Renormalization of models with radiative mass generation

Abstract

The mechanism of radiative mass generation is discussed by means of a simple model with spontaneously broken symmetry. We show how this phenomenon induces an infrared breakdown of the usual perturbative approach and proceed to identify a set of reπormalization prescriptions allowing the construction of a new perturbation theory in which the Ward identities of the model are maintained. The original pathologies are reflected in the appearance of square roots and logarithms of the expansion parameter h.

Renormalization of Feynman integrals on the lattice

“…In fact, any vertex B satisfies ω(B) = 4, and (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) shows that the divergence degrees of six-point and higher functions are smaller than zero. The four-point Green function has divergence degree ω = 0.…”

“…If massless fields occur, we have to take into account possible infrared singularities very carefully. However, they are not specific to the lattice and are expected to be tractable by the methods which were developed for continuum Feynman integrals many years ago [6][7][8][9]. We will discuss this problem in a forthcoming paper and shall see that one only has to supplement the ultraviolet power counting conditions by infrared power counting conditions.…”

“…Let the coupling constants g^ in (4-10) be dimensionless. Take the limit β->0 of (4)(5)(6)(7)(8) [6] satisfies ω c (B)^4, then the lattice theory is renormalizable, and its continuum limit is given by the field theory which is described by the action S C (A\ and is renormalized by the BPHZ finite part prescription.…”

“…However, ω(Δ L ) and ω(V B ) are now given by The same holds for reduced diagrams. Finally, the forest formula is changed to X f r );^a\ (6-10) (6)(7)(8)(9)(10)(11) and the only restrictions to the subtraction degrees δ(γ) are given by (4-3) and (4-4). The convergence proof of (6-10) is identical to the above, the only modifications being that divergence degrees are now determined by (6)(7)(8) and (6)(7)(8)(9).…”

Renormalizable Theories with Symmetry Breaking

“…This happens in particular if, due to tree approximation mass rules there are more massless scalar fields than Goldstone bosons (pseudo-Goldstone bosons [15]). In this case, even the construction of a Green function theory needs a deep modification of the perturbative scheme [16]. Another limiting case which is worth mentioning occurs when the breaking has dimension four and, given the direction b b b which characterizes the breaking, the most general invariant Lagrangian formed with the quantized field is not the most general Lagrangian invariant under the residual symmetry group H b b b .…”