Publications

In the search for QFTs that admit boundstates, we reinvestigate the two dimensional Gross-Neveu model, but with massive fermions. By computing the self-energy for the auxiliary boundstate field and the effective potential, we show that there are no bound states around the lowest minimum, but there is a meta-stable bound state around the other minimum, a local one. The latter decays by tunneling. We determine the dependence of its lifetime on the fermion mass and coupling constant.

Abstract: In a previous paper [1], we analyzed the theory of massive fermions in the fundamental representation coupled to a U(N) Chern-Simons gauge theory in three dimensions at level K. It was done in the large N , large K limits where (Formula presented.) was kept fixed. Among other results, we showed there that there are no high mass \u201cquark anti-quark\u201d bound states. Here we show that there are no bound states at all.

We consider a "one-current" state obtained by application of a colour current on the "adjoint" vacuum in QCD ₂ with quarks in the fundamental representation. The quarks are taken to be massless. The theory on the light-front can be "currentised" namely formulated in terms of currents only. The adjoint vacuum is obtained by applying a current derivative at zero momentum on the singlet vacuum. In general, the "one-current" states are not eigenstates of M ² = 2 P ⁺ P ^-, except in the large N _f limit. Problems with infrared regularisations are pointed out. Connection to fermionic structure is made. This talk is based on the paper of Ref. [A. Abrashkin, Y. Frishman and J. Sonnenschein, Nucl. Phys. B 703 (2004) 320, arXiv:hep-th/0405165] by the same authors.

Two-dimensional QCD has often been used as a laboratory for studying the full four-dimensional theory, providing, for example, an explicit realization of baryons as solitons. We review aspects of conventional baryons in two-dimensional QCD, including the classical and quantum contributions to their masses. We then discuss the spectrum of exotic baryons in two-dimensional QCD, commenting on the solitonic radius inferred from the excitation spectrum as well as the two-dimensional version of the Goldberger-Treiman relation relating meson couplings to current matrix elements. Our treatment of two-dimensional QCD, via the semi-classical quantization of collective coordinates, is consistent with chiral-soliton approaches to normal and exotic baryons in four dimensions, but is not able to resolve all the issues arising in four-dimensional models.

We consider a "one current" state, which is obtained by the application of a color current on the "adjoint" vacuum. This is done in QCD₂, with the underlying quarks in the fundamental representation. The quarks are taken to be massless, in which case the theory on the light-front can be "currentized", namely, formulated in terms of currents only. The adjoint vacuum is shown to be the application of a current derivative, at zero momentum, on the singlet vacuum. We apply the operator M²=2P⁺ P^- on these states and find that in general they are not eigenstates of M² apart from the large N_f limit. Problems with infra-red regularizations are pointed out. We discuss the fermionic structure of these states.

Extending previous works on the spectrum of QCD₂, we now investigate the 2D analogue of meson-baryon scattering. We use semi-classical methods, perturbing around classical soliton solutions. We start with the Abelian case, corresponding to one flavor, and find that in this case the effective potential is reflectionless. We obtain an explicit expression for the forward phase shift. In the non-Abelian case of several flavors, the method yields a potential which depends on the momentum of the incoming particle. In this case there is both transmission and reflection. In both cases no resonances appear. As a byproduct, we derive the general conditions for a 2D scalar quantum field theoretical action to yield a reflectionless effective potential when one expands in small fluctuations about the classical solution.

We discuss the spectra of multi-flavor massless QCD₂. An approximation in which the Hilbert space is truncated to two currents states is used. We write down a 't Hooft like equation for the wave function of the two currents states. We solve this equation for the lowest massive state and find an excellent agreement with the DLCQ results. In addition, the 't Hooft model and the large N_f limit spectra are re-derived by using a description in terms of currents.

We compute the string tension in massive QCD₂It is shown that the string tension vanishes when the mass of the dynamical quark is zero, with no dependence on the representations of the dynamical or of the external charges. When a small mass(m≪e)is added, a tension appears and we calculate its value as a function of the representations.

We exhibit static solutions of multi-flavour QCD in two dimensions that have the quantum numbers of baryons and mesons, constructed out of quark and anti-quark solitons. In isolation the latter solitons have infinite energy, corresponding to the presence of a string carrying the non-singlet colour flux off to spatial infinity. When N_c solitons of this type are combined, a static, finite-energy, colour singlet solution is formed, corresponding to a baryon. Similarly, static meson solutions are formed out of a soliton and an anti-soliton of different flavours. The stability of the mesons against annihilation is ensured by flavour conservation. The static solutions exist only when the fundamental fields of the bosonized lagrangian belong to U(N_c × N_f) rather than to SU(N_c) × U(N_f). Discussion of flavour-symmetry breaking requires a careful treatment of the normal-ordering ambiguity. Our results can be viewed as a derivation of the constituent quark model in QCD₂, allowing a detailed study of constituent mass generation and of the heavy-quark symmetry.

