General Renormalization Research Articles

Quantum effects and SU(2)×U(1) breaking in supergravity gauge theories

Renormalization effects are studied at the one-loop level in the context of gauge models coupled to supergravity. General renormalization group equations are derived for the parameters of the effective theory by taking advantage of the supersymmetric structure. One-loop gaugino masses, which enter these equations are calculated in grand unified models and turn out to be rather large. The results are applied to the study of the effective theory of a SU(3)×SU(2)×U(1) model which includes a gauge singlet in addition to the minimal set of fields. It is possible to induce the correct breaking of the electroweak gauge invariance when quantum corrections are taken into account, even without very heavy quarks.

Proper-time renormalization of multi-loop amplitudes in the background field method (I). φ4-theory

We develop the proper-time calculation technique for multi-loop amplitudes in the background field approach. With a judicious use of DeWitt's WKB form for the proper-time Green function, it is possible to carry out the standard renormalization program (in the coordinate space) very efficiently even beyond one-loop. The general renormalization procedure is given in the form of Zimmermann's forest formula, with Taylor subtractions now considered in terms of an appropriately chosen proper-time variable for each subgraph. We illustrate our procedure with φ 4-theory, constructing the two-loop renormalized proper-vertex generating functional and calculating two-loop renormalization group coefficients explicitly.

Probability density functions for collective coordinates in Ising-like systems

This paper describes a study of the behaviour of 'block' coordinates, measuring the spatial average, over a block of side L, of the local scalar ordering coordinates in a system undergoing a continuous phase transition. A general renormalisation group argument suggests that, in the limit in which both L and xi (the correlation length) are large compared with the lattice spacing, the block coordinate probability density function tends to a universal form, Pinfinity dependent on the ratio L/ xi . The function Pinfinity is calculated explicitly for dimension d=1. It is controlled by the statistical mechanics of kinks; intra-cluster ripples (phonons in the structural phase transition context) contribute identifiable corrections to the large L, xi limit. In the critical limit, L/ xi to 0, Pinfinity tends to a fixed point form P* with an extreme order-disorder character. Wilson's recursion formula is used to determine the form of P* m with an extreme order-disorder character. Wilson's recursion formula is used to determine the form of P* for d=3 and d=2; the latter result is substantiated by a calculation based on exactly established properties of the planar Ising model. In d=3Pinfinity is singly-peaked; in d=2 it has a strongly double-peaked character reminiscent of the d=1 system at low temperatures. The results are used to illuminate the nature of short-range order at criticality and the character of the collective excitation spectrum in the critical region, with particular reference to scattering experiments near structural phase transitions.

Conformation space renormalization of polymers. II. Single chain dynamics based on chain diffusion equation model

A polymer chain configuration space renormalization group method is developed for application to the polymer diffusion equation for a single continuous polymer chain with excluded volume and unaveraged hydrodynamic interactions. The method builds upon our chain configuration space equilibrium formulation which is based on a coarse graining procedure which utilized small chain contour length loop excluded volume interactions to define the renormalization procedure. This approach is generalized to include small loop hydrodynamic interactions within the renormalization scheme. Our coarse graining treatment is combined with the methods of Kawasaki and Gunton, designed to consider dynamical critical phenomena, to provide the chain configuration space renormalization group theory. Calculations are explicitly presented to order ε = 4−d, where d is the dimensionality of space. However, we show that the dynamical exponent can be obtained exactly from a consideration of the general renormalization scheme without the use of any ε expansion. Thus, it is demonstrated within the diffusion equation model that z = d with hydrodynamic interactions present. In the free draining limit z = 2+1/ν where ν is the exponent for the molecular weight dependence of the chain radius. In the non-free draining limit, this value of z justifies the equality of the so-called static and hydrodynamic radii exponents within the diffusion equation model.

TheO(ɛ2) scaling law fordσ/dt in the Reggeon Field Theory

We calculate the two loop contributions within the e-expansion to the Reggeon Field Theory scaling law fordσ/dt, derived using the renormalization group and a general renormalization point for the Pomeron propagator. This generalizes theO(e) work of Abarbanel, Bartels, Bronzan, and Sidhu. The invariance of the results under certain coupling constant rescalings is demonstrated. We also make some qualitative comments regarding phenomenological applications. Our amplitude in a certain limit approximates the form of the low energy diffractive amplitude advocated by Kane.

On the critical behaviour of disordered systems

It is shown by general renormalization group analysis that in disordered systems there is no quasi-homogeneous critical behaviour with positive specific-heat exponent α.

Linear combinations of radiative corrections: Renormalization group and renormalization theory

One considers first of all general renormalization transformations defined as linear combinations of radiative corrections and of their derivatives with respect to the squared mass. It is then shown that the transformations belonging to the conventional renormalization group are particular cases of these general transformations. Conditions under which such general renormalization transformations build up a group are studied, its representations are characterized and finally the connection between these transformations and the Renormalization theory are investigated.

Neutral Vector Mesons and the Hadronic Electromagnetic Current

The question of whether the entire hadronic electromagnetic current operator can be identical with a linear combination of the renormalized field operators for the known neutral vector mesons ${\ensuremath{\rho}}^{0}$, ${\ensuremath{\varphi}}^{0}$, and ${\ensuremath{\omega}}^{0}$ is investigated in the context of a Lagrangian field theory. It is found that such an identity is completely consistent with gauge invariance, provided that these mesons are coupled only to conserved currents. The general renormalization problem of the strong interactions of these vector mesons is discussed. It is shown that the proposed identity between the hadronic electromagnetic current and the renormalized meson fields can be related to the possible identity between the unrenormalized currents generating the neutral vector mesons and those generating the photon; furthermore, this proposed identity leads to an exact relation between the entire $O({e}^{2})$ hadronic contribution to the photon propagator and the renormalized propagators of the neutral vector mesons, and such a relation implies, among other consequences, that to $O({e}^{2})$ and neglecting leptonic contributions, the ratio of the unrenormalized charge ${e}_{0}$ and the renormalized charge $e$ is finite. Various experimental applications are given. In particular, the analysis of $\ensuremath{\varphi}\ensuremath{-}\ensuremath{\omega}$ mixing and their leptonic decay rates is made independently of the approximate validity of the $S{U}_{3}$ symmetry.

Expansion of the scattering matrix in terms of multiple impulsive peripheral interactions

A new perturbation expansion is developed for the S-matrix. Each term of the expansion corresponds to a Feynman graph which describes a multiple peripheral collision. The complete effects of the short range forces are included in the first approximation. The peripheral interaction expansion is based on the analytic properties of the ordinary perturbation expansion and may be looked upon as a rearrangement of the terms of that expansion. It provides a simple perturbation-theory picture for all the concrete results that have been derived by formulating dynamical theories in terms of dispersion relations. The Feynman graphs encountered in the peripheral interaction expansion are those of the ordinary expansion with all self-energy parts omitted. This omission is possible because each line of a graph represents a set of quanta which correspond not only to the discrete states, but also to all states lying in the continuum which have a limited angular momentum. Each vertex corresponds to an elementary interaction which has a strength equal to the appropriate exact S-matrix element. Renormalization of the successive terms of the peripheral interaction is the most complicated problem which is encountered, although it is similar to renormalization of the ordinary expansion. In the treatment of this problem one encounters all possible successive graphical contractions. With the aid of these topological constructions it is possible to state the general renormalization rule quite concisely.