Treatment of a System with Explicitly Broken Gauge Symmetries

A system in which the free part of the action possesses a gauge symmetry that is not respected by the interacting part presents problems when quantized. We illustrate how the Dirac constraint formalism can be used to address this difficulty by considering an antisymmetric tensor field interacting with a spinor field.Comment: 10 pages, LaTeX2e, typos correcte

The Effective Potential in Non-Conformal Gauge Theories

By using the renormalization group (RG) equation it has proved possible to sum logarithmic corrections to quantities that arise due to quantum effects in field theories. In particular, the effective potential V in the Standard Model in the limit that there are no massive parameters in the classical action (the "conformal limit") has been subject to this analysis, as has the effective potential in a scalar theory with a quartic self coupling and in massless scalar electrodynamics. Having multiple coupling constants and/or mass parameters in the initial action complicates this analysis, as then several mass scales arise. We show how to address this problem by considering the effective potential in scalar electrodynamics when the scalar field has a tree level mass term. In addition to summing logarithmic corrections by using the RG equation, we also consider the consequences of the condition V'(v)=0 where v is the vacuum expectation value of the scalar. If V is expanded in powers of the logarithms that arise, then it proves possible to show that either v is zero or that V is independent of the scalar. (That is, either there is no spontaneous symmetry breaking or the vacuum expectation value is not determined by minimizing V as V is "flat".

Renormalization Mass Scale and Scheme Dependence in the Perturbative Contribution to Inclusive Semileptonic bb Decays

We examine the perturbative calculation of the inclusive semi-leptonic decay rate $\Gamma$ for the $b$-quark, using mass-independent renormalization. To finite order of perturbation theory the series for $\Gamma$ will depend on the unphysical renormalization scale parameter $\mu$ and on the particular choice of mass-independent renormalization scheme; these dependencies will only be removed after summing the series to all orders. In this paper we show that all explicit $\mu$-dependence of $\Gamma$, through powers of ln$(\mu)$, can be summed by using the renormalization group equation. We then find that this explicit $\mu$-dependence can be combined together with the implicit $\mu$-dependence of $\Gamma$ (through powers of both the running coupling $a(\mu)$ and the running $b$-quark mass $m(\mu)$) to yield a $\mu$-independent perturbative expansion for $\Gamma$ in terms of $a(\mu)$ and $m(\mu)$ both evaluated at a renormalization scheme independent mass scale $I\!\!M$ which is fixed in terms of either the "$\overline{MS}$ mass" $\overline{m}_b$ of the $b$ quark or its pole mass $m_{pole}$. At finite order the resulting perturbative expansion retains a degree of arbitrariness associated with the particular choice of mass-independent renormalization scheme. We use the coefficients $c_i$ and $g_i$ of the perturbative expansions of the renormalization group functions $\beta(a)$ and $\gamma(a)$, associated with $a(\mu)$ and $m(\mu)$ respectively, to characterize the remaining renormalization scheme arbitrariness of $\Gamma$. We further show that all terms in the expansion of $\Gamma$ can be written in terms of the $c_i$ and $g_i$ coefficients and a set of renormalization scheme independent parameters $\tau_i$.Comment: 26 pages, 4 figures, typo correcte

Gauge Dependence in Chern-Simons Theory

We compute the contribution to the modulus of the one-loop effective action in pure non-Abelian Chern-Simons theory in an arbitrary covariant gauge. We find that the results are dependent on both the gauge parameter ($\alpha$) and the metric required in the gauge fixing. A contribution arises that has not been previously encountered; it is of the form $(\alpha / \sqrt{p^2}) \epsilon _{\mu \lambda \nu} p^\lambda$. This is possible as in three dimensions $\alpha$ is dimensionful. A variant of proper time regularization is used to render these integrals well behaved (although no divergences occur when the regularization is turned off at the end of the calculation). Since the original Lagrangian is unaltered in this approach, no symmetries of the classical theory are explicitly broken and $\epsilon_{\mu \lambda \nu}$ is handled unambiguously since the system is three dimensional at all stages of the calculation. The results are shown to be consistent with the so-called Nielsen identities which predict the explicit gauge parameter dependence using an extension of BRS symmetry. We demonstrate that this $\alpha$ dependence may potentially contribute to the vacuum expectation values of products of Wilson loops.Comment: 17 pp (including 3 figures). Uses REVTeX 3.0 and epsfig.sty (available from LANL). Latex thric

Off-Diagonal Elements of the DeWitt Expansion from the Quantum Mechanical Path Integral

The DeWitt expansion of the matrix element M_{xy} = \left\langle x \right| \exp -[\case{1}{2} (p-A)^2 + V]t \left| y \right\rangle, $(p=-i\partial)$ in powers of $t$ can be made in a number of ways. For $x=y$ (the case of interest when doing one-loop calculations) numerous approaches have been employed to determine this expansion to very high order; when $x \neq y$ (relevant for doing calculations beyond one-loop) there appear to be but two examples of performing the DeWitt expansion. In this paper we compute the off-diagonal elements of the DeWitt expansion coefficients using the Fock-Schwinger gauge. Our technique is based on representing $M_{xy}$ by a quantum mechanical path integral. We also generalize our method to the case of curved space, allowing us to determine the DeWitt expansion of \tilde M_{xy} = \langle x| \exp \case{1}{2} [\case{1}{\sqrt {g}} (\partial_\mu - i A_\mu)g^{\mu\nu}{\sqrt{g}}(\partial_\nu - i A_\nu) ] t| y \rangle by use of normal coordinates. By comparison with results for the DeWitt expansion of this matrix element obtained by the iterative solution of the diffusion equation, the relative merit of different approaches to the representation of $\tilde M_{xy}$ as a quantum mechanical path integral can be assessed. Furthermore, the exact dependence of $\tilde M_{xy}$ on some geometric scalars can be determined. In two appendices, we discuss boundary effects in the one-dimensional quantum mechanical path integral, and the curved space generalization of the Fock-Schwinger gauge.Comment: 16pp, REVTeX. One additional appendix concerning end-point effects for finite proper-time intervals; inclusion of these effects seem to make our results consistent with those from explicit heat-kernel method

On the Standard Approach to Renormalization Group Improvement

Two approaches to renormalization-group improvement are examined: the substitution of the solutions of running couplings, masses and fields into perturbatively computed quantities is compared with the systematic sum of all the leading log (LL), next-to-leading log (NLL) etc. contributions to radiatively corrected processes, with n-loop expressions for the running quantities being responsible for summing N^{n}LL contributions. A detailed comparison of these procedures is made in the context of the effective potential V in the 4-dimensional O(4) massless $\lambda \phi^{4}$ model, showing the distinction between these procedures at two-loop order when considering the NLL contributions to the effective potential V.Comment: 6 page