Gauge invariance, infrared/collinear singularities and tree level matrix element for e+ e- to nuₑ bar nuₑ gamma gamma

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One of the necessary steps in constructing high precision option of KKMC was to install the double bremsstrahlung matrix element for the process e+ e- to nuₑ bar nuₑ into the scheme of Coherent Exclusive Exponentiation. The process is also interesting because of gauge cancellation of contributions for photon emission from incoming fermion lines and t-channel W. The QED U(1) gauge properties require terms of the triple and quatric gauge couplings to be taken into considerations as well. Thanks to expansion starting from the approximation of contact interaction, good example to study the internal structure of the amplitude is available. In the developed scheme, natural separation of the complete amplitude into gauge invariant parts is straightforward. Each part has well defined physical interpretation, which after partial integration over phase space provides terms: infrared singular, leading log, next-to-leading-log, etc. Contributions related to triple and quatric gauge coupling of W (extracted with the help of expansion around contact W-interaction), have been ordered as well. The separation is also helpful, to define extrapolation/reduction procedure of CEEX exponentiation for the nuₑ channel.