Mass Singularities of Feynman Amplitudes

Abstract

Feynman amplitudes, regarded as functions of masses, exhibit various singularities when masses of internal and external lines are allowed to go to zero. In this paper, properties of these mass singularities, which may be defined as pathological solutions of the Landau condition, are studied in detail. A general method is developed that enables us to determine the degree of divergence of unrenormalized Feynman amplitudes at such singularities. It is also applied to the determination of mass dependence of a total transition probability. It is found that, although partial transition probabilities may have divergences associated with the vanishing of masses of particles in the final state, they always cancel each other in the calculation of total probability. However, this cancellation is partially destroyed if the charge renormalization is performed in a conventional manner. This is related to the fact that interacting particles lose their identity when their masses vanish. A new description of state and a new approach to the problem of renormalization seem to be required for a consistent treatment of this limit.

Quantum electrodynamics at finite temperature

“…In fact, because of the Bose-Einstein distribution function there exist no additional ultravolet divergences and dimensional regularization is not required [lo]. Thus we find (17) where…”

Two cutoff phase space slicing method

“…This is in the approximation where terms of order δ c and δ s are neglected. The cancellation of poles in n − 4 is based on the Kinoshita-Lee-Nauenberg theorem [27], or mass factorization [28,29], depending on the situation. For experimentally degenerate final states, the soft and final state hard collinear singularities cancel upon addition of the interference of the leading order diagrams with the renormalized one-loop virtual diagrams.…”

An algorithm for the high-energy expansion of multi-loop diagrams to next-to-leading logarithmic accuracy

“…This is valid for arbitrary, i.e. also positive, values of the invariants s j since mass singularities appear independently of the directions of the external momenta [26] and thus of the values (and the signs) of these invariants.…”