Study of Scheme Transformations to Remove Higher-Loop Terms in the β\beta Function of a Gauge Theory

Since three-loop and higher-loop terms in the $\beta$ function of a gauge theory are scheme-dependent, one can, at least for sufficiently small coupling, carry out a scheme transformation that removes these terms. A basic question concerns the extent to which this can be done at an infrared fixed point of an asymptotically free gauge theory. This is important for quantitative analyses of the scheme dependence of such a fixed point. Here we study a scheme transformation $S_{R,m}$ with $m \ge 2$ that is constructed so as to remove the terms in the beta function at loop order $\ell=3$ to $\ell=m+1$, inclusive. Starting from an arbitrary initial scheme, we present general expressions for the coefficients of terms of loop order $\ell$ in the beta function in the transformed scheme from $\ell=m+2$ up to $\ell=8$. Extending a previous study of $S_{R,2}$, we investigate the range of applicability of the $S_{R,3}$ scheme transformation in an asymptotically free SU($N_c$) gauge theory with an infrared zero in $\beta$ depending on the number, $N_f$, of fermions in the theory. We show that this $S_{R,3}$ scheme transformation can only be applied self-consistently in a restricted range of $N_f$ with a correspondingly small value of infrared fixed-point coupling. We also study the effect of higher-loop terms on the beta function of a U(1) gauge theory.Comment: 15 page

Higher-Loop Calculations of the Ultraviolet to Infrared Evolution of a Vectorial Gauge Theory in the Limit Nc→∞N_c \to \infty, Nf→∞N_f \to \infty with Nf/NcN_f/N_c Fixed

We consider an asymptotically free vectorial SU($N_c$) gauge theory with $N_f$ fermions in the fundamental representation and analyze higher-loop contributions to the evolution of the theory from the ultraviolet to the infrared in the limit where $N_c \to \infty$ and $N_f \to \infty$ with $r=N_f/N_c$ a fixed, finite constant. We focus on the case where the $n$-loop beta function has an infrared zero, at $\xi=\xi_{IR,n\ell}$, where $\xi=\alpha N_c$. We give results on $\xi_{IR,n\ell}$, the anomalous dimension of the fermion bilinear evaluated at $\xi_{IR,n\ell}$, denoted $\gamma_{IR,n\ell}$, and certain structural properties of the beta function, $\beta_\xi$. The approach to this limit is investigated, and it is shown that the leading correction terms are strongly suppressed, by the factor $1/N_c^2$. This provides an understanding of a type of approximate universality in calculations for moderate values of $N_c$ and $N_f$, namely that $\alpha_{IR,n\ell}N_c$, $\gamma_{IR,n\ell}$, and structural properties of the beta function are similar in theories with different values of $N_c$ and $N_f$ provided that they have similar values of $N_f/N_c$. We give results up to four loops for nonsupersymmetric theories and up to three loops for supersymmetric theories.Comment: 22 pages, latex, 2 figure

Some Recent Results on Models of Dynamical Electroweak Symmetry Breaking

We review some recent results on models of dynamical electroweak symmetry breaking involving extended technicolor.Comment: 15 pages 3 figures, review talk given at SCGT06, International Workshop on Strongly Coupled Gauge Theories, Nagoya, Nov. 2006, to appear in the Proceedings of SCGT06, eds. M. Tanabashi, M. Harada, and K. Yamawak

Studies on the UV to IR Evolution of Gauge Theories and Quasiconformal Behavior

We describe recent results from our studies of the UV to IR evolution of asymptotically free vectorial gauge theories and quasiconformal behavior. These include higher-loop calculations of the IR zero of the beta function and of the anomalous dimension of the fermion bilinear. Effects of scheme-dependence of higher-loop results are assessed in detail. Applications to models with dynamical electroweak symmetry breaking are discussed.Comment: Talk given at the KMI-GCOE Workshop Strong Coupling Gauge Theories in the LHC Perspective (SCGT 12), Nagoya University, Dec. 2012, to appear in the proceeding

Higher-Loop Structural Properties of the β\beta Function in Asymptotically Free Vectorial Gauge Theories

We investigate some higher-loop structural properties of the $\beta$ function in asymptotically free vectorial gauge theories. Our main focus is on theories with fermion contents that lead to an infrared (IR) zero in $\beta$. We present analytic and numerical calculations of the value of the gauge coupling where $\beta$ reaches a minimum, the value of $\beta$ at this minimum, and the slope of $\beta$ at the IR zero, at two-, three-, and four-loop order. The slope of $\beta$ at the IR zero is relevant for estimates of a dilaton mass in quasiconformal gauge theories. Some inequalities are derived concerning the dependence of the above quantities on loop order. A general inequality is derived concerning the dependence of the shift of the IR zero of $\beta$, from the $n$-loop to the $(n+1)$-loop order, on the sign of the $(n+1)$-loop coefficient in $\beta$. Some results are also given for gauge theories with ${\cal N}=1$ supersymmetry.Comment: 22 pages, 4 figure