We employ the worldline formalism to derive a series representation of the low-energy limit of the N-photon amplitude in a constant background field for both scalar and spinor QED. The amplitudes are then written in terms of a single proper-time integral. The above-mentioned series representation terminates when considering a constant crossed field. This allows us to obtain even more compact expressions for these particular amplitudes for which the result of the proper-time integral, for fixed parameters, takes the form of a factorial function. In addition, we derive all helicity components of these amplitudes and express them explicitly in terms of Bernoulli numbers and spinor products.

中文翻译：

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恒定场中 N 光子振幅的低能量极限：第二部分


我们采用世界线形式来推导出标量和旋量 QED 在恒定背景场中 N 光子振幅的低能量极限的系列表示。然后，用单个适当时间的积分来写振幅。当考虑一个常数交叉场时，上述级数表示终止。这使我们能够获得这些特定振幅的更紧凑的表达式，对于固定参数，适当时间积分的结果采用阶乘函数的形式。此外，我们推导了这些振幅的所有螺旋度分量，并用伯努利数和旋量积明确表示。