Calculation of infrared-divergent Feynman diagrams with zero mass threshold

Two-loop vertex Feynman diagrams with infrared and collinear divergences are investigated by two independent methods. On the one hand, a method of calculating Feynman diagrams from their small momentum expansion [1] extended to diagrams with zero mass thresholds [2] is applied. On the other hand, a numerical method based on a two-fold integral representation is used [3], [4]. The application of the latter method is possible by using lightcone coordinates in the parallel space. The numerical data obtained with the two methods are in impressive agreement. Received: 22 April 1997 / Published online: 20 February 1998     

Spectral Flow,Maslov Index and Bifurcation of Semi-Riemannian Geodesics

We give a functional analytical proof of the equalitybetween the Maslov index of a semi-Riemannian geodesicand the spectral flow of the path of self-adjointFredholm operators obtained from the index form. This fact, together with recent results on the bifurcation for critical points of strongly indefinite functionals imply that each nondegenerate and nonnull conjugate (or P-focal)point along a semi-Riemannian geodesic is a bifurcation point.In particular, the semi-Riemannian exponential map is notinjective in any neighborhood of a nondegenerate conjugate point,extending a classical Riemannian result originally due to Morse and Littauer.     

A note on the continuous self-maps of the ladder system space

We give a partial characterization of the continuous self-maps of the ladder system space  ${K_{\mathcal{S}}}$ . Our results show that ${K_{\mathcal{S}}}$ is highly nonrigid. We also discuss reasonable notions of “few operators” for spaces C(K) with scattered K and we show that $C({K_{\mathcal{S}}})$ does not have few operators for such notions.     

A recursive reduction of tensor Feynman integrals

We perform a new, recursive reduction of one-loop n-point rank R   tensor Feynman integrals [in short: (n,R)$(n,R)$-integrals] for n?6$n?6$ with R?n$R?n$ by representing (n,R)$(n,R)$-integrals in terms of (n,R−1)$(n,R-1)$- and (n−1,R−1)$(n-1,R-1)$-integrals. We use the known representation of tensor integrals in terms of scalar integrals in higher dimension, which are then reduced by recurrence relations to integrals in generic dimension. With a systematic application of metric tensor representations in terms of chords, and by decomposing and recombining these representations, a recursive reduction for the tensors is found. The procedure represents a compact, sequential algorithm for numerical evaluations of tensor Feynman integrals appearing in next-to-leading order contributions to massless and massive three- and four-particle production at LHC and ILC, as well as at meson factories.     

An Index Theorem for Non-Periodic Solutions of Hamiltonian Systems

We consider a Hamiltonian setup M, , H, L, , P, where M, isa symplectic manifold, L is a distribution of Lagrangian subspacesin M, P is a Lagrangian submanifold of M, H is a smooth time-dependentHamiltonian function on M, and :[a,b] M is an integral curveof the Hamiltonian flow starting at P. We do not require any convexity property of the Hamiltonianfunction H. Under the assumption that (b) is not P-focal, weintroduce the Maslov index i_maslov of given in terms of thefirst relative homology group of the Lagrangian Grassmannian;under generic circumstances i_maslov() is computed as a sortof algebraic count of the P-focal points along . We prove thefollowing version of the Index Theorem: under suitable hypotheses,the Morse index of the Lagrangian action functional restrictedto suitable variations of is equal to the sum of i_maslov()and a convexity term of the Hamiltonian H relative to the submanifoldP. When the result is applied to the case of the cotangent bundleM = TM* of a semi-Riemannian manifold (M, g) and to the geodesicHamiltonian , we obtain a semi-Riemannian version of the celebrated Morse Index Theorem for geodesicswith variable endpoints in Riemannian geometry. 2000 MathematicalSubject Classification: 37J05, 53C22, 53C50, 53D12, 70H05.     

An index theory for paths that are solutions of a class of strongly indefinite variational problems

We generalize the Morse index theorem of [12,15] and we apply the new result to obtain lower estimates on the number of geodesics joining two fixed non conjugate points in certain classes of semi-Riemannian manifolds. More specifically, we consider semi-Riemannian manifolds admitting a smooth distribution spanned by commuting Killing vector fields and containing a maximal negative distribution for . In particular we obtain Morse relations for stationary semi-Riemannian manifolds (see [7]) and for the G?del-type manifolds (see [3]). Received: 4 April 2001 / Accepted: 27 September 2001 / Published online: 23 May 2002 The authors are partially sponsored by CNPq (Brazil) Proc. N. 301410/95 and N. 300254/01-6. Parts of this work were done during the visit of the two authors to the IMPA, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil, in January and February 2001. The authors wish to express their gratitude to all Faculty and Staff of the IMPA for their kind hospitality.