# Unleash the Potential of 3D Point Cloud Modeling with A Calibrated Local
Geometry-driven Distance Metric

Siyu Ren and Junhui Hou
Department of Computer Science, City University of Hong Kong
<EMAIL_ADDRESS><EMAIL_ADDRESS>This work was supported by the
Hong Kong Research Grants Council under Grants 11202320 and 11219422.
Corresponding author: Junhui Hou

###### Abstract

Quantifying the dissimilarity between two unstructured 3D point clouds is a
challenging task, with existing metrics often relying on measuring the
distance between corresponding points that can be either inefficient or
ineffective. In this paper, we propose a novel distance metric called
Calibrated Local Geometry Distance (CLGD), which computes the difference
between the underlying 3D surfaces calibrated and induced by a set of
reference points. By associating each reference point with two given point
clouds through computing its directional distances to them, the difference in
directional distances of an identical reference point characterizes the
geometric difference between a typical local region of the two point clouds.
Finally, CLGD is obtained by averaging the directional distance differences of
all reference points. We evaluate CLGD on various optimization and
unsupervised learning-based tasks, including shape reconstruction, rigid
registration, scene flow estimation, and feature representation. Extensive
experiments show that CLGD achieves significantly higher accuracy under all
tasks in a memory and computationally efficient manner, compared with existing
metrics. As a generic metric, CLGD has the potential to advance 3D point cloud
modeling. The source code is publicly available at
https://github.com/rsy6318/CLGD.

##  1 Introduction

(a) EMD [21]

(b) CD [2]

(c) Ours

Figure 1:  Visual illustration of different distance metrics for 3D point
cloud data.  and  are two point clouds under evaluation,  and  are their
underlying surfaces, and the lines with arrows indicate the correspondence. In
contrast to existing metrics measuring the difference between corresponding
points, our metric computes the difference between the surfaces underlying 3D
point clouds.

3D point cloud data, which is a set of points defined by 3D coordinates to
represent the geometric shape of an object or a scene, has been used in
various fields, such as computer vision, 3D modeling, and robotics. Measuring
the difference between 3D point clouds is critical in many tasks, e.g.,
reconstruction, rigid registration, etc. Different from 2D images, where pixel
values are structured with regular 2D coordinates, allowing us to directly
compute the difference between two images pixel-by-pixel, 3D point clouds are
unstructured, i.e., there is no point-wise correspondence naturally available
between two point clouds, posing a great challenge. The two widely used
distance metrics, namely Earth Mover’s Distance (EMD) [21] and Chamfer
Distance (CD) [2] illustrated in Figs. 1(a) and 1(b), first build point-wise
correspondence between two point clouds and then compute the distance between
corresponding points. However, EMD is both memory and time-consuming as it
involves solving a linear programming problem for the optimal bijection. For
each point in one point cloud, CD seeks its nearest point in the other point
cloud to establish the correspondence, and it could easily reach a local
minimum. Although some improved distance metrics [25, 17, 8, 22] have been
proposed, they are still either inefficient or ineffective.

Existing distance metrics for 3D point cloud data generally concentrate on the
point cloud itself, aligning points to measure the point-wise difference.
However, these metrics overlook the fact that different point clouds obtained
by different sampling could represent an identical 3D surface. In this paper,
we propose an efficient yet effective distance metric named Calibrated Local
Geometry Distance (CLGD). Unlike previous metrics, CLGD computes the
difference between the underlying 3D surfaces of two point clouds, as depicted
in Fig. 1(c). Specifically, we first sample a set of 3D points called
reference points, which we use to induce and calibrate the local geometry of
the surfaces underlying point clouds, i.e., computing the directional
distances of each reference point to the two point clouds that approximately
represent the underlying surfaces in implicit fields. Finally, we define CLGD
as the average of the directional distance differences of all reference
points. We conduct extensive experiments on various tasks, including shape
reconstruction, rigid registration, scene flow estimation, and feature
representation, demonstrating its significant superiority over existing
metrics.

In summary, the main contributions of this paper are:

  1. 1.

an efficient, effective, and generic distance metric for 3D point cloud data;

  2. 2.

state-of-the-art benchmarks of rigid registration and scene flow estimation;
and

  3. 3.

potentially advancing the field of 3D point cloud processing and analysis.

##  2 Related Work

#### Distance Metrics for 3D Point Clouds.

Most of the existing distance metrics for 3D point clouds concentrate on the
points in the point cloud. That is, they calculate the distance value based on
the point-to-point distances from different point clouds, such as CD [2] and
EMD [21]. Specifically, EMD builds a global bi-directional mapping between the
source and target point clouds. Then, the sum or mean of the distances between
corresponding points is regarded as the distance between the point clouds.
However, the computation of bijection is too expensive, especially when the
number of points is large. Differently, CD builds a local mapping between
point clouds by finding the nearest point in the other point cloud, making it
more efficient than EMD. However, such local correspondence between point
clouds may result in local minima or sub-optimal results. Hausdorff Distance
(HD) [12] is modified from CD but focuses more on the outliers. Thus, it
struggles to handle the details of point clouds and usually serves as an
evaluation metric. Considering the distribution of point clouds, Wu et al.
[25] proposed density-aware CD (DCD) by introducing density information as
weights into CD, achieving a higher tolerance to the outliers. Nguyen et al.
[17] proposed Sliced Wasserstein Distance (SWD) to measure the distance
between point clouds, making it more efficient and effective than EMD and CD
in the shape representation task. PointNetLK [1] and FMR [11] convert point
clouds to feature vectors through PointNet [18] and utilize the distance of
features to measure the difference between point clouds. However, the global
feature cannot represent the details of the point clouds, and such a kind of
distance metric relies heavily on the encoder.

The above-mentioned distance metrics concentrate on points and ignore the
geometric nature of a point cloud. In reality, the point clouds are sampled
from the surface, and differently sampled point clouds could represent the
same surface. Therefore, we should measure the difference between the
underlying surfaces as the distance of the point clouds. ARL [8] randomly
samples lines and calculates their intersections on the two point clouds’
underlying surfaces approximately. It then calculates the difference between
each line’s intersections on the two point clouds to measure the dissimilarity
of the two point clouds. However, the calculation of the intersection is time-
consuming, and the randomness of the lines could also make the measurement
unstable. To leverage the underlying surfaces of point clouds, DPDist [22]
trains a network to regress the point-to-plane distance between the two point
clouds. However, the trained network’s accuracy in regressing the point-to-
plane distance would decrease if the distribution of point cloud changes,
limiting its generalization.

#### Distance Metric-driven 3D Point Cloud Processing.

A distance metric is necessary for various 3D point cloud data processing and
analysis tasks. One of the most common examples is the rigid registration of
point clouds. The traditional registration methods, such as ICP [3] and its
variants [5, 30, 38], utilize the distance metrics between point clouds as the
objective function to optimize. Some recent learning-based registration
methods, such as DCP [24], FMR [11], and RPM-Net [32], could become
unsupervised with the distance metrics as the alignment item in their loss
functions during training. Besides, recent learning-based methods for scene
flow estimation, such as PointPWC-Net [26], NSFP [15], and SCOOP [14], adopt
distance metrics as the alignment item in the loss functions to train the
network without using ground-truth scene flow as supervision, and they have
achieve remarkable accuracy. The distance metrics are also critical in some
point cloud generation tasks, e.g., point cloud reconstruction [31, 9],
upsampling [34, 33, 19, 20], completion [36, 35, 37, 28, 29], etc., where the
difference between the generated and ground-truth point clouds is calculated
as the main loss to train the networks.

##  3 Proposed Method

###  3.1 Problem Statement and Overview

Given any two unstructured 3D point clouds $\mathbf{P}_{\rm
1}\in\mathbb{R}^{N_{1}\times 3}$ and $\mathbf{P}_{\rm
2}\in\mathbb{R}^{N_{2}\times 3}$ with $N_{1}$ and $N_{2}$ points respectively,
we aim to construct a differentiable distance metric, which can quantify the
difference between them effectively and efficiently to drive downstream tasks.
As mentioned in Section 1, the problem is fundamentally challenging due to the
lack of correspondence information between $\mathbf{P}_{1}$ and
$\mathbf{P}_{2}$. Existing metrics generally focus on establishing the point-
wise correspondence between $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ to compute
the point-to-point difference, making them either ineffective or inefficient.

In contrast to existing metrics, we address this problem by measuring the
difference between the underlying surfaces of $\mathbf{P}_{1}$ and
$\mathbf{P}_{2}$. Generally, with a set of reference points generated, we
associate each reference point with $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ to
model their local surface geometry. Then we calculate the difference between
the local surface geometry reference point-by-reference point and average the
differences of all reference points as the final distance between
$\mathbf{P}_{1}$ and $\mathbf{P}_{2}$. Such a process is memory-saving,
computationally efficient, and effective. In what follows, we will introduce
the technical details of the proposed distance metric.

