George Leibbrandt

Split Dimensional Regularization for the Coulomb Gauge

A new procedure for regularizing Feynman integrals in the noncovariant Coulomb gauge ~ r ~ A a = 0 is proposed for Yang-Mills theory. The procedure is based on a variant of dimensional regularization, called split dimensional regularization, which leads to internally consistent, ambiguity-free integrals, some of which turn out to be nonlocal .I t is demonstrated that split dimensional regularization yields a one-loop Yang-Mills self-energy, ab , that is nontransverse, but local. Despite the noncovariant nature of the Coulomb gauge, ghosts are necessary in order to satisfy the appropriate Ward/BRS identity. The computed Coulomb-gauge Feynman integrals are applicable to both Abelian and non-Abelian gauge models.

Generalized ward identity for the quark self-energy and the quark-quark-gluon vertex in the light-cone gauge

Abstract A light-cone gauge formalism, developed recently by one of the authors, is applied to the one-loop quark self-energy and quark-quark-gluon vertex function which are shown to respect the generalized Ward identity. The possible renormalization in this prescription is briefly discussed.

Split dimensional regularization for the Coulomb gauge at two loops

Abstract We evaluate the coefficients of the leading poles of the complete two-loop quark self-energy Σ(p) in the Coulomb gauge. Working in the framework of split dimensional regularization, with complex regulating parameters σ and n/2−σ for the energy and space components of the loop momentum, respectively, we find that split dimensional regularization leads to well-defined two-loop integrals, and that the overall coefficient of the leading pole term for Σ(p) is strictly local. Extensive tables showing the pole parts of one- and two-loop Coulomb integrals are given. We also comment on some general implications of split dimensional regularization, discussing in particular the limit σ→1/2 and the subleading terms in the e-expansion of noncovariant integrals.

Renormalization questions in the ligth-cone gauge

Abstract The three-gluon vertex Γ μνϱ abc ( p , 0, − p ) is calculated to one-loop order in the light-cone gauge n σ A σ =0, n 2 =0 , with one external momentum q μ set to zero. It is found that our particular choice of BRS ansatz is satisfied by the gluon self-energy, and by the local terms in Γ μνϱ abc ( p , 0, − p ). The renormalization analysis is complicated by the presence of nonlocal terms.

Nonlocal BRS counterterms in a physical gauge

Abstract The general three-gluon vertex Γ μνϱ abc ( p , q , r ) is calculated to one-loop order in the physical light-cone gauge n σ A σ = 0, n 2 = 0 . It is shown that a modified BRS ansatz for the counterterms satisfies not only the gluon self-energy Π μν ab ( p ), but also the local and nonlocal terms in Γ μνϱ abc ( p , q , r ). It is found that the number of local counterterms is finite.

Two‐loop Feynman integrals in the physical light‐cone gauge

The feasibility of doing multiloop calculations in the noncovariant light‐cone gauge nμAaμ=0, n2=0, is investigated by evaluating various Feynman integrals arising in a two‐loop Yang–Mills self‐energy. Application of a consistent prescription for (q ⋅ n)−1 is essential.

Bäcklund generated solutions of Liouville’s equation and their graphical representations in three spatial dimensions

The Backlund–Bianchi method is employed to generate, in three spatial dimensions, the following multiple solutions of Liouville’s equation ∇2α = exp α: The three‐wave interaction function α3 and the five‐wave interaction function α5. It is verified numerically that α3 satisfies Liouville’s equation to an accuracy of one part in 1014, while α5 satisfies it to one part in 106. The construction of α5 is conditional upon solving ten nonlinear constraint equations. We analyze the complicated structures of α3 and α5 with the help of a three‐dimensional plotting routine. It is found that α3 is, surprisingly enough, only characterized by a single ring singularity, while α5 exhibits three ring singularities. It is speculated that the function tanh α3 represents a ring soliton whose shape appears to be preserved in the nonlinear superposition of similar ring solitons. The derivation of Liouville’s solutions α3 and α5 is intimately connected with the auxiliary functions β2 and β4 which solve Laplace’s equation. The ...

Non-transversality of the Yang-Mills self-energy in the planar gauge

Abstract It is shown that the Yang-Mills self-energy in the planar gauge is non-transverse, even though this self-energy satisfies the appropriate Ward identity. The non-transversality implies that the one-loop counterterm is no longer proportional to ( F μν a ) 2 .

Nonlinear superposition for Liouville's equation in three spatial dimensions

AbstractWe derive in 3+0 dimensions exact solutions of Liouvilles equation