# Semantic Diffusion Network for Semantic Segmentation

$\textbf{Haoru Tan}^{\ast}$
Sitong Wu

Baidu Research
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Contribute equallyCorresponding author Jimin Pi

###### Abstract

Precise and accurate predictions over boundary areas are essential for
semantic segmentation. However, the commonly-used convolutional operators tend
to smooth and blur local detail cues, making it difficult for deep models to
generate accurate boundary predictions. In this paper, we introduce an
operator-level approach to enhance semantic boundary awareness, so as to
improve the prediction of the deep semantic segmentation model. Specifically,
we first formulate the boundary feature enhancement as an anisotropic
diffusion process. We then propose a novel learnable approach called semantic
diffusion network (SDN) to approximate the diffusion process, which contains a
parameterized semantic difference convolution operator followed by a feature
fusion module. Our SDN aims to construct a differentiable mapping from the
original feature to the inter-class boundary-enhanced feature. The proposed
SDN is an efficient and flexible module that can be easily plugged into
existing encoder-decoder segmentation models. Extensive experiments show that
our approach can achieve consistent improvements over several typical and
state-of-the-art segmentation baseline models on challenging public
benchmarks. The code will be released soon.

##  1 Introduction

Semantic segmentation aims at assigning pixel-wise semantic labels to an
image. It plays an important role in various applications such as autonomous
driving [9, 13], medical diagnostics [31] and virtual reality [4]. A key
challenge in semantic segmentation is how to improve the boundary quality of
segmentation results (Figure 1). In general, people have been working on this
problem in three different ways: (i) adopting a post-processing module to
refine the boundary [26, 23, 43, 39, 58], (ii) jointly learned the edge
detection and semantic segmentation task for boundary-awareness [60, 7, 30],
and (iii) making training more focused on the boundary by designing a
boundary-aware loss with higher sensitivity to the boundary changes [3, 20,
45].

Nevertheless, all these methods explored the boundary information via either
post-processing refinement or additional supervision, rather than directly
address the intrinsic reason for fuzzy boundary and detail degradation, that
is, vanilla convolution [24] naturally tends to smooth the feature [41, 53,
54, 55, 56], making it difficult to capture the boundary cues (see Figure 2b).
Although traditional gradient operators [19, 22] and difference convolutions
[53, 54, 55, 56, 41] have the ability to perceive boundary, they are sensitive
to all the boundaries including intra-class texture edges, which makes it not
suitable for semantic segmentation. This motivates us to explore a
segmentation-friendly operator with only inter-class boundary awareness to
improve the boundary ambiguity in semantic segmentation.

Our method. In this paper, we are committed to improving the boundary quality
of semantic segmentation from the operator level. Inspired by the anisotropic
diffusion [33] in the signal processing field, we formulate the inter-class
boundary enhancement as a modified anisotropic diffusion process guided by
semantics, which can be solved by the finite difference method. While such a
traditional solver can hardly be integrated into deep segmentation networks
due to the hyper-parameter sensitivity, numerical instability, and high
computational complexity caused by its iterative formulation. To solve this
problem, we propose a learnable semantic diffusion network (SDN) to
approximate the diffusion process, which parameterizes the traditional solver
and only requires only one forward instead of multiple iterations. The SDN
aims to construct a differentiable mapping from the original feature to the
inter-class boundary-enhanced feature, which consists of a semantic difference
convolution and a feature fusion operation. Our proposed SDN is an efficient
and flexible module that can be easily plugged into the existing encoder-
decoder segmentation models. To evaluate its effectiveness, we integrate our
SDN into multiple baseline segmentation models and conduct experiments on two
challenging benchmarks (i.e., ADE20K [62] and Cityscapes [9]). Extensive
experiments show that our approach can bring consistent improvements for
existing segmentation methods with few additional computational costs. In
particular, our SDN can bring +2.57% and +1.43% mIoU improvements for
Segmenter [40] with ViT-B [11] backbone on ADE20K [62] and Cityscapes [9],
respectively. Qualitative visualization verifies the obvious boundary quality
improvement of our approach.

Figure 1:  Qualitative analysis of the segmentation error maps. The baseline
denotes OCRNet [57], and the non-black pixels indicate the wrong predictions.
It can be seen that segmentation error is particularly obvious in the boundary
region. Our method achieves significant boundary improvements over the
baseline model. The examples are selected and cropped from Cityscapes-val [9].

##  2 Related Work

Semantic Segmentation. In 2015, FCN [18] first proposed to use a fully
convolutional network to generate dense predictions, which is regarded as a
milestone for semantic segmentation. Since then, lots of works have been
exploring larger receptive fields for better scene understanding based on CNN
architecture. For example, PSPNet [15] proposed a general pyramid pooling
module to fuse features at different scales. DeepLab family [26, 27, 25, 28]
used atrous convolution to enlarge the receptive fields while keeping the
resolution. Recent works focused on how to capture more precise context [57,
17, 59, 29, 52] and better boundary [58, 3]. Considering that the attention
operation [46] has a large receptive field and can adaptively aggregate
features, [12, 16, 63, 2] employed the attention-based modules on the top of
CNN segmentation networks to adaptively combine local features with their
global dependencies. Inspired by the success of Transformer [44] in NLP tasks,
SETR [61] and DPT [35] attempted to use Vision Transformer [11] as the
backbone. They demonstrated the effectiveness of global receptive fields for
semantic segmentation tasks. Inspired by [5], Tran2Seg [51] explored a hybrid
encoder and prototype-based transformer decoder. FTN [49] and Segmenter [40]
further designed fully transformer networks. SegFormer [50] proved that a
lightweight MLP decoder can also achieve satisfactory results under a strong
transformer encoder. Recently, MaskFormer [8] proposed a simple but effective
mask classification scheme to predict a set of binary masks, which unified the
semantic, instance, and panoptic segmentation.

Boundary-aware Segmentation. Methods for promoting boundary quality in
semantic segmentation can be divided into three categories: (1) Adopting the
post-processing module. Most methods applied a post-processing module on the
top of coarse segmentation results to refine the boundary and recover the
details [26, 23, 43, 39, 58]. DenseCRF model [23] is the most well-known post-
processing method, which achieves fine boundary results by encouraging
assigning neighboring pixels with similar colors to the same semantic label.
SegFix [58] proposed a model-agnostic boundary refinement model, which aims to
replace the ambiguous predictions at boundaries with predictions at the intra-
class areas. (2) Auxiliary branch for boundary-awareness. [60, 7, 30] utilized
the multi-task learning strategy by jointly learning boundary detection and
semantic segmentation together. For example, RPCNet [60] introduced the
auxiliary branch for boundary enhancement to the semantic segmentation model.
(3) Boundary-aware loss. These works guide the model to focus more on boundary
accuracy by designing the boundary-aware loss function with high sensitivity
to the boundary changes [3, 20, 45]. Inverseform [3] introduced a boundary
distance-based measure into the popular segmentation loss functions. Kervadec
et.al.[20] proposed a boundary loss taking the form of a distance metric on
the space of contours for highly unbalanced situations. ABL [45] designed an
active boundary loss function for progressively encouraging the predicted
boundaries matching the ground-truth boundaries. Instead of imposing various
boundary-related constraints, we consider the intrinsic reason for fuzzy
boundary from the operator level, and design a boundary-sensitive operator to
make the network naturally capable of modeling and perceiving boundaries.

Related Operators. The convolution operator, which calculates the kernel’s
response to each local patch of the input feature maps, is not good at
modeling boundary information. To solve this, people try to combine the
traditional image gradient operator with convolution operators. Local binary
convolution (LBC) [19] used a series of pre-defined binary filters to replace
learnable kernels in convolution. Subsequently, a series of difference
convolution operators represented by central difference convolution (CDC) [53,
54, 55, 56] used learnable kernels to capture useful boundary information on
the central difference map. PDC [41] further extended the flexibility of CDC
and achieved excellent results in boundary detection tasks. In addition, the
local self-attention (LSA) [34] operator replaces the fixed kernel with the
content-adaptive attentive weights to achieve local feature fusion, making it
good at capturing contextual dependence. But it is still not good at
perceiving the boundary. Different from existing methods, we are committed to
designing a novel and efficient operator-level solution with a high
sensitivity to the inter-class boundaries rather than all the edges.

