Contemts

# Contemts

## Idea

In the formulation of (perturbative) quantum field theory various naive mathematical constructions give rise to ill-defined “divergent” expressions.

Some of these occur in the limit that interaction points are taken close to each other. Since the short distances involved in this limit translate, under Fourier transform, to high frequencies, these are called ultraviolet divergences. The correct way to deal with them is called renormalization.

Other divergences occur in the limit that the spacetime support of the interaction (the support of the “adiabatic switching”) tends without bounds to all of spacetime, also called the adiabatic limit. Since the large distances involved in this limit translate, under Fourier transform, to low frequencies, these are called infrared divergences.

In perturbative AQFT the issue is dealt with by realizing that, despite superficial appearance, the adiabatically switched S-matrix already makes physical sense, namely, for the computation of those quantum observables whose spacetime support is such that its causal closure lies inside the support of the switched S-matrix (this prop.). This gives a well-defined meaning to causal perturbation theory without having to consider the adiabatic limit.

## Properties

The Lee-Nauenberg theorem is a fundamental quantum mechanical result which provides the standard theoretical response to the problem of collinear and infrared divergences. Its argument, that the divergences due to massless charged particles can be removed by summing over degenerate states, has been successfully applied to systems with final state degeneracies such as LEP processes. If there are massless particles in both the initial and final states, as will be the case at the LHC, the theorem requires the incorporation of disconnected diagrams which produce connected interference effects at the level of the cross-section. However, this aspect of the theory has never been fully tested in the calculation of a cross-section. We show through explicit examples that in such cases the theorem introduces a divergent series of diagrams and hence fails to cancel the infrared divergences. It is also demonstrated that the widespread practice of treating soft infrared divergences by the Bloch-Nordsieck method and handling collinear divergences by the Lee-Nauenberg method is not consistent in such cases.

## References

Infrared divergences in QED were first observed in

• N. F. Mott, On the influence of radiative forces on the scattering of electrons, Proc. Camb. Phil. Soc. 27, 255 (1931).

• Felix Bloch and Arnold Nordsieck, Note on the Radiation Field of the electron, Phys. Rev. 52 , 54 (1937)

A famous approach to the problem is due to

• Ludwig Faddeev, P. P. Kulish, Asymptotic conditions and infrared divergences in quantum electrodynamics, Theoretical and Mathematical Physics June 1970, Volume 4, Issue 2, pp 745–757 (mathnet.ru, springer)

But see

• Wojciech Dybalski, From Faddeev-Kulish to LSZ. Towards a non-perturbative description of colliding electrons (arXiv:1706.09057)

which says:

the Faddeev-Kulish approach is justified at best by working out some test cases in perturbation theory. The question if the infrared finite S-matrix has any non-perturbative meaning is left completely open. Secondly, the relation between the Faddeev-Kulish approach to the more standard LSZ scattering theory? has never been clarified. While a naive application of the LSZ ideas clearly fails in the presence of infrared problems, a careful LSZ description of a bare electron accompanied by real and virtual photons is in fact possible.

In the present work we outline a bridge from the Faddeev-Kulish formalism to this LSZ descriptionin the massless Nelson model.

On the Lee-Nauenberg theorem:

• Martin Lavelle, David McMullan, Collinearity, convergence and cancelling infrared divergences, JHEP 0603 (2006) 026 (arXiv:hep-ph/0511314)

Discussion from the point of view of causal perturbation theory / perturbative AQFT:

Further developments include