pooling kernels of the VGG-16 architecture are
inflated, remodeling squared filters into cubic filters. The bottom of Table 2
shows the results of these two models in comparison with the other experiments
performed.

$\text{SF}_{G}$(PMs,N) is seen as having good overall performance and is
chosen as the proposed model. The model is tested with a previously unseen
test set, and the performance is compared with manual annotations from two
different experts. Table 3 presents the test results and the inter-observer
variability in comparison with two expert neuroradiologists. Fig. 3 shows
examples of predictions from six patients from the test set using the proposed
model.

##  5 Discussion

We developed a multi-input CNN with early and slow fusion using transfer
learning in this work. The proposed network aims to simultaneously segment
dead (core) and salvageable tissues (penumbra) in AIS patients. The proposed
method is learned on patients with or without ischemic stroke and for
different vessel occlusion severities. This generalization helps the model
correctly segment most ischemic regions, regardless of the patient group.
After a series of experiments, multi-input including NIHSS, by a slow fusion
approach, $\text{SF}_{G}$(PMs,N), is chosen as the proposed method based on
the total of the results.

Figure 3: Prediction results for six test patients using the selected model,
the set of all possible inputs, and the relative ground truth images.

A hyper-parameter search was performed for the first experiment. Table 1 shows
the average Dice coefficient for selecting the optimal hyper-parameters for
the FTL function. Our observations showed that $\gamma=4/3$, $\alpha=0.7$, and
$\beta=1-\alpha$ give satisfactory results for all the classes in the two
groups. Thus we apply these parameters in the following experiments. From the
validation results of Table 2 it can be seen that unfreezing the weights of
the VGG-16 feature extractors improves the models but gives an overestimation
of the volume for both the classes and also that the gradual fine-tuning
approach gives a slight improvement when compared to unfreezing all weights
from the start. The choice of freezing the parameters reduces training time
since a smaller set of parameters needs to be learned; however, the
statistical results are less than satisfactory; this could be since PMs are
not included in the ImageNet dataset, then the weights are not optimized for
these images. Therefore, at the cost of a longer training time, fully
unfreezing the weights or using a gradual fine-tuning technique will allow the
model to familiarize and learn this particular dataset more accurately.

From the validation results of the different gradual fine-tuned models, it is
clear that multi-input fusion, including the NIHSS or MIP images, is better
than only PMs. However, it is not entirely clear if including both NIHSS and
MIP images improves the models compared to only including NIHSS in addition to
PMs. Based on the results presented in Table 2, we select
$\text{SF}_{G}$(PMs,N) as the proposed model. One can argue that
$\text{SF}_{U}$(PMs,N) yields similar results, but $\text{SF}_{G}$(PMs,N) is
chosen because of its lower $\Delta V$ for the LVO group, high Dice
coefficient over the entire dataset, and satisfactory results for all the
other metrics. Furthermore, Exp-3 favored the proposed slow fusion approach
over the two early fusion approaches, $\text{EF}_{G}$(PMs,N) and
$\text{EFI}_{G}$(PMs,N).

Results from the 33 randomly selected patients constituting the test set using
our selected proposed model (Table 3) show an average Dice coefficient of 0.34
for core and 0.63 for penumbra over the entire test set and, on the LVO set,
0.81 and 0.52, respectively. These results present higher or analogous values
compared to the inter-observer variability in most of the metrics, regardless
of the stroke’s severity. A separate comparison of the predicted outputs with
both the neuroradiologists’ annotations demonstrates that the proposed
architecture can achieve high statistic values, regardless of the
neuroradiologist. This achievement can be considered valuable throughout the
first stages of an AIS.

Predictions from six brain slices by six patients in the test set are shown in
Fig. 3. Results display comparable regions as the ground truth images, both
with LVO and Non-LVO groups, showing high Dice coefficient and promising
results with the model. However, the third example in Fig. 3 shows false-
positive regions in the brain: this is possibly due to artifacts present in
the generated color-coded PMs. We have noticed similar false-positive with all
the other architectures and hyper-parameters as well. These false-positive
might be avoided using 4D raw CTP datasets instead of the pre-generated PMs.

Several researchers have proposed methods with promising results to segment
the ischemic core [2, 5, 9]. They all use PMs derived from CTP studies for
their architectures. Nevertheless, their only focus was to segment the
ischemic core without considering the penumbra; thus, excluding a critical
aspect for medical treatment decisions. Furthermore, there was no
differentiation in the severity level of AIS for the patients involved in
their research.

##  6 Conclusion

The ability to achieve comparable results with two expert neuroradiologists
for all the segmented classes (core and penumbra) is valuable to
neuroradiologists during the first stages of an ischemic stroke. The proposed
model might be a supportive tool that can help doctors to make treatment
decisions.

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2.5cm2cm2.5cm2.5 cm

# Contribution of $\bm{SU(3)}$ quadratic Casimir squared to rotational bands
in the interacting boson model

Víctor Miguel Banda Guzmán  Instituto de Física y Matemáticas, Universidad
Michoacana de San Nicolás de Hidalgo, Edificio C-3, Apdo. Postal 2-82,
Morelia, Michoacán, 58040, México  Rubén Flores-Mendieta  Instituto de Física,
Universidad Autónoma de San Luis Potosí, Álvaro Obregón 64, Zona Centro, San
Luis Potosí, S.L.P. 78000, Mexico  Johann Hernández  Instituto de Física,
Universidad Autónoma de San Luis Potosí, Álvaro Obregón 64, Zona Centro, San
Luis Potosí, S.L.P. 78000, Mexico

###### Abstract

Rotational bands are commonly used in the analysis of the spectra of atomic
nuclei. The early version of the interacting boson model of Arima and Iachello
has been foundational to the description of rotations in nuclei. The model is
based on a unitary spectrum generating algebra $U(6)$ and an orthogonal
(angular momentum) symmetry algebra $SO(3)$. A solvable limit of the model
contains $SU(3)$ in its dynamical symmetry chain. The corresponding
Hamiltonian is written as a linear combination of linear and quadratic Casimir
invariants of all the algebras in the chain. Prompted by these facts, a
Hamiltonian containing the $SU(3)$ quadratic Casimir squared is proposed to
evaluate its effects on rotational bands. The additional term yields three
undetermined parameters into the theory, which need be obtained from
experiment. The lack of data does not allow one to perform a detailed
numerical analysis, but a rather restricted one in terms of a one-parameter
fit. Nevertheless, these additional terms provide a good description of
rotational bands of two nuclei of interest, ${}^{156}\mathrm{Ga}$ and
${}^{234}\mathrm{U}$.

##  I Introduction

The atomic nucleus is a non-trivial many-body quantum system which has
collective properties resulting in various deformed shapes. A strongly
deformed nucleus rotates, exhibiting characteristic rotational band structures
with noticeable regularity. A number of different methods have been used to
study many-body quantum systems. One of the oldest methods is the shell model
which assumes that a single nucleon behaves under the influence of a nuclear
mean field. Formally, in this description nucleons populate shells that
represent energy levels, going from lower to higher energies. When the number
of nucleons equals 2, 8, 20, 28, 50, 82, and 126 the shells are completely
filled. These so-called magic numbers represent nuclei with great stability.