###  3.2 Generation of Reference Points

We generate a set of 3D points
$\mathbf{Q}=\\{\mathbf{q}_{m}\in\mathbb{R}^{3}\\}_{m=1}^{M}$ named reference
points, which will be used to indirectly establish the correspondence between
$\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ and induce the local geometry of the
surfaces underlying $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$. Technically, after
selecting either one of $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$111For point
cloud reconstruction-based tasks, where one point cloud is used to supervise a
network to generate the other point cloud, the one used as supervision will be
selected to generate $\mathbf{Q}$ because the reconstructed one is messy,
lacking sufficient geometric meaning. we add Gaussian noise to each point,
where the standard deviation is $T$ times the distance to its nearest point in
the point cloud, and repeat the noise addition process $R$ times randomly to
generate $R$ reference points that are distributed near to the underlying
surface. Additionally, to ensure the local surface geometry induced by
$\mathbf{Q}$ covers each point of $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$, we
also include the non-selected point cloud into $\mathbf{Q}$. See Table 7 for
the ablative studies on this process.

Figure 2: Illustration of the reference point-induced local surface geometry
of a 3D point cloud. (a) $f(\mathbf{q},\mathbf{P})$. (b)
$\mathbf{v}(\mathbf{q},\mathbf{P})$.

###  3.3 Calibrated Local Surface Geometry

Let $\mathbf{P}\in\\{\mathbf{P}_{1},\mathbf{P}_{2}\\}$,
$\mathbf{q}\in\mathbf{Q}$, and
$\Omega(\mathbf{q},\mathbf{P}):=\\{\mathbf{p}_{k}\\}_{k=1}^{K}$ be the set of
$\mathbf{q}$’s $K$-NN ($K$-Nearest Neighbor) points in $\mathbf{P}$. Note that
we sort all points in $\Omega(\mathbf{q},\mathbf{P})$ according to their
distances to $\mathbf{q}$, i.e.,
$\|\mathbf{q}-\mathbf{p}_{1}\|_{2}\leq\|\mathbf{q}-\mathbf{p}_{2}\|_{2}\leq...\leq\|\mathbf{q}-\mathbf{p}_{K}\|_{2}$,
where $\|\cdot\|_{2}$ is the $\ell_{2}$ norm of a vector. Let $\mathcal{S}$ be
the underlying surface of $\mathbf{P}$, and $\mathbf{p}^{*}\in\mathcal{S}$ be
the closest point to $\mathbf{q}$. As illustrated in Fig. 2, we could simply
approximate the distance from $\mathbf{q}$ to $\mathcal{S}$, i.e., the
distance between $\mathbf{q}$ and $\mathbf{p}^{*}$, through the weighted
averaging of $\\{\|\mathbf{q}-\mathbf{p}_{k}\|_{2}\\}_{k=1}^{K}$:

$f(\mathbf{q},\mathbf{P}):=\|\mathbf{q}-\mathbf{p}^{*}\|_{2}\approx\frac{\sum_{k=1}^{K}w(\mathbf{q},\mathbf{p}_{k})\cdot\|\mathbf{q}-\mathbf{p}_{k}\|_{2}}{\sum_{k=1}^{K}w(\mathbf{q},\mathbf{p}_{k})},\
{\rm where}\
w(\mathbf{q},\mathbf{p}_{k})=\frac{1}{\|\mathbf{q}-\mathbf{p}_{k}\|_{2}^{2}}.$
(1)

Similarly, as shown in Fig. 2, we could also approximate the vector from
$\mathbf{p}^{*}$ to $\mathbf{q}$ with the weighted averaging of
$\\{\mathbf{q}-\mathbf{p}_{k}\\}_{k=1}^{K}$:

$\mathbf{v}(\mathbf{q},\mathbf{P}):=\mathbf{q}-\mathbf{p}^{*}\approx\frac{\sum_{k=1}^{K}w(\mathbf{q},\mathbf{p}_{k})\cdot(\mathbf{q}-\mathbf{p}_{k})}{\sum_{k=1}^{K}w(\mathbf{q},\mathbf{p}_{k})}.$
(2)

Then, we concatenate $f(\mathbf{q},\mathbf{P})$ and
$\mathbf{v}(\mathbf{q},\mathbf{P})$ to form a 4D vector
$\mathbf{g}(\mathbf{q},\mathbf{P})=[f(\mathbf{q},\mathbf{P})||\mathbf{v}(\mathbf{q},\mathbf{P})]\in\mathbb{R}^{4}$
named directional distance, which characterizes the local geometry of the
surface underlying $\mathbf{P}$ induced by $\mathbf{q}$ in implicit fields.

###  3.4 Local Surface Geometry-driven Distance Metric

Calibrated by the reference point $\mathbf{q}_{m}\in\mathbf{Q}$, the
difference between its directional distances to $\mathbf{P}_{1}$ and
$\mathbf{P}_{2}$ can reflect the difference in their local surface geometry,
i.e., the difference between $\Omega(\mathbf{q}_{m},~{}\mathbf{P}_{1})$ and
$\Omega(\mathbf{q}_{m},~{}\mathbf{P}_{2})$. Specifically, we calculate the
difference between the directional distances of $\mathbf{q}_{m}$ as

$d(\mathbf{q}_{m},\mathbf{P}_{\rm 1},\mathbf{P}_{\rm
2})=\|\mathbf{g}(\mathbf{q}_{m},\mathbf{P}_{\rm
1})-\mathbf{g}(\mathbf{q}_{m},\mathbf{P}_{\rm 2})\|_{1},$ (3)

where $\|\cdot\|_{1}$ computes the $\ell_{1}$ norm of a vector. Note that
$\mathbf{g}(\mathbf{q}_{m},\mathbf{P}_{\rm 1})$ and
$\mathbf{g}(\mathbf{q}_{m},\mathbf{P}_{\rm 2})$ share identical weights
$\\{w\\}_{k=1}^{K}$, i.e., the weights of the point cloud selected to generate
reference points are also applied to the other point cloud. The proposed
distance metric for 3D point clouds is finally defined as the weighted sum of
$d(\mathbf{q}_{m},\mathbf{P}_{1},\mathbf{P}_{2})$:

$\mathcal{D}_{\rm CLGD}(\mathbf{P}_{\rm 1},\mathbf{P}_{\rm
2})=\frac{1}{M}\sum_{\mathbf{q}_{m}\in\mathbf{Q}}s(\mathbf{q}_{m})\cdot
d(\mathbf{q}_{m},\mathbf{P}_{\rm 1},\mathbf{P}_{\rm 2}),$ (4)

where $s(\mathbf{q}_{m})=\texttt{Exp}(-\beta\cdot
d(\mathbf{q}_{m},\mathbf{P}_{1},\mathbf{P}_{2}))$ is the confidence score of
$d(\mathbf{q}_{m},~{}\mathbf{P}_{1},~{}\mathbf{P}_{2})$ with $\beta\geq 0$
being a hyperparameter. $s(\mathbf{q}_{m})$ is introduced to cope with the
case where $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ are partially overlapped.

##  4 Experiments

To demonstrate the effectiveness and superiority of the proposed CLGD, we
applied it to a wide range of downstream tasks, including shape
reconstruction, rigid point cloud registration, scene flow estimation, and
feature representation, and compared it with EMD [21], CD [2], and ARL [8]222
ARL was primarily designed for rigid point cloud registration, and thus, we
compared our CLGD with ARL only in the registration task.. In all experiments,
we set $R=10$, $T=3$, and $K=5$. We set $\beta=3$ in rigid registration due to
the partially overlapping point clouds, and $\beta=0$ in the other three
tasks. We conducted all experiments on an NVIDIA RTX 3090 with Intel(R)
Xeon(R) CPU.

###  4.1 3D Shape Reconstruction

We consider a learning-based point cloud shape reconstruction task.
Technically, we followed FoldingNet [31] to construct a reconstruction
network, where regular 2D grids distributed on a square area
$[-\delta,~{}\delta]^{2}$ are fed into MLPs to regress 3D point clouds. The
network was trained by minimizing the distance between the reconstructed point
cloud and the given point cloud in an overfitting manner, i.e., each shape has
its individual network parameters. Additionally, based on the mechanism of the
reconstruction network and differential geometry, we could obtain the normal
vectors of the reconstructed point cloud through the backpropagation of the
network. Finally, we used SPSR [13] to recover the mesh from the resulting
point cloud and its normal vectors. We refer the readers to the Supplementary
Material for more details.

#### Implementation Details.

We utilized three categories of the ShapeNet dataset [6], namely chair, sofa,
and table, each containing 200 randomly selected shapes. For each shape, we
normalized it within a unit cube and sampled 4096 points from its surface
uniformly using PDS [4] to get the point cloud. As for the input regular 2D
grids, we set $\delta=0.3$ and the size to be $64\times 64$. We used the ADAM
optimizer to optimize the network for $10^{4}$ iterations with a learning rate
of $10^{-3}$.

