One occasionally encounters the misconception that two-component spinor notation is somehow inherently ill-suited or unwieldy for practical use. Perhaps this is due in part to a lack of examples of calculations using two-component language in the pedagogical literature. In this chapter, we seek to dispel this idea by presenting Feynman rules for external fermions using two-component spinor notation, intended for practical calculations of cross sections, decays, and radiative corrections.

Consider a collection of $N$ two-component left-handed $(\frac{1}{2},0)$ fermions. The corresponding free-field Lagrangian is invariant under a global $\text{U}\left(N\right)$ symmetry. When a mass term and interactions are added to the theory, the global $\text{U}\left(N\right)$ symmetry is broken down to a discrete ${\mathbb{Z}}_2$ symmetry that reflects the fact that any term in the Lagrangian must contain an even number of fermion fields.

In Chapter 1, we focused on quantum field theories of free fermions. In order to construct renormalizable interacting quantum field theories, we must introduce additional fields. The requirement of renormalizability imposes two constraints. First, the couplings in the interaction Lagrangian must have nonnegative mass dimension.

In this chapter, we devise a set of Feynman rules to describe matrix elements of processes involving spin-1/2 fermions. The rules are developed for two-component fermions and are then applied to tree-level decay and scattering processes and the fermion self-energy functions in the one-loop approximation.

Despite the successes of the Standard Model, it seems very likely that new physics is present at the TeV scale.

In this chapter, we present example Feynman diagrammatic calculations of supersymmetric decay and scattering processes, employing the two-component fermion techniques developed in . We present the first calculations in some detail to get the reader acquainted with the technical details.

In this chapter, we present example one-loop Feynman diagrammatic calculations in the Standard Model and MSSM, employing the two-component fermion techniques developed in Chapter 2.

Despite the inherent beauty of supersymmetric field theories, we know that supersymmetry cannot be an exact symmetry of Nature. The observed spectrum of fundamental particles does not consist of mass-degenerate supermultiplets. Hence, supersymmetry must be broken. In this chapter, we shall discuss how supersymmetry (SUSY) breaking can arise.

So far, the experimental study of supersymmetry has unfortunately been confined to setting limits. As noted in Section 13.6, there can be indirect signals for supersymmetry from processes that are rare or forbidden in the Standard Model but have contributions from sparticle loops.

Quantum fields possess definite transformation properties under the Lorentz and Poincaré groups. In the 1960s, with the discovery of new global internal symmetry groups such as the flavor SU(3) group of the quark model (based on the three quarks $u$, $d$, and $s$ that were known at that time), the following question was considered.

There are a number of reasons to consider realistic models of supersymmetry that go beyond the MSSM. The first reason is connected to the $\mu$ problem, which was introduced in Section 14.6. Naively, one would expect the dimensionful parameter $\mu$ to be of the same order of magnitude as the largest possible mass scale.