Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry

TL;DR

This work provides a comprehensive, practical framework for two-component spinor techniques in quantum field theory and supersymmetry. It develops complete Feynman rules for fermions in two-component notation, covering external lines, propagators, and interactions, and connects these to four-component formalisms and helicity methods. The mass-diagonalization formalism, Majorana/Dirac classifications, and extensive SM/MSSM examples establish the method’s utility for calculating cross-sections, decays, and loop corrections. Appendices extend the formalism to Euclidean and dimensional regularization contexts, and to a dictionary translating between two- and four-component languages, making it a versatile toolkit for particle physics computations.

Abstract

Two-component spinors are the basic ingredients for describing fermions in quantum field theory in four space-time dimensions. We develop and review the techniques of the two-component spinor formalism and provide a complete set of Feynman rules for fermions using two-component spinor notation. These rules are suitable for practical calculations of cross-sections, decay rates, and radiative corrections in the Standard Model and its extensions, including supersymmetry, and many explicit examples are provided. The unified treatment presented in this review applies to massless Weyl fermions and massive Dirac and Majorana fermions. We exhibit the relation between the two-component spinor formalism and the more traditional four-component spinor formalism, and indicate their connections to the spinor helicity method and techniques for the computation of helicity amplitudes.

Figures (69)

• Figure 5.1: The two-component field labeling conventions for external Dirac fermion lines in a Feynman diagram for a physical process. The top row corresponds to an initial state electron, the second row to an initial state positron, the third row to a final state electron, and the fourth row to a final state positron. The labels above each line are the two-component field names. The corresponding conventions for a massless neutrino are obtained by deleting the diagrams with $\bar{e}$ or ${\bar{e}}^\dagger$, and changing $e$ and ${e}^\dagger$ to $\nu$ and $\nu^\dagger$, respectively.
• Figure 5.2: The two-component field labeling conventions for external Majorana fermion lines in a Feynman diagram for a physical process. The top row corresponds to an initial state neutralino, and the second row to a final state neutralino. The labels above each line are the two-component field names. (The neutralino is its own antiparticle.)
• Figure 5.3: The two-component Feynman rules for the QED vertex. Following the conventions outlined in Section \ref{['sec:nomenclature']}, we label these rules with the $({\frac{1}{2}}$12$,0)$ [left-handed] fields $e$ and $\bar{e}$, which comprise the Dirac electron. Note that $Q_e=-1$, and the electromagnetic coupling constant $e$ (not to be confused with the two-component electron field that is denoted by the same letter) is conventionally defined such that $e>0$ [cf. Fig. \ref{['SMintvertices']}].
• Figure 5.4: Tree-level $s$-channel Feynman diagrams for $e^- e^+\to e^- e^+$, with the external lines labeled according to the particle names. The initial state is on the left, and the final state is on the right. Thus, the physical momentum flow of the external particles, as well as the flow of the labeled charges, are indicated by the arrows adjacent to the corresponding fermion lines in the upper left diagram.
• Figure 5.5: Tree-level $s$-channel Feynman diagrams for $e^+ e^-\to e^+ e^-$. These diagrams are the same as in Fig. \ref{['fig:Bhabhaparticlelabels']}, but with the external lines relabeled by the two-component fermion fields according to the conventions of Fig. \ref{['labelconvention']}.