A second method which has been widely used in the understanding of the
collective behavior of nuclei is the interacting boson model (IBM) originally
introduced by Arima and Iachello iach75 ; iach1 ; iach2 ; iach3 ; iach4 ; ibm
. The earliest version of the model, applied to even-even nuclei, describes
the collective properties in terms of pairs of valence nucleons, which conform
a set of interacting $s$ and $d$ bosons with positive parity and with angular
momenta $L=0$ and $L=2$, respectively. The simplest version of the model,
usually referred to as IBM-1, treats both types of nucleons the same so it
assumes that there is only a single kind of boson, unlike IBM-2, which treats
protons and neutrons separately.

From its very construction, the IBM possesses a symmetry-based formulation and
in particular, dynamical symmetries are essential to it. Dynamical symmetries
arise when the Hamiltonian of a system can be written in terms of the Casimir
(or invariant) operators of a chain of groups (or algebras). A remarkable
application of the group (or algebra) chains is in the construction of bases
in which the Hamiltonian can be diagonalized, or equivalently, in the
construction of bases that transform as representations of the appropriate
groups, labeling those states with the corresponding quantum numbers iach75 .

In the IBM-1, with $s$ and $d$ bosons, the spectrum generating algebra is
$U(6)$ and it has $SU(3)$ as a subalgebra generating rotational spectrum
iach75 ; iach2 . Rotational bands lie within irreducible representations
(irreps) of $SU(3)$, which are subspaces invariant under the generators of the
group. The IBM-1 Hamiltonian in thus written as a linear combination of linear
and quadratic Casimir invariants of all the algebras in a given chain iach75 .
Those terms are usually refer to as $1$\- and $2$-body terms, respectively. In
most calculations only up to two-body terms are retained.

The aim of the present paper is to introduce a $4$-body contribution to the
IBM-1 Hamiltonian to study its effects on the rotational bands of nuclei. This
contribution will thus be proportional to the $SU(3)$ quadratic Casimir
squared.

This paper is organized as follows. In Sec. II some necessary material on the
IBM is reviewed in order to introduce notation and conventions. This includes
elementary definitions about dynamical symmetries. In Sec. III the Hamiltonian
for 2-body structures is reviewed, to immediately discuss the $4$-body
Hamiltonian in Sec. IV. In Sec. V a fit to experimental data for rotational
bands is presented, via a one-parameter fit. Results and conclusions are given
in Sec. VI. The paper is complemented by two appendices, where some useful
material can be found.

##  II A survey on the IBM

In this section, an elementary survey on the IBM is provided in order to
introduce notation and conventions. For this purpose, the first concept to be
reviewed is that of a dynamical symmetry. A comprehensive introduction on the
subject can be found in Ref. leviatan , so only a few salient facts will be
repeated here.

Dynamical symmetries arise when the Hamiltonian $H$ of a system can be written
as a sum of commuting operators in the form

$H=\sum_{G}a_{G}C_{G},$ (1)

where $C_{G}$ are the Casimir operators of a chain of nested algebras111For
ease of notation, no distinction between a group, for instance $U(6)$, and its
algebra, $\mathfrak{u}(6)$, will be made, but denote both by a capital letter,
$U(6)$.

$G_{\mathrm{dyn}}\supset G_{1}\supset G_{2}\supset\ldots\supset
G_{\mathrm{sym}}.$ (2)

If this is the case, the spectrum can be solved in an explicit analytic form:
The eigenstates
$|\lambda_{\mathrm{dyn}},\lambda_{1},\lambda_{2},\ldots,\lambda_{\mathrm{sym}}\rangle$
and eigenvalues
$E(\lambda_{\mathrm{dyn}},\lambda_{1},\lambda_{2},\ldots,\lambda_{\mathrm{sym}})$
are labeled by quantum numbers
$\lambda_{\mathrm{dyn}},\lambda_{1},\lambda_{2},\ldots,\lambda_{\mathrm{sym}}$,
which characterize irreps of the algebras in the chain. Note that the
condition of the nesting of the algebras in (2) is indispensable for
constructing a set of commuting operators and hence for obtaining an analytic
solution. For definiteness, $G_{\mathrm{dyn}}$ stands for the spectrum
generating algebra of the system so operators corresponding to physical
observables can be written in terms of its generators and $G_{\mathrm{sym}}$
is the symmetry algebra iach95 . Note also that the symmetry
$G_{\mathrm{dyn}}$ is broken and the only remaining symmetry is
$G_{\mathrm{sym}}$ which is the true symmetry of the problem. A
$G_{\mathrm{dyn}}$ algebra can have several chains so analytic solutions can
only be obtained whenever the Hamiltonian can be written in terms only of the
Casimir operators of a given chain.

As for the IBM-1, it is based on a unitary spectrum generating algebra
$G_{\mathrm{dyn}}=U(6)$ and an orthogonal (angular-momentum) symmetry algebra
$G_{\mathrm{sym}}=SO(3)$. The Hamiltonian is expanded in the elements of
$U(6)$ and consists of Hermitian, rotational-scalar interactions which
conserve the total number of $s$ and $d$ bosons,
$\hat{N}=\hat{n}_{s}+\hat{n}_{d}=s^{\dagger}s+\sum_{m}d_{m}^{\dagger}d_{m}$
leviatan . The IBM-1 admits three chain algebras, namely,

$U(6)\supset\left\\{\begin{array}[]{c}U(5)\supset SO(5)\\\ SU_{\pm}(3)\\\
SO_{\pm}(6)\supset SO(5)\end{array}\right\\}\supset SO(3),$ (3)

which are known as the vibrational $U(5)$ iach1 , the rotational $SU(3)$ iach2
, and the $\gamma$-unstable $SO(6)$ limits iach4 . The $\pm$ labels attached
to $SU(3)$ and $SO(6)$ in chains (3) will serve as a reminder that those
algebras have two different realizations depending on the phase choices for
the $s$ and $d$ bosons shi .

The present analysis is concerned with the study of rotational bands, so the
chain algebra under consideration is

$U(6)\supset SU_{\pm}(3)\supset SO(3),$ (4)

where the algebras $SU_{+}(3)$ and $SU_{-}(3)$ correspond to prolate and
oblate shapes, respectively

A Hamiltonian exhibiting the chain algebra (4) can be constructed as

$H^{SU(3)}=c_{1}+c_{2}C^{(1)}_{U(6)}+c_{3}C^{(2)}_{U(6)}+c_{4}C^{(2)}_{SU_{\pm}(3)}-c_{5}C^{(2)}_{SO(3)},$
(5)

where $C^{(1)}_{U(6)}$ and $C^{(2)}_{U(6)}$ are the linear and quadratic
Casimir operators of $U(6)$ and $C^{(2)}_{SU_{\pm}(3)}$ and $C^{(2)}_{SO(3)}$
are the quadratic Casimir operators of $SU_{\pm}(3)$ and $SO(3)$,
respectively. Since a quadratic Casimir operator is obtained from the product
of two creation and two annihilation operators, it is usually referred to as a
$2$-body operator. In general, $n$-body operators are constructed from the
product of $n$ creation and $n$ annihilation operators.

