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The vector boson dark matter particles which stem from some broken gauge symmetries usually requires some unbroken symmetries to keep themselves stable. In the previous literature, some simplest cases have been discussed, in which the unbroken symmetry is provided by a remnant subgroup of the gauge group. It would be interesting to ask whether all the possible remnant subgroups as well as all the possible coupling forms can be enumerated. Classifying all the Higgs components into different mass degenerate representations to simplify the diagonalization processes is also necessary. Rather than the ambitious target of providing a general solution to all kinds of gauge groups configured with all forms of the Higgs multiplets, in this paper, we concentrate on the case of $\text{SU(2)}_{\text{D}}$ gauge group together with a single Higgs multiplet. We enumerate all possible discrete subgroups that can survive up to $n=21$, where $n$ is the dimension of the Higgs multiplet. We also provide the general algorithms to enumerate all possible renormalizable operators, to write down the general forms of the vacuum expectation value (VEV) configurations, and to give the detailed results of all the mass degenerate irreducible representations embedded in the Higgs multiplet.

In this paper, we discuss the possibility to probe a nearly-degenerate $Z^{\prime}$-$Z$ system by analyzing the $Z$-lineshape at an electron-positron collider. Compared with the usual $Z^{\prime}$ in the literature well separated with the standard model (SM) $Z$ boson in mass, the nearly-degenerate $Z^{\prime}$-$Z$ mixing affects the observed effective ``oblique parameters'' $\tilde{S}$, $\tilde{T}$, $\tilde{U}$, and the effective deviation of ``number of neutrino species'' $\delta \tilde{N}_{\nu}$ in a more complicated way and cannot be simply computed perturbatively up to a particular order. Aiming at solving this problem, we write down a general simplified effective Lagrangian and enumerate some parameter spaces corresponding to some typical models, and suggest a method to extract the constraints by looking into the line-shape of the $Z$-like resonance at an electron-positron collider.

In this paper, we present an approach to image enhancement with diffusion model in underwater scenes. Our method adapts conditional denoising diffusion probabilistic models to generate the corresponding enhanced images by using the underwater images and the Gaussian noise as the inputs. Additionally, in order to improve the efficiency of the reverse process in the diffusion model, we adopt two different ways. We firstly propose a lightweight transformer-based denoising network, which can effectively promote the time of network forward per iteration. On the other hand, we introduce a skip sampling strategy to reduce the number of iterations. Besides, based on the skip sampling strategy, we propose two different non-uniform sampling methods for the sequence of the time step, namely piecewise sampling and searching with the evolutionary algorithm. Both of them are effective and can further improve performance by using the same steps against the previous uniform sampling. In the end, we conduct a relative evaluation of the widely used underwater enhancement datasets between the recent state-of-the-art methods and the proposed approach. The experimental results prove that our approach can achieve both competitive performance and high efficiency. Our code is available at \href{mailto:https://github.com/piggy2009/DM_underwater}{\color{blue}{https://github.com/piggy2009/DM\_underwater}}.

In this paper, we would like to compute the muon anomalous precession frequency through solving the wave functions of the Dirac equations straightforwardly. The precession of a (anti-)muon with an anomalous magnetic momentum term is calculated together with the quantum corrections. Lorentz violation terms up to the lowest non-trivial order is introduced, and their effects on anomalous precession are evaluated perturbatively.

Given an image of a white shoe drawn on a blackboard, how are the white pixels deemed (say by human minds) to be informative for recognizing the shoe without any labeling information on the pixels? Here we investigate such a ``white shoe'' recognition problem from the perspective of tensor network (TN) machine learning and quantum entanglement. Utilizing a generative TN that captures the probability distribution of the features as quantum amplitudes, we propose an unsupervised recognition scheme of informative features with the variations of entanglement entropy (EE) caused by designed measurements. In this way, a given sample, where the values of its features are statistically meaningless, is mapped to the variations of EE that statistically characterize the gain of information. We show that the EE variations identify the features that are critical to recognize this specific sample, and the EE itself reveals the information distribution of the probabilities represented by the TN model. The signs of the variations further reveal the entanglement structures among the features. We test the validity of our scheme on a toy dataset of strip images, the MNIST dataset of hand-drawn digits, the fashion-MNIST dataset of the pictures of fashion articles, and the images of brain cells. Our scheme opens the avenue to the quantum-inspired and interpreted unsupervised learning, which can be applied to, e.g., image segmentation and object detection., Comment: 7 pages, 8 figures. In the updated version, we added the unsupervised segmentation of images of cells, and the results with multi-qubit measurements