Publications

We analyze the theory of massive fermions in the fundamental representation coupled to a U(N) Chern-Simons gauge theory at level K. It is done in the large N, large K limits where lambda = N/K is kept fixed. Following [1] we obtain the solution of a Schwinger-Dyson equation for the two point function, the exact expression for the fermion propagator and the partition function at finite temperature. We prove that in the large K limit there exists an infinite set of classically conserved high spin currents also when a mass is introduced, breaking the conformal invariance. In analogy to the seminal work of 't Hooft on two dimensional QCD, we write down a Bethe-Salpeter equation for the wave function of a "quark anti-quark" bound state. We show that unlike the two dimensional QCD case, the three dimensional Chern-Simons theory does not admit a confining spectrum.

Extending earlier work on strong-coupling meson-baryon scattering in QCD(2), we derive the effective meson-baryon action for any value of the coupling constant, in the large-N-C limit. Colour degrees of freedom play an important role, and we show that meson-baryon scattering can be formulated as a relativistic potential problem. We distinguish two cases that are non-trivial for unequal quark masses, and present the resulting equations for meson-baryon scattering amplitudes. (C) 2003 Published by Elsevier B.V.

We review and elaborate on properties of the string tension in two-dimensional gauge theories. The first model we consider is massive QED in the m

We analyze the question of screening versus confinement in bosonized massless QCD in two dimensions. We deduce the screening behavior of massless SU(N-c) QCD with flavored fundamental fermions and fermions in the adjoint representation. This is done by computing the potential between external quarks as well, as by bosonizing also the external sources and analyzing the states of the combined system, We write down novel ''non-Abelian Schwinger like'' solutions of the equations of motion, compute their masses and argue that an exchange of massive modes of this type is associated with the screening mechanism. (C) 1997 Elsevier Science B.V.

It is shown that in a formulation of Yang-Mills theory in two dimensions in terms of A = if(-1)partial derivative f, (A) over bar = if partial derivative (f) over bar(-1) with f(z, (z) over bar>, (f) over bar(z, (z) over bar) is an element of [SU(N-C)](c) the complexification of SU(N-C), reveals certain subtleties. ''Physical'' massive color singlet states seem to exist. When coupled to N-F quarks the coupling constant is renormalized in such a way that it vanishes for the pure Yang-Mills case. This renders the above states massless and unphysical. In the abelian case, on the other hand, the known results of the Schwinger model are reproduced with no need of such a renormalization. The massless QCD(2) theory is analyzed in similar terms and peculiar massive states appear, with a mass of e(c) root N-F/2 pi.

Quantum Chromodynamics (QCD) is believed to describe the strong interactions. In the asymptotic domain of large momenta, improved perturbation theory describes phenomena by use of point-like quarks and gluons. But the spectrum and wave functions are in the nonperturbative domain, for which not much can be done analytically in four dimensions. In order to develop analytical methods physicists turned to simpler models, like QCD2, the theory in one space and one time dimensions. This review is devoted to the application of bosonization techniques to two-dimensional QCD. We start with a description of the ''abelian bosonization''. The methods of the abelian bosonization are applied to several examples like the Thirring model, the Schwinger model and QCD2. The failure of this scheme to handle flavored fermions is explained. Witten's non-abelian bosonization rules are summarized including the generalization to the case of fermions with color and flavor degrees of freedom. We discuss in detail the bosonic version of the mass bilinear of colored-flavored fermions in various schemes. The color group is gauged and the full bosonized version of massive multiflavor QCD2 is written down. The strong coupling limit is taken in the ''product scheme'' and then in the U(N(F) x N(C)) scheme. Once the multiflavor QCD2 action in the interesting region of the low energies is written down, we extract the semiclassical low-lying baryonic spectrum. First, classical soliton solutions of the bosonic action are derived. Quantizing the flavor space around those classical solutions produces the masses as well as the flavor properties of the two-dimensional baryons. In addition, low-lying multibaryonic solutions are presented, as well as wave functions and matrix elements of interest, like qqBAR content.

The operator product expansion, of appropriate products of quark fields, is used to find the anomalous dimensions which control the short distance behavior of hadronic wave functions. This behavior in turn controls the high-Q2 limit of hadronic form factors. In particular, we relate each anomalous dimension of the nonsinglet structure functions to a corresponding logarithmic correction factor to the nominal αS(Q2)/Q2 fall off of meson form factors. Unlike the case of deep inelastic lepton-hadron scattering, the operator product necessary here involves extra terms which do not contribute to forward matrix elements.