The eigenstates of the Hamiltonian (5) can be represented by

$|N,(\lambda_{\pm},\mu_{\pm}),K_{\pm},L\rangle,$ (6)

where $N$, $(\lambda_{+},\mu_{+})$, $(\lambda_{-},\mu_{-})$, and $L$ label the
relevant irreps of $U(6)$, $SU_{+}(3)$, $SU_{-}(3)$, and $SO(3)$,
respectively, and $K_{+}$ and $K_{-}$ are multiplicity labels leviatan . The
quantum numbers for $SU_{-}(3)$ and $SU_{+}(3)$ are not identical but are
obtained from each other under the interchange $\lambda\leftrightarrow\mu$.
Hereafter, only the $SU_{-}(3)$ labels are involved, so they will be loosely
denoted by $(\lambda,\mu)$ for simplicity. No confusion should arise because
irreps with $\lambda>\mu$ correspond to prolate shapes whereas irreps with
$\lambda<\mu$ correspond to oblate shapes bon .

Therefore, for a given $N$, there are several $SU(3)$ irreps with
$(\lambda,\mu)$ defined as Pfeifer

$\displaystyle\mu$ $\displaystyle=$ $\displaystyle 0,2,4,\dots$ (7a)
$\displaystyle\lambda$ $\displaystyle=$ $\displaystyle 2N-6l-2\mu,\qquad
l=0,1,\dots,N,$ (7b)

and for a given $SU(3)$ irrep there are eigenstates with different $L$
expressed in terms of Elliott’s quantum number $K$ elliott1 ; elliott2

$\displaystyle K$ $\displaystyle=$ $\displaystyle
0,2,4,\dots,\text{min}(\lambda,\mu),$ (8a) $\displaystyle L$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lll}0,2,4,\dots,\text{max}(\lambda,\mu),&&\mbox{for}\quad
K=0,\\\\[8.53581pt]
K,K+1,K+2,\dots,K+\text{max}(\lambda,\mu),&&\mbox{for}\quad
K>0.\end{array}\right.$ (8d)

Since

$C^{(2)}_{SO(3)}=\mathbf{L}^{2},$ (9)

where $\mathbf{L}\equiv(L_{1},L_{2},L_{3})$ is the angular momentum operator
for a fixed value of $(\lambda,\mu)$, then for a given $SU(3)$ irrep there is
a rotational band with energy levels spaced according to the eigenvalues of
$\mathbf{L}^{2}$, that is to say $L(L+1)$. In the $2$-body Hamiltonian (5),
the energy difference between these rotational bands is given by the $SU(3)$
quadratic Casimir $C^{(2)}_{SU(3)}$. In general, the differences would be
controlled by higher-order $n$-body $SU(3)$ invariants in the IBM Hamiltonian.
Therefore, in order to obtain a more accurate phenomenological description of
the nuclear spectra and be able to understand the role that higher-order terms
in the chain group (4) play, the aim of the present work is to propose a
general $4$-body $SU(3)$ Hamiltonian. For this task, the $SU(3)$ projection
operators introduced in Ref. banda  come in handy.

##  III IBM-1 in the rotational limit for $2$-body operators

The building blocks of the IBM-1 are the creation and annihilation operators
of the $s$ and $d$ bosons. Creation operators will be denoted as

$s^{\dagger},\quad d_{m}^{\dagger},\qquad(m=0,\pm 1,\pm 2)$

so that the commutator of the creation and annihilation operators that are
associated with the same boson state equals one, while all other commutators
vanish.

Defining $b^{\dagger}_{l,m}$, with $l=0,2$ and $-l\leq m\leq l$, as ibm

$b^{\dagger}_{0,0}=s^{\dagger},\quad b^{\dagger}_{2,2}=d^{\dagger}_{2},\quad
b^{\dagger}_{2,1}=d^{\dagger}_{1},\quad
b^{\dagger}_{2,0}=d^{\dagger}_{0},\quad b^{\dagger}_{2,-1}=d_{-1},\quad
b^{\dagger}_{2,-2}=d^{\dagger}_{-2},$ (10)

the canonical commutation relations are now written as

$[b_{l,m},b^{\dagger}_{l^{\prime},m^{\prime}}]=\delta_{ll^{\prime}}\delta_{mm^{\prime}},\quad[b^{\dagger}_{l,m},b^{\dagger}_{l^{\prime},m^{\prime}}]=[b_{l,m},b_{l^{\prime},m^{\prime}}]=0.$
(11)

In the IBM-1, a Hamiltonian that conserves the total number of bosons is
constructed from $n$-body operators, which represent the interactions between
bosons. Up to $2$-body operators, the most general Hamiltonian reads

$H=E_{0}+\sum_{\alpha\beta}c_{\alpha\beta}b_{\alpha}^{\dagger}b_{\beta}+\sum_{\alpha\beta\gamma\delta}c_{\alpha\beta\gamma\delta}b_{\alpha}^{\dagger}b_{\beta}^{\dagger}b_{\gamma}b_{\delta},$
(12)

where the Greek indices are a short-hand notation for the two indices in the
creation and annihilation operators; for example, $\alpha=(l,m)$, and so on.

The eigenvalue problem for $H$ can be solved analytically for some particular
cases. To illustrate this point, consider the algebra of the bilinear
operators ibm

$G_{\kappa}^{(k)}(l,l^{\prime})=[b_{l}^{\dagger}\times\tilde{b}_{l^{\prime}}]_{\kappa}^{(k)},$
(13)

where $l,l^{\prime}=0,2$, $\tilde{b}_{l,m}=(-1)^{l+m}b_{l,-m}$, and the right-
hand side of Eq. (13) can be written as

$[b_{l}^{\dagger}\times\tilde{b}_{l^{\prime}}]_{\kappa}^{(k)}=\sum_{mm^{\prime}}(lml^{\prime}m^{\prime}|k\kappa)B_{lm;l^{\prime}m^{\prime}},$
(14)

where $(lml^{\prime}m^{\prime}|k\kappa)$ denote Clebsch-Gordan coefficients
and
$B_{lm;l^{\prime}m^{\prime}}=b^{\dagger}_{lm}\tilde{b}_{l^{\prime}m^{\prime}}$,
whose commutation relations are obtained as

$[B_{lm;l^{\prime}m^{\prime}},B_{\tilde{l}\tilde{m};\tilde{l}^{\prime}\tilde{m}^{\prime}}]=(-1)^{l^{\prime}+m^{\prime}}\delta_{l^{\prime}\tilde{l}}\delta_{-m^{\prime}\tilde{m}}B_{lm;\tilde{l}^{\prime}\tilde{m}^{\prime}}-(-1)^{\tilde{l}^{\prime}+\tilde{m}^{\prime}}\delta_{l\tilde{l}^{\prime}}\delta_{m-\tilde{m}^{\prime}}B_{\tilde{l}\tilde{m};l^{\prime}m^{\prime}}.$
(15)

The 36 operators defined in Eq. (13) satisfy the Lie algebra of the $U(6)$
group. There are also linear combinations of these operators that satisfy the
Lie algebras of the $SU(3)$ and $SO(3)$ groups ibm . This is referred to as
the rotational limit of the IBM ibm ; iach2 . Those linear combinations are
given by the following angular momentum and quadrupole operators,

