Preprint 0.925 1

# Causality-Driven One-Shot Learning for Prostate Cancer Grading from MRI
††thanks: Preprint accepted on Aug 07, 2023 for ICCV-CVAMD 2023

Gianluca Carloni †1,2 Eva Pachetti †1,2 Sara Colantonio1

1Institute of Information Science and Technologies (ISTI-CNR), Italy
2University of Pisa, Italy
†These authors contributed equally
{gianluca.carloni, eva.pachetti<EMAIL_ADDRESS>
###### Abstract

In this paper, we present a novel method to automatically classify medical
images that learns and leverages weak causal signals in the image. Our
framework consists of a convolutional neural network backbone and a causality-
extractor module that extracts cause-effect relationships between feature maps
that can inform the model on the appearance of a feature in one place of the
image, given the presence of another feature within some other place of the
image. To evaluate the effectiveness of our approach in low-data scenarios, we
train our causality-driven architecture in a One-shot learning scheme, where
we propose a new meta-learning procedure entailing meta-training and meta-
testing tasks that are designed using related classes but at different levels
of granularity. We conduct binary and multi-class classification experiments
on a publicly available dataset of prostate MRI images. To validate the
effectiveness of the proposed causality-driven module, we perform an ablation
study and conduct qualitative assessments using class activation maps to
highlight regions strongly influencing the network’s decision-making process.
Our findings show that causal relationships among features play a crucial role
in enhancing the model’s ability to discern relevant information and yielding
more reliable and interpretable predictions. This would make it a promising
approach for medical image classification tasks.

##  1 Introduction

Building models that automatically perform diagnoses from medical data could
revolutionize a patient’s clinical pathway, especially in oncology, minimizing
invasive procedures and maximizing the chances of cures for the most severe
cases. The implementation of such models in the medical field is severely
limited by data availability and the heavy domain shifts that plague the data,
which makes the models non-generalizable. Especially in the magnetic resonance
imaging (MRI) domain, different vendors yield images with different features,
and that prevents the model from being able to generalize well. In a classical
fully-supervised manner, the problem can be overcome by training a model with
lots of data that covers all the possible data distribution. In practice,
however, this is not always possible, due to the limited amounts of
labeled/annotated data that affect the medical imaging domain.

Especially when dealing with limited data, one may wish to make an automated
recognition model focus on the most discriminative regions of the image
instead of paying attention to side details. Traditional convolutional neural
networks (CNN) and transformer networks, for instance, can discover features
hidden within input data together with their mutual co-occurrence. However,
they are weak at discovering and making explicit hidden causalities between
the features, which could be the reason behind a particular outcome. Indeed,
image classification models are expected to distinguish between the classes of
images taking into account the causal relationships between the features from
the images, in a way that might resemble how humans accomplish the task. For
this reason, several bridges between the field of causality and computer
vision are emerging in the literature [14, 1].

A particular case would be discovering hidden causalities among objects in an
image dataset. Unlike tabular data, which have a structured nature, when it
comes to images, their representation does not include any explicit
indications regarding objects or patterns. Instead, individual pixels convey a
particular scene visually, and image datasets do not provide labels describing
the objects’ dispositions. Due to this, supervised machine learning as such
cannot approach them. Additionally, unlike video frames, from a single image
one may not see the dynamics of appearance and change of the objects in the
scene. Therefore, a priori information as a hint for causal discovery is
absent.

To approach the problem of learning hidden causalities within images, Lopez-
Paz _et al_. [7] suggest the “causal disposition” concept as more primitive
than interventional causation (do-calculus) and causal graphs from Pearl’s
approach [11, 12]. However, it could be the only way to proceed with limited a
priori information. In their view, by counting the number $C(A,B)$ of images
in which the causal dispositions of artifacts $A$ and $B$ are such that $B$
disappears if one removes $A$, one can assume that artifact $A$ causes the
presence of artifact $B$ when $C(A,B)$ is greater than the converse $C(B,A)$.
As a trivial example, imagine the image of a car on a bridge. Now, if we were
to remove the car, then this would keep the image realistically looking (i.e.,
scene consistency), since an observer may see similar scenes among other
images. Conversely, if we were to remove the bridge, then this would make the
scene inconsistent, as that scenario is likely never seen among other images
(i.e., flying cars). Therefore, we may assume that the presence of the bridge
has some effect on the presence of the car. This concept leads to the
intuition that any causal disposition induces a set of asymmetric causal
relationships between the artifacts from an image (features, object
categories, etc.) that represent (weak) causality signals regarding the real-
world scene. To automatically infer such an asymmetric causal relationship
from the statistics observed in an image dataset would be a meeting point with
a machine vision system.

In this work, we combine a regular CNN with a causality-extraction module to
investigate the features and causal relationships between them extracted
during training. We build on ideas from [18] who suggest a way to compute such
an asymmetric measure for possible causal relationships within images, and we
propose a new scheme based on feature maps enhancement to enable “causality-
driven” CNNs to classify images taking into account hidden causalities within
them. Our hypothesis is that it would be possible and reasonable to get some
weak causality signals from the individual images of some medical datasets
without adding primary expert knowledge, and leveraging them to better guide
the learning phase. Ultimately, a model trained in such a manner would be able
to exploit weak causal dispositions of objects in the image scene to
distinguish lesion grades even with limited data and possibly domain shift on
the test set.

To evaluate how these causality-driven networks could behave in a low-data
regime, we perform the training in a Few-shot learning (FSL) manner, and in
particular in One-shot learning (OSL). Here, we propose a novel training
scheme in which we design meta-training and meta-testing tasks having related
classes (i.e., the same clinical problem is addressed) but at different
granularity levels. To perform such experiments, we exploit the Deep Brownian
Distance Covariance (DeepBDC) [19] method.

Our paper is structured as follows. After citing relevant works related to our
topic in Sec. 2, we present the rationale behind causality-driven CNNs and our
network proposition in Sec. 3.1, together with a description of the DeepBDC
method of our choice in Sec. 3.2. Later, in Sec. 4, we dive into the details
of our experiments settings, regarding both the dataset used and the meta-
training and meta-testing schemes. Finally, in Sec. 5 and Sec. 6, we provide
the results of our experiments and discuss our findings, summarizing our key
conclusions.

##  2 Related works

Several approaches to integrating causality into FSL have been proposed in the
literature, leading to different directions and applications. One notable
example is the work by Yue _et al_. [22], where they leverage causality to
demonstrate that pre-training in FSL can act as a confounder, resulting in
spurious correlations between samples in the support set and their
corresponding labels. To address this issue, they propose a novel FSL paradigm
in which they perform causal interventions on the Structural Causal Model
(SCM) of many-shot learning employing a backdoor adjustment approach. Based on
that work, Li _et al_. [6] propose a method to mitigate the influence of
confounders during the pre-training phase of the prototypical network [17].
They accomplish this by stratifying the pre-trained knowledge using a backdoor
adjustment based on causal intervention. Specifically, the backdoor adjustment
operations are applied in the metric layer of the prototypical network. The
feature vectors of the class prototype and the query set are divided into N
equal-size disjoint subsets, and the corresponding subsets are fed into their
respective classifiers. The final prediction result is obtained by averaging
the prediction probabilities from the N classifiers. Furthermore, in the work
by Yang _et al_. [20], the authors propose a method to enhance the robustness
and generalizability of few-shot text classification. They achieve this by
extracting causal associations from text using a causal representation
framework for FSL. The process involves performing causal interventions to
generate new data with label-relevant information for each input. The original
and augmented texts turned into feature representations are fed into a
factorization module, which enforces the separation and joint independence of
the representations from non-causal factors. Finally, the feature
representations are utilized by a classification module for making
predictions.

