Field theory techniques: (I) The background geodesic expansion and heat kernel methods; (II) The 1/N expansion and the Nambu-Jona-Lasinio model

Keith M Costa, Purdue University

Abstract

The reparameterization invariant one-loop radiative corrections for non-polynomial scalar field theories are found using the background geodesic expansion together with heat kernel techniques regulated with an ultraviolet cutoff. These theories have dynamics invariant under a symmetry group ${\cal G}$ with fields that transform linearly only under a subgroup ${\cal H}$ of ${\cal G}$. The fields are coordinates on the coset manifold ${\cal G}$/${\cal H}$, generated by the action of the group on the vacuum state identified with the origin. In cases where the fields are complex, the coset manifold is Kahlerian, guaranteeing hermiticity of the lagrangian and greatly simplifying the counterterm structure. This approach is ideally suited to the study of radiative corrections of the non-linear sigma model lagrangian describing the spontaneous breakdown of a symmetry group ${\cal G}$ to an unbroken subgroup ${\cal H}$. In Part One, it is applied to the heavy Higgs limit of the Glashow-Weinberg-Salam model where new higher dimensional longitudinal gauge boson-fermion interactions are found with effects linear in the Yukawa couplings. In pp-collisions at SSC energies, these interactions result in larger than expected yields of heavy fermions and longitudinally polarized gauge bosons. The complex fields case is studied in the CP$\sp{n}$ model. In Part Two, the 1/N expansion is employed to study a model field theory outside of weak coupling perturbation theory and search for collective states in spinor theories with four-fermi interactions. These bosonic states are introduced via lagrange multipliers into the generating functional and become dynamic in the 1/N expansion. The globally invariant Nambu-Jona-Lasinio (NJL) model in 2+1 dimensions is shown to be equivalent to local gauge theory. This equivalence is demonstrated for the renormalized theories in the 1/N expansion utilizing an unconventional, cutoff dependent bare coupling constant to study the limit of weak or strong bare couplings. The weakly coupled abelian NJL model is renormalized to order 1/N and compared to a renormalized strongly coupled QED3. Next the U(2) globally invariant NJL model is studied in the broken phase and renormalized to leading order. The resulting $U$(1) $\times$ $U$(1) gauge invariant theory is shown to be equivalent to a spontaneously broken $U$(2) gauge theory analyzed in the 1/N expansion.

Degree

Ph.D.

Advisors

Clark, Purdue University.

Subject Area

Particle physics

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