$\displaystyle L_{m}$ $\displaystyle=$
$\displaystyle\sqrt{10}G_{m}^{(1)}(d,d),\quad m=-1,0,1,$ (16a) $\displaystyle
Q_{m}$ $\displaystyle=$ $\displaystyle
G_{m}^{(2)}(d,s)+G_{m}^{(2)}(s,d)-\dfrac{\sqrt{7}}{2}G_{m}^{(2)}(d,d),\quad
k=-2,-1,\dots,2.$ (16b)

Using the commutation relations (15), it is straightforward to show that

$\displaystyle[L_{0},L_{1}]$ $\displaystyle=$ $\displaystyle L_{1},$ (17a)
$\displaystyle[L_{0},L_{-1}]$ $\displaystyle=$ $\displaystyle-L_{-1},$ (17b)
$\displaystyle[L_{1},L_{-1}]$ $\displaystyle=$ $\displaystyle L_{0},$ (17c)

which correspond to the commutation relations for the spherical components of
the angular momentum operator. Thus, defining the Cartesian components as

$\displaystyle L^{1}$ $\displaystyle=$
$\displaystyle\dfrac{\sqrt{2}}{2}\left(L_{-1}-L_{1}\right),$ (18a)
$\displaystyle L^{2}$ $\displaystyle=$
$\displaystyle\dfrac{\sqrt{2}}{2}i\left(L_{1}+L_{-1}\right),$ (18b)
$\displaystyle L^{3}$ $\displaystyle=$ $\displaystyle L_{0},$ (18c)

the Lie algebra of $SO(3)$, namely,
$[L^{i},L^{j}]=i\sum_{k}\epsilon^{ijk}L^{k}$, with $i,j,k=1,2,3$, is
recovered. Henceforth, the sum over repeated indices will be implicit, so
expressions as $X^{a}Y^{a}$ should be understood as $\sum_{a}X^{a}Y^{a}$.

Following similar steps, the commutation relations of the Lie algebra of
$SU(3)$ can be obtained. Starting from the definitions

$\displaystyle\begin{array}[]{ll}Y=\dfrac{2\sqrt{2}}{3}Q_{0},&T^{3}=\dfrac{1}{2}L_{0},\\\\[11.38109pt]
T_{+}=\dfrac{2}{\sqrt{3}}Q_{2},&T_{-}=\dfrac{2}{\sqrt{3}}Q_{-2},\\\\[11.38109pt]
U_{+}=i\left[\dfrac{1}{2}L_{-1}-\sqrt{\dfrac{2}{3}}Q_{-1}\right],&U_{-}=i\left[\dfrac{1}{2}L_{1}-\sqrt{\dfrac{2}{3}}Q_{1}\right],\\\\[11.38109pt]
V_{+}=-i\left[\dfrac{1}{2}L_{1}+\sqrt{\dfrac{2}{3}}Q_{1}\right],&V_{-}=-i\left[\dfrac{1}{2}L_{-1}+\sqrt{\dfrac{2}{3}}Q_{-1}\right],\end{array}$
(23)

it is quite easy to verify that

$\displaystyle[Y,T^{3}]$ $\displaystyle=$ $\displaystyle
0,\quad[Y,U_{\pm}]=\pm U_{\pm},\quad[Y,V_{\pm}]=\pm
V_{\pm},\quad[Y,T_{\pm}]=0,$ $\displaystyle[T^{3},U_{\pm}]$ $\displaystyle=$
$\displaystyle\mp\dfrac{1}{2}U_{\pm},\quad[T^{3},V_{\pm}]=\pm\dfrac{1}{2}V_{\pm},\quad[T^{3},T_{\pm}]=\pm
T_{\pm},$ $\displaystyle[T_{+},T_{-}]$ $\displaystyle=$ $\displaystyle
2T^{3},\quad[T_{+},V_{+}]=[T_{+},U_{-}]=0,\quad[T_{+},V_{-}]=-U_{-},\quad[T_{+},U_{+}]=V_{+},$
$\displaystyle[T_{-},V_{-}]$ $\displaystyle=$
$\displaystyle[T_{-},U_{+}]=0,\quad[T_{-},V_{+}]=U_{+},\quad[T_{-},U_{-}]=-V_{-},$
$\displaystyle[U_{+},U_{-}]$ $\displaystyle=$
$\displaystyle\dfrac{3}{2}Y-T^{3},\quad[U_{+},V_{+}]=0,\quad[U_{+},V_{-}]=T_{-},$
$\displaystyle[U_{-},V_{-}]$ $\displaystyle=$ $\displaystyle
0,\quad[U_{-},V_{+}]=-T_{+},\quad[V_{+},V_{-}]=\dfrac{3}{2}Y+T^{3}.$ (24)

Therefore, the Lie algebra of the $SU(3)$ group,
$[T^{a},T^{b}]=if^{abc}T^{c}$, with $a,b,c=1,\dots,8$, and

$\displaystyle\begin{array}[]{ll}T^{1}=\dfrac{1}{2}\left[T_{+}+T_{-}\right],&T^{2}=-\dfrac{i}{2}\left[T_{+}-T_{-}\right],\\\\[11.38109pt]
T^{4}=\dfrac{1}{2}\left[U_{+}+U_{-}\right],&T^{5}=\dfrac{-i}{2}\left[U_{+}-U_{-}\right],\\\\[11.38109pt]
T^{6}=\dfrac{1}{2}\left[V_{+}+V_{-}\right],&T^{7}=\dfrac{-i}{2}\left[V_{+}-V_{-}\right],\\\\[11.38109pt]
T^{8}=\dfrac{\sqrt{3}}{2}Y,&\end{array}$ (29)

is found greiner .

Using the operators introduced in Eqs. (18c) and (29), the quadratic Casimir
$C^{(2)}_{SU(3)}$ in the Hamiltonian (5) reads

$C^{(2)}_{SU(3)}=T^{a}T^{a},$ (30)

whose eigenvalue for the irrep $(\lambda,\mu)$ is greiner

$\dfrac{1}{3}\left[\lambda^{2}+\mu^{2}+\lambda\mu+3(\lambda+\mu)\right].$ (31)

The eigenvalues for the other Casimir operators in (5) can be found in Refs.
ibm  and Pfeifer  and will not be given here.

In the $2$-body Hamiltonian with the dynamical symmetry of the chain algebra
(4), the energy difference between the rotational bands is given by the
eigenvalue (31). In order to gain more control on this difference, a $4$-body
$SU(3)$-invariant term will be added to Hamiltonian (5); this will be
discussed in the following section.

##  IV Higher-order $SU(3)$-invariant terms in the IBM-1 Hamiltonian

The $T^{a}$ operators listed in Eq. (29) are the generators of the Lie algebra
of $SU(3)$ so they transform in the $8$-dimensional adjoint representation.
The tensor product of two adjoint representations can be separated into the
symmetric product $(8\otimes 8)_{S}$ and the antisymmetric product $(8\otimes
8)_{A}$ as

$\displaystyle(8\otimes 8)_{S}=1\oplus 8\oplus 27,$ (32a)
$\displaystyle(8\otimes 8)_{A}=1\oplus 10\oplus\overline{10}.$ (32b)

where the different $SU(3)$ representations are denoted by their dimensions.