Non-Abelian gauge theory of quarks and gluons with SU(n) symmetry, in one-space and one-time dimensions, is investigated in the large-n limit. Special attention is given to the infrared regularization. The fermion propagator turns out to be cut-off independent. It is shown that in the n→∞ approximation, tachyons appear in the two-point function of the scalar density for mass m=0 of the quarks. This is not the case when m≠0 with m→0 is taken. Thus m=0 is singular. It is also shown that the singlet vector current has a free two-point function for m=0, which entails a consistency condition on bound state wave functions.

Four-fermion interactions of the current-current type with U(n) symmetry, in one space and one time dimension, are investigated. It is shown that the equations of motion yield scale-invariant solutions only for two values of the coupling gv of the SU(n) currents, namely gv=0 and gv=4π(n+1). This holds for any value of the coupling gB of the U(1) currents. For the above two values of gv and any gB the theory is solved completely. Operator products of spinor fields are shown to be equal to c-number functions singular on the light cone times analytic bilocal operators expressed in terms of currents and free spinor fields. The currents are free for the above two values of gv. The connection with the coupling as defined through four-point functions is discussed, and it turns out that the combination corresponding to SU(n) coupling is zero for both solutions. However, the solution for gv=4π(n+1) exhibits nontrivial four-point functions also for gB=0. It is shown, in an expansion around gv=0, that there is only one Callan-Symanzik function β which depends only on gv and that gv=0 is relevant to the ultraviolet limit of the gv>0 theories. When mass terms are introduced, this still holds in an infinite interval for gB, which is bounded below by a certain negative value and in which the mass term is soft.

Generalizations of the Thirring model to Fermi fields with U(n) symmetry are treated. When interactions quadratic in the SU(n) currents are introduced, scale invariance (with anomalous dimensions) is maintained only for values of the coupling gv=0 or gv=4φ(n+1).

The recently proposed expansion of products of local operators when their space-time distance approaches the light cone is motivated and discussed. Attempts to prove the expansion from Wilson's short distance expansion are analyzed. Applications to processes involving currents in certain asymptotic domains are reviewed and difficulties are pointed out.

Free fields of massless particles transforming covariantly under the Poincaré group are constructed. The allowed infinite- and finite-dimensional representations of the Lorentz group are obtained. The wave functions are calculated in these representations in various bases. The commutation rules are computed, and turn out to be nonlocal for any infinite-dimensional fields. The transformation law of a certain irreducible infinite-dimensional representation is shown to coincide, for its lowest-spin component, with the usual, radiation-gauge, vector-potential transformation law, as already discovered by Bender.

We discuss the possibility of constructing, out of particle creation and destruction operators, local quantum fields that transform as representations of the homogeneous Lorentz group. Our immediate goal is to write down a consistent local quantum field theory which can simultaneously describe many particles with different masses and spins. In the case that the field is a finite-dimensional irreducible Lorentz tensor, we are able to carry through our program with no restrictions on the masses considered as functions of the spin, provided the usual connection between spin and statistics is satisfied. However, when the field transforms as a unitary irreducible representation of the homogeneous Lorentz group (an infinite-dimensional representation), the requirement of locality, along with the physical assumption that the masses are bounded below, m(j)≥m0>0, leads to the restriction that the masses are independent of the spin. This property is shown to hold when the transformation law of the field is taken to be an irreducible finite-dimensional representation ⊗ a unitary irreducible representation. The physical consequences of this result and possible methods for evading it are discussed. Finally, an Appendix is included, where the related problem of orthogonality properties of timelike solutions to infinite-component wave equations is examined. In particular, we show that when the solutions of such wave equations transform as unitary irreducible representations of the homogeneous Lorentz group, only the Majorana representations support a scalar product, which is orthogonal for different spins.

It is shown, directly from proper Lorentz invariance and a positive Hilbert-space metric, that the vacuum expectation value ⟨0∣∣[j0(x),jk†(y)]∣∣0⟩ cannot vanish unless jμ(x)∣∣0⟩=jμ†(x)∣∣0⟩≡0. Neither locality nor Källén-Lehmann—type representations are needed. The same is demonstrated for ⟨0∣∣[Skl(x),S0m†(y)]∣∣0⟩, for any antisymmetric tensor Sμν. The explicit dependence of jμ and Sμν on the fields with which they interact is an immediate consequence in our approach. Similarly, it is easy to show that χ(x), χ(x)=∂μjμ(x), does not commute with j0†(y) for y0=x0, unless χ(x)∣∣0⟩=χ†(x)∣∣⟩≡0.

A simple connection between the radial Schrödinger equation for the bound states of a hydrogen atom with angular momentum l and that of an isotropic harmonic oscillator of even dimension from 2 to 4l + 4 is noted. The case of highest dimensionality 4l + 4 is shown to lead in a very simple way to the energy levels and degeneracies of the hydrogen atom, once the energy levels of a one‐dimensional harmonic oscillator are known.