Soft Divergences Research Articles

Introduction: the Asia-Pacific HRM model revisited

This article introduces a symposium which 'revisits' the Asia-Pacific HRM model, much discussed in recent literature. It argues that, while the IR/HRM systems of the countries in the region are prima facie heterogeneous, there is both commonality and diversity. It posits the four logical cases of 'hard convergence', 'soft convergence', 'soft divergence' and 'hard divergence'. It argues that the most probable outcomes are likely to be the middle options. 'Soft convergence' may have occurred as a result of broader responses to broad economic trends such as globalization; 'soft divergence' may still be de rigueur , as the devil is always 'in the details'. Institutional and legal changes may also take place more slowly than enterprise-level or organizational ones.

The perturbative proton form factor reexamined

We recalculate the proton Dirac form factor based on the perturbative QCD factorization theorem, which includes Sudakov suppression. The evolution scale of the proton wave functions and the infrared cutoffs for the Sudakov re-summation are carefully chosen such that the soft divergences from large coupling constants are diminished and perturbative QCD predictions are stabilized. We find that the King–Sachrajda model for the proton wave function leads to results which are in better agreement with experimental data than those from the Chernyak–Zhitnitsky wave function.

The perturbative proton form factor reexamined

We recalculate the proton Dirac form factor based on the perturbative QCD factorization theorem which includes Sudakov suppression. The evolution scale of the proton wave functions and the infrared cutoffs for the Sudakov resummation are carefully chosen, such that the soft divergences from large coupling constants are diminished and perturbative QCD predictions are stablized. We find that the King-Sachrajda model for the proton wave function leads to results which are in better agreement with experimental data compared to the Chernyak-Zhitnitsky wave function.

Antenna factorization of gauge-theory amplitudes

I derive a single factorization formula which summarizes all soft and collinear divergences of a tree-level gauge theory amplitude. The singular factor squared is in a certain sense the generalization of the Catani-Seymour dipole factorization formula.

Soft dynamics and gauge theories

Infrared divergences obscure the underlying soft dynamics in gauge theories. They remove the pole structures associated with particle propagation in the various Green's functions of gauge theories. Here we present a solution to this problem. We give two equations which describe how charged particles must be dressed by gauge degrees of freedom. One follows from gauge invariance, the other, which is new, from velocity superselection rules familiar from the heavy quark effective theory. The solution to these equations in the Abelian theory is proven to lead to on-shell Green's functions that are free of soft divergences at all orders in perturbation theory.

Higher-order corrections to jet cross sections in e+e- annihilation.

A general method to calculate next-to-leading order multijet cross sections is presented. The emphasis is on how to isolate the soft and collinear divergences in multiparton matrix elements. As an example, the method is used to isolate the divergences at leading order in the number of colours in the processes e+e- + qQ + n gluons and e+e- -t qcjqQ + n gluons. The usual algebraic complexity of calculating next-to-leading order corrections in QCD is avoided, especially the d-dimensional squaring of the real matrix elements and the hard phase space integrals. Some remarks about the stucture of the virtual contributions are made. As a first application, and to examine the feasibility of the approach, explicit Monte Carlo programs are constructed which contain the next-to-leading order corrections to e+e- -+ 2 jets and e+e- + 3 jets. It is demonstrated that the method works and can be readily applied to a variety of processes.

Gluonic contribution to g1 and its relationship to the spin-dependent parton distributions.

We examine the suggestion of Altarelli and Ross and of Carlitz, Collins, and Mueller that there is a hard gluonic contribution to the first moment of the proton's spin-dependent structure function ${g}_{1}$. We find that if the soft (collinear) divergence in the gluonic contribution is regulated dimensionally or with a quark mass, then the first moment vanishes. More generally, we suggest that the hard gluonic contribution to ${g}_{1}$ be identified by subtracting certain contributions that are attributable to the spin-dependent quark distributions. We show that the first moment of the resulting hard gluonic contribution vanishes, provided that the UV regulator for the spin-dependent quark distributions respects gauge invariance, Lorentz invariance, and the analyticity structure of the unregulated distributions. However, by relaxing the Lorentz-invariance requirement, we are able to construct quark distributions such that the hard gluonic contribution to the first moment of ${g}_{1}$ is nonzero. The corresponding quark distributions are related to matrix elements of Lorentz-variant operators. Hence, they have no analogue in the standard operator-product expansion and do not satisfy the usual forms of the quark sum rules. We conclude that the size of the gluonic contribution to the first moment of ${g}_{1}$ is entirely a matter of the convention used in defining the quark distributions.

