Masses and electric charges: gauge anomalies and anomalous thresholds
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Regular Article - Theoretical Physics
Masses and electric charges: gauge anomalies and anomalous thresholds César Gómeza , Raoul Letschkab Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco 28049, Madrid, Spain
Received: 27 September 2019 / Accepted: 15 September 2020 © The Author(s) 2020
Abstract We work out in the forward limit and up to order e6 in perturbation theory the collinear divergences. In this kinematical regime we discover new collinear divergences that we argue can be only cancelled using quantum interference with processes contributing to the gauge anomaly. This rules out the possibility of a quantum consistent and anomaly free theory with massless charges and long range interactions. We use the anomalous threshold singularities to derive a gravitational lower bound on the mass of the lightest charged fermion.
1 Introduction
in addition collinear divergences that contribute logarithmically to Weinberg’s B factor [5,6].1 The standard recipe used to cancel these divergences requires to include in the definition of the inclusive cross section not only soft emission but also collinear hard emission and absorption (i.e. photons with energy bigger than the energy resolution scale) and to set an angular resolution scale. In [9] a unifying picture to the problem was suggested on the basis of degenerations. The idea is to define, for a given amplitude Si, f associated with a given scattering process i → f an inclusive cross section formally defined as |Si , f |2 , (1) i ∈D(i), f ∈D( f )
For theories with long range gauge forces as QED the IR completion problem goes around the quantum consistency of a quantum field theory with massless charged particles in the physical spectrum. This is an old problem that has been considered from different angles along the years (see [1–4] for an incomplete list). As a matter of fact in Nature we don’t have any example of massless charged particles. In the Standard Model this is the case both for spin 1/2 as well as for the spin 1 charged vector bosons. In the particular case of charged leptons the potential inconsistency of a massless limit should imply severe constraints on the consistency of vanishing Yukawa couplings. Technically the infrared (IR) origin of the problem is easy to identify. In the case of massless charged particles radiative corrections due to loops of virtual photons lead to two types of infrared problems. One can be solved, in principle, using the standard Bloch–Nordiesk-recipe [5] that leads to infrared finite inclusive cross sections at each order in perturbation theory depending on an energy resolution cutoff. In this case the infrared finite cross section is defined taking into account soft radiation. In the massless case we have a e-mail:
[email protected]
b e-mail:
[email protected] (corresponding author)
where D(i) is the set of asymptotic states degenerate with the asymptotic state i. For the case of massless electrically charged particles the degeneration used in [9] for the case where the
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