Table 1: Quantitative comparisons of reconstructed point clouds and surfaces under different loss functions. The best results are highlighted in bold. $\downarrow$ (resp. $\uparrow$) indicates the smaller (resp. the larger), the better. Shape | Loss | Point Cloud | Triangle Mesh
---|---|---|---
CD ($\times 10^{-2}$) $\downarrow$ |  HD ($\times 10^{-2}$) $\downarrow$ |  P2F ($\times 10^{-3}$) $\downarrow$ |  NC $\uparrow$ |  F-0.5% $\uparrow$ |  F-1% $\uparrow$
Chair | EMD | 2.935 | 12.628 | 9.777 | 0.781 | 0.277 | 0.524
CD | 2.221 | 9.430 | 4.543 | 0.839 | 0.465 | 0.721
Ours | 1.898 | 6.787 | 2.591 | 0.908 | 0.709 | 0.913
Sofa | EMD | 2.534 | 7.355 | 7.433 | 0.879 | 0.459 | 0.717
CD | 1.972 | 5.323 | 3.392 | 0.920 | 0.668 | 0.887
Ours | 1.770 | 4.191 | 2.040 | 0.940 | 0.806 | 0.949
Table | EMD | 2.996 | 11.374 | 8.921 | 0.768 | 0.243 | 0.473
CD | 2.272 | 8.881 | 4.353 | 0.824 | 0.403 | 0.658
Ours | 1.974 | 6.302 | 2.480 | 0.900 | 0.643 | 0.873
Average | EMD | 2.821 | 10.452 | 8.720 | 0.809 | 0.326 | 0.571
CD | 2.155 | 7.878 | 4.096 | 0.861 | 0.521 | 0.755
Ours | 1.880 | 5.760 | 2.370 | 0.916 | 0.719 | 0.911
Figure 3: Visual comparisons of reconstructed shapes in the form of point
clouds and surfaces under different distance metrics. From top to bottom: EMD,
CD, Ours, and GT.

(a) Running time

(b) GPU Memory

Figure 4: Efficiency comparison under 3D shape reconstruction with various
numbers of points.

#### Comparisons.

To quantitatively compare different distance metrics, we employed CD,
Hausdorff distance (HD), and the point-to-surface distance (P2F) to evaluate
the accuracy of reconstructed point clouds. As for the reconstructed triangle
meshes, we used Normal Consistency (NC) and F-Score with thresholds of 0.5%
and 1%, denoted as F-0.5% and F-1%, as the evaluation metrics. Table 1 and
Fig. 3 show the numerical and visual results, respectively, where both
quantitative accuracy and visual quality of reconstructed shapes by the
network trained with our CLGD are much better. Besides, due to the difficulty
in establishing the optimal bijection between a relatively large number of
points, i.e., 4096 points in our experiments, the network trained with EMD
produces even worse shapes than that with CD.

#### Efficiency Analysis.

We compared the running time in Fig. 4(a), where it can be seen that with the
number of points increasing, the reconstruction driven by EMD requires much
more time to optimize than that by CD and our CLGD. Fig. 4(b) compares the GPU
memory consumption, showing that these three distance metrics are comparable
when the number of points is relatively small, but when dealing with a large
number of points, EMD requires more GPU memory.

###  4.2 Rigid Registration

Given source and target point clouds, denoted as $\mathbf{P}_{\rm
src}\in\mathbb{R}^{N_{\rm src}\times 3}$ and $\mathbf{P}_{\rm
tgt}\in\mathbb{R}^{N_{\rm tgt}\times 3}$, respectively, rigid registration is
to seek a spatial transformation $[\mathbf{R},\mathbf{t}]$ to align
$\mathbf{P}_{\rm src}$ with $\mathbf{P}_{\rm tgt}$, where
$\mathbf{R}\in\mathbb{R}^{3\times 3}$ is the rotation matrix and
$\mathbf{t}\in\mathbb{R}^{3}$ is the translation vector. The optimization-
based registration directly solves the following problem:

$\\{\hat{\mathbf{R}},\hat{\mathbf{t}}\\}=\mathop{\arg\min}_{\mathbf{R},\mathbf{t}}\mathcal{D}\left(\mathbf{RP}_{\rm
src}+\mathbf{t},\mathbf{P}_{\rm tgt}\right),$ (5)

where $\mathcal{D}(\cdot,~{}\cdot)$ is the distance metric that can be EMD,
CD, ARL, or our CLGD. We also consider unsupervised learning-based rigid
registration. Specifically, following [8], we modify RPM-Net [32], a
supervised learning-based registration method, to be unsupervised by using a
distance metric to drive the learning of the network. See the Supplementary
Material for more details. We also selected two common rigid registration
methods as baselines, i.e., ICP [3] and FGR [38].

Table 2: Quantitative comparisons of rigid registration on the Human dataset [8]. | Method |  ${\rm RE}$ (∘) $\downarrow$ |  TE (m) $\downarrow$
---|---|---|---
Optimization-based |  ICP [3] | 1.276 | 0.068
FGR [38] | 19.684 | 0.384
EMD [21] | 7.237 | 0.642
CD [2] | 10.692 | 0.421
ARL [8] | 1.245 | 0.085
Ours | 1.040 | 0.040
Unsupervised learning (RPM-Net [32]) |  EMD [21] | 7.667 | 0.638
CD [2] | 2.197 | 0.287
ARL [8] | 1.090 | 0.075
Ours | 0.793 | 0.053

#### Implementation Details.

We used the Human dataset provided in [8], containing 5000 pairs of source and
target point clouds for training and 500 pairs for testing. The source point
clouds are 1024 points, while the target ones are 2048 points, i.e., they are
partially overlapped. We used Open3D [39] to implement ICP and FGR. For the
optimization-based methods, we utilized Lie algebraic to represent the
transformation and optimized it with the ADAM optimizer for 1000 iterations
with a learning rate of 0.02. For the unsupervised learning-based methods, we
kept the same training settings as [8]. We refer the reader to Supplemental
Material for more details.

Initial
---
RE ($\ {}^{\circ}$)
TE (m)
ICP
---
37.455
0.401
FGR
---
73.212
1.358
Opt-EMD
---
2.311
0.745
Opt-CD
---
9.044
0.311
Opt-ARL
---
0.804
0.074
Opt-Ours
---
0.425
0.039
RPM-EMD
---
13.799
0.848
RPM-CD
---
109.364
0.921
RPM-ARL
---
1.260
0.223
RPM-Ours
---
0.867
0.088
Figure 5: Visual comparisons of rigid registration results. The blue and green
points represent the source and target point clouds, respectively.  Zoom in to
see details.

#### Quantitative Comparison.

Following previous registration work [7], we adopted Rotation Error (RE) and
Translation Error (TE) as the evaluation metrics. As shown in Table 2 and Fig.
5, our distance metric outperforms the baseline methods in both optimization-
based and unsupervised learning methods. ICP, FGR, as long as EMD and CD, can
easily get local optimal solutions since they struggle to properly handle the
outliers in the non-overlapping regions. ARL [8] and our CLGD do not have such
a disadvantage, but the randomly sampled lines in ARL decrease the
registration accuracy.

Table 3: Running time (s) and GPU memory (GB) costs of different distance metrics under the optimization-based rigid registration task. | Time | GPU Memory
---|---|---
EMD [21] | 11 | 1.680
CD [2] | 11 | 1.809
ARL [8] | 281 | 7.202
Ours | 13 | 1.813

#### Efficiency Analysis.

Table 3 lists the running time and GPU memory costs of the optimization-based
registration driven by different metrics333Note that for the unsupervised
learning-based methods, different distance metrics are only used to train the
network, and the inference time of a trained network with different metrics is
equal. . Since the number of points is relatively small, EMD has comparable
running time and GPU memory consumption to CD and our CLGD. In contrast, ARL
[8] is much less efficient because of the calculation of the intersections.

###  4.3 Scene Flow Estimation

This task aims to predict point-wise offsets $\mathbf{F}\in\mathbb{R}^{N_{\rm
src}\times 3}$, which can align source point cloud $\mathbf{P}_{\rm src}$ to
target point cloud $\mathbf{P}_{\rm tgt}$. The optimization-based methods
directly solve

$\hat{\mathbf{F}}=\mathop{\arg\min}_{\mathbf{F}}\mathcal{D}(\mathbf{P}_{\rm
src}+\mathbf{F},\mathbf{P}_{\rm tgt})+\alpha\mathcal{R}_{\rm
smooth}(\mathbf{F}),$ (6)

where $\mathcal{R}_{\rm smooth}(\cdot)$ is the spatial smooth regularization
term and the hyperparameter $\alpha>0$ balances the two iterms. Besides, we
also evaluated the proposed CLGD by incorporating it into unsupervised
learning-based frameworks, i.e., replacing the distance metrics of existing
unsupervised learning-based frameworks with our CLGD to train the network, We
adopted two SOTA unsupervised scene flow estimation methods named NSFP [15]
and SCOOP [14].

#### Implementation Details.

We used the Flyingthings3D dataset [16] preprocessed by [10], where $N_{\rm
src}=N_{\rm tgt}=8192$. For the optimization-based methods, we used
$\ell_{2}$-smooth regularization, set $\alpha=50$, and optimized the scene
flow directly with the ADAM optimizer for 500 iterations with a learning rate
of 0.01. For the unsupervised learning-based methods, we adopted the same
training settings as their original papers.

