# Uncertainty Quantification using Variational Inference for Biomedical Image
Segmentation

Abhinav Sagar
Vellore Institute of Technology
Vellore, Tamil Nadu, India
<EMAIL_ADDRESS>
Website of author - https://abhinavsagar.github.io/

###### Abstract

Deep learning motivated by convolutional neural networks has been highly
successful in a range of medical imaging problems like image classification,
image segmentation, image synthesis etc. However for validation and
interpretability, not only do we need the predictions made by the model but
also how confident it is while making those predictions. This is important in
safety critical applications for the people to accept it. In this work, we
used an encoder decoder architecture based on variational inference techniques
for segmenting brain tumour images. We compare different backbones
architectures like U-Net, V-Net and FCN as sampling data from the conditional
distribution for the encoder. We evaluate our work on the publicly available
BRATS dataset using Dice Similarity Coefficient (DSC) and Intersection Over
Union (IOU) as the evaluation metrics. Our model outperforms previous state of
the art results while making use of uncertainty quantification in a principled
bayesian manner.

##  1 Introduction

Medical image segmentation is a challenging task for medical practitioners. It
is costly, takes time and is prone to error. Hence there is a need to automate
the manually done segmentation. Lately Neural Networks have shown great
potential on a variety of medical image segmentation problems. The challenge
with the approaches used in literature is that the model doesn’t predict the
uncertainty associated. This is where Bayesian methods come into play as it
gives a principled way of measuring uncertainty from the model predictions.
Measuring uncertainty in the output predictions made by neural networks is
important for interpretation and validation. Rather than learning the point
estimates, Bayesian Neural Networks (BNN) learns the distribution over the
weights. The training process of BNN involves first initializing the
parameters of the neural network. Next the weights are sampled from some
distribution (like gaussian with zero mean and unit variance) and both the
forward pass and backward pass is done to update the weights using the
conventional backpropagation algorithm.

Monte Carlo dropout networks (Kingma et al., 2015) use dropout layers to
approximate deep Gaussian processes which still lack theoretical
understanding. Bayesian Convolutional Neural Network (Gal and Ghahramani,
2015) use variational inference to learn the posterior distribution over the
weights given the dataset. The problem with this approach is that it requires
a lot of computation involving a lot of parameters, making this technique not
scalable in practice.

Variational Autoencoder (Kingma et al., 2015) which is based on generative
models solves the above problems and has been successful in a number of tasks
like generating images, texts, recommender systems etc. This approach comes
with several challenges in its own right which have been successfully tackled
in the literature. A random variable sampled from posterior distribution has
no gradient so the conventional backpropagation techniques can’t be applied to
it. Local Reparameterization Trick (Kingma et al., 2015) was proposed to
tackle this by converting the random variable to a deterministic one for
computation. The second challenge was the huge computational requirement since
it required weight updates in every iteration. Bayes by Backprop algorithm
(Blundell et al., 2015) tackled this by calculating gradients in back-
propagation using a scale and shift approach by updating the posterior
distribution in the backward pass.

##  2 Related Work

###  2.1 Medical Image Segmentation

The problem of segmenting medical images have been successfully tackled in
literature using mainly two techniques, first using a Fully Convolutional
Network (FCN) (Long et al., 2015) and second those which are based on U-Net
(Ronneberger et al., 2015). The main characteristic of FCN architectures is
that it doesn’t use fully connected layers at the end which have been used
successfully for image classification problems. U-Net methods on the other
hand uses an encoder-decoder architecture with pooling layers in the encoder
and upsampling layers in the decoder. Skip connections connect the encoder
layers to the decoder layer to create an additional path for the flow of
gradients back in the backpropagation step. This helps in reducing overfitting
due to many parameters involved while training the network.

###  2.2 Bayesian Neural Network

Lately, there has been a revival of interest in bayesian methods as some of
the inherent problems with deep learning could be solved using it. It is a
scalable approach of avoiding overfitting in neural networks and at the same
time gives us a measure of uncertainty. This is very important in critical
applications where not only do we require the predictions made from the model,
but also how confident it is while making its predictions. BNN can be
considered as an ensemble of neural networks (Gal, 2016). It has two
advantages over the standard neural networks, first it avoids overfitting and
second it gives a measure of uncertainty involved.

Instead of point estimates, the neural network learns posterior distribution
over the weights given the dataset as defined in Equation 1.

$p(\omega|\mathcal{D})=\frac{p(\mathcal{D}|\omega)p(\omega)}{p(\mathcal{D})}=\frac{\prod_{i=1}^{N}p\left(y_{i}|x_{i},\omega\right)p(\omega)}{p(\mathcal{D})}$
(1)

The predictive distribution can be calculated by approximating the integral as
defined in Equation 2.

$p\left(y^{*}|x^{*},\mathcal{D}\right)=\int_{\Omega}p\left(y^{*}|x^{*},\omega\right)p(\omega|\mathcal{D})d\omega$
(2)

The challenge is that the posterior is often intractable in nature. To combat
this, (Neal, 1993) used Markov Chain Monte Carlo (MCMC) for learning the
weights over the bayesian neural networks. Also (Graves, 2011), (Blundell et
al., 2015) and (Louizos and Welling, 2016) proposed independently a technique
using variational inference techniques for approximating the posterior
distribution. KL Divergence between the posterior and the true distribution
can be calculated using Equation 3.