##  3 Preliminaries

###  3.1 Diffusion Process

Diffusion is a physical model aimed at minimizing the spatial concentration
difference [37], which is widely used in image processing [33, 6, 48, 47].
Given an image tensor $U$ to be smoothed, this process needs to solve the
well-known second-order partial differential equation:

$\frac{\partial U}{\partial t}=\textit{Div}(D\cdot\nabla U),$ (1)

where $\frac{\partial U}{\partial t}$ is the time derivative of the solution
tensor, Div is the divergence operator, $D$ is the diffusivity tensor, which
describes the speed of the diffusion process, $\nabla$ is the gradient
operator. With increasing time $t$, the solution $U_{t}$ of this process will
correspond to increasingly smoothed versions of the original tensor.

The concrete form of $D$ determines the properties of the PDE. For $D=1$, it
is called homogeneous linear diffusion, where the diffusion velocity of
spatial concentration is exactly the same in all directions. For a spatial-
dependent $D=D(x)$ (e.g. learnable kernel), the process is linear and
inhomogeneous. However, the two kinds of simple linear diffusion processes not
only smooth the noise but also blur the edges. An ideal diffusivity function
allows more smoothing parallel to image edges and less smoothing perpendicular
to these edges, which is actually anisotropic and nonlinear. For this reason,
some classic milestones [33, 6, 48, 47] increase the nonlinear modeling
ability by designing complex diffusivity functions $D=D(U)$ depending on the
input $U$.

###  3.2 Rethinking Operator-level Boundary Awareness

Figure 2:  Visualization of the feature maps from (b) vanilla convolution
[24], (c) central difference convolution (CDC) [55]. We choose three points
$A_{1},A_{2},B$ on the image, corresponding to car-body, car-glass and plant,
respectively. In (b), due to the local smoothness of the vanilla-conv
operator, we can see that there is no significant difference in the feature
responses at the three coordinates in the feature map. According to (c), the
CDC operator can hardly distinguish the true semantic boundary and the pseudo
boundary caused by textures or noise, for example, we can see that the
features captured by CDC have cluttered false boundary responses between these
two coordinates $A_{1}$ (car-body) and $A_{2}$ (car-glass).

Here we discuss the operator-level causes of potential CNN failures in
boundary predictions. Given the input feature map $\bm{x}$, the vanilla
convolution calculates the output tensor $\bm{y}$ through the inner product
between the kernel $\bm{w}$ and each local feature patch. Without loss of
generality, we assume that the channel number of both input and output is $1$.
The vanilla convolution with kernel $\bm{w}\in\mathcal{R}^{h\times w}$ over a
input feature $\bm{x}\in\mathcal{R}^{H\times W}$ could be written as:
$\bm{y}_{p}=\langle\bm{w},\bm{x}_{p}\rangle$, where $\bm{x}_{p}$ is the local
patch centering at the position $p$. Intuitively, the vanilla convolution is
just a local weighted average operation, which tends to smooth the local
detailed information [55]. Especially when entries within the kernel are non-
negative, the smoothness will be particularly significant as it forms a low-
pass filter. This results in a poor capacity for modeling the semantic
boundary information, see Figure 2b.

Recently, people try to combine the traditional gradient operator (edge-
detector) with convolution operators to enhance its edge awareness [19, 53,
54, 55, 56, 41]. Local binary convolution (LBC) [19] used a series of pre-
defined binary filters to replace learnable kernels in convolution.
Subsequently, a series of difference convolution operators represented by
central difference convolution (CDC) [53, 54, 55, 56] used learnable kernels
to capture useful boundary information on the central difference map, that is,
$\bm{y}_{p}=\langle\bm{w},(\bm{x}_{p}-\bm{x}_{\text{center}})\rangle$, where
$\bm{x}_{\text{center}}$ indicates the center entry of the current patch, and
the rest symbols are defined in the same way as that of the vanilla
convolution. It aggregates the center-oriented gradient within the local
region instead of the feature values. Even though this kind of operator has
better edge awareness, however, because the gradient operator cannot
distinguish between real semantic boundary and pseudo boundary caused by
textures or noise (see Figure 2c), it is difficult for us to directly apply
them to tasks such as semantic segmentation.

##  4 Methodology

This section introduces the semantic diffusion network (SDN) for semantic
segmentation tasks to compensate for the detail blur caused by the vanilla
convolution operator. It is combined with CNN backbone when used, so as to
strengthen semantic boundary information and filter out pseudo boundary
information caused by texture or noise.

###  4.1 Formulation

Considering a CNN backbone model with $L$ stacked neural stages mapping an
image $I$ to pixel-wise feature maps $\bm{X}$. Let $\bm{X}^{\ell}$ denote the
feature map from the $\ell$-th stage. We define the semantic diffusion process
over the feature map $\bm{X}^{\ell}$ as:

$\frac{\partial U_{t}}{\partial t}=\textit{Div}\Big{(}g(|\nabla V|^{2})\nabla
U_{t}\Big{)},$ (2)

where $\frac{\partial U}{\partial t}$ is the time derivative of the solution
tensor, Div is the divergence operator, $\nabla$ is the gradient operator. $V$
is the guidance feature map with the same resolution of $U$. The diffusivity
term is a monotonically decreasing function of the square of the gradient
$|\nabla V|^{2}$, for example, $g(|\nabla V|^{2})=1/{\sqrt{1+|\nabla
V|^{2}/\lambda^{2}}}$. Ideally, the diffusion close to the semantic boundary
areas will be suppressed, while the diffusion far away from the semantic
boundary will be accelerated. Therefore, $V$ should contain sufficient
semantic information, so the gradient norm $|\nabla V|^{2}$ can be used as a
reliable semantic boundary indicator to construct the diffusivity. The PDE of
the non-linear diffusion process defined by Eq.(2) could be approximately
solved by performing the finite difference update rule for $T$ steps:

$\displaystyle\widetilde{U}^{t+1}_{(m,n)}=\sum_{(i,j)\in\mathcal{N}_{(m,n)}}g(|V_{(i,j)}-V_{(m,n)}|^{2})\cdot(U^{t}_{(i,j)}-U^{t}_{(m,n)}),$
(3) $\displaystyle U^{t+1}_{(m,n)}=\alpha^{t}\cdot
U^{t}_{(m,n)}+\beta^{t}\cdot\widetilde{U}^{t+1}_{(m,n)},$ (4)

where the step-index $t\in\\{0,...,T-1\\}$, the tuple $(i,j)$ enumerates the
index of every entry within the local neighbor region $\mathcal{N}_{(m,n)}$ of
the position $(m,n)$, and the initial state is $U^{0}=\bm{X}^{\ell}$. In fact,
it is obvious that the diffusivity $g(|V_{(i,j)}-V_{(h,w)}|^{2})$ calculates
the similarity between semantic guidance features at $(i,j)$ and $(m,n)$. The
solver builds a differentiable map from the input feature map $\bm{X}^{\ell}$
to the boundary-enhanced feature.

##### Remark.

The finite difference based solver can hardly be directly integrated with deep
segmentation models for several reasons: (1). the stability of the finite
difference method depends on the setting of boundary conditions and the
selection of parameters $(\alpha,\beta)$; (2). the solver requires multiple
iterations, which will consume high computational complexity. (3). integrating
the iterative solver with the deep networks would result in numerical
instability to the training process, such as gradient explosion [39].

###  4.2 Semantic Diffusion Network

This subsection describes a novel learnable approach called semantic diffusion
network (SDN) for approximating the diffusion process, which contains a
parameterized semantic difference convolution operator followed by a feature
fusion module and constructs a differentiable mapping from original backbone
features to advanced boundary-aware features. As a learnable PDE approximate
solver, the well-trained SDN could approximate the diffusion process in only
one forward computation, reducing the computational complexity and improving
the stability. Overall, we formulate the SDN as
$\bm{F}^{\text{sdn}}=\textbf{SDN}\big{(}\bm{U},\bm{V}\big{)}$, where $\bm{U}$
is the input feature map to be processed and $\bm{V}$ is the semantic guidance
map.

