Chapter 9 - Neutrino and Electron Scattering from Point Particles

Summary

Introduction

We have seen in Chapter 8, that the standard model for the leptons developedby Salam [37] and Weinberg [157], which was later extended to the quarksector by Glashow [64], unifies weak and electromagnetic interactions. Itpredicts, in a unique way, the interaction Lagrangian for charge changing(CC) weak interactions of leptons and neutrinos of all flavors with chargedgauge vector bosons, W^± and theelectromagnetic interactions of charged leptons with photons. It alsopredicts the existence of neutral current(NC) weak interactions of chargedleptons and neutrinos of all flavors with the neutral gauge boson,Z⁰. The strength of the interaction of thecharged, neutral, and electromagnetic currents with theW^±,Z⁰, and A gauge bosons aredescribed in terms of the weak coupling constants g,electromagnetic coupling constant e, and a free parameterθ_W called the weak mixing angle.Specifically, the interaction Lagrangian discussed in Chapter 8 is writtenhere again as:

where Z_μ, andA_μ are the charged, neutral andelectromagnetic gauge fields and

with is the fine structure constant. In the following sections, we use theinteraction Lagrangian in Eq. (9.1) to calculate the cross sections for someweak and electromagnetic processes using point particles, that is, chargedleptons and neutrinos.

Scattering

This scattering process can take place through an electromagnetic processmediated by a virtual photon as well as by the weak neutral current mediatedby a Z⁰ boson in the standard model [400].

When an electron interacts with a photon field (Figure 9.1(a)), theinteraction Lagrangian is given by:

and when it interacts with the Z⁰ boson field (Figure 9.2(a)), theLagrangian is given by:

Using the aforementioned Lagrangians corresponding to Figure and followingthe Feynman rules the transition amplitude for the process

mediated through virtual photon exchange of momentum in the lowest order, maybe written as:

and for the process mediated through the virtual Z⁰ exchange,shown by the Feynman diagram in Figure 9.2(b) as:

where. The process proceeding through Z⁰ is highly suppressed ascompared to the photon exchange therefore, in the present case, we presentthe cross section for the process given in Eq. (9.7) mediating through-exchange.