Renormalized Effective Actions in Radially Symmetric Backgrounds I: Partial Wave Cutoff Method

Abstract

The computation of the one-loop effective action in a radially symmetric background can be reduced to a sum over partial-wave contributions, each of which is the logarithm of an appropriate one-dimensional radial determinant. While these individual radial determinants can be evaluated simply and efficiently using the Gel'fand-Yaglom method, the sum over all partial-wave contributions diverges. A renormalization procedure is needed to unambiguously define the finite renormalized effective action. Here we use a combination of the Schwinger proper-time method, and a resummed uniform DeWitt expansion. This provides a more elegant technique for extracting the large partial-wave contribution, compared to the higher order radial WKB approach which had been used in previous work. We illustrate the general method with a complete analysis of the scalar one-loop effective action in a class of radially separable SU(2) Yang-Mills background fields. We also show that this method can be applied to the case where the background gauge fields have asymptotic limits appropriate to uniform field strengths, such as for example in the Minkowski solution, which describes an instanton immersed in a constant background. Detailed numerical results will be presented in a sequel.

35 pages

References in corpus (21)

• On the metastability of the Standard Model vacuum
• Heisenberg-Euler Effective Lagrangians : Basics and Extensions
• Functional determinants by contour integration methods
• Functional determinants for general Sturm-Liouville problems
• Beyond the thin-wall approximation : precise numerical computation of prefactors in false vacuum decay
• Canonical Transformations in Quantum Mechanics
• Quantum Fluctuations around the Electroweak Sphaleron
• Functional determinants for radial operators
• Two-loop self-dual Euler-Heisenberg Lagrangians (I): Real part and helicity amplitudes
• Quantum fluctuations of the electroweak sphaleron: Erratum and Addendum
• One-loop corrections to the metastable vacuum decay
• A Consistent Calculation of Bubble-Nucleation Rates
• An Exact QED_{3+1} Effective Action
• Effective Energy for QED${}_{2+1}$ with Semi-Localized Static Magnetic Fields: A Solvable Model
• Precise Quark Mass Dependence of Instanton Determinant
• Calculation of QCD Instanton Determinant with Arbitrary Mass
• Fluctuations about Cosmological Instantons
• Massive field contributions to the QCD vacuum tunneling amplitude
• Instanton Determinant with Arbitrary Quark Mass: WKB Phase-shift Method and Derivative Expansion
• One-loop fermionic corrections to the instanton transition in two dimensional chiral Higgs model
• Confining Properties of the Homogeneous Self-Dual Field and the Effective Potential in SU(2) Yang-Mills Theory

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