Perturbative renormalization with flow equations in Minkowski space.

*(English)*Zbl 0876.35097Prior to 1974 renormalization was an ad hoc technique used mainly in quantum field theory, and related statistical mechanics. The article of Wilson and Kogut [Phys. Rep. 12C, 75 (1975)] and subsequent articles by Wilson and his collaborators established the theory of renormalization group and of an effective Lagrangian. The subsequent article by Polchinski [Nucl. Phys. B 231, 269-295 (1984)] applied these techniques to the renormalization of perturbative field theory, bypassing previously used studies of convergence or divergence of Feynman diagram representations. In a series of articles two of the authors (Keller and Kopper) extended Polchinski’s theory and showed how the flow equations technique can be applied to various problems of perturbative field theory. In particular, their interest was concentrated on the renormalization of both massless and massive \(\Phi^4_4\) complementing Symanzik’s construction.

The present paper is part of this program. The article begins with the renormalization at zero momentum, similarly to the Bogoliubov-Parasiuk and Hepp subtraction technique. The authors derive renormalized \(n\)-point functions. They observe that several generalizations of their technique can be derived routinely. First, flow equations are derived for one-particle irreducible Green’s functions in Euclidean theory. Relativistic effects are taken care by transition to Minkowski space. Specifically, a regularized propagator analytic in momenta is given by: \[ \int^\alpha_{\alpha_0} d\alpha'e^{-\alpha'(p\eta p+(i+\varepsilon)m^2)}, \] where \(\eta\) is the \(\varepsilon\)-regularized Minkowski diagonal \(4\times 4\) matrix, whose diagonal entries are \(-i+\varepsilon\), \(i+\varepsilon\), \(i+\varepsilon\), \(i+\varepsilon\). Because this regularization the breaks the Lorentz invariance, the interaction Lagrangian contains a countering term. Differentiating with respect to the time scale \(\alpha\) yields flow equations of the relativistic theory. Green’s functions become Lorentz invariant as \(\varepsilon\to 0\).

The present paper is part of this program. The article begins with the renormalization at zero momentum, similarly to the Bogoliubov-Parasiuk and Hepp subtraction technique. The authors derive renormalized \(n\)-point functions. They observe that several generalizations of their technique can be derived routinely. First, flow equations are derived for one-particle irreducible Green’s functions in Euclidean theory. Relativistic effects are taken care by transition to Minkowski space. Specifically, a regularized propagator analytic in momenta is given by: \[ \int^\alpha_{\alpha_0} d\alpha'e^{-\alpha'(p\eta p+(i+\varepsilon)m^2)}, \] where \(\eta\) is the \(\varepsilon\)-regularized Minkowski diagonal \(4\times 4\) matrix, whose diagonal entries are \(-i+\varepsilon\), \(i+\varepsilon\), \(i+\varepsilon\), \(i+\varepsilon\). Because this regularization the breaks the Lorentz invariance, the interaction Lagrangian contains a countering term. Differentiating with respect to the time scale \(\alpha\) yields flow equations of the relativistic theory. Green’s functions become Lorentz invariant as \(\varepsilon\to 0\).

Reviewer: V.Komkov (Roswell)