Table 4: Quantitative comparisons of scene flow estimation on the Flyingthings3D dataset . | Method |  EPE3D(m)$\downarrow$ |  Acc-0.05 $\uparrow$ |  Acc-0.1$\uparrow$ |  Outliers $\downarrow$
---|---|---|---|---|---
Optimization-based |  EMD [21] | 0.3681 | 0.1894 | 0.4226 | 0.7838
CD [2] | 0.1557 | 0.3489 | 0.6581 | 0.6799
Ours | 0.0843 | 0.5899 | 0.8722 | 0.4707
Unsupervised Learning |  NSFP [15] | 0.0899 | 0.6095 | 0.8496 | 0.4472
NSFP [15] \+ Ours | 0.0662 | 0.7346 | 0.9107 | 0.3426
SCOOP [14] | 0.0839 | 0.5698 | 0.8516 | 0.4834
SCOOP [14] \+ Ours | 0.0742 | 0.6134 | 0.8858 | 0.4497
Table 5: Running time (s) and GPU memory (GB) costs of different distance metrics under the optimization-based scene flow estimation task. | Time | GPU Memory
---|---|---
EMD | 1011 | 2.204
CD | 7 | 1.704
Ours | 14 | 1.725

#### Comparison.

Following [15, 14], we employed End Point Error (EPE), Flow Estimation
Accuracy (Acc) with thresholds 0.05 and 0.1 (denoted as Acc-0.05 and Acc-0.1),
and Outliers as the evaluation metrics. From the results shown in Table 4 and
Fig. 6, it can be seen that our CLGD drives much more accurate scene flows
than EMD and CD under the optimization-based framework, and our CLGD further
boosts the accuracy of SOTA unsupervised learning-based methods to a
significant extent, demonstrating its superiority and the importance of the
distance metric in 3D point cloud modeling.

GT
---
EPE3D (m)
Acc-0.05
Acc-0.1
Outliers
Opt-EMD
---
0.247
0.437
0.672
0.361
Opt-CD
---
0.079
0.627
0.963
0.069
Opt-Ours
---
0.067
0.875
0.973
0.044
NSFP
---
0.086
0.527
0.974
0.057
NSFP + Ours
---
0.050
0.859
0.991
0.010
SCOOP
---
0.071
0.648
0.989
0.040
SCOOP + Ours
---
0.047
0.892
0.995
0.019
Figure 6: Visual comparisons of scene flow estimation. The blue and green
points represent the source and target point clouds, respectively, and the red
points are the warped source point cloud with estimated scene flows.  Zoom in
to see details.

#### Efficiency Analysis.

We also compared the running time and GPU memory cost of different metrics in
the optimization-based framework. As shown in Table 5, EMD consumes much more
time and GPU memory than CD and our CLGD.

###  4.4 Feature Representation

In this experiment, we trained an auto-encoder with different distance metrics
used as the reconstruction loss to evaluate their abilities. Technically, an
input point cloud is encoded into a global feature through the encoder, which
is further decoded to reconstruct the input point cloud through a decoder.
After training, we used the encoder to represent point clouds as features for
classification by an SVM classifier.

#### Implementation Details.

We built a auto-encoder with MLPs and used the ShapeNet [6] and ModelNet40
[27] datasets for training and testing, respectively. We trained the network
for 300 epochs using the ADAM optimizer with a learning rate of $10^{-3}$. We
refer the readers to the Supplemental Material for more details.

Table 6: Classification accuracy by SVM on ModelNet40 [27].  Loss Function | EMD | CD | Ours
---|---|---|---
Accuracy (%) | 78.12 | 78.89 | 81.28

#### Comparison.

As listed in Table 6, the higher classification accuracy by our CLGD
demonstrates the auto-encoder driven by our CLGD can learn more discriminative
features. Besides, we also used T-SNE [23] to visualize the 2D embeddings of
the global features of 1000 shapes from 10 categories in Fig. 7 to show the
advantages of our method more intuitively.

(a) EMD

(b) CD

(c) Ours

Figure 7: The T-SNE clustering visualization of the features obtained from the
auto-encoder trained with different distance metrics.

###  4.5 Ablation Study

We carried out comprehensive ablation studies on shape reconstruction and
rigid registration to verify the rationality of the design in our CLGD.

#### Directional Distance.

We removed $f$ or $\mathbf{v}$ from the directional distance $\mathbf{g}$ when
modeling the local surface geometry in Section 3.3. The accuracy of
reconstructed 3D mesh shapes is listed in Table 7, where it can be seen that
removing either one of $f$ and $\mathbf{v}$ would decrease the reconstruction
accuracy, especially $\mathbf{v}$, because using $f$ or $\mathbf{v}$ alone
could hardly characterize the underlying surfaces of point clouds.

#### Size of $\Omega(\mathbf{q},\mathbf{P})$.

We changed the size of $\Omega(\mathbf{q},\mathbf{P})$ by varying the value of
$K$ used in Eqs. (1) and (2). As shown in Table 7, a small or large value of
$K$ would decrease the performance of CLGD because a small $K$ only covers a
tiny region on the point cloud, resulting in that
$\mathbf{g}(\mathbf{q},\mathbf{P})$ cannot represent the local surface
geometry induced by $\mathbf{q}$, while a large $K$ includes too many points,
making $\mathbf{g}(\mathbf{q},\mathbf{P})$ would become smooth and ignore the
details of the local surface geometry induced by $\mathbf{q}$.

#### Reference Points.

We studied how the number of reference points affects CLGD by varying the
value of $R$. As shown in Table 7, too few reference points cannot
sufficiently capture the local surface geometry difference, while too many
reference points are not necessary for improving performance but compromise
efficiency. We also studied how the distribution of reference points affects
CLGD by varying the value of $T$, concluding that the reconstruction accuracy
decreases if the reference points are either too close to or too far from the
underlying surface. Finally, from Table 7, it can be seen that after removing
the source point cloud $\mathbf{P}_{\rm src}$ from $\mathbf{Q}$, the
reconstruction accuracy decreases, demonstrating the necessity of
$\mathbf{P}_{\rm src}$ as reference points.

Table 7: Results of the ablative studies about our CLGD under the 3D shape reconstruction task.  Geometry | $K$ | $R$ | $T$ | $\mathbf{P}_{\rm src}\subset\mathbf{Q}$ |  NC $\uparrow$ |  F-0.5% $\uparrow$ |  F-1% $\uparrow$ |  Time (s) $\downarrow$
---|---|---|---|---|---|---|---|---
$f$ | 5 | 10 | 3 | ✓ | 0.791 | 0.611 | 0.776 | 123
$\mathbf{v}$ | 5 | 10 | 3 | ✓ | 0.907 | 0.675 | 0.888 | 122
$\mathbf{g}$ | 1 | 10 | 3 | ✓ | 0.913 | 0.687 | 0.888 | 119
$\mathbf{g}$ | 3 | 10 | 3 | ✓ | 0.916 | 0.717 | 0.908 | 124
$\mathbf{g}$ | 10 | 10 | 3 | ✓ | 0.913 | 0.712 | 0.907 | 132
$\mathbf{g}$ | 5 | 1 | 3 | ✓ | 0.899 | 0.610 | 0.831 | 110
$\mathbf{g}$ | 5 | 5 | 3 | ✓ | 0.912 | 0.686 | 0.886 | 112
$\mathbf{g}$ | 5 | 20 | 3 | ✓ | 0.913 | 0.693 | 0.893 | 149
$\mathbf{g}$ | 5 | 10 | 1 | ✓ | 0.916 | 0.722 | 0.909 | 126
$\mathbf{g}$ | 5 | 10 | 5 | ✓ | 0.911 | 0.692 | 0.896 | 126
$\mathbf{g}$ | 5 | 10 | 10 | ✓ | 0.891 | 0.610 | 0.836 | 126
$\mathbf{g}$ | 5 | 10 | 3 | ✗ | 0.915 | 0.714 | 0.909 | 123
$\mathbf{g}$ | 5 | 10 | 3 | ✓ | 0.916 | 0.719 | 0.911 | 126
Table 8: Effect of the value of $\beta$ on rigid registration accuracy.  $\beta$ |  RE (∘) | TE (m)
---|---|---
0 | 5.637 | 0.286
1 | 1.598 | 0.080
3 | 1.040 | 0.040
5 | 1.290 | 0.037
10 | 1.769 | 0.046

#### Effectiveness of $s(\mathbf{q})$ in Rigid Registration.

We set various values of $\beta$ in Eq. (4) in the rigid registration task. As
shown in Table 8, when $\beta=0$, the registration accuracy decreases
significantly because the source and target point clouds are partially
overlapped, and those reference points, which correspond to the non-
overlapping regions and negate the optimization process, are still equally
taken into account. On the contrary, if the value of $\beta$ is large, the
registration accuracy only drops slightly. The reason is that the weights of
the reference points with similar local surface geometry differences would be
quite different, and the details of the point clouds may be ignored, making
the registration accuracy slightly affected.