##  3 Methods

###  3.1 Causality-driven CNNs

#### Preliminaries.

In automatic image recognition, deep neural network classifiers obtain the
essential features required for classification not directly from the pixel
representation of the input image but through a series of convolution and
pooling operations. These operations are designed to capture meaningful
features from the image. Convolution layers are responsible for summarizing
the presence of specific features in the image and generating a set of feature
maps accordingly. As these maps are sensitive to the spatial location of
features in the image, pooling is employed to consolidate the presence of
particular features within groups of neighboring pixels in square-shaped sub-
regions of the feature map.

#### Causality signals in images.

When a feature map $F^{i}$ contains only non-negative numbers (e.g., thanks to
ReLU functions) and is normalized in the interval $[0,1]$, we can interpret
its values as probabilities of that feature to be present in a specific
location, for instance, $F^{i}_{r,c}$ is the probability that the feature $i$
is recognized at coordinates ${r,c}$. By assuming that the last convolutional
layer outputs and localizes to some extent the object-like features, we may
modify the architecture of a CNN such that the $n\times n$ feature maps
($F^{1},F^{2},\dots F^{k}$) obtained from that layer are fed into a new module
that computes pairwise conditional probabilities of the feature maps. The
resulting $k\times k$ causality map would represent the causality estimates
for the features.

#### Computing causality maps.

Given a pair of feature maps $F^{i}$ and $F^{j}$ and the formulation that
connects conditional probability with joint probability,
$P(F^{i}|F^{j})=\frac{P(F^{i},F^{j})}{P(F^{j})}$, following [18], we
heuristically estimate this quantity regarding the pairs of features by
adopting two possible methods, namely Max and Lehmer. The Max method considers
the joint probability to be the maximal presence of both features in the image
(each one in its location):

$P(F^{i}|F^{j})=\frac{(\max_{r,c}F^{i}_{r,c})\cdot(\max_{r,c}F^{j}_{r,c})}{\sum_{r,c}F^{j}_{r,c}}$
(1)

On the other hand, the Lehmer method entails computing

$P(F^{i}|F^{j})_{p}=\frac{LM_{p}(F^{i}\times F^{j})}{LM_{p}(F^{j})}$ (2)

where $F^{i}\times F^{j}$ is a vector of $n^{4}$ pairwise multiplications
between each element of the two $n\times n$ feature maps, while $LM_{p}$ is
the generalized Lehmer mean function [2] with parameter $p$, which is an
alternative to power means for interpolating between minimum and maximum of a
vector $x$ via harmonic mean ($p=-2$), geometric mean ($p=-1$), arithmetic
mean ($p=0$), and contraharmonic mean ($p=1$):
$LM_{p}(x)=\frac{\sum_{k=1}^{n}x_{k}^{p}}{\sum_{k=1}^{n}x_{k}^{p-1}}$.
Equations 1 and 2 could be used to estimate asymmetric causal relationships
between features $F^{i}$ and $F^{j}$, since, in general, $P(F^{i}|F^{j})\neq
P(F^{j}|F^{i})$. By computing these quantities for every pair $i$ and $j$ of
the $k$ feature maps, the $k\times k$ causality map is obtained. We interpret
asymmetries in such probability estimates as weak causality signals between
features, as they provide some information on the cause-effect of the
appearance of a feature in one place of the image, given the presence of
another feature within some other places of the image. Accordingly, a feature
may be deemed to be the reason for another feature when
$P(F^{i}|F^{j})>P(F^{j}|F^{i})$, that is ($F^{i}\rightarrow F^{j}$), and vice
versa.

#### Embedding causality in regular CNNs.

Once the causality map is computed, it can be embedded into the basic CNN
architecture. Terziyan and Vitko [18] flatten these suggested causality
estimates, concatenate them to the set of flattened feature maps, and let the
CNN learn how these estimates might influence image classification.
Differently from them, we exploit the causality map in a new way and get a
weighting vector to enhance or penalize the single feature maps during
training.

#### Causality weights.

At each epoch, as training progresses, we look for asymmetries between
elements opposite the main diagonal of the causality map. Some features may be
more often found on the left side of the arrow (i.e., $F\rightarrow$) than on
the right side (i.e., $\rightarrow F$). Therefore, we use such learned
causalities to compute causality weights to assign a degree of importance to
each feature map. Specifically, for each feature map, we take as its weighting
factor the difference between the number of times it was found to cause other
feature maps and the number of times it was found to be caused by another
feature map. Computing such quantity for every feature results in a vector of
causality factors, which is then passed through a ReLU activation to set to
zero all the negative elements.

#### Models.

We propose two variants of the model:

  * •

mulcat (multiply and concatenate). The non-negative causality factors multiply
the corresponding feature maps, resulting in a causality-driven version of
these feature maps. In this enhanced version, each feature map is strengthened
according to its causal influence within the image’s scene. Those features are
merged with the original features by concatenation along the channel axis and
form the final feature set that influences the classification outcomes.

  * •

mulcatbool. Same as the previous, but before multiplication, the factors
undergo boolean thresholding where all the non-zero factors are assigned a new
weight of $1$, while $0$ otherwise.

The first method weighs features more according to their causal importance (a
feature that is cause $10$ times more than another receives $10$ times more
weight). In contrast, the second method is more conservative and assigns all
features that are most often causes the same weight. We experiment with both
of them and compare their results.

###  3.2 One-shot learning

In standard FSL, the training process occurs in episodes or tasks. Each task
is formulated as an _$N$ -way $K$-shot_ classification problem, where $N$
represents the number of classes, and $K$ is the number of support images per
class. We refer to $Q$ as the number of query images per class. For our
experiments, we specifically focused on _N-way 1-shot_ classification, and we
employed the DeepBDC method introduced by Xie _et al_. [19]. In particular, we
utilized the meta-learning implementation of DeepBDC, known as Meta DeepBDC.
DeepBDC is a metric-based FSL method that employs the BDC as the distance
measure between prototypes. The BDC is defined as the Euclidean distance
between the joint characteristic function and the product of the marginal of
two random variables $X\in\mathbb{R}^{p}$ and $Y\in\mathbb{R}^{q}$. Following
[19], we provide a more formal definition of BDC:

$\rho(X,Y)=\int_{\mathbb{R}^{p}}\int_{\mathbb{R}^{q}}\frac{|\Phi_{XY}(t,s)-\Phi_{X}(t)\Phi_{Y}(s)|^{2}}{c_{p}c_{q}||t||^{1+p}||s||^{1+q}}dtds,$
(3)

where $\Phi_{X}(t)$ and $\Phi_{Y}(s)$ are the marginal distributions of X and
Y, respectively, $\Phi_{XY}(t,s)$ is the joint characteristic function of the
two random variables and $c_{p}$ is defined as
$c_{p}=\pi^{(1+p)/2}/\Gamma((1+p)/2)$, where $\Gamma$ is the complete gamma
function. DeepBDC has demonstrated higher performance compared to state-of-
the-art methods while being straightforward to deploy since it can be
implemented as a parameter-free spatial pooling layer that accepts feature
maps as input and provides a BDC matrix.