####  4.2.1 Semantic Difference Convolution

Inspired by Eq.(3), we develop the semantic difference convolution (SDC)
operator. We set the $\mathcal{N}_{m,n}$ as a $h\times w$ local region
centered at position $(m,n)$. SDC takes the feature map $\bm{U}$ and the
semantic guidance $\bm{V}$ as inputs and calculates the output feature map
$\bm{Y}$. Specifically, each component of $\bm{y}$ indexed by $(m,n)$ is
calculated via

$\displaystyle\bm{Y}_{(m,n)}=\sum_{(i,j)\in\mathcal{N}_{(m,n)}}{\bm{W}_{(i,j)}}\cdot\bm{\mathcal{S}}\big{(}\bm{V}_{(i,j)},\bm{V}_{(m,n)}\big{)}\cdot\big{(}\bm{U}_{(i,j)}-\bm{U}_{(m,n)}\big{)},$
(5)

where the tuple $(i,j)$ enumerates the index of every entry within the local
neighbor region $\mathcal{N}_{m,n}$. In the right hand of Eq.(5), the first
term is the learnable convolutional kernel $\bm{W}$, of which $\text{(height,
width)}=(h,w)$. The kernel introduces a more powerful modeling capability than
the naive non-linear diffusion process in Eq.(2). The second term
$\bm{\mathcal{S}}\big{(}\bm{V}_{(i,j)},\bm{V}_{(m,n)}\big{)}$, called the
semantic similarity term, is a scalar measuring the similarity between
semantic guidance features at $(i,j)$ and $(m,n)$. This term corresponds to
the diffusivity $g(|V_{(i,j)}-V_{(h,w)}|^{2})$ in Eq.(3). The third term
$(\bm{U}_{(i,j)}-\bm{U}_{(m,n)})$, called the pixel difference term,
calculates the feature difference between features at the position $(i,j)$ and
$(m,n)$. Note that Eq.(5) can be implemented efficiently using CUDA. By
default, the kernel size of $\bm{W}$ is set to $3\times 3$.

Figure 3:  Comparisons between the vanilla convolution, the central difference
convolution, and our semantic difference convolution.

Figure 3 provides the comparison between SDC with other well-known convolution
operators. As defined in Eq.(5), the proposed SDC operator contains two
sequential steps: (i) calculating the semantic similarity, and (ii)
calculating the output feature. The first step consumes a complexity of
$\mathcal{O}(HWhwC_{f})$, and the second step consumes a complexity of
$\mathcal{O}(HWhwC_{i}C_{o})$. Thus, the overall complexity is
$\text{Complexity}=\mathcal{O}(HWhw(C_{i}C_{o}+C_{f}))$, where $H,W$ are the
height and width of the input feature, and $h,w$ are the height and width of
the kernel tensor. $C_{i},C_{o}$, and $C_{f}$ are the channel number of input,
output, and semantic features, respectively.

####  4.2.2 Feature Fusion

This step aims to fuse the semantic difference convolution output $\bm{Y}$ and
the module’s input feature map $\bm{U}$ into the final boundary-enhanced
feature map $\bm{F}^{\text{sdn}}$. This module is actually corresponding to
Eq.(4) in the classic finite differential solution. Here, we extend the simple
weighted addition operation in Eq.(4) to a learnable and more flexible way,
that is,
$\bm{F}^{\text{sdn}}=\textbf{Conv}\Big{(}\big{[}\bm{U},\bm{{Y}}\big{]}\Big{)}$.
It first concatenates the original input feature $\bm{U}$ with the semantic
difference convolution output $\bm{Y}$ and then uses a simple but effective
$1\times 1$ convolution to achieve channel-wise feature fusion, resulting in
the boundary-enhanced feature $\bm{F}^{\text{sdn}}$. Note that both $\bm{U}$
and $\bm{Y}$ should have the same shape. Otherwise, we need to perform the
bilinear interpolation upsampling operation on the smaller one to align the
shapes.

###  4.3 Segmentation Model with SDN

Here, we introduce how to integrate our SDN into the existing encoder-decoder-
based segmentation models. For better compatibility, we treat SDN as a neck
part between the encoder and decoder, without breaking the original design of
the baseline network. According to the feature scales required by the decoder,
existing segmentation models can be divided into two categories: (i) encoder
with single-scale decoder (i.e., ViT+Segmenter and ResNet+FCN); (ii) encoder
with single-scale decoder, such as ResNet+SemanticFPN. Next, we discussed the
two cases separately.

Figure 4:  Illustration of how to combine the proposed semantic diffusion
network (SDN) with the baseline segmentation model with single-scale decoder
and multi-scale decoder in (a) and (b), respectively. The SDN is regarded as a
neck part in order not to break the original encoder and decoder design of the
baseline model.

SDN with single-scale decoder. As shown in Figure 4(a), given an input image
$\bm{I}\in\mathbb{R}^{3\times H\times W}$, the encoder is first applied to it
to extract the feature, resulting in $\bm{F}\in\mathbb{R}^{C^{\prime}\times
H^{\prime}\times W^{\prime}}$, where $H^{\prime}$, $W^{\prime}$ and
$C^{\prime}$ denote the height, width and channel numbers, respectively. We
then incorporate our SDN on the encoder output feature $\bm{F}$ to enhance the
boundary details, and the output of SDN $\bm{F}^{\text{sdn}}$ is sent to the
decoder. Note that $\bm{F}^{\text{sdn}}$ has the same size of $\bm{F}$. As
mentioned in Sec. 4.2.1, SDN requires an additional feature with rich
semantics as guidance. To construct such a semantic feature $\bm{F}^{s}$ for
SDN, we stack a lightweight layer $\Phi$ on the top of the encoder, where
$\Phi$ is instantiated as a $3\times 3$ convolution with stride 2. Thus, we
have $\bm{F}^{\text{sdn}}=\textbf{SDN}\big{(}\bm{F},\bm{F}^{s}\big{)}$.

SDN with multi-scale decoder. As Figure 4(b), the input image
$\bm{I}\in\mathbb{R}^{3\times H\times W}$ is first passed through a backbone
with totally $L$ scales (generally $L=4$), resulting in multi-scale features
$\\{\bm{F}_{i}\in\mathbb{R}^{C_{i}\times H_{i}\times W_{i}}\\}_{i=1}^{L}$.
$H_{i}$, $W_{i}$ and $C_{i}$ denote the height, width, and channel numbers,
respectively. We then incorporate the SDN after all the backbone stages to
strengthen the boundaries comprehensively. In particular, for the $i$-th SDN,
it takes $\bm{F}_{i}$ and $\bm{F}_{i}^{s}$ as the inputs, and generates the
feature:
$\bm{F}_{i}^{\text{sdn}}=\textbf{SDN}\big{(}\bm{F}_{i},\bm{F}_{i}^{s}\big{)},1\leq
i\leq L$, where $\bm{F}_{i}^{s}$ denotes the semantic feature at scale $i$,
which is defined as follows:

$\displaystyle\bm{F}_{i}^{s}=\begin{cases}\bm{F}_{i+1},&1\leq
i<L{}{}{}{}{}{}{}{}{}{}{}\\\ \Phi\big{(}\bm{F}_{L}\big{)},&i=L\end{cases}$ (6)

Similarly, $\Phi$ is also implemented by a $3\times 3$ convolution with stride
2. Note that the SDNs applied on different stages have the same architecture
but do not share the same parameters. Finally, the boundary-enhanced features
are fed into the decoder to generate the pixel-wise probability map.

##  5 Experiments

In this section, we first introduce the datasets and implementation details in
Sec. 5.1. Then, we evaluate our approach over several baseline segmentation
models on these public benchmarks in Sec. 5.2. Finally, in Sec. 5.3, we
conduct ablation studies to analyze the effect of key designs and the
compatibility of our approach.

###  5.1 Experimental Setup

Datasets. Experiments are conducted on two widely-used public benchmarks:
ADE20K [62] is a very challenging benchmark including 150 categories, which is
split into 20000 and 2000 images for training and validation. Cityscapes [9]
is a real-world scene parsing benchmark, which contains over 5000 urban scene
images with 19 classes. The number of images for training, validation, and
testing are 2975, 500, and 1525, respectively.

Baseline models. To evaluate the effectiveness of our SDN, we integrate it
into two typical CNN-based baseline methods (FCN [18] and SemanticFPN [21])
and a recent Transformer-based segmentation model (Segmenter [40]).