$KL\left\\{q_{\theta}(\omega)\|p(\omega|\mathcal{D})\right\\}:=\int_{\Omega}q_{\theta}(\omega)\log\frac{q_{\theta}(\omega)}{p(\omega|\mathcal{D})}d\omega$
(3)

Alternatively minimizing the KL divergence can be written in another form by
maximizing the Evidence Lower Bound (ELBO) which is tractable. This is shown
in Equation 4.

$-\int_{\Omega}q_{\theta}(\omega)\log
p(y|x,\omega)d\omega+KL\left\\{q_{\theta}(\omega)\|p(\omega)\right\\}$ (4)

###  2.3 Variational Inference

Variational inference finds the parameters of the distribution by maximizing
the Evidence Lower Bound. ELBO consists of sum of two terms Kullback-Leibler
(KL) divergence between two distributions and the negative log-likelihood
(NLL) as defined in Equation 5.

$\left.\min\mathrm{KL}\left(q_{\theta}(w)\right)\|p(w|\mathcal{D})\right)$ (5)

The KL divergence is defined in equation 6.

$\mathrm{KL}(q(x))\|p(x))=-\int q(x)\log\left(\frac{p(x)}{q(x)}\right)$ (6)

The posterior in the above equation contains an integral which is intractable
in nature. The equation can be re written in Equation 7.

$\displaystyle\left.\operatorname{KL}\left(q_{\theta}(w)\right)\|p(w|\mathcal{D})\right)$
$\displaystyle=\mathbb{E}_{q_{\theta}(w)}\log\frac{q_{\theta}(w)p(\mathcal{D})}{p(\mathcal{D}|w)p(w)}=$
(7) $\displaystyle=\log
p(\mathcal{D})+\mathbb{E}_{q_{\theta}(w)}\log\frac{q_{\theta}(w)}{p(w)}-\mathbb{E}_{q_{\theta}(w)}\log
p(\mathcal{D}|w)$ $\displaystyle=\log p(\mathcal{D})-\mathcal{L}(\theta)$

The above equation can be decomposed into two parts one of which is the KL
divergence between the exact posterior and its variational approximation which
needs to be minimized and the second is ELBO term which needs to be maximized.
This is shown in Equation 8.

$\left.\max_{\theta}\log
p(\mathcal{D})=\max_{\theta}\left[\operatorname{KL}\left(q_{\theta}(w)\right)\|p(w|\mathcal{D})\right)+\mathcal{L}(\theta)\right]$
(8)

KL divergence is zero if exact posterior is equal to variational
approximation. Since the KL divergence is always greater than zero hence the
equation can be approximated by maximizing only the ELBO (Kingma et al., 2015)
as defined in equation 9.

$\mathcal{L}(\theta)=\mathbb{E}_{q_{\theta}(w)}\log
p(\mathcal{D}|w)-\mathbb{E}_{q_{\theta}(w)}\log\frac{q_{\theta}(w)}{p(w)}=\mathcal{L}_{\mathcal{D}}-\mathrm{KL}\left(q_{\theta}(w)\|p(w)\right)$
(9)

###  2.4 Aleatoric uncertainty and epistemic uncertainty

There are two types of uncertainty - aleatory and epistemic uncertainty where
variance is the sum of both these. Bayesian Neural Networks can be considered
an ensemble of neural networks initialized randomly which averages the test
results in parallel (Gal, 2016). For final predictions, single mean and
variance can be estimated as shown in Equation 10 and Equation 11
respectively.

$\mu_{c}(x)=\frac{1}{M}\sum_{i=1}^{M}\hat{\mu}_{i}(x)$ (10)

$\hat{\sigma}_{c}^{2}(x)=\frac{1}{M}\sum_{i=1}^{M}\tilde{\sigma}_{i}^{2}(x)+\left[\frac{1}{M}\sum_{i=1}^{M}\hat{\mu}_{i}^{2}(x)-\hat{\mu}^{2}(x)\right]$
(11)

The first term in variance denotes aleatoric uncertainty while the second
denotes epistemic uncertainty. Bayesian Neural Network model for uncertainty
estimation was done by (Kendall and Gal, 2017) with the last layer
representing the mean and variance of logits. The predictive distribution
approximating the posterior distribution which gives a measure of uncertainty
is defined in Equation 12.

$q_{\hat{\theta}}\left(y^{*}|x^{*}\right)=\int_{\Omega}p\left(y^{*}|x^{*},\omega\right)q_{\hat{\theta}}(\omega)d\omega$
(12)

Aleatoric uncertainty is a measure of the variability of the predictions from
the dataset hence it is inherent in the data present. Epistemic uncertainty on
the other hand is a measure of the variability of predictions from the model
which is tied to various metrics used for evaluation like accuracy, loss etc.

##  3 Proposed method

The prior distribution helps to incorporate learning of the weights over the
network. Variational Autoencoder has been used successively as a kind of
generative model by sampling from the prior distribution in the encoder. The
decoder uses the mean vector and standard deviation vector from the latent
space to reconstruct the input.

Our model uses a similar encoder decoder architecture as that used in VAEs
with the input to the encoder coming from a pre trained image segmentation
architecture. We tried different backbones which have previously enjoyed
success and found original U-Net gave the best results. The input to the
encoder only needs the mean vector, the standard deviation vector of the
conditional distribution expressing the confidence with which the pixels are
correctly predicted.

After passing through the encoder, the parameters get converted to a latent
representation which is again sampled in a mean and standard deviation vector.
The decoder later recovers this back to the original distribution. The
conventional backpropagation algorithm is used for training the model with
gradient descent. The network architecture is shown in Fig 1.