##  4 Experiments

###  4.1 Dataset and pre-processing

In our study, we conducted meta-training, meta-validation, and meta-testing
using the publicly available $1500$-acquisition dataset from the PI-CAI
challenge [13]. This dataset comprises mpMRI acquisitions of the prostate, and
for our experiments, we focused exclusively on cancerous patients. In
particular, we selected only T2-weighted (T2w) images containing lesions by
exploiting the expert annotations provided in the dataset. The dataset
contained biopsy reports expressing the severity of each lesion as Gleason
Score (GS). The pathologists assign a score of $1$ to $5$ to the two most
common patterns in the biopsy specimen based on the tumor severity. The two
grades are then added together to determine the GS, which can assume all the
combinations of scores from ”$1$+$1$” to ”$5$+$5$”. Additionally, the dataset
included the assigned GS’s group affiliation, defined by the International
Society of Urological Pathology (ISUP) [4], ranging from $1$ to $5$, which
provides the tumor severity information at a higher granularity level. From an
even more high-level perspective, lesions with a GS $\leq 3+3$ (ISUP = $1$)
and with GS $=3+4$ (ISUP = $2$) are considered low-grade (LG) tumors, as
patients with such lesions typically undergo active surveillance [9].
Conversely, lesions with GS $>3+4$ (ISUP $>2$) are high-grade (HG) tumors, as
treatment is foreseen [9]. In this study, we considered only lesions whose GS
was $\geq 3+4$ (ISUP $\geq 2$), as lesion annotations were not provided for
ISUP-$1$ lesions. As a result, we had eight classes of GS and four classes of
ISUP in our dataset. The total number of images we used was $2049$ (from $382$
patients), which we divided into training, validation, and testing subsets.
Specifically, we used $1611$ images for training, $200$ for validation, and
$238$ for testing. During the splitting process, we ensured patient
stratification, i.e., all the images of the same patient were grouped in the
same subset, avoiding any data leakage. To replicate a realistic scenario
involving distinct distributions in training and testing data, we utilized
data from two different vendors: SIEMENS vendor data for meta-training and
Philips vendor data for both meta-validation and meta-testing. Indeed, we
chose the same validation and test distributions since, as highlighted by
Setlur _et al_. [16], using validation samples that are not independent and
identically distributed with the test samples can lead to unreliable results
when determining the optimal model, specifically the one that maximizes
performance on the test set.

As for the data pre-processing, we utilized the provided whole prostate
segmentation to extract the mask centroid for each slice. We standardized the
field of view (FOV) at $100$ mm in both $x$ ($FOV_{x}$) and $y$ ($FOV_{y}$)
directions to ensure consistency across all acquisitions and subsequently
cropped each image based on this value around the found centroid. To determine
the number of rows ($N_{rows}$) and columns ($N_{cols}$) corresponding to the
fixed FOV, we utilized the pixel spacing in millimeters along the x-axis
(denoted as $px$) and the y-axis (denoted as $py$). The relationships used to
derive the number of columns and rows are $N_{cols}=\frac{FOV_{x}}{px}$ and
$N_{rows}=\frac{FOV_{y}}{py}$, respectively. Additionally, we resized all
images to a uniform matrix size of $128\times 128$ pixels to maintain a
consistent pixel count. Finally, we performed image normalization using an in-
volume method. This involved calculating the mean and standard deviation (SD)
of all pixels within the volume acquisition and normalizing each image based
on these values using a z-score normalization technique.

###  4.2 Classification experiments

In our study, we conducted experiments on two classification scenarios: (i)
distinguishing between LG and HG lesions and (ii) ISUP grading. For each
scenario, we carefully designed meta-training and meta-testing tasks. Most FSL
approaches typically involve using unrelated classes between meta-training and
meta-testing tasks [3, 8, 10]. Instead, we propose a different approach where
we focus on the same clinical problem but at varying levels of granularity.
Specifically, during meta-training, we designed more challenging tasks,
requiring the model to distinguish between classes with higher levels of
granularity. Conversely, during meta-testing, we provided the model with
easier classification tasks involving higher-level classification. Our
rationale is that this approach would lead to higher performance on the
specific task of interest performed during meta-testing, as the model would
find it relatively easier to execute due to its exposure to more complex tasks
during meta-training. Below we provide a detailed explanation of how we
designed our meta-training and meta-testing tasks for both experiments.

In the first scenario, we labeled the meta-training data according to the four
ISUP classes. The model performed binary classification in each meta-training
task between two randomly selected classes from the four provided. However,
during meta-testing, the model was tasked with a higher-level classification,
namely, distinguishing between LG and HG lesions. For ease of reference, we
will refer to this experiment as the _2-way_ experiment. In the second
scenario, we labeled the meta-training data based on the GS. Each training
task required the model to distinguish between four randomly selected GS
classes out of the total eight. For meta-validation and meta-testing, we
labeled each patient based on the ISUP tumor severity score and made the model
distinguish across these four classes. Henceforth, we will refer to this
experiment as _4-way_ experiment. We summarized the labeling procedure for the
two experiments in Table 1.

Experiment | Splitting | Labels
---|---|---
2-way | Meta-training |  ISUP $2$, ISUP $3$, ISUP $4$, ISUP $5$
Meta-validation |  LG (ISUP $2$) - HG (ISUP $3$, ISUP $4$, ISUP $5$)
Meta-test |  LG (ISUP $2$) - HG (ISUP $3$, ISUP $4$, ISUP $5$)
4-way | Meta-training |  GS $3+4$, GS $4+3$, GS $4+4$, GS $3+5$, GS $5+3$, GS $4+5$, GS $5+4$, GS $5+5$
Meta-validation | ISUP $2$, ISUP $3$, ISUP $4$, ISUP $5$
Meta-test | ISUP $2$, ISUP $3$, ISUP $4$, ISUP $5$
Table 1: A summary of the labeling procedure according to our training
approach. ISUP = International Society of Urological Pathology, LG = Low
Grade, HG = High Grade, GS = Gleason Score.

In both scenarios, we employed a one-shot setting for both meta-training and
meta-testing. This means that the model only observed a single example per
class in the support set of each task. However, during the evaluation phase,
we expanded the query set by utilizing ten samples per class in both meta-
training and meta-testing tasks.

