Reformulation Instead of Renormalizations

Nuclear Physics B, 1970

The connection between the field theory and the perturbation expansion of quantum electrodynamics is studied. As a starting point the usual Lagrangian is taken but with bare electron mass and the renormalization constant Z3 set equal to zero. This theory is essentially equivalent to the usual one; however, it does not contain any constant of nature and is dilatational and gauge invariant, both invariances being spontaneously broken. The various limiting procedures implied by the differentiation, the multiplication and the renormalization of the field operators in the Lagrangian are combined in a gauge invariant way to a single limit. Propagator equations are derived which are the usual renormalized ones, except for: (i) a natural cancellation of the quadratic divergence of the vacuum polarization; (ii) the presence of an effective cutoff at p ≈ ɛ-1; (iii) the replacement of the renormalization constants Z1 and Z2 by one gauge dependent function Z( ɛ2); (iv) the limit ɛ → 0 which has to be taken. The value Z(0) corresponds to the usual constants Z1 and Z2. It is expected that in general Z(0) = 0, but this poses no problem in the present formulation. It is argued that the function Z( ɛ2), which is determined by the equations, may render the vacuum polarization finite. One may eliminate the renormalization function from the propagator equations and then perform the limit ɛ → 0; this results in the usual perturbation series. However, the renormalization function is essential for an understanding of the high momentum behaviour and of the relation between the field theory and the perturbation expansion.

The European Physical Journal C, 2001

In several preceding studies, the explicitly covariant formulation of light front dynamics was developed and applied to many observables. In the present study we show how in this approach the renormalization procedure for the first radiative correction can be carried out in a standard way, after separating out the contributions depending on the orientation of the light-front plane. We calculate the renormalized QED vertices γe − → e − and e + e − → γ and the electron self-energy and recover, in a straightforward way, the well known analytical results obtained in the ¿Feynman approach.

In this paper the nonperturbative analysis of the spectrum for one-particle excitations of the electronpositron field (EPF) is considered in the paper. A standard form of the quantum electrodynamics (QED) is used but the charge of the "bare" electron e0 is supposed to be of a large value (e0 ≫ 1). It is shown that in this case the quasi-particle can be formed with a non-zero averaged value of the scalar component of the electromagnetic field (EMF). Self-consistent equations for the distribution of charge density in the "physical" electron (positron) are derived. A variational solution of these equations is obtained and it defines the finite renormalization of the charge and mass of the electron (positron). It is found that the coupling constant α0 between EPF and EMF and mass m0 of the "bare" electron can be connected with the observed values of the fine structure constant α and the mass of the "physical" electron m as follows (h = c = 1): 2

2002

This paper proves that it is possible to build a Lagrangian for quantum electrodynamics which makes it explicit that the photon mass is eventually set to zero in the physical part on observational ground. Gauge independence is achieved upon considering the joint effect of gauge-averaging term and ghost fields. It remains possible to obtain a counterterm Lagrangian where the only non-gauge-invariant term is proportional to the squared divergence of the potential, while the photon propagator in momentum space falls off like k −2 at large k which indeed agrees with perturbative renormalizability. The resulting radiative corrections to the Coulomb potential in QED are also shown to be gauge-independent.

Nature, 2010

Lettere Al Nuovo Cimento Series 2, 1976

Journal of Mathematical Physics, 2005

In nonrelativistic QED the charge of an electron equals its bare value, whereas the self-energy and the mass have to be renormalized. In our contribution we study perturbative mass renormalization, including second order in the fine structure constant α, in the case of a single, spinless electron. As well known, if m denotes the bare mass and m eff the mass computed from the theory, to order α one has which suggests that m eff /m = (Λ/m) 8α/3π for small α. If correct, in order α 2 the leading term should be 1 2 ((8α/3π) log(Λ/m)) 2 . To check this point we expand m eff /m to order α 2 . The result is Λ/m as leading term, suggesting a more complicated dependence of m eff on m.

Physics Letters B, 1988

Incorporation of effective masses into negative energy states (nucleon loop corrections) gives rise to repulsive many-body forces, as has been known for some time. Rather than renormalizing away the three-and four-body terms, we introduce medium corrections into the effective t~-exchange, which roughly cancel the nucleon loop terms for densities P~Pnm, where P,m is nuclear matter density. Going to higher densities, the repulsive contributions tend to saturate whereas the attractive ones keep on growing in magnitude. The latter is achieved through use of a density-dependent effective mass for the o-particle, m~ = mo (p), such that mo (p) decreases with increasing density. Such a behavior is seen e.g. in the Nambu-Jona-Lasinio model. It is argued that a smooth transition to chiral restoration implies a similar behavior. The resulting nuclear equation of state is, because of the self-consistency in the problem, immensely insensitive to changes in the mass or coupling constant of the o-particle.

In contrast with the paradigm of effective Quantum Field Theory (EFT), realistic Renormalization Group (RG) flows approaching fixed points are neither perturbative nor linear. We argue that overlooking these limitations is necessarily linked to many unsolved puzzles challenging the Standard Model of particle physics (SM). Here we show that the analysis of non-linear attributes of RG flows near the electroweak scale can recover the full mass and flavor structure of the SM. It is also shown that this analysis brings closure to the "naturalness" puzzle without impacting the cluster decomposition principle of EFT.

Physical Review D, 1994

In this work the determination of low-energy bound states in Quantum Chromodynamics is recast so that it is linked to a weak-coupling problem. This allows one to approach the solution with the same techniques which solve Quantum Electrodynamics: namely, a combination of weak-coupling diagrams and many-body quantum mechanics. The key to eliminating necessarily nonperturbative effects is the use of a bare Hamiltonian in which quarks and gluons have nonzero constituent masses rather than the zero masses of the current picture. The use of constituent masses cuts off the growth of the running coupling constant and makes it possible that the running coupling never leaves the perturbative domain. For stabilization purposes an artificial potential is added to the Hamiltonian, but with a coefficient that vanishes at the physical value of the coupling constant. The weak-coupling approach potentially reconciles the simplicity of the Constituent Quark Model with the complexities of Quantum Chromodynamics. The penalty for achieving this perturbative picture is the necessity of formulating the dynamics of QCD in light-front coordinates and of dealing with the complexities of renormalization which such a formulation entails. We describe the renormalization process first using a qualitative phase space cell analysis, and we then set up a precise similarity renormalization scheme with cutoffs on constituent momenta and exhibit calculations to second order. We outline further computations that remain to be carried out. There is an initial nonperturbative but nonrelativistic calculation of the hadronic masses that determines the artificial potential, with binding energies required to be fourth order in the coupling as in QED. Next there is a calculation of the leading radiative corrections to these masses, which requires our renormalization program. Then the real struggle of finding the right extensions to perturbation theory to study the strong-coupling behavior of bound states can begin.