In this paper we analyze for the first time the γNΔ excitation vertex from the point of view of the dynamics of the Δ field, ${\mathrm{\Psi }}_{\mu }$. That is, we look for the value of the Z parameter, present in all contact invariant Δ-field interaction Lagrangians and usually regarded as redundant, and shift it to contact background non-Δ resonant amplitudes (in this sense it is called “redundant”) by imposing that the ${\mathrm{\Psi }}_0$ has no dynamics, instead of readjusting the background coupling constants. We do this within an unitarized model that comprises the Δ-direct amplitude plus background contributions including the Δ-cross term, nucleon Born and meson exchange ones, already implemented in previous works of πN scattering, photo-production and weak-π production. Also we analyze the use of a πNΔ decay vertex interaction containing both first ($I_1$) and second ($I_2$) order derivative contributions, as required by renormalization and power counting considerations, in building the $\gamma N\to \mathrm{\Delta }\to \pi N$ amplitude, in contrast with other works where only $I_1$ or $I_2$ is adopted. It is shown that the description of the $\gamma N\to \pi N$ process, following these prescriptions, is improved. This is a first step: we plan to introduce final state interactions (FSI) in the future, following our previous work in which FCI were introduced using the $I_1$ interaction alone.

在本文中，我们首次从 Δ 场动力学的角度分析γN Δ 激励顶点，${\mathrm{\Psi }}_{\mu }$。也就是说，我们寻找Z参数的值，该值存在于所有接触不变 Δ 场相互作用拉格朗日量中，通常被视为冗余，并将其转移到接触背景非 Δ 共振振幅（在这个意义上，它被称为“冗余”） ）通过强加${\mathrm{\Psi }}_0$没有动力学，而不是重新调整背景耦合常数。我们在一个酉化模型中做到这一点，该模型包括 Δ-直接振幅加上背景贡献，包括 Δ-交叉项、核子玻恩和介子交换项，已经在之前的 πN 散射、光产生和弱 π 产生工作中实现。我们还分析了πN Δ 衰减顶点相互作用的使用，其中首先包含两者 (${我}_1$）和第二个（${我}_2$）阶导数贡献，根据重正化和幂计数考虑的要求，在构建$\gamma \mathrm{不是}\to \mathrm{\Delta }\to \pi \mathrm{不是}$幅度，与其他作品相比，只有${我}_1$金子${我}_2$被采纳。可以看出，描述的$\gamma \mathrm{不是}\to \pi \mathrm{不是}$遵循这些规定，流程得到改进。这是第一步：我们计划在未来引入最终状态交互（FSI），继我们之前的工作中使用 FCI 引入${我}_1$单独互动。