##  5 Conclusion and Discussion

We have introduced CLGD, a novel, robust, and generic distance metric for 3D
point clouds. CLGD measures the difference between the local surfaces
underlying 3D point clouds under evaluation, which is significantly different
from existing metrics. Our experiments have demonstrated the significant
superiority of CLGD in terms of both efficiency and effectiveness for various
tasks, including shape reconstruction, rigid registration, scene flow
estimation, and feature representation. Besides, comprehensive ablation
studies have validated its rationality. We believe CLGD will advance the field
of 3D point cloud processing and analysis.

Our proposed CLGD measures the differences between two point clouds by
modeling the local geometry on their underlying surfaces. As a result, it is
important that the points of a point cloud are distributed on the surfaces
rather than completely random, as the local geometry induced by reference
points in a randomly distributed point cloud would be meaningless.
Consequently, in some generation tasks, CLGD requires a relatively good
initialization in the beginning. Otherwise, it may perform worse than distance
metrics that focus on the point cloud, such as EMD and CD.

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Veltman Criteria in Beyond Standard Model Effective Field Theory
of Complex Scalar Triplet

aJaydeb Das<EMAIL_ADDRESS>, bNilanjana
<EMAIL_ADDRESS>
a Department of Physics and Astrophysics, University of Delhi, Delhi-110007,
India

bCentre For Cosmology and Science Popularization (CCSP),

SGT University, Gurugram, Haryana-122006, India

###### Abstract

The Standard Model Higgs mass, not being protected by any symmetry, suffers
from large correction terms due to quadratic divergence coming from the self
energy corrections. Veltman Condition (V.C.) ensures that the coefficient of
the quadratic divergent term either vanishes or becomes negligible. If the
Standard Model is valid upto a scale ($\Lambda$) and new physics takes over
beyond that; Veltman condition demands $\Lambda\lesssim 760$ GeV. But the non-
observation of new physics has pushed $\Lambda$ to be $\geq 1$ TeV already,
making it impossible to satisfy the Veltman condition in the Standard Model
without very large fine tuning. Attempts has been made to satisfy the V.C. in
many Beyond Standard Model theories but they fail to satisfy the V.C. at large
$\Lambda$ including the theory of complex scalar triplet with hypercharge $1$.
Hence, an alternate scenario can be considered where the new physics appears
much above the Electroweak scale. Then, the effect of the new physics can be
observed by integrating out the new degrees of freedoms in terms of the Wilson
Coefficients of the Standard Model Effective Field Theory (SMEFT) operators.
Here we consider some specific new physics models to appear at a very large
scale, and name it Beyond Standard Model Effective Field Theory (BSM-EFT). We
found that the V.C. satisfies easily in the type-II seesaw model with complex
scalar triplet ($Y=1$) compared to the other extensions in the BSM-EFT
framework. We examine the model parameter dependence of the Wilson
Coefficients in detail and show how the cancellation of the Wilson
Coefficients appearing in the V.C. is highly dependent on some specific values
of the model parameters.

##  1 Introduction

The smallness of the observed Higgs mass is confirmed by the experiments [1,
2] at the Large Hadron Collider (LHC). However, in the Standard Model (SM) of
particle physics, the scalar mass (mass of Higgs boson) is not protected by
any symmetry. Hence, if SM is valid upto a large scale, Planck scale, the
Higgs mass suffers from quadratic divergence ($\sim\Lambda^{2}$). In order to
ensure that the mass of the Higgs boson is small, one has to consider a very
large fine tuning in the SM. A way to ensure that the Higgs mass does not get
large correction at a higher scale is coined as Veltman Condition (V.C.) [3].
V.C. checks if the sum of all quadratically divergent terms coming from the
self energy diagrams of the Higgs boson are either zero or very small. On the
other hand, experiments such as the Large Hadron Collider (LHC) is pushing the
New Physics (NP) scale towards $>1$ TeV and the Veltman Criteria is not
possible to satisfy in the SM, as it demands $\Lambda$ to be less than 760 GeV
[4].

Simple extensions of SM has been studied in literature [5, 6, 7, 8, 9, 4, 10,
11, 12], where the V.C. is valid but only in some region of parameter space.
Overall, there are two main concerns in these models: (1) These theories
encounter different problems at large scale, such as the potential becomes
unstable leading to the invalidity of the theory beyond that scale. (2) The
non observation of the Beyond Standard Model (BSM) particles at LHC is pushing
their masses above TeV scale [13].

One may assume that SM is valid upto certain scale ($\Lambda$) and above that
scale some unknown symmetry appears to protect the Higgs mass, then the Higgs
mass can be stabilized and the fine tuning problem can also be addressed. For
examples, in the Composite Higgs Scenario [14], where the Higgs is dissolved
in higher degrees of freedom above the symmetry breaking scale or in
Supersymmetric theories [15], where the bosonic and fermionic degrees of
freedom cancels out exactly – the Higgs mass is maintained to be finite and
small. However these theories also can not avoid certain amount of fine tuning
[16, 17] coming from several sources. However, the none of these theories are
observed at LHC and other experiments so far.

These observations raise the question that what if the new physics lie at a
very large scale. In such a scenario, SM can emerge as an Effective Field
Theory (SMEFT) [18] by integrating out the dynamics of the larger theory. The
Information of the heavy particles appearing in the loop are absorbed in the
higher dimensional operators in the Effective Field Theory (EFT) and the
theory is invariant under SM symmetries. Ref [19] has shown that the V.C. can
be satisfied in the SMEFT framework by including the higher dimensional
operators and their Wilson Coefficients. Only a few of the operators are
relevant to the V.C. and they play a major role in satisfying the V.C.

In this paper, we take one step forward and ask this question what if the
theory at a very high scale ($\Lambda$) is governed by the larger symmetries,
and how they affect the V.C.? We adopt the Beyond Standard Model Effective
Field theory (BSM-EFT) [20] approach, which has been studied previously in Ref
[20, 21, 22, 23, 24, 25, 26, 27]. In BSM-EFT the Lagrangian becomes invariant
under the particular BSM model in consideration. The motivation to study the
V.C. in BSM-EFT framework is two fold. 1) We can specifically check how many
higher dimensional operators are allowed by the model. 2) We can express the
Wilson Coefficients in terms of the model parameters. Hence, the sign
dependence of the W.C.’s come naturally.

We begin with simple BSM scenarios (in BSM-SMEFT) such as, scalar singlet,
doublet and triplets (real or complex). Remarkably, we found that among these
models, it is possible to generate all the SMEFT operators that contributes to
the V.C. in the complex scalar triplet model with $Y=1$. This model makes the
cancellation easier in V.C. with less fine tuning than the other
scenarios333Even type I and type II seesaw models does not generate all these
operators. Moreover a recent study [24] has also shown that these models are
also not favored from the fact that the radiative electroweak symmetry
breaking can not be triggered even at the Planck scale.. This particular model
is well motivated in literature from different aspects such as: 1) Neutrino
mass generation through the see-saw mechanism [28], 2) type-II Leptogenesis
scenario [29] 3) Enhancement of the $h\rightarrow\gamma\gamma$ branching ratio
[30] etc, among many other[31]. Hence we pursue this model only in detail.

In Section 2, we show how the V.C. depends on the SMEFT operators in the
WARSAW basis [32]. In Section 3, we discuss some specific models in the BSM-
EFT scenario, and express the Wilson coefficients of $Y=1$ complex scalar
triplet model in terms of the model parameters. In Section 4, we show how it
is possible to satisfy the V.C. by exact cancellation of the Wilson
Coefficients at different scale and interpret the result in terms of the model
parameter space. Then in Section 5 we conclude.

##  2 SMEFT operators in Veltman Criteria

The physical mass of the Higgs in the Standard Model can be written in terms
of the bare mass term $m_{h}(0)$ and the higher-order self-energy corrections:

$m^{2}_{h}=m^{2}_{h}(0)+\delta m^{2}_{h}=m^{2}_{h}(0)+\text{Log Div. Term +
Quadratic Div. Term + Finite terms},$ (1)

where the assumption is that the SM is valid upto the scale $\Lambda$ and the
correction terms are coming from the loop diagrams involving scalars, fermions
and bosons in the loop. The $d=4$ potential in the Standard Model in terms of
Higgs doublet ($H$) is

$V(H)=-m_{H}^{2}H^{\dagger}H+\lambda(H^{\dagger}H)^{2}.$ (2)

This leads to the correction to the higgs mass and the quadratic divergent
contribution is,

$\displaystyle(\delta m_{h}^{2})_{\rm SM}$ $\displaystyle=$
$\displaystyle\frac{\Lambda^{2}}{16\pi^{2}}\left(6\lambda+\frac{9}{4}g_{W}^{2}+\frac{3}{4}g_{Y}^{2}-6y_{t}^{2}\right),$
(3)

where, $g_{Y}$ and $g_{W}$ are the $U(1)_{Y}$ and $SU(2)_{L}$ gauge couplings
respectively and $g_{t}=2m_{t}/v$ is the top quark Yukawa coupling. Here we
neglect couplings of the lighter quarks and $\Lambda$ is the cut-off scale.
The Veltman Condition (V.C.) demands that $\delta m_{h}^{2}\sim 0$ or at least
controllably small. With the observed Higgs mass at 125 GeV, the condition to
make $\delta m_{h}\sim 0$ demands $\Lambda<760$ GeV, which is already ruled
out by LHC. One way to solve this problem is to introduce new particles, which
can contribute in the loops and soften the fine tuning by ensuring exact
cancellation or partial as we have already discussed in the introduction.

