We present a direct calculation for the first derivative of the isovector nucleon form factors with respect to the momentum transfer $q^2$ using the lower moments of the nucleon 3-point function in the coordinate space. Our numerical simulations are performed using the $N_f=2+1$ nonperturbatively $O(a)$-improved Wilson quark action and Iwasaki gauge action near the physical point, corresponding to the pion mass $M_{\pi }=138\mathrm{MeV}$, on a $(5.5\mathrm{fm}{)}^4$ lattice at a single lattice spacing of $a=0.085\mathrm{fm}$. In the momentum derivative approach, we can directly evaluate the mean square radii for the electric, magnetic, and axial-vector form factors, and also the magnetic moment without the $q^2$ extrapolation to the zero momentum point. These results are compared with the ones determined by the standard method, where the $q^2$ extrapolations of the corresponding form factors are carried out by fitting models. We find that the new results from the momentum derivative method are obtained with a larger statistical error than the standard method, but with a smaller systematic error associated with the data analysis. Within the total error range of the statistical and systematic errors combined, the two results are in good agreement. On the other hand, two variations of the momentum derivative of the induced pseudoscalar form factor at the zero momentum point show some discrepancy. It seems to be caused by a finite volume effect, since a similar trend is not observed on a large volume, but seen on a small volume in our pilot calculations at a heavier pion mass of $M_{\pi }=510\mathrm{MeV}$. Furthermore, we discuss an equivalence between the momentum derivative method and the similar approach with the point splitting vector current.

中文翻译：

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计算Nf=2+1lattice QCD atMπ=138MeVon a(5.5fm)3volume 中核子形状因子的导数

我们提出了关于动量传递的等向量核子形状因子的一阶导数的直接计算 $q^2$使用坐标空间中核子三点函数的低矩。我们的数值模拟是使用$N_F=2+1$ 无扰动地 $哦(\mathrm{一种})$- 改进了物理点附近的威尔逊夸克作用和岩崎规范作用，对应于π质量 ${米}_{\pi }=138\mathrm{电动汽车}$, 在 $(5.5\mathrm{调频}{)}^4$ 单晶格间距的晶格 $\mathrm{一种}=0.085\mathrm{调频}$. 在动量导数方法中，我们可以直接计算电、磁和轴矢量形状因子的均方半径，以及磁矩，无需$q^2$外推到零动量点。将这些结果与通过标准方法确定的结果进行比较，其中$q^2$相应形状因子的外推是通过拟合模型进行的。我们发现动量导数法获得的新结果比标准方法具有更大的统计误差，但与数据分析相关的系统误差更小。在统计误差和系统误差相结合的总误差范围内，两者的结果吻合良好。另一方面，在零动量点诱导伪标量形状因子的动量导数的两个变化显示出一些差异。这似乎是由有限体积效应引起的，因为在大体积上没有观察到类似的趋势，但在我们的试点计算中，在较小的体积上看到了更重的 pion 质量${米}_{\pi }=510\mathrm{电动汽车}$. 此外，我们讨论了动量导数方法与点分裂矢量电流的类似方法之间的等效性。