###  4.3 Architecture and training

Many widely used architectures for large-scale image recognition incorporate
an adaptive average pooling layer with an output size of $1\times 1$ placed
just before the classifier. Its primary advantage is the ability to
accommodate input images of varying sizes, as it automatically adjusts its
parameters (such as kernel size, stride, and padding) to ensure that the
output is consistently $1\times 1$ in shape. This dimensionality reduction,
however, conflicts with the 2D nature of feature maps for computing
causalities. Therefore, since we chose the ResNet18 as the backbone
architecture in our work, we substituted its AdaptiveAvgPool2D layer with an
identity layer in our experiments. We performed an optimization over the
method by which computing the causality maps (i.e., Max or Lehmer) and, for
the Lehmer case, over six different values of its parameter p:
[$-100,-2,-1,0,1,100$]. Accordingly, we trained seven models for each
causality setting (i.e., mulcat or mulcatbool), resulting in $14$ causality-
driven models plus one baseline model for each experiment (i.e., 2-way and
4-way). For each of the two causality settings, we chose the best-performing
model on the meta-validation set.

Given an input image, the causality-driven ResNet18 extracts $512$
bidimensional feature maps of shape $4\times 4$. While those features proceed
along the main branch of the network, a copy of them enters the causality
module where for each pair, we extract their conditional probabilities by
either applying Eq. 1 or Eq. 2 depending on the causality method of choice.
Starting from the resulting $512\times 512$ causality map, the vector of $512$
causality factors is obtained according to the model variant of choice (i.e.,
mulcat or mulcatbool) and then multiplied for the corresponding feature maps.
Then, after concatenation of two such feature sets, we obtain a set of $1024$
feature maps of shape $4\times 4$ for each input image. Figure 1 shows the
proposed causality-driven network. Although the training is performed task by
task, here we represented the functioning of our method for just one input
image.

At this point, the final set of feature maps is used to calculate the image
representations. Following the Prototypical Networks [17] approach, the
classification is performed by computing the BDC between the prototypes of
each supported class, calculated as the mean of the BDC matrix of each support
image of that class and each query image representation. To infuse the model
with robustness to different data selections, we performed $600$ meta-training
tasks, $600$ meta-validation tasks, and $600$ meta-testing tasks for each
experiment.

Figure 1: Causality-Driven ResNet18 for prostate cancer Grading from MRI.
Here, the causality map can be computed with one of Max and Lehmer options,
while the causality factors can be computed using either mulcat or mulcatbool
methods. For visualization purposes, the size of the box representing the
causality map has been reduced.

As our loss function and optimizer, we employed the AUC margin loss (AUCM)
[21] and the Proximal epoch stochastic method (PESG) [5], respectively. These
were employed to maximize the Area Under the ROC curve (AUROC), which is more
stable w.r.t accuracy to the dataset unbalancing. In addition, we performed
our experiments with the following hyperparameters: initial learning rate =
$1e-2$, weight decay = $1e-2$, number of epochs = $100$, decay epochs:
[$20$,$80$]. At each decay epoch, the learning rate value is divided by $10$.

###  4.4 Evaluation metrics

In our evaluation, we utilized the AUROC as the performance metric for all our
experiments. For the $2$-way experiment, we computed the classical binary
AUROC by considering the HG as the positive class. In the case of the $4$-way
experiment, instead, we calculated the AUROC using the _One-vs-rest_ setting.
This approach involves computing the AUROC for each class against the rest of
the classes independently. In addition, we evaluated the binary classification
performance of the models in the $4$-way experiment by computing the AUROC of
ISUP class $2$ versus all the rest.

###  4.5 Visualizing the impact of causality

To test the hypothesis that a causally trained model can learn more
discriminative representations for image classes, we performed post hoc
explainability evaluations. Investigating the variability of visualization
results with different choices of post hoc explainable AI (XAI) methods is
beyond the scope of our work, therefore, we employed the popular Grad-CAM
[15], which belongs to the broad literature on class activation maps (CAM) The
basic idea behind Grad-CAM is to utilize the gradients flowing back from a
chosen layer of a CNN to understand which parts of the image contribute the
most to the activation of a particular class. In this case, we chose to
compute the Grad-CAM heatmaps at the BDC module level, which takes as input
the final set of feature maps. In addition, to make a fair comparison, we
computed the heatmaps w.r.t. the ground-truth target of each image and only in
the cases for which the prediction was performed correctly by both the non-
causality-driven model and the mulcat and mulcatbool models.

##  5 Results

###  5.1 Performance of causality-driven CNNs

The main results of our analysis are reported in Table 2. We reported all
those values as mean and SD AUROC across all the $600$ meta-test tasks.
Concerning the $2$-way experiment (i.e., LG vs HG), the baseline model
achieved $0.539$ ($0.141$), and embedding the causality module led the model
to improve, obtaining $0.550$ ($0.144$) and $0.556$ ($0.141$) AUROC for the
mulcat and mulcatbool variants, respectively. In particular, both these
causality-driven variant results were obtained with Lehmer causality setting,
using a Lehmer parameter of -$100$. Similar behaviour, with more pronounced
improvement, was observed for the $4$-way experiment (i.e., ISUP $2-5$), where
we obtained $0.585$ ($0.068$) multi-class AUROC for the non-causality-driven
model, whereas the mulcat and mulcatbool variants achieved $0.611$ ($0.069$)
and $0.614$ ($0.067$), respectively. Again, both best-performing mulcat and
mulcatbool variants were obtained employing the Lehmer setting, with Lehmer
parameters equal to $1$ and -$2$, respectively.

Table 2 also shows the results of an ablation study. Indeed, since in the
causally-driven implementations is the causality factors vector that
ultimately determines which (and how) feature maps are enhanced, we modify
that vector to weigh features in a random manner rather than based on a
principled way based on the causality map. The ablation variants of the mulcat
and mulcatbool models that we realized are:

  * •

ablation mulcat. The $1\times k$ vector of causality factors (i.e., weights)
is replaced with a random vector of the same size with integer values ranging
from $0$ (a feature map is never cause of another feature) and $k-1$ (it is
cause of every other feature).

  * •

ablation mulcatbool. Similar to the previous, the values of the weights are
randomly assigned either to $0$ or to $1$.

Setting | 2-way 1-shot | 4-way 1-shot | 4-way 1-shot*
---|---|---|---
Main results
Non causality-driven | 0.539 (0.141) | 0.585 (0.068) | 0.586 (0.118)
Causality-driven mulcat | 0.550 (0.144) | 0.611 (0.069) | 0.712 (0.118)
Causality-driven mulcatbool |  0.556 (0.141) |  0.614 (0.067) |  0.713 (0.119)
Ablation study
Ablation mulcat | 0.535 (0.143) | 0.557 (0.063) | 0.557 (0.111)
Ablation mulcatbool | 0.540 (0.139) | 0.571 (0.068) | 0.612 (0.119)
Table 2: Results of the best-performing models w.r.t the causality setting and
causality method compared to the non-causality choice for 2-way and 4-way
experiments. *: trained to distinguish four classes (ISUP $2-5$), but the
AUROC is computed between ISUP $2$ vs. rest.

###  5.2 Visualizing activation maps

As a result of the post hoc explainability evaluations, we obtain the
visualizations shown in Figure 2. The first row (a to e) regards a test case
from models trained in the $2$-way setting, while the second row (f to j)
pertains to a case from $4$-way models. From left to right, the columns
represent the input test image, the annotation of the lesion in terms of
binary masks, the Grad-CAM activation for the baseline models (not causality-
driven), the Grad-CAM activation for the mulcat models, and the Grad-CAM
activation for the mulcatbool models.