A popular way to address this problem is to consider the effects of the higher
dimensional operators in the EFT framework. Let us assume that the New Physics
(NP) exists at a very high scale $\Lambda$. The effect of NP can be
effectively integrated out at $\Lambda$ and this will effectively give us SM,
plus some effective operators involving only the SM fields, which holds
through out the low energy scale, otherwise known as the Standard Model
Effective Field Theory (SMEFT) [18]. The Lagrangian, which incorporates
dimension six SMEFT operators in addition to the Standard Model dimension four
operators, can be expressed as,

$\mathcal{L}=\sum_{i}\mathcal{C}_{4i}\mathcal{Q}_{4i}+\frac{1}{\Lambda^{2}}\sum_{i}\mathcal{C}_{6i}\mathcal{Q}_{6i}.$
(4)

In contrast to $\mathcal{C}_{4i}$, which is the only function of the
parameters linked to the degrees of freedom in the Standard Model,
$\mathcal{C}_{6i}$ are the Wilson Coefficients, which are functions of the
integrated out dynamics at $\Lambda$. These operators can be expanded at any
choice of basis, for example, HISZ basis [33], Warsaw basis [32, 35], SILH
basis [34] etc. The set of dimension six operators that involves Higgs in
Warsaw basis are:

$\displaystyle\mathcal{Q}_{H}=(H^{\dagger}H)^{3},\,\,\
\mathcal{Q}_{HD}=(H^{\dagger}\mathcal{D}_{\mu}H)^{\ast}(H^{\dagger}\mathcal{D}^{\mu}H),\,\,\
\mathcal{Q}_{H\square}=(H^{\dagger}H)\square(H^{\dagger}H)\ $
$\displaystyle\mathcal{Q}_{HB}=(H^{\dagger}H)B_{\mu\nu}B^{\mu\nu},\,\,\
\mathcal{Q}_{HW}=(H^{\dagger}H)W_{\mu\nu}^{a}W^{a,\mu\nu},\,\,\
\mathcal{Q}_{GG}=(H^{\dagger}H)G^{A}_{\mu\nu}G^{A,\mu\nu}\ $
$\displaystyle\mathcal{Q}_{HWB}=(H^{\dagger}\tau_{a}H)B^{\mu\nu}W^{a}_{\mu\nu}$
(5)

It can be shown that the last operator does not contribute Higgs self energy
correction[19]. The first operator will also not contribute at one-loop level
as the Higgs does not develop a vev at $\Lambda$. There can be the appearance
of the operators involving the gluons of the form
$\mathcal{Q}_{GG}=(H^{\dagger}H)G^{A}_{\mu\nu}G^{A,\mu\nu}$. However, while
considering BSM-EFT framework with heavy scalars, this operator does not
contribute as scalars do not carry any color charge. Note that, these
operators can be written in any basis, for example Ref [19] choose the HISZ
basis. We choose the Warsaw basis because it is self consistent at one loop
[35, 36] and easier to check the running of the Wilson coefficients in Warsaw
basis.

The correction to the Higgs mass from the higher order terms in the Lagrangian
is given by

$\displaystyle(\delta m_{h}^{2})_{\rm total}$ $\displaystyle=$
$\displaystyle\frac{\Lambda^{2}}{16\pi^{2}}\sum_{i}f_{i}(\mathcal{C}_{4i},\mathcal{C}_{6i})+\frac{\Lambda^{2}}{(16\pi^{2})^{2}}\sum_{i}g_{i}(\mathcal{C}_{4i},\mathcal{C}_{6i})$
(6)

Here $f_{i}$ and $g_{i}$ are one loop and two loop correction to the Higgs
mass. The V.C., $\delta m_{h}^{2}\sim 0$ translates into

$f(\mathcal{C}_{4i},\mathcal{C}_{6i}),g(\mathcal{C}_{4i},\mathcal{C}_{6i})\sim
0$ (7)

The coefficients, $\mathcal{C}_{4i}$ and $\mathcal{C}_{6i}$ are function of
$\Lambda$ and the model parameters. Hence Eq:6 can be written in terms of the
SM and higher dimension operators contribution as,

$\displaystyle(\delta m_{h}^{2})_{\rm total}\equiv(\delta m_{h}^{2})_{\rm
SM}\big{(}f_{i}(\mathcal{C}_{4i}),g_{i}(\mathcal{C}_{4i})\big{)}+(\delta
m_{h}^{2})_{\rm
HO}\big{(}f_{i}(\mathcal{C}_{6i}),g_{i}(\mathcal{C}_{6i})\big{)}$ (8)

Also, it has been shown in Ref [19] that at $d\geq 8$, the SMEFT operators are
not able to produce any $\Lambda^{6}$ divergence, which will produce any
effective $\Lambda^{2}$ divergence while calculating the self energy
correction of Higgs mass. There are studies in the literature, where the V.C
in terms of EFT has been studied in detail [19, 37, 38]. In particular it has
been shown in Ref: [19] that it is possible to satisfy the V.C for appropriate
values and sign of the Wilson coefficients at large $\Lambda$.

##  3 BSM-EFT with Complex Scalar Triplet

In the above section, we see that only four operators are involved in the V.C.
Now, we assume that the new physics at a high scale follow certain symmetries
of a BSM model which effectively produces SM as an EFT. In this BSM-EFT
framework, these 4 operators may or may not be possible to generate at one
loop, depending on the underlying symmetry of the model at scale $\Lambda$. In
Table: 1 we present if these 4 operators can be generated at one loop in some
simple BSM-EFT cases with additional scalar(s) or not 444Note that we are not
checking non scalar extensions of SM because, the sign of the top-loop
contribution (dominant contribution) or rather fermionic contribution is
opposite to the other diagrams with a gauge boson or a scalar in the loop.
Therefore, V.C. is hard to solve by adding non scalar particles such as
vector-like quarks or fermions, additional gauge bosons etc.. For the
calculation, we have implemented the Lagrangian of each model in CoDEx [39,
40] and generated the Wilson coefficients as an output 555We have also cross
checked our result with Matchmakereft[41]..

Model | Quantum No | $\mathcal{Q}_{HD}$ | $\mathcal{Q}_{HB}$ | $\mathcal{Q}_{HW}$ | $\mathcal{Q}_{H\square}$
---|---|---|---|---|---
Real Scalar Singlet | (1,1,0) | ✓ | ✗ | ✗ | ✗
Real Scalar Triplet | (1,3,0) | ✗ | ✗ | ✓ | ✓
Complex Scalar Triplet | (1,3,1) | ✓ | ✓ | ✓ | ✓
Complex scalar doublets (2HDM) | (1,2,$\pm$1/2) | ✓ | ✗ | ✓ | ✓
Real Scalar Singlet + | (1,1,0) | ✓ | ✗ | ✓ | ✓
Real Scalar Triplet | (1,3,0) |  |  |  |
Complex Scalar Triplet + | (1,3,1) | ✓ | ✓ | ✓ | ✓
Complex Scalar Doublet | (1,2,1/2) |  |  |  |
Table 1: SMEFT operators in Warsaw basis in different BSM-EFT scenarios.

Among all popular SM extensions, we have found that BSM-EFT with complex
scalar triplet will easily address the V.C, as it generates all four Wilson
Coefficients at one loop. In other models, the cancellation will be harder to
achieve as the number of operators are less than four. For example in 2HDM
scenario and real scalar singlet + triplet model, only three operators can be
generated. Whereas, in complex scalar singlet model, only 2 operators are
generated and the cancellation will be hard to obtain (hence large fine
tuning) in these models compared to the complex scalar triplet model. Even
larger fine tuning will be unavoidble for the real scalar singlet model as it
generates only one operator. The complex scalar triplet with additional
doublet also can generate these four operators but we examine the minimal
scenario only with complex scalar triplet model in the following.

Let us consider that beyond the scale $\Lambda$, there exists a heavy complex
triplet, $\Delta$, with weak hypercharge $Y=1$. The most general
renormalizable tree-level scalar potential of such a model is given by

$\displaystyle V(H,\Delta)$ $\displaystyle=$ $\displaystyle-
m_{H}^{2}\left(H^{\dagger}H\right)+M^{2}\mathrm{Tr}\left[\Delta^{\dagger}\Delta\right]+\left(\mu_{\Delta}H^{T}i\sigma^{2}\Delta^{+}H+h.c\right)+\lambda\left(H^{\dagger}H\right)^{2}+\lambda_{1}\left(H^{\dagger}H\right)\mathrm{Tr}\left[\Delta^{\dagger}\Delta\right]$
$\displaystyle+$
$\displaystyle\lambda_{2}\left(\mathrm{Tr}[\Delta^{\dagger}\Delta]\right)^{2}+\lambda_{3}\mathrm{Tr}\left[(\Delta^{\dagger}\Delta)^{2}\right]+\lambda_{4}\left(H^{\dagger}\Delta\Delta^{\dagger}H\right).$
(9)

The extra Yukawa term for neutrino mass generation is,

$\displaystyle\mathcal{L}_{Y}$ $\displaystyle=$ $\displaystyle
y_{\Delta}\ell^{T}iCi\sigma^{2}\Delta\ell+h.c.$ (10)

Here the trilinear coupling $\mu_{\Delta}$ can be taken as positive by
absorbing its phase into $\Phi$ and $\Delta$. The total Lagrangian is,

$\mathcal{L}=\mathcal{L}_{Y}-V(H,\Delta),$ (11)

The detail of this model is summarized in ‘Model Description’ part of the
Appendix.

