物理学报2026，Vol.75Issue(17)：71-78,8.DOI:10.7498/aps.75.20260493
基于神经网络波函数的粲偶素势模型
Charmonium potential models based on neural network wave functions
摘要
Abstract
This study presents a comprehensive framework for fitting the phenomenological potential model parameters of charmonium by integrating neural network wavefunctions.This approach overcomes the limitations of traditional fitting methods,which adapt slowly to new experimental data.In this method,the radial Schrödinger equation is solved variationally using a neural network wavefunction ansatz.The wavefunction incorporates an explicitly constructed node factor that accurately represents the nodal structure of excited states.It also employs a residual network architecture to enhance expressiveness and training stability.The number of nodes is specified according to the target radial excitation,while the node positions are treated as trainable parameters,enabling the neural network to optimize the excited-state wavefunctions without preassigning their nodal locations.The energy variance is used as the training objective,and automatic differentiation is employed to calculate the derivatives required in the variational optimization and subsequent parameter inference.After determining the eigenwavefunctions and eigenenergies,the potential parameters are inferred via maximum likelihood estimation with the Laplace approximation,yielding optimal parameter values and their uncertainties.In the likelihood construction,a fixed theoretical uncertainty is introduced together with the experimental errors so that the fitting procedure reflects both experimental precision and the intrinsic approximation of the potential model.The method is applied to a phenomenological charmonium potential that includes a Coulomb term,a linear confinement term,a smeared spin-spin interaction,and spin-orbit and tensor forces treated perturbatively.The fitted parameters are the charm quark mass mc=m(c)=(1.4801±0.0055)GeV,the strong coupling constant αs=0.5459±0.0120,the string tension σlin=(0.1442±0.0028)GeV2,and the smearing radius σsmear=(1.1047±0.0568)GeV.Compared with the original parameters,the optimized model improves the agreement with the 16 known charmonium states from the Particle Data Group.The root-mean-square deviation is reduced from 26.26 MeV to 24.62 MeV.Based on the optimized parameters,the mass spectrum of charmonium is predicted up to high excitation levels,including S,P,D,and F waves.The predicted energy level spacings and fine structures are consistent with the expected behavior of the confining potential,indicating that the optimized parameters retain the main physical features of the conventional potential model.Notably,the framework provides predictions for unobserved high-angular-momentum states such as the n3FJ and n1F3 series,which may serve as references for future experimental searches at facilities such as BESⅢ,Belle Ⅱ and LHCb.The main innovation of this work is to combine neural network wavefunctions with statistical parameter inference,so that the solution of the Schrödinger equation,the fitting of potential parameters,and the quantification of parameter uncertainties are incorporated into an automated and extensible workflow.The proposed framework can be readily extended to other heavy quark systems such as bottomonium and Bc mesons,as well as to more complex spectroscopic problems involving coupled-channel effects.关键词
粲偶素/势模型/量子蒙特卡罗/神经网络波函数Key words
charmonium/potential model/quantum Monte Carlo/neural network wavefunction引用本文复制引用
史焕辰,崔二亮,周丹..基于神经网络波函数的粲偶素势模型[J].物理学报,2026,75(17):71-78,8.基金项目
国家自然科学基金(批准号:12005172)资助的课题. Project supported by the National Natural Science Foundation of China(Grant No.12005172). （批准号:12005172）
