Instituto de Física Matemáticas

We study the dynamical generation of masses for fundamental fermions in quenched quantum electrodynamics, in the presence of magnetic fields of arbitrary strength, by solving the Schwinger-Dyson equation (SDE) for the fermion self-energy in the rainbow approximation. We employ the Ritus eigenfunction formalism which provides a neat solution to the technical problem of summing over all Landau levels. It is well known that magnetic fields catalyze the generation of fermion mass m for arbitrarily small values of electromagnetic coupling α. For intense fields it is also well known that m ∝ √ eB. Our approach allows us to span all regimes of parameters α and eB. We find that m ∝ √ eB provided α is small. However, when α increases beyond the critical value αc which marks the onslaught of dynamical fermion masses in vacuum, we find m ∝ Λ, the cut-off required to regularize the ultraviolet divergences. Our method permits us to verify the results available in literature for the limiting cases of eB and α. We also point out the relevance of our work for possible physical applications.

We discuss and compare two alternative perturbation approaches for the calculation of the period of nonlinear systems based on the Lindstedt-Poincaré technique. As illustrative examples we choose one-dimensional anharmonic oscillators and the Van der Pol equation. Our results show that each approach is better for just one type of model considered here.

Working in the linear sigma model with quarks, we compute the finite-temperature effective potential in the presence of a weak magnetic field, including the contribution of the pion ring diagrams and considering the sigma as a classical field. In the approximation where the pion selfenergy is computed perturbatively, we show that there is a region of the parameter space where the effect of the ring diagrams is to preclude the phase transition from happening. Inclusion of the magnetic field has small effects that however become more important as the system evolves to the lowest temperatures allowed in the analysis.

We study the dynamical generation of masses for fundamental fermions in quenched quantum electrodynamics in the presence of weak magnetic fields using Schwinger-Dyson equations. Contrary to the case where the magnetic field is strong, in the weak field limit the coupling should exceed certain critical value in order for the generation of masses to take place, just as in the case where no magnetic field is present. The weak field limit is defined as eB ≪ m(0) 2 , where m(0) is the value of the dynamically generated mass in the absence of the field. We carry out a numerical analysis to study the magnetic field dependence of the mass function above critical coupling and show that in this regime the dynamically generated mass and the chiral condensate for the lowest Landau level increase proportionally to (eB) 2 .

Gauge theories have been a corner stone of the description of the world at the level of fundamental particles. The Lagrangian or the action describing the corresponding interactions is invariant under certain gauge transformations and contains the dependence on a covariant gauge parameter. This symmetry is reflected in terms of the Ward-Green-Takahashi (or the Slavnov-Taylor) identities (WGTI) which relate various Green functions among each other, and the Landau-Khalatnikov-Fradkin transformations (LKFT) which relate a Green function in a particular gauge to it in an arbitrary covariant one. As an outcome, all physical observables should be independent of the choice of gauge. The most systematic scheme to solve quantum field theories (QFT) is perturbation theory where the above-mentioned identities are satisfied at every order of approximation. This feature is exploited highly usefully as a verification of the results obtained after time-and effort-consuming exercises.

Through an explicit calculation for a Lagrangian in quantum electrodynamics in (2+1)-space-time dimensions (QED 3 ), making use of the relativistic Kubo formula, we demonstrate that the filling factor accompanying the quantized electrical conductivity for massive Dirac fermions of a single species in two spatial dimensions is a half (in natural units) when time reversal and parity symmetries of the Lagrangian are explicitly broken by the fermion mass term. We then discuss the most general form of the QED 3 Lagrangian, both for irreducible and reducible representations of the Dirac matrices in the plane, with emphasis on the appearance of a Chern-Simons term. We also identify the value of the filling factor with a zero field quantum Hall effect (QHE).