The dimension six operators the Warsaw basis, as listed in Eq:2, can be
expanded and the calculation of the symmetry factors are shown in the
‘Calculation’ part of the Appendix. Hence, the Higgs mass correction in terms
of W.C.’s is found to be,

$\displaystyle(\delta m_{h}^{2})_{\rm BSM}$ $\displaystyle=$
$\displaystyle\frac{\Lambda^{2}}{16\pi^{2}}\Big{(}-3\mathcal{C}_{HD}+12\mathcal{C}_{H\square}+9\mathcal{C}_{HW}+3\mathcal{C}_{HB}\Big{)}\
$ (12) $\displaystyle+$
$\displaystyle\frac{\Lambda^{2}}{(16\pi^{2})^{2}}\left(54\mathcal{C}_{H}-\frac{9}{2}(g_{Y}^{2}+3g_{W}^{2})\mathcal{C}_{HD}+108g_{W}^{2}\mathcal{C}_{HW}\right)$

Which leads to the total correction to the Higgs mass to be,

$\displaystyle\delta m_{h}^{2}=(\delta m_{h}^{2})_{\rm SM}+(\delta
m_{h}^{2})_{\rm BSM}$ (13)

We found the the following expressions of the Wilson Coefficients appearing in
one loop contribution in $(\delta m_{h}^{2})_{\rm BSM}$:

$\displaystyle\mathcal{C}_{HD}$ $\displaystyle=$
$\displaystyle-\frac{g_{Y}^{4}}{320\pi^{2}}+\frac{4\mu_{\Delta}^{2}}{M^{2}}-\frac{\lambda_{4}^{2}}{24\pi^{2}}+\frac{11g_{Y}^{2}\mu_{\Delta}^{2}}{24\pi^{2}M^{2}}-\frac{8\mu_{\Delta}^{4}}{3\pi^{2}M^{4}}+\frac{\lambda_{4}\mu_{\Delta}^{2}}{6\pi^{2}M^{2}}+\frac{3\lambda\mu_{\Delta}^{2}}{8\pi^{2}M^{2}}$
(14) $\displaystyle\mathcal{C}_{H\square}$ $\displaystyle=$
$\displaystyle-\frac{g_{W}^{4}}{1920\pi^{2}}+\frac{2\mu_{\Delta}^{2}}{M^{2}}-\frac{\lambda_{1}^{2}}{16\pi^{2}}-\frac{\lambda_{1}\lambda_{4}}{16\pi^{2}}-\frac{\lambda_{4}^{2}}{192\pi^{2}}-\frac{g_{W}^{2}\mu_{\Delta}^{2}}{96\pi^{2}M^{2}}+\
$ (15) $\displaystyle+$
$\displaystyle\frac{11g_{Y}^{2}\mu_{\Delta}^{2}}{96\pi^{2}M^{2}}-\frac{49\mu_{\Delta}^{4}}{12\pi^{2}M^{4}}+\frac{\lambda_{1}\mu_{\Delta}^{2}}{8\pi^{2}M^{2}}+\frac{\lambda_{4}\mu_{\Delta}^{2}}{48\pi^{2}M^{2}}+\frac{3\lambda\mu_{\Delta}^{2}}{4\pi^{2}M^{2}}$
$\displaystyle\mathcal{C}_{HB}$ $\displaystyle=$
$\displaystyle\frac{g_{Y}^{2}\lambda_{1}}{32\pi^{2}}+\frac{g_{Y}^{2}\lambda_{4}}{64\pi^{2}}+\frac{11g_{Y}^{2}\mu_{\Delta}^{2}}{64\pi^{2}M^{2}}$
(16) $\displaystyle\mathcal{C}_{HW}$ $\displaystyle=$
$\displaystyle\frac{g_{W}^{2}\lambda_{1}}{48\pi^{2}}+\frac{g_{W}^{2}\lambda_{4}}{96\pi^{2}}+\frac{25g_{W}^{2}\mu_{\Delta}^{2}}{192\pi^{2}M^{2}}.$
(17)

Here, $M$ is the mass of the heavy triplet. For the theory to be valid, it is
sufficient to assume that $M$ is greater than $\Lambda$. We assume the order
of magnitude to be the same for $M$ and $\Lambda$ in our calculation as a
limiting scenario. For $M>>\Lambda$, the W.C.’s will obtain smaller values.

##  4 Result

We consider the one loop correction to the Higgs mass at first and fix two
benchmark scenarios at large scales, such as $100$ TeV, $10^{6}$ TeV. In Fig:
1 we show the model parameter space of $\lambda_{1}$ and $\lambda_{4}$, for
which quadratic divergence cancels out exactly, making $\delta m_{h}^{2}=0$.
The SM input parameters, such as ($g_{W}$, $y_{t}$, $g_{Y}$, $\lambda$) are
determined at the benchmark scales by solving two loop Renormalized Group
Equation (RGE)’s. $\lambda_{1}$ and $\lambda_{4}$ are varied in such a way
that the Wilson Coefficients obey the perturbative limit and the running of
the Wilson Coefficients from $\Lambda$ to the EW scale is not varied much.
Note that, the tree level couplings ($\lambda$ and $\mu_{H}$) also get shifted
due to the higher dimensional operators. The parameter $\lambda$ can not be
more than O(1) and this puts an upper limit on the quantity,
$\frac{\mu_{\Delta}^{2}}{2M^{2}}<O(1)$, where
$\mu_{\Delta}=\frac{\sqrt{2}v_{\Delta}M^{2}}{v_{H}^{2}}$, in the limit of
large masses of the triplet ($M$). Also, recent precission measurements of the
$\rho$ parameter gives $\rho=1.00038\pm 0.00020$, resulting in
$v_{\Delta}<2.56$ [42].

Figure 1: Variation of $\lambda_{1}$ and $\lambda_{4}$ with $\mu_{\Delta}$ at
two benchmark values of $\Lambda$.

From Fig: 1, we can see that the parameter space of ($\lambda_{1}$,
$\lambda_{4}$) is very much constrained from the V.C. Note that, both positive
and negative values of $\lambda_{1}$ and $\lambda_{4}$ are allowed. The green
line represents the highest possible value of $\mu_{\Delta}$, which comes from
$\frac{\mu_{\Delta}^{2}}{2M^{2}}\sim O(1)$. The V.C. only satisfies over the
thin lines for different values of $\mu_{\Delta}$. The nature of the plots is
highly dependent on the values of $\mu_{\Delta}$, because the Wilson
coefficients have $(\mu_{\Delta}/M)^{2}$ and $(\mu_{\Delta}/M)^{4}$ dependence
with additional suppression of $1/16\pi^{2}$. The freedom to choose the W.C’s
in terms of parameters $\lambda_{1}$, $\lambda_{4}$ and $\mu_{\Delta}$, allows
the exact cancellation even at a very large scale ( $\Lambda=10^{6}$ TeV). The
small change in parameter space is due to the running of the SM parameters.
Hence we found that the V.C is insensitive to the scale $\Lambda$, and the
fine tuning is appearing only due to the precision of the numbers that the
parameters take, which is negligible666Note that, for other models, where the
number of Wilson Coefficient is less than 4 can be generated, the exact
cancellation will be harder to obtain and the amount of fine tuning will also
be very high compared to this model..

Figure 2: Variation of the Wilson Coefficients with model parameter
$\lambda_{4}$

In Fig: 2, we show the variation of the Wilson Coefficients with the model
parameter $\lambda_{4}$ at 100 TeV. The corresponding value(s) of
$\lambda_{1}$ can be inferred form Fig: 1. The Wilson coefficients show
similar behaviour for the othe benchmark case. For $\mathcal{C}_{HD}$ and
$\mathcal{C}_{H\Box}$, negative values are more preferred, whereas, for
$\mathcal{C}_{HW}$ and $\mathcal{C}_{HB}$, both positive and negative values
are allowed. However, when $\lambda_{4}$ is negative, all coefficients are
negative mostly, except for some values of the $\mathcal{C}_{HD}$ and
$\mathcal{C}_{H\Box}$. Again, when $\lambda_{4}$ is positive,
$\mathcal{C}_{HW}$ and $\mathcal{C}_{HB}$ are always positive but
$\mathcal{C}_{HD}$ and $\mathcal{C}_{H\Box}$ are mostly negative except for
some values as shown in Fig: 2. Thus, it is clearly visible that the
cancellation among the Wilson coefficients are not ad-hoc in V.C., but are
controlled by the model parameters. We have also implemented the two loop
contribution to Higgs mass correction and in V.C. Due to the extra suppression
by ($1/16\pi^{2}$), the effect is not visible, hence we do not show that.

