Non-exceptional Momenta Research Articles

The on-shell expansion: from Landau equations to the Newton polytope

We study the application of the method of regions to Feynman integrals with massless propagators contributing to off-shell Green’s functions in Minkowski spacetime (with non-exceptional momenta) around vanishing external masses, {p}_i^2to 0 . This on-shell expansion allows us to identify all infrared-sensitive regions at any power, in terms of infrared subgraphs in which a subset of the propagators become collinear to external lightlike momenta and others become soft. We show that each such region can be viewed as a solution to the Landau equations, or equivalently, as a facet in the Newton polytope constructed from the Symanzik graph polynomials. This identification allows us to study the properties of the graph polynomials associated with infrared regions, as well as to construct a graph-finding algorithm for the on-shell expansion, which identifies all regions using exclusively graph-theoretical conditions. We also use the results to investigate the analytic structure of integrals associated with regions in which every connected soft subgraph connects to just two jets. For such regions we prove that multiple on-shell expansions commute. This applies in particular to all regions in Sudakov form-factor diagrams as well as in any planar diagram.

Renormalizability proof for QED based on flow equations

We prove the perturbative renormalizability of Euclidean QED4 using flow equations, i.e. with the aid of the Wilson renormalization group adapted to perturbation theory. As compared toΦ 4 4 the additional difficulty to overcome is that the regularization violates gauge invariance. We prove that there exists a class of renormalization conditions such that the renormalized Green functions satisfy the QED Ward identities and such that they are infrared finite at nonexceptional momenta. We give bounds on the singular behaviour at exceptional momenta (due to the massless photon) and comment on the adaptation to the case when the fermions are also massless.

Perturbative infrared finiteness of Yang-Mills theory from renormalization group flow

It is commonly believed that a Yang-Mills theory (and in general a massless theory) with a nonvanishing subtraction point is infrared finite, i.e. the vertex functions at non-exceptional momenta are finite. We give a simple perturbative proof of this fact by using the Wilson renormalization group formulation. The proof requires the control of the singular behaviour of vertex functions only for a restricted class of exceptional configurations of momenta.

On the absence of logarithmic growths in mass parameters in renormalized perturbation theory. II: Generalized conditions

In an earlier paper rules were developed which guarantee the absence of logarithmic growths in mass parameters in renormalized perturbation theory in Euclidean space with nonexceptional momenta. In the present paper, the rules are generalized thus the latter being applicable to a large class of graphs. The rules are easy to apply and to verify in terms of the structure of the proper and connected graph in question. All subtractions of renormalization are carried out directly in momentum space, about the origin, with the degree of divergence of a subtraction coinciding with the dimensionality of the corresponding subdiagram.

Local existence of the Borel transform in euclidean massless Φ 4 4

We extend the methods of [1] to prove large order estimates on the renormalized Feynman amplitudes of massless Φ44 euclidean field theory, at non-exceptional momenta. The Borel transform of the perturbative series is analytic in a disk centered at the origin of the complex plane. This result is a step towards the rigorous investigation of the infra-red singularities in the Borel plane, for theories containing massless particles, like the gauge theories.

Asymptotic behaviour from the exact Callan-Symanzik equations

The exact Callan-Symanzik equations written in the functional formalism are solved without dropping the mass-insertion terms. We find, from the exact solution, the asymptotic behaviour of Green functions at non-exceptional momenta without referring to Weinberg's power-counting theorem. In the case of asymptotically free theories, we also compute the non-leading asymptotic terms without using Wilson's operator product expansion. Finally, we discuss the case of exceptional momenta and study the asymptotics of theories with broken symmetry.