Figure 1: Our network architecture

Let dataset be denoted by
$\mathcal{D}:\left\\{\left(x_{i},y_{i}\right)\right\\}_{i=1}^{N}$, variational
approximation of the posterior distribution by $q_{\theta}(w)$, encoder as
$r_{\psi}(z|w)$ and decoder as $p_{\phi}(w)$.

The objective function used in this work is defined in Equation 13:

$\left.\mathcal{L}^{\text{approx }}=\sum_{i}\left[-\log
q_{\theta_{i}}\left(\widehat{w}^{(i)}\right)\right)-\log
r_{\psi^{(i)}}\left(\widehat{z}|\widehat{w}^{(i)}\right)+\log
p_{\phi^{(i)}}\left(\widehat{w}^{(i)}|\widehat{z}\right)\right]$ (13)

The network is trained using Gradient Descent as defined in Equation 14 and
Equation 15:

$\theta=\theta+\alpha\nabla_{\theta}\mathcal{L}^{\text{ }}\text{ and
}\psi=\psi+\beta\nabla_{\psi}\mathcal{L}^{\text{}}$ (14)

$g_{\phi}\leftarrow\frac{1}{m}\sum_{k=1}^{m}\nabla_{\phi}\log
p_{\phi}\left(x^{(k)}|z_{\theta}\left(x^{(k)},\epsilon^{(k)}\right)\right)$
(15)

We need as predictions the posterior distribution of the model parameters
which is denoted as $q_{\theta}(w)$.

The complete algorithm used in our work is shown below:

Input: Dataset
$\mathcal{D}:\left\\{\left(x_{i},y_{i}\right)\right\\}_{i=1}^{N}$

Input: Variational approximation of the posterior distribution $q_{\theta}(w)$

Input: encoder $r_{\psi}(z|w)$ and decoder $p_{\phi}(w)$

while  _not converged_ do

Sample minibatch:
$\mathcal{D}^{*}:\left\\{\left(x_{i},y_{i}\right)\right\\}_{i=1}^{M}$

Sample weights with reparametrization: $\widehat{w}^{(i)}\sim
q_{\theta_{i}}\left(w^{(i)}\right)$

Sample latent variables with reparametrization: $\widehat{z}^{(i)}\sim
r_{\psi^{(i)}}\left(z|\widehat{w}^{(i)}\right)$

Compute stochastic gradients of the objective:
$\left.\mathcal{L}^{\text{approx }}=\sum_{i}\left[-\log
q_{\theta_{i}}\left(\widehat{w}^{(i)}\right)\right)-\log
r_{\psi^{(i)}}\left(\widehat{z}|\widehat{w}^{(i)}\right)+\log
p_{\phi^{(i)}}\left(\widehat{w}^{(i)}|\widehat{z}\right)\right]$

$\text{ Update parameters
}\theta=\theta+\alpha\nabla_{\theta}\mathcal{L}^{\text{ }}\text{ and
}\psi=\psi+\beta\nabla_{\psi}\mathcal{L}^{\text{}}$

$g_{\phi}\leftarrow\frac{1}{m}\sum_{k=1}^{m}\nabla_{\phi}\log
p_{\phi}\left(x^{(k)}|z_{\theta}\left(x^{(k)},\epsilon^{(k)}\right)\right)$

end while

Output: $q_{\theta}(w)-$ posterior distribution of the model parameters

Algorithm 1 Uncertainty Quantification using Variational Inference for
Biomedical Image Segmentation

###  3.1 Datasets

To validate the performance of our proposed approach to generalization,
publicly available datasets were used for brain tumour segmentation BRATS18
(Menze et al., 2015) and (Bakas et al., 2018). It contains MRI scans of 175
patients with glioblastoma and lower grade glioblastoma. The images were of
resolution 240$\times$240$\times$155 pixels. The ground truth labels were
created by expert neuroradiologists. The sample from the dataset is shown in
Fig 2.

Figure 2: Example of MRI slices and ground truth segmentation

###  3.2 Hyperparameters

The hyperparameters used in our model are specified in Table 1.

Table 1: Hyperparameters details Parameter | Value
---|---
Batch Size | 16
Optimizer | Adam
Learning Rate | 0.001
LR scheduler patience | 10
LR scheduler factor | 0.1
Latent Variable Size | 10
Max epochs | 500

In addition to the above hyperparamaters, cyclical learning rate schedulers
and ReduceLROnPlateau was used. In gradient descent, the value of momentum was
taken as 0.9, $\gamma$ value of 0.1 and weight decay of 0.0005.

###  3.3 Evaluation

Evaluation metrics for semantic segmentation problems which have often been
used in literature are Dice Similarity Coefficient (DSC) also known as
F1-score and Intersection over union (IoU). The corresponding equations are
shown in Equation 16 and Equation 17 respectively.

$\mathrm{DSC}=\frac{2TP}{2TP+FN+FP}$ (16)

$\mathrm{IoU}=\frac{TP}{TP+FN+FP}$ (17)

True positive (TR), false negative (FN) and false positive (FP) number of
pixels is calculated separately for each image and averaged over the test set.
The ground truth is labelled manually by experts which are compared against.

###  3.4 Loss Functions

A combination of binary cross entropy and dice losses have been used to train
the network. The first part binary cross entropy is a commonly used loss
function for classification problems as shown by (Goodfellow et al., 2016).
Every pixel in the image needs to be classified and hence loss function can be
written as shown in Equation 18.

