4 - Elementary processes

Summary

The preceding chapter showed that the coupling constant must be renormalized. In the present chapter we shall describe, to lowest order in the coupling, the other basic or elementary processes that give divergent results. These divergencies are compensated by renormalizing other parameters of the theory, the mass and field strength. To render the theory completely well defined, another renormalization is needed — the addition of an (infinite) constant Λ₀ to the Lagrange function. This constant corresponds to an infinite renormalization of the “zero-point energy” — in rough terms, to zeroth order in the coupling, the field theory corresponds to an infinite collection of harmonic oscillators with the a-th oscillator having a zero-point or ground state energy given by ½ω_a, and the sum Σ_a½ω_a diverges. Higher order terms in the coupling give additional zero-point divergencies. This divergence was previously hidden in a re-definition of the measure [dϕ]. However, the divergence depends upon the scalar field's mass, and it is best to define the functional integral, and thereby the theory, in a mass-independent fashion. This requires that the zero-point divergence be displayed explicitly. Moreover, when gravitational couplings are introduced into theory, this additional constant in the Lagrange function appears as a contribution to the “cosmological constant” Λ₀. The exceedingly small value of this constant on the scale set by elementary particles is an outstanding problem in contemporary physics, and this puzzle further motivates our discussion of Λ₀.