Training. All the experiments are implemented with PyTorch [1] and conducted
on 4 NVIDIA V100 GPUs. For fair comparisons, we apply the same settings as the
baseline methods. Specifically, for CNN baselines (i.e., FCN and SemanticFPN),
we use the SGD optimizer with momentum 0.9 and weight decay 0.0005. The
learning rate is initialized at 0.01 and decays until 1e-4 by a polynomial
strategy with power 0.9. For Transformer baselines (i.e., Segmenter), AdamW is
used as the optimizer without weight decay. The learning rate is initialized
as 6e-5 and decays until 0 via a polynomial strategy with power 1.0. We train
the models with 160k and 80k iterations for ADE20K and Cityscapes,
respectively. For ADE20K, we train the models with 160k iterations, each of
which involves 16 images. For Cityscapes, the training contains 80k iterations
with a batch size of 8. Synchronized BN [32] is used to synchronize the mean
and standard deviation of BN [38] across multiple GPUs. The backbone is
initialized by the ImageNet-1K [36] and ImageNet-22K [10] pre-trained weights
for ResNet-50 [14] and ViT-B [11], respectively.

Evaluation. We adopt the widely-used mean intersection of union (mIoU) to
measure the overall segmentation performance. In order to measure the boundary
quality of the prediction, we compute F-score only on the 1px and 3px boundary
region, which denotes the 1 pixel and 3 pixels width area along the ground-
truth boundary. For the multi-scale inference, we follow the previous works
[57, 8] to perform horizontal flip and average the predictions at multiple
scales [0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0].

Table 1:  Experimental results on ADE20K-val and Cityscapes-val. “s.s.” and “m.s.” denote the single-scale and multi-scale mIoU, respectively. Method | Encoder | ADE20K | Cityscapes
---|---|---|---
mIoU (s.s.) | mIoU (m.s.) | mIoU (s.s.) | mIoU (m.s.)
FCN [18] | ResNet-50 | 36.10 | 38.08 | 72.64 | 73.32
FCN+Ours | ResNet-50 | 38.12 (+2.02) | 39.36 (+1.28) | 74.75 (+2.11) | 75.79 (+2.47)
SemanticFPN [21] | ResNet-50 | 37.49 | 39.09 | 74.10 | 75.98
SemanticFPN+Ours | ResNet-50 | 38.79 (+1.30) | 40.27 (+1.18) | 75.97 (+1.87) | 77.31 (+1.33)
Segmenter [40] | ViT-B | 48.48 | 50.00 | 77.97 | 80.07
Segmenter+Ours | ViT-B | 51.05 (+2.57) | 52.18 (+2.18) | 79.42 (+1.45) | 81.38 (+1.31)
Table 2:  Boundary F-score comparisons on Cityscapes-val.  Method | Encoder | with SegFix [58] | with SDN (ours) | F-score
---|---|---|---|---
1px | 3px
OCRNet[57] | HRNet-48[42] | ✗ | ✗ | 65.2 | 76.7
✓ | ✗ | 66.7 (+1.5) | 77.1 (+0.4)
✗ | ✓ | 69.5 (+4.3) | 78.2 (+1.5)
Figure 5:  Visualization comparisons of the segmentation results on
Cityscapes-val, where the baseline is OCRNet.

###  5.2 Main Results

To verify the effectiveness and generalization of our approach, we apply the
proposed SDN to several baseline segmentation models on various datasets.
Table 1 reports the quantitative comparisons.

ADE20K. Table 1 shows that our approach brings significant mIoU improvements
for all the baselines on the challenging ADE20K dataset, i.e., improves FCN by
+2.02% mIoU (s.s.), improves SemanticFPN by +1.30% mIoU (s.s.), and improves
Segmenter by +2.57% mIoU (s.s.). The improvement could be attributed to the
fact that explicit boundary information is particularly useful for ADE20K
images with complex scenarios. With the HRNet-48, ResNet-101, and Swin-B
backbone, we contribute +2.6%, +1.7%, and +1.1% mIoU gain over OCRNet and
MaskFormers, respectively.

Cityscapes. Table 1 shows that our approach brings nontrivial mIoU
improvements for all the baselines, i.e., improves FCN by +2.47% mIoU (m.s.),
improves SemanticFPN by +1.87% mIoU (s.s.), and improves Segmenter by +1.45%
mIoU (s.s.). Additionally, with the OCRNet(HRNet-w48) [57] baseline, SDN
brings +0.9% mIoU improvement to the baseline OCRNet (81.9% to 82.8% mIoU) and
outperforms the SegFix model by +0.3% mIoU (the OCRNet’s results are not
listed in Table 1). Moreover, our approach (84.7% mIoU) outperforms the
OCRNet+SegFix (84.5% mIoU) by +0.2% on Cityscapes-test, where the detailed
results can be found in the web-link. We also compare the boundary-promoting
ability of SDN and the classic SegFix [58] model in Table 2, where our method
brings significant improvement, i.e., +4.3 and +1.5 for OCRNet in terms of the
1px and 3px F-score. Figure 5 provides some typical visualizations to show the
boundary improvement effect of our method.

###  5.3 Further Study

Compare SDC with other operators. For direct comparisons with other related
operators, we replace the semantic difference convolution (SDC) in SDN with
other two related operators, vanilla convolution [24] and CDC [55]. Table 3(a)
compares their boundary performance on Cityscapes with OCRNet [57] as the
baseline model. It can be found that SDC outperforms its vanilla convolution
counterpart by +4.3 F-score, and surpasses CDC by a large margin, which
demonstrates the irreplaceability of SDC.

Effect of kernel settings of SDC. We conduct experiments under different
choices of the kernel size and dilation rate of SDC in Table 3(b). It can be
found that: (1). Increasing the size of the convolution kernel in SDC does not
bring significant changes. (2). Excessive dilation rate leads to negative
effects. In fact, large kernels and dilated convolution are mainly used for
enlarging the receptive fields. While, the goal of our SDC is to infer the
inter-class boundary cue in a local region, which may not require more
contexts brought by large kernels. Similarly, a large dilation rate may lead
to the lack of local information, where, however, the local information is
critical for local boundary exploration.

Table 3:  Further studies on (a) comparison with other operators, where V-Conv
denotes the vanilla convolution, (b) the effect of the kernel size and
dilation rate settings of the SDC operator in the SDN module, and (c)
compatibility with other boundary-promoting methods in semantic segmentation.

(a) Operator | F-score
---|---
Baseline | 65.2
V-Conv [24] | 65.2
CDC [53] | 60.1
SDC (ours) | 69.5

(b) Kernel-size | Dilation-rate | F-score
---|---|---
$3\times 3$ | 1 | 69.5
$5\times 5$ | 1 | 69.6
$3\times 3$ | 3 | 69.1
$3\times 3$ | 5 | 68.8

(c) Model | F-score
---|---
Baseline | 65.2
\+ Ours | 69.5
\+ Ours + CRF [23] | 69.7
\+ Ours + SegFix [58] | 70.1
\+ Ours + IF [3] | 70.5

Compatibility with other boundary promoting methods. Our approach can be
incorporated with other boundary improvement methods, for example, using our
SDN and a post-processing boundary refinement module simultaneously in a
network. Table 3(c) presents the good compatibility of our approach with three
well-known methods on Cityscapes-val. Specifically, the additional DenseCRF
[23], SegFix [58] and InverseForm [3] further bring $+0.2$, $+0.6$ and $+1.0$
boundary F-score gains, respectively. It is worth noting that, our method can
directly achieve $+4.4$ boundary F-score improvement over the baseline without
any other methods. Benefiting from the intrinsic boundary exploration,
segmentation networks with our SDN can relieve the dependence on other
boundary improvement techniques.

Figure 6:  Visualization of the feature mapping in SDN. The second row shows
the output feature of each encoder stage, which is also the input feature of
SDNs. The output features of SDN according to each stage are shown in the
third row. Obviously, our SDN enhances inter-class boundaries and suppresses
the intra-class pseudo boundaries. The example is selected from Cityscapes.

##  6 Conclusion

In this paper, we have summarized the internal reasons why the deep semantic
segmentation model is difficult to produce accurate boundary prediction, that
is, the commonly used convolutional operators tend to smooth and blur local
detail cues, making it difficult for deep models to capture precise boundary
information. To solve this problem, this paper introduces an operator-level
approach to enhance semantic boundary awareness, so as to improve the
prediction of the deep semantic segmentation model. Specifically, we formulate
the boundary feature enhancement process as an anisotropic diffusion process
and propose the novel semantic diffusion network (SDN) to approximate the
anisotropic diffusion process. The proposed SDN is an efficient, stable, and
flexible module that can be plugged into existing encoder-decoder segmentation
models. Extensive experiments show that our approach can achieve consistent
improvement over various baseline semantic segmentation models and
significantly improve the boundary quality of segmentation results.