Figure 2: Impact of causality on the learned models. From left to right, the
columns represent the input test images, the binary masks for the annotated
lesion, the Grad-CAM activations for the baseline models (not causality-
driven), the Grad-CAM activations for the mulcat models, and the Grad-CAM
activations for the mulcatbool models. Each row represents a different test
case. This image is best seen in colour.

##  6 Discussion and Conclusion

In this study, we investigated the impact of integrating a new causality-
extraction module into traditional CNNs to enhance classification performance.
We trained this causality-driven model using an OSL approach, leveraging meta-
learning conditions with the MetaDeepBDC model [19]. We aimed to assess the
effectiveness of such a model in situations where only a few samples are
available for training, a challenge frequently encountered in medical image
analysis.

In Pearl’s terms, our work regards the first rung of the ladder of causation,
where reasoning is limited to conditional probabilities based on observational
datasets. However, we aimed to explore whether this approach could yield
improvements in a scenario involving image data (rather than structured
tabular data), and no prior knowledge of the data generation process. Our
findings demonstrate that incorporating a causality-driven module into our
model leads to enhanced performance compared to the baseline. This behavior is
evident in both the $2$-way and $4$-way experiments. In particular, in the
$4$-way experiment, the causality module provided a $3$% improvement over the
baseline in terms of the multi-class AUROC and about $13$% improvement in
terms of ISUP $2$ vs. rest AUROC.

We additionally validated our numerical results both quantitatively and
qualitatively. Quantitatively, we performed ablation studies on the actual
impact of the causality factors on producing valuable causality-driven feature
maps. As expected, when the causal weights are replaced with random vectors,
the accuracy of the final model is worse than its causally-driven counterpart
(see Table 2). This seems to suggest that, albeit weak, the causality signals
learned during training help the network. Qualitatively, we generated Grad-CAM
heatmaps to highlight the regions of the input image that strongly influence
the network’s output. Figure 2 presents examples for both the $2$-way and the
$4$-way experiments. In both cases, we calculated the heatmaps w.r.t. the
ground truth target when all three types of models correctly classified the
images. The heatmaps reveal distinct patterns between the baseline model and
the causality-driven models. The former tends to focus on a larger area of the
image, including regions outside the prostate and lesion boundaries.
Conversely, the causality-driven models concentrate on smaller areas,
predominantly encompassing the prostate and the lesion. Figure 2 (c-e), which
depicts the $2$-way experiment, shows that the baseline model (Figure 2 c)
primarily attends to the left half of the image, encompassing the lesion as
well as non-prostate tissues. In contrast, the mulcat version (Figure 2 d)
exhibits a more focused heatmap, highlighting mainly the prostate and a
portion of the lesion. The mulcatbool case (Figure 2 e) further refines the
focus by emphasizing the prostate and a larger portion of the lesion.
Similarly, as for the $4$-way experiment, the baseline model (Figure 2 h) pays
attention to the left half of the image. In contrast, the mulcat and
mulcatbool versions (Figure 2 i-j) prioritize the lesion’s immediate
surroundings. Although all three models produce accurate predictions, the
heatmaps demonstrate that the causality-driven module enables better
localization of relevant regions, providing more reliable explanations for the
model’s predictions.

Comparing the $4$-way experiment to the $2$-way experiment, the former
produced better classification results. Indeed, despite being a more complex
task, the mean AUROC across all classes is higher. We argue that two factors
contribute to this outcome, both associated with the meta-training phase.
Firstly, in the $4$-way experiment, the model encounters a more diverse range
of tasks, as the four classes can be selected from a pool of eight distinct
options. In contrast, the $2$-way experiment encompasses only four classes in
total. Secondly, for the $4$-way experiment, in each meta-training task, the
model is trained to distinguish a higher number of classes, representing a
more challenging task w.r.t. the binary case. Consequently, the model is
better equipped to handle the testing phase, resulting in improved
performance. This superiority becomes even more evident when examining the
models’ performance in the $4$-way experiment in classifying ISUP class $2$
(representing LG lesions) against all other classes (representing HG lesions).
Notably, when the mulcat and mulcatbool causality-driven modules are embedded
into the model, the AUROC value for this particular task increases by almost
$16$%.

Our work comes with several limitations. We only used ResNet18 as our backbone
network, and this might have limited the opportunity to find better-suited
architectures able to detect finer details in the image and consequently
extract more informative latent representations. In addition, we performed our
experiments only in an OSL setting, limiting the classification performance of
our models. In fact, we must note that the performance values obtained are not
yet sufficient for clinical practice. Finally, we validated our method on only
one dataset.

Despite that, our findings indicate that integrating a causality-driven module
into a classification model can enhance performance, even with severe data
limitations, which are common in medical imaging. The causality-driven
approach not only improves overall classification results but also helps the
model focus more accurately on the critical regions of the image, leading to
more reliable and robust predictions. This aspect is particularly critical in
medical imaging, where precise and reliable classification is crucial for
effective diagnosis and treatment planning.

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#  Explaining the $\gamma\gamma+X$ Excesses at $\approx$151.5 GeV
via the Drell-Yan Production of a Higgs Triplet

Saiyad Ashanujjaman ID<EMAIL_ADDRESS>Institute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China  Kaiping Neutrino
Research Center, Jiangmen 529399, China  Sumit Banik ID<EMAIL_ADDRESS>Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH–8057 Zürich,
Switzerland  Paul Scherrer Institut, CH–5232 Villigen PSI, Switzerland
Guglielmo Coloretti ID<EMAIL_ADDRESS>Physik-Institut,
Universität Zürich, Winterthurerstrasse 190, CH–8057 Zürich, Switzerland  Paul
Scherrer Institut, CH–5232 Villigen PSI, Switzerland  Andreas Crivellin ID
<EMAIL_ADDRESS>Physik-Institut, Universität Zürich,
Winterthurerstrasse 190, CH–8057 Zürich, Switzerland  Paul Scherrer Institut,
CH–5232 Villigen PSI, Switzerland  Siddharth P. Maharathy ID
<EMAIL_ADDRESS>Institute of Physics, Bhubaneswar, Sachivalaya Marg,
Sainik School, Bhubaneswar 751005, India  Homi Bhabha National Institute,
Training School Complex, Anushakti Nagar, Mumbai 400094, India  Bruce Mellado
<EMAIL_ADDRESS>School of Physics and Institute for Collider Particle
Physics, University of the Witwatersrand, Johannesburg, Wits 2050, South
Africa  iThemba LABS, National Research Foundation, PO Box 722, Somerset West
7129, South Africa

###### Abstract

The multi-lepton anomalies and searches for the associated production of a
narrow resonance indicate the existence of a $\approx$151 GeV Higgs with a
significance of $>5\sigma$ and $>3.9\sigma$, respectively. On the one hand,
these anomalies require a sizable branching fraction of the new scalar to
$WW$, while on the other hand, no $ZZ$ signal at this mass has been observed.
This suggests that the new boson is the neutral component of an $SU(2)_{L}$
triplet with zero hypercharge. This field leads to a positive definite shift
in the $W$ mass, as preferred by the current global fit, and is produced via
the Drell-Yan process $pp\to W^{*}\to\Delta^{0}\Delta^{\pm}$. We use the side-
bands of the ATLAS analysis [1] of the associated production of the Standard
Model Higgs in the di-photon channel to search for this production mode of the
triplet. Since the dominant decays of $\Delta^{\pm}$ depend only on its mass,
the effect in the 22 signal categories considered by ATLAS is completely
correlated. We find that the ones most sensitive to the Drell-Yan production
of the triplet Higgs show consistent excesses at a mass of $\approx$151.5 GeV.
Combining these channels in a likelihood ratio test, a non-zero
Br$[\Delta^{0}\to\gamma\gamma]=0.66\%$ is preferred by $\approx$3$\sigma$,
supporting our conjecture.