$\mathcal{L}_{CE}=-\sum_{i,j}y_{i,j}\log\widehat{y}_{i,j}+\left(1-y_{i,j}\right)\log\left(1-\widehat{y}_{i,j}\right)$
(18)

The problem with binary cross entropy loss is that it doesn’t take into
account the class imbalance as the background is the dominant class. This is
one of fundamental challenges in semantic segmentation problems. Dice Loss is
robust to the aforementioned problem which is based on Dice Similarity
Coefficient as defined in Equation 19.

$\mathcal{L}_{DICE}=\sum_{i=1}^{N}\frac{FN_{i}+FP_{i}}{2TP_{i}+FN_{i}+FP_{i}}=\sum_{i=1}^{N}\left(1-DSC^{(i)}\right)$
(19)

Both the loss terms were combined in a single term with more weight given to
the Dice Loss term since it handles the class imbalance problem better. This
is shown in Equation 20.

$\mathcal{L}=0.9\cdot\mathcal{L}_{DICE}+0.1\cdot\mathcal{L}_{CE}$ (20)

##  4 Results

The Mean Dice Similarity value for various backbone architectures compared
against different train size values are shown in Table 2.

Table 2: Mean Dice Similarity metrics for the experiments Train Size | UNet | VNet | FCN
---|---|---|---
5 | 53.1 | 50.6 | 50.2
10 | 56.6 | 51.3 | 52.1
15 | 60.8 | 53.8 | 54.4
20 | 64.3 | 56.5 | 58.9

The IOU value for various backbone architectures compared against different
train size values are shown in Table 3.

Table 3: Intersection over Union metrics for the experiments Train Size | UNet | VNet | FCN
---|---|---|---
5 | 48.4 | 46.7 | 47.2
10 | 50.6 | 48.8 | 50.4
15 | 53.1 | 50.8 | 52.6
20 | 55.8 | 52.8 | 54.3

The predicted segmentation along with uncertainty involved in segmentation is
shown in Fig 3:

Figure 3: Examples of models predictions on test samples, compared to ground
truth segmentation. First column: input image, second column: ground truth
segmentation, third column: predicted segmentation, fourth column: aleatoric
uncertainty and fifth column: epistemic uncertainty.

The darker color denotes more confidence while the lighter means the model is
less confident in those areas.

##  5 Conclusions

In this work, we proposed a way to quantify uncertainty in the context of
medical image segmentation. Our network is based on an encoder decoder
framework similar to that used by VAEs. The weights of the network represent
distributions instead of point estimates and thus give a principled way of
measuring uncertainty at the same time while making the predictions. Our model
uses bayesian neural networks for both the encoder and decoder. The inputs to
encoder come from pre trained backbones like U-Net, V-Net and FCN sampled from
conditional distribution representing the confidence with which pixels are
labelled correctly. We evaluated our network on publicly available BRATS
dataset with our model outperforming previous state of the art using DSC and
IOU evaluation metrics.

#### Acknowledgments

We would like to thank Dhruv Raj and Shreya Singh for their helpful comments
on this work. We would also like to thank Nvidia for providing the GPUs.

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# The Generalized Scalar Weak Gravity Conjecture and its Implications

Fayez Abu-Ajamieh<EMAIL_ADDRESS>Centre for High Energy Physics,
Indian Institute of Science, Bangalore 560012, India  Nobuchika Okada
<EMAIL_ADDRESS>Department of Physics and Astronomy; University of Alabama;
Tuscaloosa; Alabama 35487; USA  Sudhir K. Vempati<EMAIL_ADDRESS>Centre
for High Energy Physics, Indian Institute of Science, Bangalore 560012, India

###### Abstract

We propose a generalized formulation of the Scalar Weak Gravity Conjecture
(SWGC) based on the analogy with the derivation of the Gauge Weak Gravity
Conjecture (GWGC). We discuss some phenomenological implications of this
Generalized SWGC, including the scale of New Physics (NP) when applied to the
SM Higgs sector, and the bounds on the axion’s couplings to fermions and the
photon when applied to the axion. The Generalized SWGC constraints rule out
most of the parameter space of axion-nucleon couplings, leaving only a tiny
parameter space.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

Although gravity is the most familiar interaction that we experience on a day-
to-day basis, it remains the least understood out of all the fundamental
interactions. Quantum theories for the electromagnetic, strong and weak
interactions exist and have proven successful, as evidenced by the many
successes of the Standard Model (SM). However, a quantum theory for gravity is
still lacking in spite of the existence of several candidates, most notable of
which is string theory.

The swampland program Vafa:2005ui ; Ooguri:2006in  attempts at qualifying
Quantum Field Theories (QFTs) that are consistent with quantum gravity
represented by string theory. According to the Swampland Conjecture, QFTs that
can be successfully UV-completed to include quantum gravity are said to belong
to the landscape, whereas those that cannot are said to belong to the
swampland. Only the former QFTs are considered candidates for quantum gravity.

One of the main principles of the swampland program is that gravity should be
the weakest force, as embodied by the GWGC Arkani-Hamed:2006emk . The
conjecture states that for any $U(1)$ gauge group with charge $q$, there must
exist a particle of mass $m$ such that

$m\leq qM_{\text{Pl}},$ (1)

where $M_{\text{Pl}}=2.4\times 10^{18}$ GeV is the reduced Planck scale. The
main motivation for this conjecture arises from extremal black holes: If a
particle satisfying eq. (1) does not exist, then charged black holes cannot
evaporate, and we are faced with the problem of remnants Susskind:1995da .