Erratum: Factorization of soft and collinear divergences in QCD in the Feynman gauge via a background field gauge

We prove Ward-type identities which are needed in the proof of factorization of the Drell-Yan process. Previous treatments prove them only for an axial gauge, and the proofs are diagrammatic in nature. We, on the other hand, establish the identities for the Feynman gauge and through symmetry considerations at the Lagrangian level. Our strategy is to first derive exact results in a background field gauge and then to show that to leading order in the mass divergences the background-field-gauge results can be used in the Feynman gauge.

Factorization of soft and collinear divergences in QCD in the Feynman gauge via a background field gauge.

We prove Ward-type identities which are needed in the proof of factorization of the Drell-Yan process. Previous treatments prove them only for an axial gauge, and the proofs are diagrammatic in nature. We, on the other hand, establish the identities for the Feynman gauge and through symmetry considerations at the Lagrangian level. Our strategy is to first derive exact results in a background field gauge and then to show that to leading order in the mass divergences the background-field-gauge results can be used in the Feynman gauge.

Soft gluons and the eikonal approximation with massless quarks

In perturbative QCD, non-cancelling soft divergences are power-suppressed at high energies up to order g 4 . Previous proofs of this property have used the eikonal approximation, which with massless quarks is not correct for individual graphs (in the Feynman gauge). Nevertheless, the above property is here shown to be correct for the sum of all graphs.

Soft divergences in perturbative QCD

Abstract We demonstrate simply and directly that Bloch-Nordsieck inclusive cross sections are not in general infrared finite in non-abelian gauge theories treated by perturbation theory. We take the example q q →λ ∗ + (gluons) to order α s 2 , and use the Coulomb gauge in the lab frame, which greatly reduces the work. The relation to the Lee-Nauenberg theorem is discussed. In a very oversimplified model of bound quarks, the infrared divergence is replaced by a logarithm of the size of the bound state.

High-energy behavior of field-strength interactions

It is known that spontaneously broken gauge theories are the only renormalizable theories of massive spin-one particles with mass dimension less than or equal to 4. This paper describes a search for renormalizable interactions with higher mass dimension. Specifically, we examine the high-energy behavior of a class of models which involve field-strength interactions. Power counting shows that the high-energy behavior of these models is no worse than the naively estimated high-energy behavior of a gauge theory in the $U$ gauge. Therefore, there may be a "soft" symmetry-breaking mechanism (for instance, a soft divergence of an antisymmetric tensor current) which enforces renormalizable high-energy behavior in the same way that spontaneously broken gauge invariance guarantees the renormalizability of gauge theories. This hope is supported by the existence of "gauge theories" of strings, which describe analogous interactions of strings and field strengths. Unfortunately, this idea is tarnished by explicit calculations in which renormalizability is imposed in the form of unitarity bounds. These unitarity bounds imply that all possible field-strength couplings must be zero and that the remaining interactions describe a spontaneously broken gauge theory. Thus this result supports an earlier conjecture that gauge theories are the only renormalizable theories of massive vector bosons.

Apparent soft mode linewidth divergence in PbTiO3

The apparent soft mode linewidth divergence observed in Raman scattering from PbTiO 3 in the vicinity of the transition temperature is explained by an enhancement of scattering processes that are allowed due to finite acoustic phonon lifetimes. Die in Raman Streuung an PbTiO 3 auftretende Divergenz der ‘soft mode’ Linienbreite wird mit dem Einsatz hydrodynamischer Streuung erklärt.