Figure 3: Running of the Wilson Coefficients from $\Lambda=10^{6}$ TeV to the
cut-off scale for one set of model parameters. We have kept the value of
$\mu_{\Delta}$ to be fixed at $1$ TeV.

We have also checked the running of the Wilson coefficients from the effective
scale $\Lambda$ to the electroweak scale. We show the running of the Wilson
coefficients in Fig: 3 for a particular choice of the model parameters,
$\lambda_{1}$ and $\lambda_{4}$. We choose $\lambda_{1}=4.0$ and
$\lambda_{4}=40$ as an input parameter. This particular choice of parameter
represents the maximum possible value of the model parameters as can be seen
in Fig: 2. We found that, the values of these W.C.’s do not change much and
also the sign does not change. The conclusion remains same for other allowed
values of $\lambda_{1}$ and $\lambda_{4}$. The values of W.C’s. ($C_{i}(1\rm
TeV)^{2}/\Lambda^{2}$) are highly constrained at the EW scale [43] from
various experiments. The values of Wilson coefficients (as in Fig: 2, Fig: 3),
for which the V.C. is satisfied, is well within the current experimental
limits.

##  5 Conclusion

The Veltman Condition can not be satisfied within the framework of the
Standard Model because of significant quadratic divergences to the Higgs self-
energy correction if the cutoff scale $\Lambda$ is $\sim 1$ TeV or higher.
However, in addition to the dimension four operators from the Standard Model,
we have also included dimension six operators whose contributions to the Higgs
mass correction result from integrating out the heavy triplet scalar with
hypercharge one in terms of the SMEFT operators. We show how the quadratic
divergence of the Higgs self-energy vanishes in this particular model due to
the cancellation among the SM parameters and the Wilson Coefficients.

We have shown the relevant SMEFT operators which contributes in the V.C., and
expressed them in terms of the model parameters. Hence, the sign of the Wilson
Coefficients are not ad-hoc, it is driven by the larger theory, which is a
heavy triplet scalar in our case. We found that, in other models the
cancellation is harder to achieve because some of the operators are absent. In
other words, one has to allow for a minimum fine tuning in order to generate
the model parameter space which is allowed by the V.C. However, the values of
the Wilson Coefficients will be different in every model, as it is controlled
by the specific model parameters.

In order to achieve the Veltman Condition, it should be noted that the
contributions from two particular dimension six operators $\mathcal{Q}_{HD}$
and $\mathcal{Q}_{H\square}$ play a dominating role in canceling out the
quadratic divergences. However, this may or may not be the case in other
models. We have observed that for energy scales $\Lambda$ = 100 TeV and
$10^{6}$ TeV, the cancellation is almost similar, when the W.C’s are expressed
in terms of $\lambda_{1}$ and $\lambda_{4}$ for a given $\mu_{\Delta}$. The
minimal change in the parameter space is mainly due to the running of the SM
parameters. If we introduce some relaxation in the V.C., by allowing some
amount of fine tuning, the model parameter space will surely enlarge, but it
will get narrower with the increasing values of $\Lambda$. Thus, the Veltman
Condition can be easily satisfied in the framework of effective field theory,
when a scalar triplet exists at a very large scale. The study of this model as
an Effective Field Theory can also be useful to revisit the Type II
leptogenesis scenario, where it will be possible to generate specific
dimension six terms which are allowed by the symmetries of the model.

Acknowledgements: JD acknowledges the Council of Scientific and Industrial
Research (CSIR), Government of India, for the SRF fellowship grant with File
No. 09/045(1511)/ 2017-EMR-I. JD also would like to acknowledge Research Grant
No. SERB/CRG/004889/SGBKC/2022/04 of the SERB, India, for partial financial
support. The work of NK is supported by Department of Science and Technology,
Government of India under the SRG grant, Grant Agreement Number
SRG/2022/000363. We also thank Prof. Anirban Kundu and Dr. Supratim Das Bakshi
for useful discussion.

##  6 Appendix

### Model Description

In the type-II seesaw model, the scalar sector is extended by a complex scalar
triplet($\Delta$) with hypercharge $1$, in addition to the Higgs doublet
($H$). Explicitly,

$H\left(1,2,+1/2\right)=\begin{pmatrix}\phi^{+}\\\ \phi^{0}\end{pmatrix},\,\,\
\Delta\left(1,3,+1\right)=\begin{pmatrix}\Delta^{+}/\sqrt{2}&\Delta^{++}\\\
\Delta^{0}&-\Delta^{+}/\sqrt{2}\end{pmatrix}$ (18)

with the neutral components:

$\phi^{0}=\frac{v_{H}+h+i\phi_{3}}{\sqrt{2}},\,\,\
\Delta^{0}=\frac{v_{\Delta}+\delta+i\xi}{\sqrt{2}}$ (19)

The numbers in the parentheses represent the charges of $SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}$ gauge group of the SM. The kinetic terms
corresponding to the scalar fields are given as

$\mathcal{L}_{\rm
kin}\supset(D^{\mu}H)^{\dagger}D_{\mu}H+\mathrm{Tr}\left[(D^{\mu}\Delta)^{\dagger}(D_{\mu}\Delta)\right],$
(20)

with the covariant derivatives

$\displaystyle D_{\mu}H$ $\displaystyle=$
$\displaystyle\partial_{\mu}H-i\frac{g_{Y}}{2}W_{\mu}^{a}\sigma^{a}H-i\frac{g_{W}}{2}B_{\mu}H,\
$ $\displaystyle D_{\mu}\Delta$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Delta-i\frac{g_{Y}}{2}\mathrm{Tr}\left[W_{\mu}^{a}\sigma^{a},\Delta\right]-i\frac{g_{W}}{2}B_{\mu}\Delta.$
(21)

Here $\sigma^{a}$ ($a$ = 1, 2, 3) are the Pauli spin matrices and $g_{W}$ and
$g_{Y}$ are the gauge couplings associated with $SU(2)_{L}$ and $U(1)_{Y}$
gauge group respectively.

### Calculation

The dimension six SMEFT operators which contribute Higgs mass correction
either at one-loop or two-loop level in this model can be written upto a total
derivative as,

$\displaystyle\mathcal{Q}_{HD}$ $\displaystyle=$
$\displaystyle(H^{+}\mathcal{D}_{\mu}H)^{\ast}(H^{+}\mathcal{D}^{\mu}H)\supset\left(\partial_{\mu}H^{\dagger}\right)HH^{\dagger}\left(\partial^{\mu}H^{\dagger}\right)+\Big{[}\frac{g_{W}^{2}}{4}\sigma^{a}\sigma^{b}H^{\dagger}W_{\mu}^{a}HH^{\dagger}W^{\mu
b}H\ $ $\displaystyle+$
$\displaystyle\frac{g_{Y}^{2}}{4}H^{\dagger}B_{\mu}HH^{\dagger}B^{\mu}H\Big{]}\
$ $\displaystyle\mathcal{Q}_{H\square}$ $\displaystyle=$
$\displaystyle(H^{+}H)\square(H^{+}H)=-\partial_{\mu}\left(H^{\dagger}H\right)\partial^{\mu}\left(H^{\dagger}H\right)\
$ $\displaystyle\mathcal{Q}_{HW}$ $\displaystyle=$
$\displaystyle(H^{+}H)W_{\mu\nu}^{a}W^{a,\mu\nu}\supset
2H^{\dagger}\Big{[}\sigma^{a}\left(\partial_{\mu}W_{\nu}^{a}\right)\sigma^{b}\left(\partial^{\mu}W^{\nu
b}\right)-\sigma^{a}\left(\partial_{\mu}W_{\nu}^{a}\right)\sigma^{b}\left(\partial^{\nu}W^{\mu
b}\right)\Big{]}H\ \ $ $\displaystyle+$ $\displaystyle
g_{W}^{2}\sigma^{a}f_{abc}\sigma^{p}f_{pqr}H^{\dagger}W_{\mu}^{b}W_{\nu}^{c}W^{\mu
q}W^{\nu r}H\ $ $\displaystyle\mathcal{Q}_{HB}$ $\displaystyle=$
$\displaystyle(H^{+}H)B_{\mu\nu}B^{\mu\nu}\supset
2H^{\dagger}\Big{[}\partial_{\mu}B_{\nu}\partial^{\mu}B^{\nu}-\partial_{\mu}B_{\nu}\partial^{\nu}B^{\mu}\Big{]}H\
$ $\displaystyle\mathcal{Q}_{H}$ $\displaystyle=$
$\displaystyle(H^{\dagger}H)^{3}.$ (22)

Note that only momentum dependent vertices can generate quartic divergence at
one-loop level. Possible Feynman diagrams originating from these terms are
similar to Ref.[19].

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