Eq. (1) is usually called the electric WGC and should also apply to magnetic
monopoles

$m_{\text{mag}}\lesssim
g_{\text{mag}}M_{\text{Pl}}\sim\frac{1}{q}M_{\text{Pl}}.$ (2)

As monopoles have mass that is at least of the order of the magnetic field it
generates, which is linearly divergent, one has

$m_{\text{mag}}\sim\frac{\Lambda}{q^{2}}.$ (3)

Plugging eq. (3) in eq. (2), we find

$\Lambda\lesssim qM_{\text{Pl}}.$ (4)

Eq. (4) is called the magnetic WGC and it implies that there is a natural
cutoff for any $U(1)$ gauge theory where the Effective Field Theory (EFT)
breaks down.111The magnetic WGC was challenged in Saraswat:2016eaz , although
the electric form was argued to continue to hold in the same reference.

The natural question that arises is: What about scalar interactions? In other
words, is gravity a weaker force than that generated by scalar interactions?
There have been several attempts at formulating a Scalar WGC Lust:2017wrl ;
Gonzalo:2019gjp ; Shirai:2019tgr ; Heidenreich:2019zkl ; Freivogel:2019mtr ;
Benakli:2020pkm . In this letter, we attempt at formulating a generalized form
of the SWGC and we investigate some of its phenomenological implications.

This paper is organized as follows: In Section II we review the several
attempts at formulating a SWGC. In Section III we present our Generalized
SWGC. In Section IV we discuss some of its phenomenological implications and
then we conclude in Section V.

##  II The Different Forms of the SWGC

The first attempt at answering this question came in Palti:2017elp ;
Lust:2017wrl . There, it was conjectured that the particle with the largest
charge to mass ratio should not form gravitationally bound states, which
implies that

$q^{2}\geq\frac{m^{2}}{M^{2}_{\text{Pl}}}+\sum_{i,j}g^{ij}(\partial_{\varphi_{i}}m)(\partial_{\varphi_{j}}m),$
(5)

where $g^{ij}$ is the metric in the field space of $\varphi_{i}$, which are
the scalar mediators of the force between WGC particles of mass $m$. The
mediating scalars $\varphi_{i}$ should be massless, as massive scalars will
develop a Yukawa potential exponentially suppressed by its mass ($\sim
e^{-mr}/r$), which means that at sufficiently large distances, the
corresponding force will eventually becomes weaker than the gravitational
force.

If the WGC particle has no charge, then one can recast eq. (5) into a
statement about the magnitude of the forces

$|\partial_{\varphi}m|^{2}\equiv\sum_{i,j}g^{ij}(\partial_{\varphi_{i}}m)(\partial_{\varphi_{j}}m)\geq\frac{m^{2}}{M^{2}_{\text{Pl}}}.$
(6)

The physical content of eq. (6) is that gravity is a weaker force than that
mediated by a (massless) scalar force

$F_{\text{garv}}=\frac{m^{2}}{8\pi M_{\text{Pl}}^{2}r^{2}}\leq
F_{\text{scalar}}=\frac{|\partial_{\varphi}m|^{2}}{4\pi r^{2}}.$ (7)

The masslessness assumption of $\varphi$ was relaxed in Shirai:2019tgr  into
$m_{\varphi}\lesssim 10^{-33}$ eV equivalent to the Hubble radius, which means
that gravity should be weaker than a scalar force within the observable
universe. Also, Shirai:2019tgr  showed that the above form of the SWGC was in
tension with fifth force searches and with the de Sitter Swampland Conjecture
Obied:2018sgi .

In Gonzalo:2019gjp  a Strong SWGC was introduced. There, it was stated that
the potential of any canonically normalized real scalar field must satisfy the
constraint

$2(V^{\prime\prime\prime})^{2}-V^{\prime\prime}V^{\prime\prime\prime\prime}\geq\frac{(V^{\prime\prime})^{2}}{M_{Pl}^{2}},$
(8)

where the primes indicate derivatives with respect to the field. The above
form was motivated by the desire to extend the original SWGC to all scalar
fields (and not just scalar field whose masses are functions of $\varphi$),
and to accommodate the periodic potential of axions. However, a (possible)
counterexample was proposed in Freivogel:2019mtr .

Heidenreich:2019zkl  introduced the Repulsive Force Conjecture (RFC), a
refinement of the the original SWGC in eq. (5) which states that the force
between two copies of a charged particle must be repulsive. In other words,
since gravitational and scalar interactions are both attractive, the repulsive
$U(1)$ force should be larger than both forces combined, essentially leading
to eq. (5).

Another refinement of the SWGC was introduced in Benakli:2020pkm , where it
was postulated that in the low energy (non-relativistic) limit, the
gravitational contribution to the leading interaction must be subleading. The
conjecture was applied to a massive $\phi^{4}$ theory, where it was found that
the SWGC can be violated in a small region of size
$\frac{\Delta\phi^{2}}{m^{2}}\sim\frac{m^{2}}{M_{\text{Pl}}^{2}}$.

We should point out that all attempts to formulate a SWGC stand on a weaker
footing compared to the GWGC due to the lack of a black hole evaporation
argument in the scalar case. Therefore, all forms of the SWGC should be
considered with this caveat in mind.