We exhibit soliton solutions of QCD in two dimensions that have the quantum numbers of quarks. They exist only for quarks heavier than the dimensional gauge coupling, and have infinite energy, corresponding to the presence of a string carrying the non-singlet color flux off to spatial infinity. The quark solitons also disappear at finite temperature, as the temperature, as the temperature-dependent effective quark mass is reduced in the approach to the quark/hadron phase transition.

We find multi-baryon states in two-dimensional quantum chromodynamics. We show that to bind two baryons the model must involve at least seven flavours. All calculations are performed in the strong coupling limit and the semiclassical approximation is employed in the derivation of the results. We briefly comment on the properties of the derived states.

We present bosonic actions which are equivalent to various chiral fermion theories. For the case of one chiral fermion coupled to an abelian gauge field, we present two bosonized actions, one corresponding to regularizing in the vector conserving scheme and the other in the left-right scheme. We then propose an action for the non-abelian bosonization of Weyl fermions which is a WZW action coupled to a fixed curved background. The chiral WZW action is then coupled to non-abelian gauge fields. We derive the anomalies of the axial current (in the vector conserving scheme) and the left-right currents (in the left-right regularization scheme), both for the abelian and non-abelian bosonized actions. The expressions for the anomalies are identical to those derived in the corresponding fermionic theories.

We show directly the connection between two- and four-dimensional soliton theories coupled to fermions, thus explaining the equality of their respective fractional charges. The dependence of these charges on the boundary conditions of the "half fermions" is also computed.

Vector SU(N_c) gauge theories with n_f flavors in the fundamental representation are considered. We prove that if the persistent mass condition is assumed, the two anomaly equations are i dentical and flavor independent for n_f{slanted equal to or greater-than} 3. Integer solutions exist only for n_f=2. The necessity of a separate discussion for 2≤n_f≤N_c is explained.

The effects of vacuum copies on perturbation theory are studied. The usual procedure of perturbation expansion around the zero copy leads to incorrect results in the Abelian case. However, in non-Abelian gauge theory the above usual procedure reproduces the correct perturbation series.

It is shown that the standard infrared-cutoff procedures are inconsistent with the general axial gauge in 't Hooft's two-dimensional model of confinement.

Spontaneous mass generation is discussed. General arguments are given for the case of two zeros of β, implying no mass generation between the zeros. This is used to question the 1N expansion in a four-fermion two-dimensional model. General comments regarding dynamical mass and effective potential are also included.

In this article short distance and almost light-like distance behaviour of operator products are discussed. In particular, products of electromagnetic and weak currents are treated and applications made to the region of deep inelastic lepton-hadron scattering. This is motivated by the observed scaling behaviour in the deep inelastic region. We review briefly the kinematics and light cone dominance, and then discuss the structure of operator products at nearly light like distances. Light cone expansions are postulated and bilocal operators are introduced. These are generalization of Wilson's short distance expansion and were abstracted from studies in model field theories. Applications include, among others, treatment of Regge behaviour in relation to sum rules and fixed poles implied by scaling. Regarding models, we discuss the Thirring model and its generalization to include U(n) symmetry. The latter shows scale invariance only for one value of the coupling constant. In both cases anomalous dimensions occur. Regarding field theories in four dimensions, gauge theories are "nearest" to canonical scaling, for which only logarithmic violations occur. We review the quark algebra on the light cone and discuss the applications to structure functions and sum rules. In e⁺e^- annihilation into hadrons we review the various quark schemes. In single particle inclusive annihilation the singularity structure and multiplicity are analyzed. Finally, we comment on various other problems and approaches.

Exact expansions of operator products, in terms of c-number functions singular on the light cone and regular operators, are exhibited explicitly in the Thirring model. For the products ψ1(x)ψ1(x) and ψ2(x) ψ2(x) of fermion fields the expansion reduces to one term only, with the c-number function having a singularity on the light cone which depends on the coupling constant, and the regular operator depending only on the currents, which are free. The resulting formula allows one to calculate all Wightman functions in terms of current matrix elements and thereby provides a simple and complete solution to the Thirring model. The different charge sectors are realized as inequivalent irreducible representation spaces of the canonical current commutation relations, on which the charged field ψ acts as an intertwining operator.

Consequences of the assumption that current commutators have power-type singularities near the light cone are derived. An extension of Wilson's short-distance expansion of commutators is suggested for almost lightlike distances. Consequences for power-type singularities are derived for processes involving currents. Especially, generalized scaling laws are obtained for inelastic electron-nucleon scattering. The connection with Regge behavior is discussed. An application to electron-positron annihilation into hadrons is also made.