##  7 Limitations and Social Impact

We believe that SDN is a general approach that can benefit deep learning
models in various popular visual tasks. However, this paper only verifies the
effectiveness of SDN on a single task (semantic segmentation). In the recent
future, we will explore the application of SDN in other 2D/3D tasks, such as
instance segmentation, panoptic segmentation, flow estimation, image
denoising, image super-resolution, and so on. SDN has potential value in
important applications such as autonomous driving and medical image analysis.
But it can also deploy inhumane surveillance. The potential negative effects
can be avoided by implementing strict and secure regulations.

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EPJ Web of Conferences Lattice2017  11institutetext: HISKP (Theory) and BCTP,
University of Bonn, Germany

# Vector-Vector Scattering on the Lattice

Fernando Romero-LÃpez Speaker, 11<EMAIL_ADDRESS>Carsten Urbach 11
Akaki Rusetsky 11

###### Abstract

In this work we present an extension of the LÃscher formalism to include the
interaction of particles with spin, focusing on the scattering of two vector
particles. The derived formalism will be applied to Scalar QED in the Higgs
Phase, where the U(1) gauge boson acquires mass.

##  1 Introduction

The study of scattering in Lattice Field Theory starts with the original work
of LÃscher Luscher . In this first work, he derived equations for the
scattering length and phase shift of spinless particles in the rest frame. The
formalism has been extended to include moving frames Gottt , $\pi-N$
scattering Bernard:2008ax , $N-N$ scattering Briceno:2013lba , different
masses Bernard:2012bi ; ZiwenFu , moving frames with different masses and one
particle with spin $1/2$ Gockeler:2012yj  and any multichannel system with
arbitrary spin, momentum and masses Briceno:2014oea .

In this work, we derive a general LÃscher equation for scattering of particles
with arbitrary spin through the matching to a non-relativistic effective
theory. The results obtained here are in agreement with Ref. Briceno:2014oea .
We will focus on the case of two identical vector particles and we will make
use of the spatial symmetries of the lattice to factorize the LÃscher
equation. By means of operators that transform under a certain irreducible
representation of the spatial symmetry group, we gain access to the different
phase shifts of the theory. The equations will be tested in Scalar QED, for
which first numerical results will be shown.

##  2 Scattering of two vector particles

###  2.1 Derivation of LÃscher Equation for Arbitrary Spin

Let us consider a system of two particles with masses $M_{i}$, $i=1,2$ and
described by the effective non-relativistic Lagrangian:

$\mathcal{L}=\phi_{1}^{\dagger}2W_{1}(i\partial_{t}-W_{1})\phi_{1}+\phi_{2}^{\dagger}2W_{2}(i\partial_{t}-W_{2})\phi_{2}+\mathcal{L}_{I}.$
(1)

Here, $\phi_{i}$ are the non relativistic fields with spin $s_{i}$ and
$W_{i}=(M_{i}^{2}-\nabla^{2})^{1/2}$. The corresponding non-relativistic
propagators, with $\omega_{i}=(M_{i}^{2}+\mathbf{p}^{2})^{1/2}$, read

$S_{i}(p)=\frac{1}{2\omega_{i}(\mathbf{p})}\frac{1}{\omega_{i}(\mathbf{p})-p^{0}-i\epsilon},$
(2)

The scattering T matrix is defined through the Lippman-Schwinger (LS)
equation:

$T(z)=(-H_{I})+(-H_{I})(-G_{0}(z))T(z),$ (3)

where $G_{0}(z)=(z-H_{0})^{-1}$ is the free resolvent and the two particle
states coupled to a spin S are

$\displaystyle\ket{\mathbf{k_{1}},\mathbf{k_{2}},S,\nu}\equiv\ket{\mathbf{P},\mathbf{k},S,\nu},$
(4)
$\displaystyle\braket{\mathbf{P^{\prime}},\mathbf{k^{\prime}},S^{\prime},\nu^{\prime}}{\mathbf{P},\mathbf{k},S,\nu}=4\omega_{1}\omega_{2}(2\pi)^{d}\delta^{d}(\mathbf{P^{\prime}}-\mathbf{P})(2\pi)^{d}\delta^{d}(\mathbf{k^{\prime}}-\mathbf{k})\delta_{S^{\prime}S}\delta_{\nu^{\prime}\nu},$
(5) $\displaystyle\mathbf{P}=\mathbf{k_{1}}+\mathbf{k_{2}},\ \ \
\mathbf{k}=\mu_{2}\mathbf{k_{1}}-\mu_{1}\mathbf{k_{2}},\ \ \
\mu_{1,2}=\frac{1}{2}\left(1\pm\frac{m_{1}^{2}-m_{2}^{2}}{P^{2}}\right),$ (6)

where $S$, $\nu$ label the total spin of the two particle system and
$\mathbf{P}$, $\mathbf{k}$ are the total and relative momentum in the
“laboratory frame”. Now define the matrix elements:

$t^{S^{\prime}S}_{\nu^{\prime}\nu}(\mathbf{k^{\prime}},\mathbf{k},\mathbf{P},z)=\int\frac{d^{d}\mathbf{P^{\prime}}}{(2\pi)^{d}}\braket{\mathbf{P^{\prime}},\mathbf{k^{\prime}},S^{\prime},\nu^{\prime}}{T(z)}{\mathbf{P},\mathbf{k},S,\nu},$
(7)
$h^{S^{\prime}S}_{\nu^{\prime}\nu}(\mathbf{k^{\prime}},\mathbf{k},\mathbf{P})=\int\frac{d^{d}\mathbf{P^{\prime}}}{(2\pi)^{d}}\braket{\mathbf{P^{\prime}},\mathbf{k^{\prime}},S^{\prime},\nu^{\prime}}{(-H_{I})}{\mathbf{P},\mathbf{k},S,\nu}.$
(8)

Additionally, $G_{0}$ can be written in terms of an elementary diagram using
the Feynman rules and one can perform the integration over $q^{0}$ using the
Cauchy integration formula. One may rewrite the Lippman-Schwinger Equation in
terms of matrix elements using Equations 7 and 8:

$\displaystyle
t^{S^{\prime}S}_{\nu^{\prime}\nu}(\mathbf{k^{\prime}},\mathbf{k},\mathbf{P},z)=h^{S^{\prime}S}_{\nu^{\prime}\nu}(\mathbf{k^{\prime}},\mathbf{k},\mathbf{P})+\int\frac{d^{d}\mathbf{q_{1}}}{(2\pi)^{d}}\sum_{\nu^{\prime\prime}}\frac{h^{S^{\prime}S^{\prime\prime}}_{\nu^{\prime}\nu^{\prime\prime}}(\mathbf{k^{\prime}},\mathbf{q_{1}},\mathbf{P})t^{S^{\prime\prime}S}_{\nu^{\prime\prime}\nu}(\mathbf{q_{1}},\mathbf{k},\mathbf{P},z)}{4\omega_{1}(\mathbf{q_{1}})\omega_{2}(\mathbf{P}-\mathbf{q_{1}}).(\omega_{1}(\mathbf{q_{1}})+\omega_{2}(\mathbf{P}-\mathbf{q_{1}})-z)},$
(9)

where $\mathbf{q}=\mu_{2}\mathbf{q_{1}}-\mu_{1}\mathbf{q_{2}}$, as in Equation
6. A key point here is that the elementary bubble is diagonal in spin, because
also the single particle propagators are. However, the scattering amplitude
need not be diagonal. Now define the projectors to the partial waves in the CM
frame, whose momenta are $\mathbf{k^{*}}$:

$\displaystyle\Pi_{\nu^{\prime}\nu}^{A^{\prime}A}(\mathbf{k^{\prime*}},\mathbf{k^{*}})=\sum_{\rho,\rho^{\prime}}U^{S}_{\nu^{\prime}\rho^{\prime}}(\mathbf{k^{\prime*}})^{*}U^{S}_{\nu\rho}(\mathbf{k^{*}})(\mathcal{Y}_{J^{\prime}l^{\prime}S^{\prime}\mu^{\prime}}(\mathbf{k^{\prime*}},\rho^{\prime}))^{*}\mathcal{Y}_{JlS\mu}(\mathbf{k^{*}},\rho),$
(10) $\displaystyle A=(J,l,S,\mu),\ \
A^{\prime}=(J^{\prime},l^{\prime},S^{\prime},\mu^{\prime}),$ (11)

where $U^{S}_{\nu\rho}(\mathbf{k^{*}})$ is the unitary transformation of the
spin indices under a boost and the spin spherical harmonics, with
$\mathbf{\hat{k}}=\mathbf{k}/|\mathbf{k}|$, read