##  III The Generalized SWGC

Let’s first consider the magnetic GWGC in eq. (4). Notice that this bound can
be obtained by requiring that gravity be a weaker interaction than that of a
gauged $U(1)$. More specifically, if we require gravitational interactions to
be always weaker than $U(1)$ interactions at any energy and not just in the
non-relativistic limit (see Fig. 1) and impose

$|\mathcal{M}_{\text{grav}}(s)|\lesssim|\mathcal{M}_{\text{U(1)}}(s)|,$ (9)

then we can see that in the limit $\sqrt{s}\gg m$, where $m$ is the mass of
the WGC particle, eq. (9) implies that $q^{2}\lesssim\frac{s}{M_{Pl}^{2}}$.
Setting $\sqrt{s}=\Lambda$ as the scale of NP, we retrieve eq. (4).

Figure 1: Extracting the GWGC.

One can also argue that even if the charge carriers inside black holes have
high energy, black holes should still be able to evaporate, and thus the
electric WGC should also extend beyond the non-relativistic regime. This
motivates us to generalize the SWGC beyond the non-relativistic regime by
requiring that

$|\mathcal{M}_{\text{grav}}(s)|\lesssim|\mathcal{M}_{\text{scalar}}(s)|.$ (10)

In applying eq. (10), we should keep in mind that massless gravity suffers
from a divergence in the $t$\- and $u$\- channels. Thus, to be conservative,
we only consider the (finite) $s$\- channel when evaluating the gravitational
amplitude. Notice that unlike the original formulation of the SWGC in eqs. (5)
and (6), the Generalized SWGC applies to both massless and massive scalar
mediator and not just massless ones.

##  IV Some Phenomenological Implications of the WGC

Beyond qualifying candidate theories for quantum gravity, both the GWGC and
the SWGC have some interesting phenomenological implications. The implications
of the GWGC were discussed in Abu-Ajamieh:2024gaw . As shown there in detail,
one can use the magnetic form of the GWGC to set a limit on the scale of NP
that corresponds to a certain $U(1)$ gauge theory if the charge is known. For
example, the cutoff scale that corresponds to $U(1)_{\text{EM}}$ is
$\Lambda\sim 10^{17}$ GeV. Conversely, if a lower limit on the scale of NP is
known (for example from null collider searchers), then the magnetic GWGC can
be used to set a lower limit on the charge of that $U(1)$.

A similar argument holds for the SWGC. Here, we discuss some pehomenological
implications of the SWGC in two case: When the mediator is the SM Higgs boson
and when the mediator is the axion. In our calculation, we limit ourselves to
amplitudes at tree-level.

###  IV.1 The SM Higgs

To the best of our knowledge, the SM Higgs is the only fundamental scalar that
exists. If we apply the SWGC where the Higgs is the scalar mediator and the
other SM particles are the WGC states, we can set limits on the scale of NP in
each state.

####  IV.1.1 Fermion WGC States

The Higgs-mediated $\overline{f}f\rightarrow\overline{f}f$ proceeds via the
$s$\- and $t$\- channels. The amplitude reads

$|\mathcal{M}_{h}|^{2}=y_{f}^{4}\frac{\Big{(}3(1+\tau+\tau^{2})+(3-3\tau+\tau^{2})\cos^{2}{\theta}-6\cos{\theta}\Big{)}}{(1-\tau)^{2}(1-\cos{\theta}+2\tau)^{2}},$
(11)

where $\tau=\frac{m_{h}^{2}}{s}$ and we have dropped the masses of the initial
and final state fermions. On the other hand, the gravitational amplitude reads

$|\mathcal{M}_{\text{grav}}|^{2}=\frac{s^{2}}{32M_{\text{Pl}}^{4}}(2+\cos{(2\theta)}+\cos(4\theta)),$
(12)

Applying the Generalized SWGC in eq. (10), it is easy to see that the
strongest bound arises in the forward region $\theta\rightarrow 0$. In the
high energy limit $\tau\rightarrow 0$, we have

$\frac{s}{2\sqrt{2}M^{2}_{\text{Pl}}}\lesssim y_{f}^{2}.$ (13)

Setting $\sqrt{s}=\Lambda$ as the scale of NP, we find that
$\Lambda\lesssim\sqrt[4]{8}y_{f}M_{\text{Pl}}$. Obviously the scale of NP is
set by the lightest fermion. If neutrinos are Majorana fermions, then the
electron will be the lightest SM fermion and $\Lambda\lesssim 1.2\times
10^{13}$ GeV. On the other hand, if neutrinos are Dirac fermions, then the
scale of NP is set by the mass of the lightest neutrino. If we naively use the
upper limit on $m_{\nu_{e}}=1.1$ eV as indicated by the PDG
ParticleDataGroup:2020ssz , then the scale on NP becomes $\Lambda\lesssim
2.6\times 10^{7}$ GeV. However, much stronger bounds can be obtained using the
neutrino oscillation data. Fig. 2 shows the scale of NP as a function of the
mass of the lightest neutrino for both the normal and the inverted
hierarchies. Fig. 2 was created using neutrino oscillation data as input to
the neutrino mixing (PMNS) matrix, where the input values are
$\sin^{2}{(2\theta_{12})}=0.87$, $\sin^{2}{(2\theta_{23})}=1.0$,
$\sin^{2}{(2\theta_{13})=0.092}$, $\delta_{CP}=\frac{3}{2}\pi$, $\Delta
m_{21}^{2}=7.6\times 10^{-5}$ $\text{eV}^{2}$ for both the normal and the
inverted hierarchies, and $|\Delta m_{31}^{2}(\Delta m_{32}^{2})|=2.4\times
10^{-3}$ $\text{eV}^{2}$ for the normal (inverted hierarchy).

