Jacques Bloch

PDFfit2 and PDFgui: computer programs for studying nanostructure in crystals

PDFfit2 is a program as well as a library for real-space refinement of crystal structures. It is capable of fitting a theoretical three-dimensional (3D) structure to atomic pair distribution function data and is ideal for nanoscale investigations. The fit system accounts for lattice constants, atomic positions and anisotropic atomic displacement parameters, correlated atomic motion, and experimental factors that may affect the data. The atomic positions and thermal coefficients can be constrained to follow the symmetry requirements of an arbitrary space group. The PDFfit2 engine is written in C++ and is accessible via Python, allowing it to inter-operate with other Python programs. PDFgui is a graphical interface built on the PDFfit2 engine. PDFgui organizes fits and simplifies many data analysis tasks, such as configuring and plotting multiple fits. PDFfit2 and PDFgui are freely available via the Internet.

Propagators and running coupling from SU(2) lattice gauge theory

We perform numerical studies of the running coupling constant αR(p2) and of the gluon and ghost propagators for pure SU(2) lattice gauge theory in the minimal Landau gauge. Different definitions of the gauge fields and different gauge-fixing procedures are used, respectively, for gaining better control over the approach to the continuum limit and for a better understanding of Gribov-copy effects. We find that the ghost–ghost–gluon vertex renormalization constant is finite in the continuum limit, confirming earlier results by all-order perturbation theory. In the low momentum regime, the gluon form factor is suppressed while the ghost form factor is divergent. Correspondingly, the ghost propagator diverges faster than 1/p2 and the gluon propagator appears to be finite. Precision data for the running coupling αR(p2) are obtained. These data are consistent with an IR fixed point given by limp→0αR(p2)=5(1).

Running coupling in nonperturbative QCD : Bare vertices and y-max approximation

A recent claim that in quantum chromodynamics tin the Landau gauge) the gluon propagator Vanishes in the infrared limit, while the ghost propagator is more singular than a simple pole, is investigated analytically and numerically. This picture is shown to be supported even at the level in which the vertices in the Dyson-Schwinger equations are taken to be bare. The gauge invariant running coupling is shown to be uniquely determined by the equations and to have a large finite infrared limit. [S0556-2821(98)04421-X].

QCD IN THE INFRARED WITH EXACT ANGULAR INTEGRATIONS

In a previous paper we have shown that in quantum chromodynamics the gluon propagator vanishes in the infrared limit, while the ghost propagator is more singular than a simple pole. These results were obtained after angular averaging, but here we go beyond this approximation and perform an exact calculation of the angular integrals. The powers of the infrared behavior of the propagators are changed substantially. We nd the very intriguing result that the gluon propagator vanishes in the infrared exactly like p2, whilst the ghost propagator is exactly as singular as 1=p4. We also nd that the value of the infrared xed point of the QCD coupling is much decreased: it is now equal to 4=3. Following a recent study by von Smekal et al., 1 we analyzed in Ref. 2 the coupled Dyson{Schwinger equations for the gluon and ghost form factors F and G .T he approximations were twofold: rst the vertices were taken bare, and second angular averaging was introduced. Here we seek to remove the deciency of the angular averaging. On the one hand, the results might be regarded simply as quantitative adjustments to the calculations, on the other hand, they are far from negligible. The numerical value of the infrared xed point is reduced by a factor of almost three; and the nding that the gluon propagator has a simple zero, while the ghost propagator has double poles, might perhaps be deemed a qualitatively new result. As an improvement on the approximation used in Ref. 2, we now solve the coupled integral equations for the gluon and ghost propagators with an exact treatment of the angular integrals. Although the angular averaging, the so-called y-max approximation, a is good in the ultraviolet region where the form factors run logarithmically, we will see that it is a crude approximation in the infrared region where the form factors exhibit power behavior. The approximation apparently leads to a diculty in the ghost equation, as the asymmetry in the treatment of the gluon and ghost momenta in the loop gives an ambiguous result. In our previous study

Two-Loop Improved Truncation of the Ghost-Gluon Dyson-Schwinger Equations: Multiplicatively Renormalizable Propagators and Nonperturbative Running Coupling

Abstract. The coupled Dyson-Schwinger equations for the gluon and ghost propagators are investigated in the Landau gauge using a two-loop improved truncation that preserves the multiplicative renormalizability of the propagators. In this truncation all diagrams contribute to the leading-order infrared analysis. The infrared contributions of the nonperturbative two-loop diagrams to the gluon vacuum polarization are computed analytically, and this reveals that infrared power-behaved propagator solutions only exist when the squint-diagram contribution is taken into account. For small momenta the gluon and ghost dressing functions behave like (p2)2κ and (p2)−κ, respectively, and the running coupling exhibits a fixed point. The values of the infrared exponent and fixed point depend on the precise details of the truncation. The coupled ghost-gluon system is solved numerically for all momenta, and the solutions have infrared behaviors consistent with the predictions of the infrared analysis. For truncation parameters chosen such that κ = 0.5, the two-loop improved truncation is able to produce solutions for the propagators and running coupling which are in very good agreement with recent lattice simulations.