$\mathcal{Y}_{JlS\mu}(\mathbf{k},\nu)=\sum_{m,\sigma}\braket{lmS\sigma}{J\mu}|\mathbf{k}|^{l}Y_{lm}(\mathbf{\hat{k}})\chi^{S}_{\sigma}(\nu)\equiv|\mathbf{k}|^{l}Y_{JlS\mu}(\mathbf{\hat{k}},\nu).$
(12)

Using the projectors, Equations 7 and 8 take the form

$\displaystyle
t^{S^{\prime}S}_{\nu^{\prime}\nu}(\mathbf{k^{\prime}},\mathbf{k},\mathbf{P},z)$
$\displaystyle=4\pi\sum_{A^{\prime},A}\Pi_{\nu^{\prime}\nu}^{A^{\prime}A}(\mathbf{k^{\prime*}},\mathbf{k^{*}})t_{A^{\prime}A}(s,\mathbf{P},z),$
(13) $\displaystyle
h^{S^{\prime}S}_{\nu^{\prime}\nu}(\mathbf{k^{\prime}},\mathbf{k},\mathbf{P})$
$\displaystyle=4\pi\sum_{A^{\prime},A}\Pi_{\nu^{\prime}\nu}^{A^{\prime}A}(\mathbf{k^{\prime*}},\mathbf{k^{*}})h_{A^{\prime}A}(s,\mathbf{P}).$
(14)

If the system is placed in a box, the integral may be replaced by an sum:

$\int\frac{d^{d}\mathbf{q_{1}}}{(2\pi)^{d}}\rightarrow\frac{1}{L^{3}}\sum_{\mathbf{q_{1}}},$
(15)

and by plugging Equations 13 and 14 into Equation 9 on the mass-shell, one
arrives to

$\displaystyle
t_{A^{\prime}A}(s,\mathbf{P})-h_{A^{\prime}A}(s,\mathbf{P})=\frac{k^{*}}{8\pi\sqrt{s}}\sum_{B,B^{\prime}}h_{A^{\prime}B}(s,\mathbf{P})((\mathbf{k^{*}})^{l+l^{\prime}}i^{l-l^{\prime}}\delta_{S_{B}S_{B^{\prime}}}\mathcal{M}_{BB^{\prime}}(s,\mathbf{P}))t_{B^{\prime}A}(s,\mathbf{P}),$
(16)

with $s=P^{2}$ and $S_{B}$ being the spin of the multi-index $B$ as in
Equation 11. Now, using unitarity of the transformation of the spin indices,
one arrives at

$\displaystyle\mathcal{M}_{JlS\mu,J^{\prime}l^{\prime}S^{\prime}\mu^{\prime}}(s,\mathbf{P})=\frac{32\pi^{2}}{\mathbf{k^{*}}}\frac{\sqrt{s}}{L^{3}}i^{l^{\prime}-l}\delta_{S^{\prime}S}\sum_{q_{1}}\frac{\sum_{\nu}({Y}_{JlS\mu}(\mathbf{\hat{q}^{*}},\nu))^{*}{Y}_{J^{\prime}l^{\prime}S\mu^{\prime}}(\mathbf{\hat{q}^{*}},\nu)}{4\omega_{1}(\mathbf{q_{1}})\omega_{2}(\mathbf{P}-\mathbf{q_{1}})(\omega_{1}(\mathbf{q_{1}})+\omega_{2}(\mathbf{P}-\mathbf{q_{1}})-P_{0})}.$
(17)

This matrix can be related to its equivalent for scalar particles by using
Equation 12:

$\mathcal{M}_{JlS\mu,J^{\prime}l^{\prime}S^{\prime}\mu^{\prime}}=\delta_{S^{\prime}S}\sum_{m,m^{\prime},\sigma}\braket{lm,S\sigma}{J\mu}\braket{l^{\prime}m^{\prime},S\sigma}{J^{\prime}\mu^{\prime}}\mathcal{M}_{lm,l^{\prime}m^{\prime}},$
(18)

where we used the identity Gasser:2011ju ; Bernard:2012bi  (with
$\mathbf{q}=\mathbf{q_{1}}-\mu_{1}\mathbf{P}$)

$\displaystyle\begin{split}&\frac{1}{4\omega_{1}\omega_{2}(\omega_{1}+\omega_{2}-P_{0})}=\frac{1}{2P_{0}}\frac{1}{\mathbf{q}^{2}-\frac{(\mathbf{q}\mathbf{P})^{2}}{P_{0}^{2}}-(\mathbf{k^{*}})^{2}}\\\
&+\frac{1}{4\omega_{1}\omega_{2}}\left(\frac{1}{\omega_{1}+\omega_{2}+P_{0}}-\frac{1}{\omega_{1}-\omega_{2}+P_{0}}-\frac{1}{\omega_{2}-\omega_{1}+P_{0}}\right),\end{split}$
(19)

kept only the divergent part (first term in Equation 19) and used
$(\mathbf{q^{*}})^{2}=\mathbf{q}^{2}-\frac{(\mathbf{qP})^{2}}{P^{2}_{0}}$.
This way, and up to exponentially suppressed terms
($\mathbf{q^{*}}\rightarrow\mathbf{k^{*}}$),
$\mathcal{M}_{lm,l^{\prime}m^{\prime}}$ is given by (see Bernard:2012bi )

$\mathcal{M}_{lm,l^{\prime}m^{\prime}}(\mathbf{k^{*}},s)=\frac{(-1)^{l}}{\pi^{3/2}\gamma}\sum_{j=|l-l^{\prime}|}^{l+l^{\prime}}\sum_{s=-j}^{j}\frac{i^{j}}{\eta^{j+1}}Z^{d}_{js}(1,s)^{*}C_{lm,js,l^{\prime}m^{\prime}},\
\eta=\frac{|\mathbf{k^{*}}|L}{2\pi},$ (20)

where

$C_{lm,js,l^{\prime}m^{\prime}}=(-1)^{m^{\prime}}i^{l-j+l^{\prime}}\sqrt{(2l+1)(2l^{\prime}+1)(2j+1)}\begin{pmatrix}l&j&l^{\prime}\\\
m&s&-m^{\prime}\end{pmatrix}\begin{pmatrix}l&j&l^{\prime}\\\
0&0&0\end{pmatrix},$ (21) $Z^{d}_{lm}(1,s)=\sum_{\mathbf{r}\in
P_{d}}\frac{|\mathbf{r}|^{l}Y_{lm}(r)}{\mathbf{r}^{2}-\eta^{2}},\ \ \
P_{d}=\\{\mathbf{r_{||}}=\gamma^{-1}(\mathbf{n_{||}}-\mu_{1}\mathbf{d}),\mathbf{r_{\perp}}=\mathbf{n_{\perp}}\\},$
(22)

with $n\in\mathbb{Z}$. One can see that Equation 16 is a matrix equation, and
the poles in $t_{A^{\prime}A}$ arise when

$\det\left(\frac{8\pi\sqrt{s}}{(\mathbf{k^{*}})^{l+l^{\prime}+1}}(h^{J}_{lS,l^{\prime}S^{\prime}})^{-1}\delta_{JJ^{\prime}}\delta_{\mu\mu^{\prime}}-\delta_{S,S^{\prime}}\mathcal{M}_{JlS\mu,J^{\prime}l^{\prime}S\mu^{\prime}}\right)=0,$
(23)

where it is already implied that $J$ and $\mu$ are conserved in scattering
processes in infinite volume and mixings can be present in $l$ and $S$,
$h_{JlS\mu,J^{\prime}l^{\prime}S^{\prime}\mu^{\prime}}=h^{J}_{lS,l^{\prime}S^{\prime}}\delta_{JJ^{\prime}}\delta_{\mu\mu^{\prime}}$.
In order to express this equation in a more compact way, one uses the standard
definition of the $S$ matrix (see Briceno:2013bda , for nucleon-nucleon
scattering), $S=e^{2i\delta(s)}$, in terms of the phase shift $\delta(s)$.
This way, one can express $h^{J}_{lS,l^{\prime}S^{\prime}}$ in terms of
$\delta$:

