Renormalization Group Equations In General Gauge Field Theories

This thesis provides an analysis of the renormalization group equations(RGEs) in general gauge field theories.In Chapter 1, a brief review of history and current status of the RGEs is given.In Chapter 2, general gauge field theories and their renormalizability are briefly reviewed. The effective action provides a convenient way to demon-strate the gauge symmetry of renormalized gauge theories. With the help of BRST symmetry, Zinn-Justin equation and effective action, it is shown that the gauge invariant counterterms are sufficient to remove the divergences. Hence general gauge theories are renormalizable.In Chapter 3, a general review of renormalization group is provided. This in-cludes the Kananoff transformation on lattice, and renormalization group as the combination of Kananoff transformation and space(momentum) scaling. The renormalization group in quantum field theory(QFT), as an analogy of that in critical phenomenon, is also simply described in the Wilson approach. The form ofβ-functions andγ-functions in the minimal subtraction(MS) scheme and the modified minimal subtraction(MS) scheme is given.In Chapter 4, the complete set of two-loop RGEs in general gauge field theories is presented, where complex fermions is assumed. The two-loopβ-functions of parameters with mass dimension are derived by introducing a dummy field which has no gauge interaction and does not propagate.In Chapter 5, the two-loop RGEs for the Standard Model(SM) are recalcu-lated as a benchmark check. A new coefficient is found in theβ-function of the quartic coupling and a class of gauge invariants are found to be absent in theβ-functions of hadronic Yukawa couplings. The two-loopβ-function of the Higgs mass parameter is presented in complete form.In Chapter 6, theβ-functions of parameters without mass dimension in gauge theories with multiple U(1) groups are given. Instead of normalizing the abelian gauge fields in canonical forms, the kinetic-mixing terms are retained and the mixing coefficients are treated as free parameters. Their contribu-tions to theβ-functions of parameters without mass dimension are obtained in a straightforward manner from the Feynman diagrams. Theβ-functions of parameters with mass dimension can be derived by introducing a dummy field.In Chapter 7, we conclude with a brief discussion of the RGEs in supersym-metric theories. In supersymmetric theories one must adopt the dimensional reduction(DR) regularization to preserve supersymmetry quantum mechan-ically. Hence a translation into DR results should be performed after one gets the RGEs in MS scheme.