We can see from the plots that for the normal hierarchy, $\Lambda$ ranges
between $\sim 2.8$ TeV and $\sim 92$ TeV corresponding to a mass of $1.1$ eV
to $\lesssim 10^{-3}$ eV for the lightest neutrino respectively, whereas for
the inverted heirarchy, it ranges between $\sim 2.7$ TeV and $\sim 122$ TeV
for the same neutrino mass range. The lower range of this scale can be
potentially probed in future colliders, such as the 100 TeV FCC or the muon
collider, and can even be within the reach of the LHC.

Figure 2: The scale of NP as a function of the lightest neutrino mass from
neutrino oscillation data for the normal (top) and the inverted (bottom)
hierarchies. The different lines correspond to the eigenstates of the neutrino
mass matrix: The blue line represents $m_{11}$, the magenta is
$m_{22}=m_{33}$, the brown is $m_{23}=m_{32}$, and the red line is
$m_{12}=m_{21}=m_{13}=m_{31}$.

####  IV.1.2 Massive Gauge Boson WGC States

If we compare the amplitude of $WW(ZZ)\rightarrow WW(ZZ)$ mediated through the
Higgs, with that mediated through a graviton in the high energy limit
$\sqrt{s}\gg m_{W},m_{Z},m_{h},v$, then in the forward region, the Generalized
SWGC leads to the bound

$\Lambda\lesssim\frac{g_{V}^{2}vM_{\text{Pl}}}{\sqrt[4]{2}m_{h}},$ (14)

where $g_{W}^{2}=g^{2}$ and $g_{Z}^{2}=g^{2}+g^{\prime 2}$. This corresponds
to $\Lambda\sim 5\times 10^{17}(2.2\times 10^{18})$ GeV for the $W(Z)$.

####  IV.1.3 Massless Gauge Boson WGC States

Although the Higgs boson does not couple to photons and gluons at tree-level,
we can nonetheless integrate out the triangle loops and write the effective
coupling as

$\mathcal{L}=c_{\gamma}\frac{\alpha}{\pi
v}hA_{\mu\nu}A^{\mu\nu}+c_{g}\frac{\alpha_{s}}{12\pi
v}hG^{a}_{\mu\nu}G^{a\mu\nu},$ (15)

where for the SM, $c_{\gamma}\simeq 0.81$, $c_{g}\simeq 1.03$. Applying eq.
(10) to the photon sector in the forward region, we find

$\Lambda\lesssim\frac{2\sqrt[4]{8}\alpha
c_{\gamma}}{v\pi}m_{h}M_{\text{Pl}}\simeq 7.8\times 10^{15}\hskip
2.84526pt\text{GeV}.$ (16)

Things are a bit more subtle in the gluon sector, as we need to take into
consideration the running of $\alpha_{s}$. Applying the Generalized SWGC, we
find

$\Lambda\lesssim\frac{c_{g}m_{h}M_{\text{Pl}}}{3\sqrt[4]{2}\pi
v}\frac{\alpha_{s}(M_{Z})}{1-\frac{b_{0}\alpha_{s}(M_{Z})}{2\pi}\log(\frac{\Lambda}{M_{Z}})}\simeq
2.6\times 10^{15}\hskip 2.84526pt\text{GeV},$ (17)

where $b_{0}=-11+\frac{2}{3}n_{f}=-7$ and $\alpha_{s}(M_{Z})=0.1179$.

####  IV.1.4 Scalar WGC States

Finally, let’s consider the case where the Higgs is both the mediator and the
WGC particle. In this case, $hh\rightarrow hh$ proceeds via the $s$-, $t$\-
and $u$\- channels in addition to a contact term via the quartic coupling. In
the high energy limit, the quartic term dominates and Generalized SWGC implies

$\frac{s\sin^{2}{\theta}}{4M_{\text{P}}^{2}}\lesssim 6\lambda\hskip
5.69054pt\implies\hskip
5.69054pt\lambda\gtrsim\frac{\Lambda^{2}}{24M_{\text{P}}^{2}}.$ (18)

where we took $\theta\rightarrow\frac{\pi}{2}$ as it yields the strongest
bound. As it is well-known, in the SM, the Higgs sector suffers from an
instability when the quartic coupling runs into negative values at a scale
$\sim 10^{10}-10^{11}$ GeV, however, the Generalized SWGC suggests that the
quartic coupling must remain positive, thereby avoiding any instability. What
this means is that the Generalized SWGC (if indeed correct), informs us that
there must be NP at an energy scale before the supposed scale of instability,
that ensures the stability of the Higgs potential.

###  IV.2 Axions

We now consider the axion as the mediating particle, and cover the cases where
the WGC states are fermions and photons. The axion interaction Lagrangian is
given by

$\mathcal{L}_{\text{int}}=-\frac{1}{f}g_{a\gamma}aF_{\mu\nu}\widetilde{F}^{\mu\nu}+\frac{g_{af}}{2m_{f}}(\partial_{\mu}a)(\overline{f}\gamma^{\mu}\gamma^{5}f),$
(19)

where
$\widetilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$.
We should point out that since the axion corresponds to a $U(1)$ gauge group,
the bounds in this section can also be obtained through the GWGC (see Abu-
Ajamieh:2024gaw ), however, here we are more rigorous.