$h^{J}_{lS,l^{\prime}S}=\frac{8\pi\sqrt{s}}{(\mathbf{k^{*}})^{l+l^{\prime}+1}}(\tan\delta)^{J}_{lS,l^{\prime}S^{\prime}},$
(24)

and plugging it in Equation 23, we arrive at

$\det\left((\cot\delta)^{J}_{lS,l^{\prime}S^{\prime}}\delta_{JJ^{\prime}}\delta_{\mu\mu^{\prime}}-\delta_{SS^{\prime}}\mathcal{M}_{JlS\mu,J^{\prime}l^{\prime}S\mu^{\prime}}\right)=0.$
(25)

###  2.2 Two Vector Particles

A system of two identical vector particles can couple to total spin $S=0,1,2$.
Even spin combinations are symmetric under the exchange of two particles,
whereas odd combinations are antisymmetric. The same holds for the angular
momentum $L$. The possible combinations of $S$ and $L$ to $J^{P}$ respecting
the Bose statistics (totally symmetric state) are listed in the Table 1. The
combinations that have mixing are in the same column in the table and
correspond to same $J^{P}$ but different $L$, $S$.

$J^{P}$ | $0^{+}$ | $0^{-}$ | $1^{+}$ | $1^{-}$ | $2^{+}$ | $2^{-}$
---|---|---|---|---|---|---
$\\{S,L\\}$ | $\\{0,0\\}$ |  |  |  | $\\{0,2\\}$ |
| $\\{1,1\\}$ |  | $\\{1,1\\}$ |  |  $\\{1,1\\}$,$\\{1,3\\}$
$\\{2,2\\}$ |  | $\\{2,2\\}$ |  |  $\\{2,0\\}$,$\\{2,2\\}$,$\\{2,4\\}$ |
Table 1: Possible values of $J^{P}$ with $J<3$.

The possible mixings can be parametrized with a mixing angle and two
eigenvalues. This would be analogous to the parametrization of the mixings for
two nucleons in Reference Briceno:2013bda .

For the scattering of two spinless particles, it is well known that the phase
shift can be parametrized as a polynomial of $\mathbf{k}^{2}$. Such an
equivalent parametrization can be derived here as well:

$\mathbf{k}^{l+l^{\prime}+1}\cot\delta^{J}_{lS,l^{\prime}S^{\prime}}=\sum_{n=0}a_{n}\mathbf{k}^{2n}.$
(26)

###  2.3 Reduction of LÃscher Equation

The basis vector labeled by $\alpha$ of a symmetry group $\mathcal{G}$, in a
certain irreducible representation $\Gamma$, are constructed (up to
normalization) applying the projector

$(P^{\Gamma,J,l}_{\alpha\beta})_{\mu\mu^{\prime}}=\sum_{i\in
G}(R_{i}^{\Gamma})^{*}_{\alpha\beta}\tilde{D}^{J,l}_{\mu\mu^{\prime}}(R_{i}),$
(27)

to the basis vectors in the continuum, $\ket{J,S,l,\mu}$, for a fixed $\beta$
and $\mu$:

$\ket{\Gamma,\alpha,J,S,l,n}\propto\sum_{\mu^{\prime}}(P^{\Gamma,J,l}_{\alpha\beta})_{\mu\mu^{\prime}}\ket{J,S,l,\mu^{\prime}},$
(28)

where $n$ labels the number of occurrences of $\Gamma$ in $J$, $l$. In the
previous equations
$\tilde{D}^{J,l}_{\mu\mu^{\prime}}(R_{i})=(-1)^{l}{D}^{J}_{\mu\mu^{\prime}}(R_{i})$
if the element $i$ includes inversion, or just the standard Wigner matrix if
not. The basis vectors of the irreducible representations of the symmetry
group of the lattice can be expressed in terms of the one of the continuum:

$\ket{\Gamma,\alpha,J,l,S,n}=\sum_{\mu}c^{\Gamma
n\alpha}_{Jl\mu}\ket{JlS\mu},$ (29)

where the coefficients $c^{\Gamma n\alpha}_{Jl\mu}$ are to be read from basis
vector tables in References Gockeler:2012yj  and Bernard:2008ax , for example.
The matrix $\mathcal{M}$ can be partially diagonalized in this basis:

$\displaystyle\braket{\Gamma^{\prime},\alpha^{\prime},J^{\prime},l^{\prime},S,n^{\prime}}{\mathcal{M}}{\Gamma,\alpha,J,l,S,n}=\mathcal{M}^{\Gamma}_{J^{\prime}l^{\prime}Sn^{\prime},JlSn}\delta_{\Gamma^{\prime}\Gamma}\delta_{\alpha^{\prime}\alpha},$
(30)
$\displaystyle\mathcal{M}^{\Gamma}_{J^{\prime}l^{\prime}Sn^{\prime},JlSn}=\sum_{\mu^{\prime}\mu}(c^{\Gamma
n^{\prime}\alpha}_{J^{\prime}l^{\prime}\mu^{\prime}})^{*}c^{\Gamma
n\alpha}_{Jl\mu}\mathcal{M}_{J^{\prime}l^{\prime}S\mu^{\prime},JlS\mu}.$ (31)

Moreover, the matrix $\cot\delta$ has to be brought to the same basis as
$\mathcal{M}$:

$(\cot\delta)^{\Gamma}_{JlSn,J^{\prime}l^{\prime}S^{\prime}n^{\prime}}=\sum_{\mu^{\prime}\mu}(c^{\Gamma
n^{\prime}\alpha}_{J^{\prime}l^{\prime}\mu^{\prime}})^{*}c^{\Gamma
n\alpha}_{Jl\mu}(\cot\delta)^{J}_{lS,l^{\prime}S^{\prime}}\delta_{\mu\mu^{\prime}}\delta_{JJ^{\prime}}=\sum_{\mu}(c^{\Gamma
n^{\prime}\alpha}_{Jl^{\prime}\mu})^{*}c^{\Gamma
n\alpha}_{Jl\mu}(\cot\delta)^{J}_{lS,l^{\prime}S^{\prime}}\delta_{JJ^{\prime}}.$
(32)

The coefficients $c^{\Gamma n\alpha}_{Jl\mu}$ just depend on the total angular
momentum $J$ and the behaviour under spatial inversions $(-1)^{l}$. Moreover,
mixing can only occur between states with same $J$ and $(-1)^{l}$, so the sum
over the index $\mu$ yields either 1 or the matrix $\cot\delta$ is trivially
zero due to the symmetry considerations. Spin does not enter here at all,
because it does not influence the coefficients. This way,
$(\cot\delta)^{\Gamma}_{JlSn,J^{\prime}l^{\prime}S^{\prime}n^{\prime}}$ is
diagonal in $J$ and $n$ and the determinant factorizes:

$\prod_{S=\begin{subarray}{c}even\\\ odd\end{subarray}}\ \ \prod_{\Gamma}\
\det\left((\cot\delta)^{J}_{lS,l^{\prime}S^{\prime}}\delta_{JJ^{\prime}}\delta_{nn^{\prime}}-\delta_{SS^{\prime}}\mathcal{M}^{\Gamma}_{JlSn,J^{\prime}l^{\prime}Sn^{\prime}}\right)=0.$
(33)

##  3 Toy Model: Scalar QED

In order to test the formalism, we use Scalar QED with a Higgs mechanism,
since the vector state needs to be massive. The continuum Euclidean Lagrangian
of such a theory reads

$\mathcal{L}_{E}=(D_{\mu}\phi_{c})^{\dagger}D_{\mu}\phi_{c}+m^{2}_{0}|\phi_{c}|^{2}+\lambda_{c}|\phi_{c}|^{4},\quad
D_{\mu}\phi_{c}=\partial_{\mu}\phi_{c}+igA_{\mu}\phi_{c}.$ (34)

The discretized action is (See SCALARQED )

$\displaystyle S=\sum_{x}$
$\displaystyle\Big{(}-\frac{\beta}{2}\sum_{\mu<\nu}(U_{\mu\nu}+U_{\mu\nu}^{*})-\kappa\sum_{\mu}(\phi_{x}^{*}U_{x,\mu}\phi_{x+\mu}+cc)+\lambda(|\phi_{x}|^{2}-1)^{2}+|\phi_{x}|^{2}\Big{)},$
(35)

with

$\displaystyle\lambda_{c}=\frac{\lambda}{\kappa^{2}},\quad(am_{0})^{2}=\frac{1-2\lambda-8\kappa}{\kappa},\quad\beta=\frac{1}{g^{2}}.$
(36)

