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

TL;DR

This work develops a gauge-invariant, amplitude-level framework for the double bremsstrahlung process $e^+e^- \to \nu_e \bar{\nu}_e \gamma \gamma$ within the Coherent Exclusive Exponentiation (CEEX) scheme, enabling precise separation of infrared and collinear structures. By analyzing both $Z$-exchange and $W$-exchange channels and employing a contact-interaction expansion for $W$, the authors construct explicit gauge-invariant blocks of the spin amplitude and identify how soft-photon factors factorize from Born-like currents. The study also details extrapolation strategies for CEEX, clarifying how higher-order terms can be systematically reorganized into $\beta$-orders to improve perturbative convergence, with practical implications for KKMC implementations. The results provide a robust, modular blueprint for including multi-photon emissions in neutrino channels, with potential extensions to higher orders and broader electroweak processes.

Abstract

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 nu_e bar nu_e 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 nu_e channel.

Figures (5)

• Figure 1: The Feynman diagrams for $e^+ e^- \rightarrow \bar{\nu}_e \nu_e \gamma$.
• Figure 2: Double emission from electron
• Figure 3: Single emission from electron and positron
• Figure 4: Single and double emission from W
• Figure 5: Four boson coupling and coupling for unphysical $\chi$ field.