PINN 求解 pion：共形深度学习用于 $F_\pi(s)$ 与 $(g-2)_\mu$ 强子贡献
PINNing the pion: conformal deep learning for $F_π(s)$ and the $(g-2)_μ$ hadronic contribution
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中文总结 AI 辅助
提出一种嵌入共形 z 平面的物理信息神经网络，直接从第一性原理构建 pion 电磁形状因子，避免模型依赖，并给出电荷半径、ρ 极点参数及 muon 反常磁矩双 pion 贡献的模型无关估计。
中文摘要 AI 辅助
通过唯象曲线拟合模型提取 pion 电磁形状因子 $F_{\pi}(s)$ 会引入模型依赖性、非物理伪影和运动学不一致性。我们提出了一种嵌入共形 $z$ 平面的物理信息神经网络（PINN），该网络直接从第一性原理出发，在类空和类时域内构建 $F_{\pi}(s)$：电荷归一化和 Schwarz 反射通过构造强制执行，而 Cauchy-Riemann 解析性、色散关系、Watson 定理和微扰 QCD 渐近行为则通过损失函数引入。因此，基本的 S 矩阵原理决定了形状因子的行为，而数据仅作为约束。将割线复平面映射到单位圆盘上，可以限制 Hessian 范数，并防止神经正切核谱饥饿，这两种是深度学习优化的已知失效模式。除了 $e^+e^-$ 散射数据外，我们还通过一个开关纳入 $\tau$ 衰变数据，该开关原生地隔离纯等矢量形状因子，绕过了依赖模型的同位旋破缺预修正。该网络自然地产生了一个内部无零点的形状因子，同时该框架针对 $\rho(770)$ 峰附近的实验张力，在解析性和色散约束下进行了检验。我们获得了模型无关的 pion 电荷半径估计值 $\langle r_{\pi}^2 \rangle = 0.435 \pm 0.008_{\text{stat}} \pm 0.007_{\text{cali}}$ fm$^2$，第二黎曼面极点参数 $m_{\rho}^{\text{pole}} = 761.72\pm 1.04$ MeV 和 $\Gamma_{\rho}^{\text{pole}} = 135.99 \pm 1.20$ MeV，以及 muon 反常磁矩的双 pion 贡献 $a_{\mu}^{\pi\pi} = (506.48 \pm 2.02_{\text{stat}} \pm 1.70_{\text{cali}}) \times 10^{-10}$。
英文摘要
Extracting the pion electromagnetic form factor $F_π(s)$ through phenomenological curve-fitting models introduces model dependence, unphysical artefacts, and kinematic inconsistencies. We introduce a Physics-Informed Neural Network (PINN) embedded in a conformal $z$-plane that constructs $F_π(s)$ directly from first principles across spacelike and timelike domains: charge normalisation and Schwarz reflection are enforced by construction, while Cauchy-Riemann analyticity, dispersion relations, Watson's theorem, and perturbative QCD asymptotics enter through the loss functional. Thus, the fundamental S-matrix principles dictate the form factor's behaviour while data act as constraints. Mapping the cut complex plane onto the unit disk bounds the Hessian norm and prevents Neural Tangent Kernel spectral starvation, two known failure modes of deep-learning optimisation. Besides $e^+e^-$ scattering data, we also incorporate $τ$-decay data through a switch that isolates the pure isovector form factor natively, bypassing model-dependent isospin-breaking pre-corrections. The network organically yields an interior zero-free form factor, while the framework tests experimental tensions around the $ρ(770)$ peak against analyticity and dispersion constraints. We obtain model-independent estimates of the pion charge radius, $\langle r_π^2 \rangle = 0.435 \pm 0.008_{\text{stat}} \pm 0.007_{\text{cali}}$ fm$^2$, the second-sheet pole parameters, $m_ρ^{\text{pole}} = 761.72\pm 1.04$ MeV and $Γ_ρ^{\text{pole}} = 135.99 \pm 1.20$ MeV, and the two-pion contribution to the muon anomalous magnetic moment, $a_μ^{ππ} = (506.48 \pm 2.02_{\text{stat}} \pm 1.70_{\text{cali}}) \times 10^{-10}$.
发表机构
- International Institute of Information Technology, Hyderabad（海得拉巴国际信息技术学院）
机构由 AI 辅助整理，请以论文原文为准。
