arXiv subjects
Publications and source records attributed to Richard Hong Tuan.
An $α$-parameter representation is derived for gauge field theories.It involves, relative to a scalar field theory, only constants and derivatives with respect to the $α$-parameters. Simple rules are given to obtain the $α$-representation for a Feynman graph with an arbitrary number of loops in gauge theories in the Feynman gauge.
As the number of loops goes to infinity Feynman $α$-parameters undergo a fixing mechanism which entails a gaussian representation for propagators in scalar field theories. Here, we describe this mechanism in the fullest detail. The fixed values are in fact mean-values which can be determined via consistency conditions. The consistency conditions imply that one $α$-parameter is integrated in the usual way and the dependence of the mean-values of the other $α$-parameters on it must be determined. Here we present a method for doing this exactly which requires the solution of an equation system. We present an analytic solution for this equation system in the case of the ladder-graph topology. The Regge behaviour is obtained in a simple way as well as an analytic expression for the leading Regge trajectory. Then, the consistency equations for the two (in the ladder case) independent $α$-parameters mean-values are solved numerically. Agreement with previous determinations of the intercept $α(0)$ is obtained for $α(0) \ \gsim$ 0.3. However, we are able to calculate $α(t/m^2)$ for - 3.6 $\lsim$ $t/m^2$ $\lsim$ 1.8 and find that it is close to linear. We consider the massless limit of the theory and find that the $α$-parameters mean-values and the trajectory $α(t)$ have limits which are independent of the mass, a phenomenon which also occurs for renormalizable theories via the renormalization group equations.
Using the gaussian representation for propagators (which can be proved to be exact in the infinite number of loops limit) we are able to derive the Regge behaviour for ladder graphs of $ϕ^3$ field theory in a completely new way. An analytic expression for the Regge trajectory $α(t/m^2)$ is found in terms of the mean-values of the Feynman $α$-parameters. $α(t/m^2)$ is calculated in the range $- 3.6 < t/m^2 < 0.8$. The intercept $α(0)$ agrees with that obtained from earlier calculations using the Bethe-Salpeter approach for $α(0) \gsim 0.3$.
Recently, we proposed a new approach for calculating Feynman graphs amplitude using the Gaussian representation for propagators which was proven to be exact in the limit of graphs having an infinite number of loops. Regge behavior was also found in a completely new way and the leading Regge trajectory calculated. Here we present symmetry arguments justifying the simple form used for the polynomials in the Feynman parameters $\bar α_{\ell}$, where $\bar α_{\ell}$ is the mean-value for these parameters, appearing in the amplitude for the ladder graphs. (Taking mean-values is equivalent to the Gaussian representation for propagators).