††preprint: PSI-PR-24-06, ZU-TH 09/24, ICPP-79

##  I Introduction

The Standard Model (SM) of particle physics is the established theoretical
description of the fundamental constituents of matter and their interactions.
It has been extensively and successfully tested at the precision frontier [2,
3, 4] and its last missing particle, the Brout-Englert-Higgs boson [5, 6, 7,
8] has been discovered in 2012 at CERN [9, 10, 11]. In fact, this $125\,$GeV
scalar has properties consistent with the ones predicted by the SM [12, 13,
14, 15, 16].

However, the minimality of the SM Higgs sector, i.e. a single $SU(2)_{L}$
doublet scalar that simultaneously gives masses to the electroweak (EW) gauge
bosons and all (fundamental) fermions, is not guaranteed by any theoretical
principle or symmetry. Therefore, a plethora of extensions have been proposed
in the literature, including the addition of $SU(2)_{L}$ singlets [17, 18,
19], doublets [20, 21, 22, 23, 24] and triplets [25, 26, 27, 28, 29, 30], as
well as countless more models containing multiple new scalars.

To be viable, such extensions must respect the bounds from electroweak
precision observables (particularly the $\rho$ parameter), flavour observables
and searches at the Large Hadron Collider (LHC). While flavour observables can
be avoided if the new Higgses do not couple directly to SM fermions, the
experimental average of the $W$ mass [4] even suggests a positive new physics
(NP) effect with $3.7\sigma$ significance [31, 32, 33]. Furthermore, LHC
searches revealed increasing excesses for new scalar bosons at the EW scale
[34], most pronounced at $\approx$151.5 GeV [35] ($3.9\sigma$). This is
consistent with the mass range of 145 GeV-155 GeV predicted by a new physics
explanation of the multi-lepton anomalies [36], i.e. LHC processes involving
multiple leptons and missing transverse momentum in the final state (see Refs.
[37, 34] for recent reviews).

An explanation of the multi-lepton anomalies [38, 39, 36, 40, 41, 42, 43, 44],
in particular the very significant disagreement in the top quark differential
distributions [45, 41, 44] ($>5\sigma$), requires the new scalar to decay
dominantly to $WW$. In fact, also resonant searches show indications for this
decay mode [46, 47, 48], but no $ZZ$ excess at this mass is observed [49, 50].
This suggests that the neutral component of the $SU(2)_{L}$ triplet with
hypercharge $Y=0$, which, at tree-level in the absence of mixing, couples to
$W$ bosons but not to the $Z$, could be the $\approx$151 GeV boson. This
field, at the same time, predicts a positive definite shift in the $W$ mass
(with respect to the SM prediction) [51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63]. Furthermore, it cannot have tree-level couplings to SM fermions,
such that it naturally avoids flavour bounds and has in general suppressed
gluon and vector boson fusion production cross-sections, weakening or avoiding
LHC bounds.

The triplet Higgs is produced via the Drell-Yan process $pp\to
W^{*}\to\Delta^{0}\Delta^{\pm}$ at the LHC, i.e. together with additional
particles. This has the potential to explain the hints for the $\approx$151
GeV boson, which are most pronounced in associated production channels. Note
that these exclusive searches, in which the presence of additional particles
or signatures in the final state, together with the ones from which the
invariant mass is reconstructed, are required, can significantly improve the
new physics sensitivity by reducing the SM background.

The most extensive and complete search for the associated production of the SM
Higgs was performed in Ref. [1]. In addition to the photon pair, from which
the mass of the Higgs is reconstructed, ATLAS searched in its model-
independent analysis 22 different signatures, including leptons, jets, missing
energy, etc. While no significant excess for $m_{\gamma\gamma}=125$ GeV was
found (in a single channel), the side-bands of this analysis can be used to
search for the associated production of new Higgses in the 110 GeV–160 GeV
range.

Therefore, we consider the process $pp\to
W^{*}\to(\Delta^{0}\to\gamma\gamma)(\Delta^{\pm}\to XY)$, see Fig. 1. For our
mass range, $XY=,tb,WZ,\tau\nu$ (including that $t$, $W$ or $Z$ can be off-
shell) are the dominant decay rates. Because $\Delta^{\pm}$ and $\Delta^{0}$
turn out to be quasi-degenerate in mass and the branching ratios of
$\Delta^{\pm}$ solely depend on it, Br$(\Delta^{0}\to\gamma\gamma)$ is the
only relevant additional free parameter (for any given mass), making our setup
very predictive. We can thus perform a global fit taking the full set of
channels of Ref. [1] to test our conjecture that the neutral component of the
$SU(2)_{L}$ triplet is the (hypothetical) $\approx$151 GeV boson.

##  II Model and Analysis

$p$$p$$\gamma$$\gamma$$X$$Y$$W^{*}$$\Delta^{\pm}$$\Delta^{0}$ Figure 1:
Feynman diagram showing the Drell-Yan production of the triplet, $pp\to
W^{*}\to(\Delta^{\pm}\to XY)(\Delta^{0}\to\gamma\gamma)$, which can be
searched for in the side-bands of the ATLAS analysis of the associated
production of the SM Higgs [1]. Within our mass range, $XY$ are dominantly
$WZ,tb,cs,\tau\nu$ (see Fig. 2).

The SM extended with an $SU(2)_{L}$ triplet scalar with hypercharge $Y=0$ is
commonly referred to as the $\Delta$SM [64, 65, 66, 67, 68, 69, 70, 71]. It
contains, in addition to the SM Higgs $h$, a charged scalar ($\Delta^{\pm}$)
and a neutral one ($\Delta^{0}$). In the process of spontaneous electroweak
symmetry breaking, it acquires a vacuum expectation value (VEV) $v_{\Delta}$,
which maximally breaks custodial symmetry because it only contributes to the
$W$ mass but not to the $Z$ mass.111For details on the model and its free
parameters, see e.g. [63]. Therefore, it leads to a positive definite shift in
$m_{W}$ (at tree-level) w.r.t. the SM prediction [31, 72, 73, 74, 75, 76, 77,
78, 4]. This is in agreement with the current global average [4], which
indicates a positive effect of $\approx$20 MeV with a significance of
$3.7\sigma$ [31].222The global fit is based on the LHC [79, 80], Tevatron [81]
and LEP [82] results. While there is a tension between the most precise
measurement from CDF-II with the LHC combination, which leads to an
enlargement of the error, notice that, also excluding the CDF-II result, a
positive shift is preferred at $\approx$2$\sigma$. While the exact value is
immaterial for this work, the $W$ mass implies $v_{\Delta}=\mathcal{O}({\rm
GeV})$.