Prompted by the above decompositions, let

$Q^{ab}=\alpha_{1}Q_{(1)}^{ab}+\alpha_{8}Q_{(8)}^{ab}+\alpha_{10+\overline{10}}Q_{(10+\overline{10})}^{ab}+\alpha_{27}Q_{(27)}^{ab},$
(33)

be an $SU(3)$ $2$-body tensor operator constructed out of the product of two
adjoints, where each term $Q_{(\mathrm{irrep})}^{ab}$ transforms according to
the corresponding irrep indicated in the decomposition (32) and
$\alpha_{\mathrm{irrep}}$ are unknown coefficients.

Using the method to construct projection operators which can decompose any of
the reducible finite-dimensional representation spaces of $SU(N)$ contained in
the tensor product of two adjoint spaces into irreducible components
introduced in Ref. banda , the $2$-body operators $Q_{(\mathrm{irrep})}^{ab}$
can straightforwardly written as

$Q_{(1)}^{ab}=\dfrac{1}{n^{2}-1}\delta^{ab}T^{e}T^{e},$ (34a)
$Q_{(8)}^{ab}=\dfrac{n}{n^{2}-4}D^{abcd}T^{c}T^{d}+\dfrac{1}{n}F^{abcd}T^{c}T^{d},$
(34b)
$Q_{(10+\overline{10})}^{ab}=\dfrac{1}{2}(T^{a}T^{b}-T^{b}T^{a})-\dfrac{1}{n}F^{abcd}T^{c}T^{d},$
(34c)
$Q_{(27)}^{ab}=\dfrac{n+2}{4n}(T^{a}T^{b}+T^{b}T^{a})-\dfrac{n+2}{2n(n+1)}\delta^{ab}T^{e}T^{e}-\dfrac{n+4}{4(n+2)}D^{abcd}T^{c}T^{d}+\dfrac{1}{4}(D^{acbd}+D^{adbc})T^{c}T^{d},$
(34d)

where $n=3$ should be understood. For definiteness, $D^{abcd}$ and $F^{abcd}$
are given by

$D^{abcd}=d^{abe}d^{cde},\qquad F^{abcd}=f^{abe}f^{cde},$ (35)

where $d^{abc}$ is a symmetric rank-$3$ tensor.

Now, using the tensor operator $Q^{ab}$, Eq. (33), the $4$-body term $H^{(4)}$
in the IBM-1 Hamiltonian can readily be constructed as

$H^{(4)}=Q^{ab}Q^{ab},$ (36)

so a Hamiltonian with up to $4$-body terms that respects the dynamical
symmetry in (4) is given by

$H=c_{1}+c_{2}C^{(1)}_{U(6)}+c_{3}C^{(2)}_{U(6)}+c_{4}C^{(2)}_{SO(3)}-c_{5}C^{(2)}_{SU(3)}-H^{(4)},$
(37)

where the signs of the last two summands have been conveniently chosen to
describe the experimental spectra of the nucleus. The eigenfunctions of
Hamiltonian (37) have the same quantum numbers as the ones of Hamiltonian (5),
namely, $N$ for the boson number, $(\lambda,\mu)$ for a given $SU(3)$ irrep,
$L$ for angular momentum, and $K$ for Elliott’s quantum number. Notice that
the first three coefficients $c_{1},\ldots,c_{3}$ can be cast into a single
one $E_{0}=c_{1}+c_{2}N+c_{3}N(N+1)$ since they contribute only to binding
energies and not to excitation energies ibm ; the remaining coefficients
depend on the physical system under consideration.

Before evaluating the tensor contraction indicated in $H^{(4)}$, an extra
simplification can be exploited. Notice that

$\displaystyle Q_{(10+\overline{10})}^{ab}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}[T^{a},T^{b}]-\dfrac{1}{n}F^{abcd}T^{c}T^{d}$ (38)
$\displaystyle=$
$\displaystyle\dfrac{1}{2}[T^{a},T^{b}]-\dfrac{i}{2}f^{abe}T^{e}$
$\displaystyle=$ $\displaystyle 0,$

where use of Eq. (49) has been made.

Therefore, $Q^{ab}$ reduces to

$Q^{ab}=\alpha_{1}Q_{(1)}^{ab}+\alpha_{8}Q_{(8)}^{ab}+\alpha_{27}Q_{(27)}^{ab}.$
(39)

Now, the tensor identities listed in Appendix A of Ref. banda  and the
commutation relation $[T^{a},T^{b}]=if^{abc}T^{c}$ can be recursively used to
rewrite $H^{(4)}$ in the form

$H^{(4)}=\alpha_{1}^{2}H_{1}^{(4)}+\alpha_{8}^{2}H_{8}^{(4)}+\alpha_{27}^{2}H_{27}^{(4)},$
(40)

where

$\displaystyle H_{1}^{(4)}$ $\displaystyle=$
$\displaystyle\dfrac{1}{64}\left(T^{e}T^{e}\right)^{2},$ (41a) $\displaystyle
H_{8}^{(4)}$ $\displaystyle=$
$\displaystyle\dfrac{3}{5}D^{abcd}T^{a}T^{b}T^{c}T^{d}+\dfrac{1}{3}F^{abcd}T^{a}T^{b}T^{c}T^{d},$
(41b) $\displaystyle H_{27}^{(4)}$ $\displaystyle=$
$\displaystyle\dfrac{7}{8}\left(T^{e}T^{e}\right)^{2}+if^{abc}T^{a}T^{b}T^{c}+\dfrac{3}{4}T^{e}T^{e}-\dfrac{3}{5}D^{abcd}T^{a}T^{b}T^{c}T^{d},$
(41c)

and $n=3$ has explicitly been set. Notice that not only pure $4$-body terms
are contained in Eq. (41), but also $2$\- and $3$-body terms.

Equations (41), are they stand, can be algebraically worked out to readily
obtain their eingenvalues by means of ordinary methods. Appendix B describes
one of such methods. However, a further simplification of relations (41) is
achieved by using the definition of the quadratic Casimir operator of the Lie
algebra of $SU(3)$, Eq. (30), along with Eqs. (50) and (56). These terms
become

$\displaystyle H_{1}^{(4)}$ $\displaystyle=$
$\displaystyle\dfrac{1}{64}\left[C^{(2)}_{SU(3)}\right]^{2},$ (42a)
$\displaystyle H_{8}^{(4)}$ $\displaystyle=$
$\displaystyle\dfrac{1}{5}\left[C^{(2)}_{SU(3)}\right]^{2}-\dfrac{3}{5}\,C^{(2)}_{SU(3)},$
(42b) $\displaystyle H_{27}^{(4)}$ $\displaystyle=$
$\displaystyle\dfrac{27}{40}\left[C^{(2)}_{SU(3)}\right]^{2}-\dfrac{9}{10}\,C^{(2)}_{SU(3)}.$
(42c)

With the above expressions, the Hamiltonian (37) can now be written as

$\displaystyle H$ $\displaystyle=$ $\displaystyle
c_{1}+c_{2}C^{(1)}_{U(6)}+c_{3}C^{(2)}_{U(6)}+c_{4}C^{(2)}_{SO(3)}-c_{5}C^{(2)}_{SU(3)}-\dfrac{\alpha_{1}^{2}}{64}\left[C^{(2)}_{SU(3)}\right]^{2}$
(43)
$\displaystyle\mbox{}-\dfrac{\alpha_{8}^{2}}{5}\left(\left[C^{(2)}_{SU(3)}\right]^{2}-3\,C^{(2)}_{SU(3)}\right)-\dfrac{\alpha_{27}^{2}}{10}\left(\dfrac{27}{4}\left[C^{(2)}_{SU(3)}\right]^{2}-9\,C^{(2)}_{SU(3)}\right).$