We compute the pion inclusive transverse momentum distribution assuming thermal equilibrium together with transverse flow and accounting for finite size effects and energy loss at the time of decoupling. We compare to data on mid-rapidity pions produced in central collisions in RHIC at √ sNN = 200 GeV. We find that a finite size for the system of emitting particles results in a powerlike fall-off of the spectra that follows the data up to larger pt values, as compared to a simple thermal model. PACS numbers: 25.75.-q

We calculate the condensate and the vacuum current density induced by external static magnetic fields in (2+1)-dimensions. At the perturbative level, we consider an exponentially decaying magnetic field along one cartesian coordinate. Non-perturbatively, we obtain the fermion propagator in the presence of a uniform magnetic field by solving the Schwinger-Dyson equation in the rainbow-ladder approximation. In the large flux limit, we observe that both these quantities, either perturbative (inhomogeneous) and non-perturbative (homogeneous), are proportional to the external field, in agreement with early expectations.

We study the electron propagator in quantum electrodynamics in lower dimensions. In the case of free electrons, it is well known that the propagator in momentum space takes the simple form S F (p) = 1/(γ · p − m). In the presence of external electromagnetic fields, electron asymptotic states are no longer plane-waves, and hence the propagator in the basis of momentum eigenstates has a more intricate form. Nevertheless, in the basis of the eigenfunctions of the operator (γ · Π) 2 , where Π µ is the canonical momentum operator, it acquires the free form S F (p) = 1/(γ · p − m) where p µ depends on the dynamical quantum numbers. We construct the electron propagator in the basis of the (γ · Π) 2 eigenfunctions. In the (2+1)-dimensional case, we obtain it in an irreducible representation of the Clifford algebra incorporating to all orders the effects of a magnetic field of arbitrary spatial shape pointing perpendicularly to the plane of motion of the electrons. Such an exercise is of relevance in graphene in the massless limit. The specific examples considered include the uniform magnetic field and the exponentially damped static magnetic field. We further consider the electron propagator for the massive Schwinger model incorporating the effects of a constant electric field to all orders within this framework.

It has been recently realized that in peripheral heavy-ion collisions at high energies, a sizable magnetic field is produced in the interaction region. Although this field becomes weak at the proper times when the chiral phase transition is believed to occur, it is still significant so as to ask whether it influences such transition. We use the linear sigma model to study the chiral phase transition in the presence of weak magnetic fields.

Dynamical chiral symmetry breaking and confinement are two crucial features of Quantum Chromodynamics responsible for the nature of the hadron spectrum. These phenomena, presumably coincidental, can account for 98% of the mass of our visible universe. In this set of lectures, I shall present an introductory review of them in the light of the Schwinger-Dyson equations.

We study the analytical structure of the fermion propagator in planar quantum electrodynamics coupled to a Chern-Simons term within a four-component spinor formalism. The dynamical generation of parity-preserving and parity-violating fermion mass terms is considered, through the solution of the corresponding Schwinger-Dyson equation for the fermion propagator at leading order of the 1/N approximation in Landau gauge. The theory undergoes a first order phase transition toward chiral symmetry restoration when the Chern-Simons coefficient θ reaches a critical value which depends upon the number of fermion families considered. Parity-violating masses, however, are generated for arbitrarily large values of the said coefficient. On the confinement scenario, complete charge screening -characteristic of the 1/N approximation-is observed in the entire (N, θ)-plane through the local and global properties of the vector part of the fermion propagator.

We study the non-perturbative phenomena of Dynamical Mass Generation and Confinement by truncating at the non-perturbative level the Schwinger-Dyson equations in Maxwell-Chern-Simons planar quantum electrodynamics. We obtain numerical solutions for the fermion propagator in Landau gauge within the so-called rainbow approximation. A comparison with the ordinary theory without the Chern-Simons term is presented.

Schwinger-Dyson equations (SDE) are an ideal framework to study non perturbative phenomena, like dynamical mass generation (DGM). Loss of gauge invariance is an obstacle to achieve fully reliable predictions from these equations. Through obtaining the solution of the SDEs in various gauges, it is known that a truncation scheme of the SDEs which takes into account the Ward-Green-Takahashi identity (WGTI) is not sufficient to ensure the gauge independence of physical observables of interest. We discover that another consequence of gauge invariance, namely, the Landau-Khalatnikov-Fradkin transformations play an extremely important role in restoring the said invariance at the level of physical observables. We have checked this for the case of the chiral condensate in quantum electrodynamics in (2+1)-space-time dimensions (QED3).