Since the mass splitting between $\Delta^{\pm}$ and $\Delta^{0}$ is of the
order of $v_{\Delta}$ (for the scalar self-couplings of $\mathcal{O}(1)$), we
can assume them to be degenerate to a good approximation, i.e.
$m_{\Delta^{\pm}}\approx m_{\Delta^{0}}\approx m_{\Delta}$. The mixing angle
$\alpha$ between $h$ and $\Delta^{0}$ impacts the branching ratios of
$\Delta^{0}$. However, we are only interested in $\Delta^{0}\to\gamma\gamma$
which is loop induced and depends not only on $\alpha$ but also critically on
$m_{\Delta^{0}}-m_{\Delta^{\pm}}$ (because it is related to the
$\Delta^{0}\Delta^{\pm}\Delta^{\mp}$ coupling). Therefore, we can subsume
these parameters within Br$(\Delta^{0}\to\gamma\gamma)$ such that it is the
only free parameter in our analysis (for any given mass).

Target |  Region |  Detector level
---|---|---
High jet activity |  $\geq 4j$ |  $n_{jet}~{}\geq 4$ , $|\eta_{jet}|~{}<2.5$
Top |  $\ell b$ |  $n_{\ell=e,\mu}\geq 1$, $n_{b\text{-jet}}\geq 1$
$t_{\text{lep}}$ |  $n_{\ell=e,\mu}=1$, $n_{\text{jet}}=n_{b\text{-jet}}=1$
$t_{\text{had}}$ |  $n_{\ell=e,\mu}=0$, $n_{\text{jet}}=3$,$n_{b\text{-jet}}=1$
Lepton |  $\geq$1$\ell$ |  $n_{\ell=e,\mu}\geq 1$
$2\ell$ |  $ee,\mu\mu$ or $e\mu$
|  $\geq$3$\ell$ |  $n_{\ell=e,\mu}\geq 3$
$E_{\text{T}}^{\text{miss}}$ |  $E_{\text{T}}^{\text{miss}}>100\text{ GeV}$ |  $E_{\text{T}}^{\text{miss}}>100\text{ GeV}$
$E_{\text{T}}^{\text{miss}}>200\text{ GeV}$ |  $E_{\text{T}}^{\text{miss}}>200\text{ GeV}$
$E_{\text{T}}^{\text{miss}}>300\text{ GeV}$ |  $E_{\text{T}}^{\text{miss}}>300\text{ GeV}$
Table 1: The signal regions of the ATLAS analysis [1] that are most relevant
for the Drell-Yan production of the scalar triplet.

Figure 2: Left: Production cross-section for $pp\to\Delta^{0}\Delta^{\pm}$ and
$pp\to\Delta^{\pm}\Delta^{\mp}$ as a function of the triplet mass including
the NNLL and NLO QCD correction factor of Refs. [83, 84]. Right: Dominant
branching ratios of the charged component $\Delta^{\pm}$ as a function of the
mass.

The production cross-sections for the Drell-Yan processes, $pp\to
W^{*}\to\Delta^{0}\Delta^{\pm}$ and $pp\to
Z^{*},\gamma^{*}\to\Delta^{\mp}\Delta^{\pm}$ (where the asterisk denotes that
the particle is off-shell), as a function of $m_{\Delta}$ are given in Fig. 2
(left). Here we include the next-to-next-to leading log (NNLL) and NLO QCD
correction factor of Refs. [83, 84] (see also Ref. [85] for QCD corrections to
charged Higgs production). Note that this is the dominant production mode in
the limit of $\alpha\approx 0$ and $v_{\Delta}\ll v$ (where $v$ is the SM
VEV). These processes were studied in detail in Ref. [86] finding that
4-lepton searches can only exclude masses between $\approx 160\,$GeV and
$\approx 200\,$GeV.333In fact, Ref. [86] used
$m_{\Delta^{0}}=m_{\Delta^{\pm}}=150\,$GeV as a benchmark point. However, the
ATLAS analysis of the associated production of the SM Higgs [1] sensitive to
$pp\to W^{*}\to\Delta^{0}\Delta^{\pm}$ with $\Delta^{0}\to\gamma\gamma$ shown
in Fig. 1 was not included.

To determine the efficiency (including cuts and acceptance, etc.) we simulated
$pp\to W^{*}\to(\Delta^{\pm}\to XY)(\Delta^{0}\to\gamma\gamma)$ using
MadGraph5aMC@NLO with the parton shower performed by Pythia8.3 [87] and
carried out the detector simulation for the ATLAS detector [1] with Delphes
[88]. The UFO model file at NLO for the $\Delta$SM was built using FeynRules
[89, 90, 91].444To reduce the computational resources needed for our scan, we
performed the simulations at leading order but rescaled the production cross-
section to account for the NLO and NNLL effect. Furthermore, we simulated the
4-jet category at NLO, where gluon radiation is particularly important, and
included it via a correction factor of 1.2 in the extraction of the
efficiency. Note that the dominant branching fractions of $\Delta^{\pm}$ are
induced at tree-level by $v_{\Delta}$ such that the dependence on it cancels.
Therefore, we can show them as a function of $m_{\Delta}$ in Fig. 2 (right).
The dominant final states for the mass range of interest are
$XY=\tau\nu,cs,tb$ and $WZ$ (including implicitly that $t$, $W$ and $Z$ can be
off-shell).

Taking into account that $Z$ and $W$ bosons decay to lepton, missing energy
and jets, we expect that the ATLAS signal regions targeting leptons, missing
transverse energy ($E_{T}^{\rm miss}$) and high jet activity are the most
important ones. Furthermore, at higher values of $m_{\Delta}$, the categories
addressing top quarks become relevant.

To obtain bounds on Br$(\Delta^{0}\to\gamma\gamma)$ we perform a likelihood-
ratio test using Poisson statistics. This means that we take the theory
prediction for the number of events in a bin $i$ as the mean of the Poisson
distribution and calculate the ratio

$\mathcal{L}_{R}=\Pi_{i}\left[\mathcal{L}(N_{i}^{\rm SM},N_{i}^{\rm
exp})/\mathcal{L}(N_{i}^{\rm NP},N_{i}^{\rm exp})\right],$ (1)

where $\mathcal{L}$ is the likelihood described by the Poisson distribution,
$N_{i}^{\rm SM,NP}$ is the number of expected events (which is a real number)
in the SM (including the SM Higgs) or our $\Delta$SM and $N_{i}^{\rm exp}$ is
the number of events measured by ATLAS. Because $\mathcal{L}(N_{i}^{\rm
NP},N_{i}^{\rm exp})$ is a function of $m_{\Delta}$ and
Br$(\Delta^{0}\to\gamma\gamma)$, we can maximize it w.r.t. the latter for any
given mass. This then corresponds to the maximum likelihood
$\mathcal{L}_{R}^{\rm max}$ and thus to the best-fit value for
Br$(\Delta^{0}\to\gamma\gamma)$. Note that we allow for an unphysical negative
branching ratio to take into account the effect of downward fluctuations of
the background. The 68% and 95% confidence level (CL) regions are calculated
in the asymptotic limit (where $\chi^{2}=-2{\rm ln}(\mathcal{L}_{R}^{\rm
max})$) by requiring that $\chi^{2}=1$ and $\chi^{2}=4$, respectively.

