Modern Elementary Particle Physics Explaining and Extending the Standard Model

Now that we have learned (1) the notation of the Dirac equation in order to express the spin structure, (2) the requirements of gauge invariance that tell us to begin with a free particle Lagrangian and rewrite it with a covariant derivative, and (3) the idea of internal symmetries, we are finally ready to write down the full Lagrangian that describes the world we see. In order to describe the particles and interactions known today, three internal symmetries are needed. We do not know yet why there are three, or whether there will be more, or why these three are the ones they are, but it is a remarkable accomplishment to have discovered them. At the present time all experiments are consistent with the notion that the three symmetries are necessary and sufficient to describe the interactions of the known particles, to form our world. It is easiest to describe how these symmetries act in the language of group theory, so any reader who needs to review that way of describing invariances should turn to Appendix B before proceeding. As always in this book, we add classical gravity to the other forces.

All particles appear to have a U(1) invariance. It is like the U(1) invariance or phase invariance described in Chapters 2 and 3. That invariance was related to the electromagnetic interaction. However, since the invariance is an internal property of particles, we have no reason immediately to identify it with electromagnetism. We simply have a U(1) or phase invariance whose connection with electromagnetism will be deduced later from physical arguments. The gauge boson required by the invariance of the theory under the U(1) transformations will be called Bμ . The index μ is present since Bμ must transform under spatial rotations the same way the ordinary derivative does, thus guaranteeing the associated particle has spin one. We will reserve A μ for the name of the photon field. The connection of A μ and Bμ will be determined in Section 7.3.

All particles have a second internal invariance, under a set of transformations that form an SU(2) group, called the electroweak SU(2) invariance. These then lead to a non-Abelian gauge (phase) invariance, analogous to the strong isospin invariance of Section 4.1.