####  IV.2.1 Fermion WGC States

First, let’s consider the case $\overline{f}f\rightarrow\overline{\psi}\psi$
where $f\neq\psi$. In this case, the axion-mediated process proceeds through
the $s$-channel only. A simple calculation shows that in the forward region,
the Generalized SWGC (or Guage) implies

$g_{af}g_{a\psi}\gtrsim\frac{|s-m_{a}^{2}|}{2\sqrt{2}M_{\text{Pl}}^{2}}.$ (20)

Figure 3: The lower bound on the axion’s coupling to electrons as suggested by
the Gauge/Generalized Scalar WGC. The plot was modified from AxionLimits .

On the other hand, if $f=\psi$, then there is a $t$-channel in addition to the
$s$-channel, and $g_{af}=g_{a\psi}$. In this case, the conjecture becomes

$g_{af}\gtrsim\frac{1}{M_{\text{Pl}}}\sqrt{\frac{|s-m_{a}^{2}|}{2\sqrt{2}}}.$
(21)

Eqs. (20) and (21) can be used to set bounds on the axion’s coupling to
fermions, however, to do that, we need a lower limit on the scale of NP. As
explained in Abu-Ajamieh:2024gaw , LEP searches $e^{+}e^{-}\rightarrow
a\rightarrow\bar{f}f$ can be used, and the lower limit on the NP can be set as
$\Lambda=\sqrt{s_{\text{LEP}}}=209$ GeV. In this case, eq. (21) can be used to
set a lower bound on $g_{ae}$. Using the resources from AxionLimits , we show
in Fig. 3 the lower bound on $g_{ae}$ as suggested by the WGC (scalar or
gauge), superimposed on the other experimental bounds. As can seen from the
plot, while experimental bounds place limits on the coupling from above, the
Scalar/Gauge WGC places limits from below, and thus can help probe the entire
parameter space.

On the other hand, to set bounds on the axion couplings to other fermions, we
need to use eq. (20). To do so, we need to specify $g_{ae}$. To set
conservative bounds, we set $g_{ae}$ to be equal to the lower experimental
bounds extracted from Fig. 3. Fig. 4 shows the bounds on the axion coupling to
protons and neutrons. As the plots show, the Scalar/Gauge WGC can place very
stringent constraints on the parameter space, with only a small window
remaining open in both cases. The bounds on the axion couplings to other
fermions are identical to those to protons and neutrons as can be seen from
eq. (21).

Figure 4: The lower bound on the axion’s coupling to protons (up) and neutrons
(down) as suggested by the Gauge/Generalized Scalar WGC. The plots were
modified from AxionLimits .

####  IV.2.2 Photon WGC

Next we consider $\gamma\gamma\rightarrow a\rightarrow\gamma\gamma$. This
process proceeds via the $s$-, $t$-, and $u$-channels. Applying the
Generalized SWGC leads to the bound

$g_{\gamma
a}\gtrsim\frac{1}{M_{\text{Pl}}}\Bigg{(}\frac{8(1-m_{a}^{4}/s^{2})^{2}}{1+3m_{a}^{4}/s^{2}}\Bigg{)}^{\frac{1}{4}}.$
(22)

Here too, we need to a lower limit on the scale of NP to be able to set bounds
on $g_{a\gamma}$. As explained in detail in Abu-Ajamieh:2024gaw , the latest
Atlas results on the Light-by-Light (LBL) scattering ATLAS:2017fur  can be
used to infer a lower bound on the scale of NP. Following Ellis:2017edi ;
Ellis:2022uxv , it was found in Abu-Ajamieh:2024gaw  that
$\sqrt{s}\equiv\Lambda_{\text{LBL}}\simeq 7.8$ TeV can be obtained. Using this
in eq. (22), we plot the bound on the axion’s coupling to photons in Fig. 5,
where we can see from the plot that compared to the coupling to fermions, the
bound is weaker. This is due to the energy behavior of the amplitudes. While
at high energy, the amplitude of the fermionic WGC states becomes essentially
constant, the amplitude of the photon WGC states grows quadratically with
energy, which weakens the bound.

Figure 5: The lower bound on the axion’s coupling to photons as suggested by
the Gauge/Generalized Scalar WGC. The plot was modified from AxionLimits .

##  V Conclusions

In this paper, we proposed a generalized formulation of the SWGC, where we
suggested that gravity is always a weaker interaction than scalar interactions
at any energy scale, whether the scalar is massless or massive. We proceeded
to investigate some of the phenomenological implications of the Generalized
SWGC when the scalar is the SM Higgs and the axion. In the former case, we
found that the Generalized SWGC can set an upper bound on the scale of NP
$\sim 10^{13}$ GeV if neutrinos are Majorana fermions, and as low as $\sim
3-122$ TeV if neutrinos are Dirac fermions. We also showed that the
Generalized SWGC suggests that the Higgs quartic coupling should always remain
positive, indicating the absence of any instability in the Higgs potential.

We showed that both the Generalized SWGC and GWGC both set stringent lower
bounds on the axion couplings to fermions and to the photon, excluding much of
the parameter space. We should emphasize that due to the lack of a black hole
evaporation argument, the SWGC with its various formations, stands on weaker
grounds compared to the GWGC.

## Acknowledgments

The work of NO is supported in part by the United States Department of Energy
(DC-SC 0012447 and DC-SC 0023713). SKV is supported by SERB, DST, Govt. of
India Grants MTR/2022/000255 , “Theoretical aspects of some physics beyond
standard models”, CRG/2021/007170 “Tiny Effects from Heavy New Physics “and
IoE funds from IISC.

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