##  V Fitting experimental data for rotational bands: The one-parameter fit

The energy gaps between rotational bands are described in Hamiltonian (43) by
four parameters, namely, $c_{5}$, $\alpha_{1}$, $\alpha_{8}$, and
$\alpha_{27}$. Important information about energy gaps can be learned by
determining these parameters via a fit to data. For this task, two effective
parameters can be defined, namely, one accompanying the quadratic Casimir and
another one accompanying the quadratic Casimir squared, so these four
parameters need be recombined accordingly. Unfortunately, the lack of
experimental data does not allow one to perform an exploratory fit using the
effective parameters, much less using all four parameters. A pragmatic
approach is thus required in order to overcome this obstacle. The simplest
approach, although perhaps not the optimal one, is to consider a one-parameter
fit, either $c_{5}$, $\alpha_{1}^{2}$, $\alpha_{8}^{2}$, or $\alpha_{27}^{2}$.

The applicability of a one-parameter fit can be explored in two case examples:
156Gd and 234U. These nuclei are approximately described by the chain group
(4), provided some of their rotational bands can be treated as degenerate
iach2 . Thus, in this approximation, and taking the spacing between non-
degenerate rotational bands as the energy difference of the states with zero
angular momentum, the $SU(3)$ term in the Hamiltonian (43) that yields the
best fitting with one parameter for these spacings are

$\displaystyle\begin{array}[]{lll}c_{5}=\alpha_{8}=\alpha_{27}=0,&\quad\alpha_{1}^{2}=0.0593964\,\mbox{keV}&\qquad\mbox{for
\,}{}^{156}\mbox{Gd},\\\\[11.38109pt]
c_{5}=\alpha_{1}=\alpha_{27}=0,&\quad\alpha_{8}^{2}=0.1500500\,\mbox{keV}&\qquad\mbox{for
\,}{}^{234}\mbox{U}.\end{array}$ (46)

In Figs. 1 and 2 the comparison between the experimental energies of some
excited states with zero angular momenta for 156Gd and 234U ga ; u , and the
theoretical prediction given by the parameters in (46) is made. In these
figures, $\mathrm{Th_{2}}$ acccounts for effects of up to $2$-body terms from
Hamiltonian (5) whereas $\mathrm{Th}_{4}$ accounts for effects of up to
$4$-body terms from Hamiltonian (43), with the best-fit parameters indicated
in (46). In the degenerate approximation these states correspond to the first
excited states for each rotational band which are labelled by an irrep
$(\lambda,\mu)$ of the $SU(3)$ group.

Figure 1: Comparison between experimental and theoretical $SU(3)$ energy of
the first excited states in 156Gd ($N=12$ in the IBM-1) of the rotation bands
labelled by the indicated irreducible representation $(\lambda,\mu)$.
Theoretical values of the parameters are given in Eq. (46), where
$\mathrm{Th}_{2}$ and $\mathrm{Th}_{4}$ include up to $2$\- and up to $4$-body
contributions from Hamiltonians (5) and (43), respectively. Experimental
values are obtained from Ref. ga .

Figure 2: Comparison between experimental and theoretical $SU(3)$ energy of
the first excited states in 234U ($N=12$ in the IBM-1) of the rotation bands
labelled by the indicated irreducible representation $(\lambda,\mu)$.
Theoretical values of the parameters are given in Eq. (46), where
$\mathrm{Th}_{2}$ and $\mathrm{Th}_{4}$ include up to $2$\- and up to $4$-body
contributions from Hamiltonians (5) and (43), respectively. Experimental
values are obtained from Ref. u .

##  VI Concluding remarks

In this work, the $SU(3)$-invariant term $H^{(4)}$ with up to four-body
operators, given in Eq. (36), is introduced in the standard Hamiltonian of the
IBM-1, Eq. (5), for the chain symmetry $U(6)\supset SU(3)\supset SO(3)$. This
term contributes to the energy spacing of the rotational bands that appear
when nuclei are described by that specific symmetry.

The energy term $H^{(4)}$ originates from the most general $2$-body operator
$Q^{ab}$, which is constructed out of tensor product of two $SU(3)$ generators
as given in Eq. (39). The use of the projection operators introduced in Ref.
banda  applied to $Q^{ab}$ yields three main contributions to $H_{4}$, each
one transforming according to well defined $SU(3)$ irreps. These contributions
are $H_{1},H_{8}$, and $H_{27}$ defined in Eqs. (41).

After some simplifications, it turns out that $H_{1},H_{8}$, and $H_{27}$ can
be rewritten as linear combinations of the quadratic Casimir $C_{SU(3)}^{(2)}$
of the $SU(3)$ Lie algebra and its square as displayed in Eq. (42). Thus, the
Hamiltonian proposed in (36), with up to fourth order $d$-boson interactions
and that respects the dynamical symetry of the rotational limit of the IBM-1,
becomes the Hamiltonian given in Eq. (43). In that equation, the parameters
$c_{5}$, $\alpha_{1}^{2}$, $\alpha_{8}^{2}$ and $\alpha_{27}^{2}$ describe the
energy gaps of the rotational bands that are labeled according to distinct
irrep of $SU(3)$. These four parameters can be recombined to obtain two
effective parameters, one of them accompanying the quadratic Casimir
$C_{SU(3)}^{(2)}$, and the other one accompanying the quadratic Casimir
squared. Thus, the fitting to the experimental data can be performed using
these two effective parameters. However, because of the lack of experimental
data, a one-parameter fitting, either with $c_{5}$, $\alpha_{1}^{2}$,
$\alpha_{8}^{2}$ or $\alpha_{27}^{2}$, was done. That fitting analysis is
exemplified for the nuclei 156Gd and 234U. For the former the best fitting
arise from $H_{1}$, choosing the specific values of the parameters as in (46).
It means that, for this nuclei, instead of the standard $2$-body IBM-1
Hamiltonian given in (5) which is used in the literature, a better description
is achieved by updating this same Hamiltonian to a $4$-body version using the
Casimir $C_{SU(3)}^{(2)}$ squared instead. This situation where a higher-order
interaction introduced into the IBM Hamiltonian improves the description of
the experimental data is also reported in Refs. Stefanescu ; Heyde ; Casten .
There it is shown how cubic terms in the IBM Hamiltonian can be used to
describe features like triaxiality for some nuclei.

Similarly, for 234U the best fitting occurs from $H_{8}$, with the parameters
fixed as in (46). Again, as in the 156Gd case, the $4$-body contribution
represents a more accurate description of rotational bands than the standard
$2$-body Hamiltonian when fitting to experimental data.

A final closing remark about $H_{1}$, $H_{8}$, and $H_{27}$ is that, when they
are used independently in the fitting analysis, the resulting energies are
very close to each other, differing only in a few hundreds of eV. That
subtlety comes from the different Casimir contribution $C_{SU(3)}^{(2)}$ in
each term, as shown in (41). It means that the dominant part arises from the
squared of $C_{SU(3)}^{(2)}$.