The basic operators for the scalar and real particle are

$\displaystyle\mathcal{O}^{A_{1}}=\sum_{i}\operatorname{Re}(\phi^{\dagger}(x,t)U_{i}(x)\phi(x+ae_{i},t)),$
(37)
$\displaystyle\mathcal{O}^{T_{1}^{-}}_{n,i}=\operatorname{Im}(\phi^{\dagger}(x,t)\left(\prod_{n=0}^{N-1}U_{i}(x+ne_{i})\right)\phi(x+Ne_{i},t)),$
(38)

respectively. Notice that the second one is highly non-local, since this have
been seen to improve significantly the signal. In particular, with the vector
operator, one can build one and two particle operators that transform under
certain irreducible representations of the spatial symmetry group, in rest and
moving frames. A one particle operator transforming under a certain irrep
$\Gamma$ with invariant momentum under the spacial symmetry group
$p\in\mathcal{G}$ can be built from an arbitrary operator $\mathcal{O}(x,t)$
as follows:

$\mathcal{O}^{\Gamma}(p,t)=\sum_{x}e^{ipx}\sum_{S_{i}\in\mathcal{G}}\chi^{\Gamma}(S_{i})\
(S_{i}\mathcal{O})(S_{i}^{-1}x,t)$ (39)

Equivalently, for a two particle operator:

$\mathcal{O}^{\Gamma}(p,q,t)=\sum_{x,y}\left(\sum_{S_{i}\in\mathcal{G}}e^{ipx+S_{i}q(y-x)}\right)\sum_{S_{i}\in\mathcal{G}}\chi^{\Gamma}(S_{i})\
(S_{i}\mathcal{O})(S_{i}^{-1}x,S_{i}^{-1}y,t),$ (40)

where $p$ is the total momentum that belonging to the spatial symmetry group
$\mathcal{G}$, $q$ is a relative momentum that the two particles have and
$\mathcal{O}(x,y,t)$ is an arbitrary two particle operator.

##  4 Numerical Results

(a) Mass of the vector particle for ensemble A12 for different lengths of the
operator in Equation 38.  (b) Mass of the scalar and vector particle for
$L=12$ as a function of $\kappa$.  (c) Energy difference $\Delta E$ as a
function of $\kappa$ in the rest frame.  (d) Mass of the vector particle for
different irreps for ensemble A16.  (e) Energy difference $\Delta E$ for
ensemble A16 in the moving frame with total momentum $p=\frac{2\pi}{L}(0,0,1)$
and $q=0$ as in Equation 40 for different irreps $\Gamma$.  (f) Phase shift in
the $J^{P}=0^{+}$ channel as a function of the scattering momentum $k$. They
are calculated neglecting partial waves $J>1$ and at this level, the two
possible $L,S$ combinations cannot be distinguished.

Figure 1: Numerical results.

We use five different parameters for $L=12$ and for one parameter set $L=12$
and $L=16$ (see Table 2). In Figure 1(a) we show the dependence of mass of the
vector particle on the length of the operator in Equation 38 for ensemble A12.
The results show a clear improvement of the signal with non-local operators.
When using moving frames, the best signal is empirically seen at $N=L/(d+1)$,
being $d$ the units of momentum in that particular direction.

Moreover, we study the mass of a single vector and scalar particle in the
$L=12$ volume for the different bare parameter sets (Figure 1(b)). In the
continuum, the bare masses of the particles are $m^{2}_{\phi}=-2m_{0}^{2}$ and
$m_{V}^{2}=-\frac{g^{2}m_{0}^{2}}{\lambda_{c}}$. The mass of the vector is
suppressed by $g$ and $\lambda_{c}$, and it seems logical that it is much
smaller than the scalar mass. However, it is not clear why the scalar mass
duplicates with increasing $\kappa$, whereas the vector mass increases
slightly. NaÃvely, the increase of the vector mass should be enhanced by the
reduction of $\lambda_{c}$ with increasing $\kappa$, but there seems to be
some non-trivial behaviour.

The energy difference $\Delta E$ is defined as the difference between the two
particle energy on the lattice and the two particle energy in absence of
interactions. In Figure 1(c) we show the results for $\Delta E$ as a function
of $\kappa$. For higher $\kappa$, the interaction generates scattering states
($\Delta E>0$) with $\gtrapprox 2\sigma$ statistical significance. As $\kappa$
becomes smaller, all particles become lighter and the interaction seems to
flip sign. For the lowest values of $\kappa$, the two particle states are
bound states ($\Delta E<0$), with a similar statistical significance as in the
other case. Unfortunately, in the transition region the statistical
significance is rather small. Besides, we see the expected volume dependence
in the energy shift, when comparing $L=12$ with $L=16$.

In addition, we show the results of the mass of a single vector particle for
the different irreps in Figure 1(d). The values for the different moving
frames tend to always larger than the one in the rest frame. Since the
correlation functions are fitted to include an excited state, a contamination
of higher states is likely not the main reason. It is however true that from
the correlation functions one obtains upper bounds for the energy, which might
have an influence. In addition, the different irreps seem to split. In
particular $A_{1}$ and $E$ seem to differ by around $3\sigma$.

Furthermore, in Figure 1(e) we show $\Delta E$ for the ensemble A16 in the
first moving frame. One can see how the different irreps split. With these
values one can calculate phase shifts values (Table 3). Finally, in Figure
1(f) we show all our results for the phase shift with $J^{P}=0^{+}$. For the
highest momentum shown, the ratio between the non-relativistic kinetic energy
and the mass is quite large, $\frac{E_{k}}{m}\approx 0.8$. Hence, the
kinematic suppression of higher partial waves, though present, is not strong
any more and a corresponding systematical error is to be expected. Still, it
seems that the phase shift increases for the last points. If this increase is
physical, it could correspond to a resonance around scattering momentum
$ak\approx 0.18$.

$L^{3}\times T$ | ref. | N | $m^{2}_{0}$ | $\lambda_{c}$ | $\kappa$ | $\lambda$
---|---|---|---|---|---|---
$16^{3}\times 24$ | A16 | 138000 | -35 | 88 | 0.18425 | 2.9873
$12^{3}\times 24$ | A12 | 33000 | -35 | 88 | 0.18425 | 2.9873
B12 | 24000 | -35.5 | 90.6 | 0.18208 | 3.0036
C12 | 21500 | -35.6 | 91.6 | 0.18084 | 2.9956
D12 | 16000 | -35.85 | 93.1 | 0.17949 | 2.9994
E12 | 18800 | -36.1 | 94.7 | 0.17802 | 3.0012
Table 2: Ensembles used for the simulations. The gauge coupling is kept constant, $\beta=2.5$.  $\Gamma$ | $ak$ | $J^{P}$ | $\delta\ (^{o})$
---|---|---|---
$A_{1}$ | 0.1452(24) | $0^{+}$ | -1.13(76)
$A_{1}$ | 0.1452(24) | $1^{-}$ | -0.14(10)
$A_{2}$ | 0.1369(21) | $0^{-}$ | 1.24(67)
$A_{2}$ | 0.1369(21) | $1^{+}$ | 0.127(60)
Table 3: Obtained values for the phase shift in moving frame with the
assumption of no mixings. The results are in agreement with the kinematic
suppression expected for higher partial waves.

##  5 Summary and Outlook

We have shown that the study on the Lattice of scattering of vector particles
is possible by applying the derived framework to the toy model Scalar QED. The
complete numerical results, together with the group theoretical derivations,
will be published soon in a longer, more detailed version. On the long term,
we expect to apply the ideas of this work to study the possibility of the
Higgs boson to be a resonance of two $W$ bosons. This is the case for a model
proposed by Frezzotti et al. Frezzotti:2014wja ; Frezzotti:2016bes , where a
“superstrong interaction” together with superstrongly interacting particles
are present.

We would like to acknowledge the lattice group in Bonn and Roberto Frezzotti
for the interesting discussions and the support provided. This work was
supported in part by the DFG in the Sino-German CRC110. Finally, special
thanks to BCGS for the continuous support.

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