Pair creation: Back reactions and damping

We solve the quantum Vlasov equation for fermions and bosons, incorporating spontaneous pair creation in the presence of back reactions and collisions. Pair creation is initiated by an external impulse field and the source term is non-Markovian. A simultaneous solution of Maxwells equation in the presence of feedback yields an internal current and electric field that exhibit plasma oscillations with a period {tau}{sub pl}. Allowing for collisions, these oscillations are damped on a time scale {tau}{sub r} determined by the collision frequency. Plasma oscillations cannot affect the early stages of the formation of a quark-gluon plasma unless {tau}{sub r}>>{tau}{sub pl} and {tau}{sub pl}{approx}1/{lambda}{sub QCD}{approx}1 fm/c. (c) 1999 The American Physical Society.

Multiplicative renormalizability and quark propagator

In the standard model of the strong, weak, and electromagnetic forces, the interactions are quantitatively described by gauge field theories. Quantum chromodynamics is a nonAbelian gauge theory, and the proof of its renormalizability @1# and discovery of ultraviolet asymptotic freedom @2# have been milestones in its acceptance as the theory of the strong interaction. For large momenta the coupling becomes small, and perturbation theory seems an appropriate calculational tool. However, for small momenta the coupling grows large and adequate methods have to be used to study nonperturbative phenomena such as confinement, dynamical chiral symmetry breaking, and bound state formation. One such method is the study of the Dyson-Schwinger equations ~DSE !@ 3#, and their phenomenological applications to hadronic physics is a subject of growing interest @4#. Dynamical chiral symmetry breaking can be studied by means of the quark propagator DSE, also called the gap equation. The quark equation is part of an infinite tower of integral equations relating all the Green’s functions of the quantum field theory, and a truncation is necessary to be able to solve it. The often used Abelian approximation introduces an effective running coupling in the kernels of the quark equation, based on features of QED. A number of studies of the gap equation were performed in this approximation with effective strong couplings that are infrared enhanced @5‐7#, infrared vanishing @8,9#, infrared vanishing with large enhancements in the intermediate region @10#, and infrared finite @11,12#. These studies all find that dynamical chiral symmetry breaking can be triggered, provided some parameter in the model exceeds a critical value, and a large integration strength is needed in the self-energy kernels to achieve large enough ‘‘constituent’’ quark masses, needed for hadron phenomenology @9#. The Ansatz of a strong infrared enhancement in the kernels of the quark DSE is therefore a major ingredient in phenomenological studies, and the search for the source of such an enhancement, coming from the gluon propagator, quark-gluon vertex, or their combination is an important field of investigation. The kernel of the gap equation not only reflects the strong interaction between quarks and gluons, but also embodies the complicated structure of the QCD vacuum, which in the continuum contains self-interacting gluon fields as well as ghost fields. Understanding the infrared behavior of the gluon and ghost propagators and of the running coupling is therefore of great importance for the study of dynamical mass generation @13#. Early studies of the Dyson-Schwinger equation for the gluon propagator in the Landau gauge seemed to indicate that the gluon propagator could be highly singular in the infrared @14 ‐17#, possibly providing the above-mentioned infrared enhancement in the kernels of the gap equation. However, these gluon propagator studies neglected any contribution of the ghost fields, and required the ad hoc cancellations of certain leading terms in the equations. It is therefore far from certain that these solutions reflect the correct QCD infrared behavior. More recently, studies of the coupled set of Dyson

Overlap Dirac operator at nonzero chemical potential and random matrix theory

We show how to introduce a quark chemical potential in the overlap Dirac operator. The resulting operator satisfies a Ginsparg-Wilson relation and has exact zero modes. It is no longer gamma5 Hermitian, but its nonreal eigenvalues still occur in pairs. We compute the spectral density of the operator on the lattice and show that, for small eigenvalues, the data agree with analytical predictions of non-Hermitian chiral random matrix theory for both trivial and nontrivial topology. We also explain an observed change in the number of zero modes as a function of chemical potential.

Multiplicative renormalizability of gluon and ghost propagators in QCD

We reformulate the coupled set of continuum equations for the renormalized gluon and ghost propagators in QCD, such that the multiplicative renormalizability of the solutions is manifest, independently of the specific form of full vertices and renormalization constants. In the Landau gauge, the equations are free of renormalization constants, and the renormalization point dependence enters only through the renormalized coupling and the renormalized propagator functions. The structure of the equations enables us to devise novel truncations with solutions that are multiplicatively renormalizable and agree with the leading order perturbative results. We show that, for infrared power law behaved propagators, the leading infrared behavior of the gluon equation is not solely determined by the ghost loop, as concluded in previous studies, but that the gluon loop, the three-gluon loop, the four-gluon loop, and even massless quarks also contribute to the infrared analysis. In our new Landau gauge truncation, the combination of gluon and ghost loop contributions seems to reject infrared power law solutions, but massless quark loops illustrate how additional contributions to the gluon vacuum polarization could reinstate these solutions. Moreover, a schematic study of the three-gluon and four-gluon loops shows that they too need to be considered in more detail before a definite conclusion about the existence of infrared power behaved gluon and ghost propagators can be reached.

Nucleon form factors and a nonpointlike diquark

Nucleon form factors are calculated on q{sup 2}(set-membership sign)[0,3] GeV{sup 2} using an ansatz for the nucleons Faddeev amplitude motivated by quark-diquark solutions of the relativistic Faddeev equation. Only the scalar diquark is retained, and it and the quark are confined. A good description of the data requires a nonpointlike diquark correlation with an electromagnetic radius of 0.8 r{sub {pi}}. The composite, nonpointlike nature of the diquark is crucial. It provides for diquark-breakup terms that are of greater importance than the diquark photon absorption contribution. (c) 1999 The American Physical Society.