###### Acknowledgements.

The authors are grateful to Consejo Nacional de Ciencia y Tecnología (Mexico)
for partial support.

##  Appendix A Some useful identities of $SU(3)$ generators

In this section, some useful identities involving $SU(3)$ generators are
obtained to help simplify relations (41).

Let $T^{a}$ be the generators of the Lie algebra of the $SU(n)$ group, with
commutation relations $[T^{a},T^{b}]=if^{abc}T^{c}$.

A simple algebraic manipulation yields

$\displaystyle\dfrac{1}{n}F^{abcd}T^{c}T^{d}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2n}f^{abe}f^{cde}\left(\\{T^{c},T^{d}\\}+[T^{c},T^{d}]\right)$
(47) $\displaystyle=$ $\displaystyle\dfrac{i}{2n}f^{abe}f^{cde}f^{cdg}T^{g}.$

Using the identity djm95 ,

$f^{abc}\,f^{abd}=n\,\delta^{cd},$ (48)

allows one to rewrite Eq. (47) as

$\dfrac{1}{n}F^{abcd}T^{c}T^{d}=\dfrac{i}{2}f^{abe}T^{e}.$ (49)

In the same vein as the previous case, it can be shown that

$if^{abc}T^{a}T^{b}T^{c}=-\dfrac{n}{2}T^{e}T^{e},$ (50)

which yields a simplified version of the second summand of Eq. (41c).

Now, reducing the terms containing four $T$’s is a little bit more involved.
For this task, the projection operators $\mathcal{P}^{(3)}$ and
$\mathcal{P}^{(4)}$ for $SU(n)$ introduced in Ref. banda , namely,

$\displaystyle\left[\mathcal{P}^{(3)}\right]^{c_{1}c_{2}b_{1}b_{2}}$
$\displaystyle=$
$\displaystyle\dfrac{n+2}{4n}\left(\delta^{c_{2}b_{1}}\delta^{c_{1}b_{2}}+\delta^{c_{1}b_{1}}\delta^{c_{2}b_{2}}\right)+\dfrac{1}{4}\left(D^{c_{1}b_{1}c_{2}b_{2}}+D^{c_{2}b_{1}c_{1}b_{2}}\right)$
(51)
$\displaystyle\mbox{}-\dfrac{n+4}{4(n+2)}D^{c_{1}c_{2}b_{1}b_{2}}-\dfrac{n+2}{2n(n+1)}\delta^{c_{1}c_{2}}\delta^{b_{1}b_{2}},$

and

$\displaystyle\left[\mathcal{P}^{(4)}\right]^{c_{1}c_{2}b_{1}b_{2}}$
$\displaystyle=$
$\displaystyle\dfrac{n-2}{4n}\left(\delta^{c_{2}b_{1}}\delta^{c_{1}b_{2}}+\delta^{c_{1}b_{1}}\delta^{c_{2}b_{2}}\right)-\dfrac{1}{4}\left(D^{c_{1}b_{1}c_{2}b_{2}}+D^{c_{2}b_{1}c_{1}b_{2}}\right)$
(52)
$\displaystyle\mbox{}+\dfrac{n-4}{4(n-2)}D^{c_{1}c_{2}b_{1}b_{2}}+\dfrac{n-2}{2n(n-1)}\delta^{c_{1}c_{2}}\delta^{b_{1}b_{2}},$

are useful. First, notice that

$\left[\mathcal{P}^{(3)}\right]^{c_{1}c_{2}b_{1}b_{2}}+\left[\mathcal{P}^{(4)}\right]^{c_{1}c_{2}b_{1}b_{2}}=\dfrac{1}{2}\left(\delta^{c_{2}b_{1}}\delta^{c_{1}b_{2}}+\delta^{c_{1}b_{1}}\delta^{c_{2}b_{2}}\right)-\dfrac{n}{n^{2}-4}D^{c_{1}c_{2}b_{1}b_{2}}-\dfrac{1}{n^{2}-1}\delta^{c_{1}c_{2}}\delta^{b_{1}b_{2}}.$
(53)

For $n=3$, the $SU(3)$ case, the projector
$\left[\mathcal{P}^{(4)}\right]^{c_{1}c_{2}b_{1}b_{2}}$ vanishes. Thus, from
Eq. (53), it follows that

$D^{c_{1}b_{1}c_{2}b_{2}}+D^{c_{2}b_{1}c_{1}b_{2}}+D^{c_{1}c_{2}b_{1}b_{2}}=\dfrac{1}{3}\left(\delta^{c_{2}b_{1}}\delta^{c_{1}b_{2}}+\delta^{c_{1}b_{1}}\delta^{c_{2}b_{2}}+\delta^{c_{1}c_{2}}\delta^{b_{1}b_{2}}\right),$
(54)

where the tensor $D^{c_{1}c_{2}b_{1}b_{2}}$ is defined in the Lie algebra of
the $SU(3)$ group.

Using the commutator $[T^{a},T^{b}]=if^{abc}T^{c}$ and the identity djm95

$d^{abc}d^{ade}f^{bdf}=\dfrac{n^{2}-4}{2n}f^{cef},$ (55)

the tensor contraction indicated in Eq. (41b) reduces to

$D^{abcd}T^{a}T^{b}T^{c}T^{d}=\dfrac{1}{3}(T^{e}T^{e})^{2}+\dfrac{1}{4}T^{e}T^{e},$
(56)

which ultimately can be expressed in powers of the quadratic Casimir of
$SU(3)$.

##  Appendix B A general method to compute the eigenvalues of $H^{(4)}$

The analyses of nuclei with a large number of valence nucleons involve $SU(3)$
irreps of high dimensionality. The numerical approach to obtain the
corresponding eigenvalues becomes a rather difficult computational task. In
this section a general procedure to find the eigenvalues $\lambda_{k}$ of the
operator $H^{(4)}$ defined in Eq. (40) is outlined. Recalling that each
contribution to $H^{(4)}$ transforms according to a specific $SU(3)$
representation simplifies the analysis.

First, the eigenvalues $\lambda_{1}$ of $H_{1}^{(4)}$ for an irrep
$(\lambda,\mu)$ are the simplest to obtain. Since $T^{e}T^{e}$ is the
quadratic Casimir of the Lie algebra of $SU(3)$, from Eq. (31), $\lambda_{1}$
simply reads

$\lambda_{1}=\dfrac{1}{192}\left[\lambda^{2}+\mu^{2}+\lambda\mu+3(\lambda+\mu)\right]^{2}.$
(57)

Now, the eigenvalues $\lambda_{8}$ and $\lambda_{27}$ of $H_{8}^{(4)}$ and
$H_{27}^{(4)}$ given in Eqs. (41b) and (41c), respectively, can be obtained by
computing the eigenvalues of the explicit matrix expressions for these
operators in a given irrep $(\lambda,\mu)$. A few of those eigenvalues, mainly
for the lowest irreps, have been computed with the help of the algorithm
described in Ref. algorithm_su3  and are listed in Table 1. However, due to
the huge computational effort required in the numerical evaluations, it is
more convenient to have analytical expressions instead that can help complete
Table 1. For this purpose, a linear regression analysis using the values
formerly obtained through the algorithm of Ref. algorithm_su3  can be
performed. The conventional technique of a linear regression (see for example
Ref. bishop ) requires a linear combination of $M$ functions
$f_{k}(\lambda,\mu)$, with $f_{0}(\lambda,\mu)=1$, to fit the data set
$\mathcal{D}=\\{(\lambda_{r}^{(1)},(\lambda_{1},\mu_{1})),\dots,(\lambda_{r}^{(N)},(\lambda_{N},\mu_{N}))\\}$,
with $r=8,27$. Here $\lambda_{r}^{(n)}$ indicates the eigenvalue for the
corresponding representation $(\lambda_{n},\mu_{n})$.