Figure 3: 95% CL upper limits on Br($\Delta^{0}\to\gamma\gamma$) for the 10
most relevant categories. Note that except for the $t_{had}$ and $3\ell$
signal regions, all categories show a weakening of the limit around 151 GeV.

##  III Results

We find that among the 22 categories provided by ATLAS [1], the ten detained
in Table 1 give the most relevant constraints on
Br$(\Delta^{0}\to\gamma\gamma)$. The corresponding 95% CL bounds on
Br$(\Delta^{0}\to\gamma\gamma)$ are shown in Fig. 3. Interestingly, all
relevant distributions, except $t_{\rm had}$, show a weakening of the limit
around a mass of $\approx$150 GeV. Furthermore, a similar weakening of the
limits is observed between 125 GeV and 130 GeV. Combing all ten signal regions
in Fig. 3 by taking the product of the likelihood ratios, we obtain the best-
fit value for Br$(\Delta^{0}\to\gamma\gamma)$ as a function of the mass, as
well the 68% CL and 95% CL regions, as shown in Fig. 4.555Note that there is a
moderate correlation between the signal regions, in particular between the
different $E_{T}^{\rm miss}$ categories. However, we checked that removing two
of the three $E_{T}^{\rm miss}$ signal regions from the fit has a very small
impact on the final result. One can see that there is a preference for a non-
zero signal both at around 125–130 GeV and at $\approx$151.5 GeV. The latter
has a significance of $3\sigma$ and shows the consistency of data with the
assumption that the (hypothetical) $\approx 151.5$ GeV Higgs is the neutral
component of a triplet with $Y=0$.

Furthermore, the excess in the 125 GeV–130 GeV range, with a significance of
$3.5\sigma$ at $\approx$ 128 GeV, could point towards the enhanced associated
production of the SM Higgs, since at 125 GeV the significance is still
$1.8\sigma$. Finally, note that, in principle, the background would be lower
if it were refitted in the presence of one or more additional resonances. This
would, particularly for our two-peak structures with a deficit in events
between them, further increase the significance.

##  IV Conclusions and Outlook

The multi-lepton anomalies [38, 39, 36, 41, 48], in particular the
differential $t\bar{t}$ distributions [48] ($>5\sigma$), suggest the existence
of a new scalar with a mass in the 145–155 GeV with a sizable branching
fraction to $WW$. This mass range is in agreement with the indications for a
narrow resonance in $\gamma\gamma$ and $Z\gamma$ final states at $\approx$151
GeV ($>3.9\sigma$ [35, 92]). However, since no excess in $ZZ\to 4\ell$ is
observed [49, 50]. This motivates the hypothesis that the $\approx$151 GeV
Higgs could be the neutral component of an $SU(2)_{L}$ triplet with $Y=0$,
because it only couples to $W$ bosons at tree-level (in the absence of mixing)
but not to the $Z$. Furthermore, this field predicts a positive shift in
$m_{W}$, as preferred by the current global fit [31, 33].

For small mixing angles, the main production mechanism of $\Delta^{0}$ at the
LHC is associated production together with the charged component
$\Delta^{\pm}$ via the Drell-Yan process $pp\to
W^{*}\to\Delta^{0}\Delta^{\pm}$. In this article, we use the sidebands of the
model-independent ATLAS analysis [1] of the associated production of the SM
Higgs in the di-photon channel to search for this process. Because the leading
branching fractions of $\Delta^{\pm}$ only depend on its mass, and it must be
quasi-degenerate with the neutral component ($m_{\Delta^{\pm}}\approx
m_{\Delta^{0}}\approx m_{\Delta}$), Br$(\Delta^{0}\to\gamma\gamma)$ is the
only free parameter of interest (for a given mass). Within the mass range
110-156 GeV, ten out of the 22 signal regions of the ATLAS analysis [1] give
relevant constraints on Br$(\Delta^{0}\to\gamma\gamma)$. Furthermore, 9 of
them show a weakening of the 95% CL limit at $\approx$150 GeV. In fact,
combining all channels in a likelihood ratio test leads to the preference of
non-zero Br$(\Delta^{0}\to\gamma\gamma)$, with a significance of $3\sigma$ for
$m_{\Delta}$$\approx$151.5 GeV and a best-fit value of
Br$(\Delta^{0}\to\gamma\gamma)=0.66\%$. This shows that the indications for
the associated production of the $\approx$151 GeV scalar are consistent with
the assumption that it is the neutral component of the triplet.

Moreover, a similar excess is obtained near the SM Higgs boson mass. While it
is most pronounced at $128\,$GeV ($3.5\sigma$), it could also originate from
the associated production of the SM Higgs. In this case, the significance
reduces to $1.8\sigma$. This option is particularly interesting since it was
already pointed out in Ref. [42] that there are hints for a non-zero new
physics contribution to $h+W$, which constitutes an overlapping signal.
Furthermore, the enhanced associated production of the SM Higgs is expected in
the $\Delta$2HDMS [93], which was recently proposed as a combined explanation
of the multi-lepton anomalies, the hints for new Higgses at 152 GeV and 95 GeV
and the resonant $t\bar{t}$ excess at 400 GeV [94].

While in this article, we accumulated evidence that the (hypothetical)
$\approx$151 GeV Higgs is the neutral component of the triplet with $Y=0$,
this does not exclude a more complicated scalar sector, and additional
production mechanisms are required to explain the multi-lepton anomalies. To
disentangle them, we are eagerly waiting for the analysis of LHC run-3 data,
and a comprehensive analysis of associated Higgs production by CMS, in the
spirit of Ref. [1], would be very welcome. Furthermore, the Drell-Yan
production of $\Delta^{\pm}\Delta^{\mp}$ can lead to $WWW$-like signatures,
where an $\approx$2$\sigma$ higher cross-section than expected in the SM is
observed [95, 96]. Finally, we remark that the charged component of the
triplet could be examined with great precision at future $e^{+}e^{-}$
accelerators, such as the Circular Electron-Positron Collider (CEPC) [97, 98],
the Compact Linear Collider (CLIC) [99], the Future Circular Collider (FCC-ee)
[100, 101] and the International Linear Collider (ILC) [102, 103].

Figure 4: Statistical combination of the relevant channels. Note that a non-
zero branching ratio of $\Delta^{0}\to\gamma\gamma$ is preferred at both
$\approx$127 GeV ($3.6\sigma$) and $\approx$151 GeV ($3\sigma$).

###### Acknowledgements.

The work of A.C. is supported by a professorship grant from the Swiss National
Science Foundation (No. PP00P21_76884). B.M. gratefully acknowledges the South
African Department of Science and Innovation through the SA-CERN program, the
National Research Foundation, and the Research Office of the University of the
Witwatersrand for various forms of support. The work of S.A. is partially
supported by the National Natural Science Foundation of China under grant No.
11835013. S.A. and S.P.M. acknowledge using the SAMKHYA: High-Performance
Computing Facility provided by the Institute of Physics, Bhubaneswar.

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