Let $\bm{\lambda_{r}}$ be the vector

$\displaystyle\bm{\lambda_{r}}=\begin{pmatrix}\lambda_{r}^{(1)}\\\ \vdots\\\
\lambda_{r}^{(N)}\end{pmatrix},$ (58)

in such a way that the function

$g_{r}(\lambda,\mu)=\bm{m}_{r}\cdot\bm{f}(\lambda,\mu),$ (59)

minimizes the sum of squared errors given by

$\sum_{n=1}^{N}\left[\lambda_{r}^{(n)}-g_{r}(\lambda_{n},\mu_{n})\right]^{2}.$
(60)

Here,

$\displaystyle\bm{m}_{r}=\left(\Phi^{T}\Phi\right)^{-1}\Phi^{T}\bm{\lambda_{r}},\qquad\qquad\bm{f}(\lambda,\mu)=\begin{pmatrix}f_{0}(\lambda,\mu)\\\
\vdots\\\ f_{M-1}(\lambda,\mu)\end{pmatrix},$ (61)

and $\Phi$ represents the design matrix and $\Phi^{T}$ its transpose.
Specifically, $\Phi$ reads bishop ,

$\displaystyle\Phi=\begin{pmatrix}f_{0}(\lambda_{1},\mu_{1})&\dots&f_{M-1}(\lambda_{1},\mu_{1})\\\
\vdots&&\vdots\\\
f_{0}(\lambda_{N},\mu_{N})&\dots&f_{M-1}(\lambda_{N},\mu_{N})\end{pmatrix}.$
(62)

Since the entries of Table 1 are generated by an unknown function of
$\lambda,\mu$, the right choice of the set of functions $f_{k}(\lambda,\mu)$
can perfectly fit these data without overfitting.

The choice of

$\bm{f}(\lambda,\mu)=(1,\mu,\dots,\mu^{4},\lambda,\lambda\mu,\dots,\lambda\mu^{4},\dots,\lambda^{4},\lambda^{4}\mu,\dots,\lambda^{4}\mu^{4})^{T},$
(63)

for the regression analysis yields

$\displaystyle\lambda_{8}$ $\displaystyle=$ $\displaystyle
g_{8}(\lambda,\mu)=\dfrac{1}{45}\left[\lambda^{2}+\lambda(3+\mu)+\mu(3+\mu)-9\right]\left[\lambda^{2}+\lambda(3+\mu)+\mu(3+\mu)\right],$
(64a) $\displaystyle\lambda_{27}$ $\displaystyle=$ $\displaystyle
g_{27}(\lambda,\mu)=\dfrac{3}{40}\left[\lambda^{2}+\lambda(3+\mu)+\mu(3+\mu)\right]\left[\lambda^{2}+\lambda(3+\mu)+(-1+\mu)(4+\mu)\right].$
(64b)

The above expressions exactly reproduce the numerical values obtained with the
help of the algorithm of Ref. algorithm_su3  and are also useful to complete
Table 1. These entries also agree in full with the eigenvalues obtained from
Eqs. (42b) and (42c), so the overall analysis is consistent.

Table 1: Eigenvalues $\lambda_{8}$ and $\lambda_{27}$ of the operators $H_{8}^{(4)}$ and $H_{27}^{(4)}$ defined in Eqs. (41b) and (41c), respectively, for the corresponding $SU(3)$ irrep $(\lambda,\mu)$. $SU(3)$ irrep $(\lambda,\mu)$ |  | $\lambda_{8}$ |  | $\lambda_{27}$
---|---|---|---|---
$(0,1)$ |  | $-4/9$ |  | $0$
$(0,2)$ |  | $2/9$ |  | $9/2$
$(0,3)$ |  | $18/5$ |  | $189/10$
$(0,4)$ |  | $532/45$ |  | $252/5$
$(0,5)$ |  | $248/9$ |  | $108$
$(0,6)$ |  | $54$ |  | $405/2$
$(1,1)$ |  | $0$ |  | $27/8$
$(1,2)$ |  | $112/45$ |  | $72/5$
$(1,3)$ |  | $80/9$ |  | $315/8$
$(1,4)$ |  | $108/5$ |  | $432/5$
$(1,5)$ |  | $392/9$ |  | $1323/8$
$(1,6)$ |  | $704/9$ |  | $288$
$(2,2)$ |  | $8$ |  | $36$
$(2,3)$ |  | $170/9$ |  | $153/2$
$(2,4)$ |  | $1702/45$ |  | $1449/10$
$(2,5)$ |  | $68$ |  | $252$
$(2,6)$ |  | $5092/45$ |  | $2052/5$
$(3,3)$ |  | $36$ |  | $1107/8$
$(3,4)$ |  | $2842/45$ |  | $2349/10$
$(3,5)$ |  | $4672/45$ |  | $15111/40$
$(3,6)$ |  | $162$ |  | $1161/2$
$(4,4)$ |  | $504/5$ |  | $1836/5$

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is a primitive element of $L/K$. Moreover,
$\alpha^{p}=N_{\widetilde{L}/L}(\gamma^{p})\in L\cap M=K$, and it is clear
that $\alpha$ is an algebraic integer, so $\alpha^{p}\in\mathcal{O}_{K}$.
Finally, we prove that $M=K(\zeta_{p})$. Since $\alpha^{p}\in K$, the normal
closure of $L/K$ is $\widetilde{L}=K(\alpha,\zeta_{p})=LK(\zeta_{p})$, and
since we have also that $\widetilde{L}=LM$ with $L$ and $M$ $K$-linearly
disjoint and $K(\zeta_{p})\subseteq M$, $M=K(\zeta_{p})$. ∎

To sum up, a degree $p$ extension of $p$-adic fields $L/K$ with maximally
ramified normal closure is almost classically Galois with complement
$K(\zeta_{p})$, and in particular $L\cap K(\zeta_{p})=K$. Moreover,
$L=K(\sqrt[n]{a})$ with $v_{L}(\sqrt[n]{a})=1$, that is, $v_{K}(a)=1$. Hence
we are under the hypotheses of Proposition 7.10, and then Proposition 7.11 is
established.

## Acknowledgements

The author is thankful with Ilaria Del Corso, Paul Truman and Lorenzo
Stefanello for their insightful comments. This work was supported by Czech
Science Foundation, grant 21-00420M, and by Charles University Research Centre
program UNCE/SCI/022.

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D. Gil-Muñoz, Charles University, Faculty of Mathematics and Physics,
Department of Algebra, Sokolovska 83, 18600 Praha 8, Czech Republic

E-mail address<EMAIL_ADDRESS>