Hooft Model Research Articles

Strong decays and Adler selfconsistency condition in two-dimensional QCD

Strong mesonic decays are studied in the framework of the 't Hooft model for the two-dimensional QCD in the axial gauge. Special attention is payed to the processes with pions in the final state involved and the low energy theorems have been checked including the “Adler zero”. It is demonstrated explicitly that in the chiral limit any three-meson decay amplitude with at least one pion vertex vanishes identically for any values of the external momenta. Ward identities for the dressed vector and axial-vector currents vertices are identified and their relationship to the pionic vertex are discussed.

Decompactification of space or time in large N QCD 2

QCD 2 with fundamental quarks on a cylinder is solved to leading order in the 1/ N expansion, including the zero mode gluons. As a result of the non-perturbative dynamics of these gauge degrees of freedom, the compact space-time direction gets effectively decompactified. In a thermodynamic interpretation, this implies that there is no pressure of order N and that the chiral condensate of order N is temperature independent. These findings are consistent with confinement of quarks, rule out both chiral and deconfining phase transitions in the finite temperature 't Hooft model, and help to resolve some controversial issues in the literature.

Strongly coupled 't Hooft model on the lattice

A lattice strong coupling calculation of the spectrum and chiral condensate of the 't Hooft model is presented. The agreement with the results of the continuum theory is strikingly good even at the fourth order in the strong coupling expansions.

Strongly coupled 't Hooft model on the lattice

Strongly coupled 't Hooft model on the lattice

Heavy quark expansion and preasymptotic corrections to decay widths in the ’t Hooft model

We address nonperturbative power corrections to inclusive decay widths of heavy flavor hadrons in the context of the 't Hooft model (two-dimensional QCD at ${N}_{c}\ensuremath{\rightarrow}\ensuremath{\infty}),$ with the emphasis on the ``spectator-dependent'' effects, i.e., those sensitive to the flavor of the spectator. The summation of exclusive widths is performed analytically using the 't Hooft equation. We show that the ${1/m}_{Q}$ expansion of both the weak annihilation and Pauli interference widths coincides with the OPE predictions to the computed orders. Violation of local duality in the inclusive widths is quantified, and the new example is identified where the OPE prediction and the actual effect are completely saturated by a single final state. The qualitative aspects of quark hadronization emerging from the analysis in the 't Hooft model are discussed. Certain aspects of the summation of spectator-independent hadronic weak decay widths are given in more detail, which were not spelled out previously. We also give some useful details of the ${1/m}_{Q}$ expansion in the 't Hooft model.

Physics of the gauged four-Fermi model in 1+1 dimensions

We analyze a two dimensional model of gauged fermions with quartic couplings in the large-N limit. This combines the 't Hooft model and the Gross-Neveu model where the coupling constants of both theories are arbitrary. Analytic equations describing the meson states of the theory are derived and are solved systematically using various methods. The physics of the model is investigated.

Can DLCQ test the Maldacena conjecture?

We consider the Maldacena conjecture applied to the near horizon geometry of a D1-brane in the supergravity approximation and consider the possibility of testing the conjecture against the boundary field theory calculation using DLCQ. We propose the two point function of the stress energy tensor as a convenient quantity that may be computed on both sides of the correspondence. On the supergravity side, we may invoke the methods of Gubser, Klebanov, Polyakov, and Witten. On the field theory side, we derive an explicit expression for the two point function in terms of data that may be extracted from a DLCQ calculation at a given harmonic resolution. This gives rise to a well defined numerical algorithm for computing the two point function, which we test in the context of free fermions and the 't Hooft model. For the supersymmetric Yang-Mills theory with 16 supercharges that arises in the Maldacena conjecture, the algorithm is perfectly well defined, although the size of the numerical computation grows too fast to admit any detailed analysis at present, and our results are only preliminary. We are, however, able to present more detailed results on the supersymmetric DLCQ computation of the stress energy tensor correlators for two dimensional Yang Mills theories with (1,1) and (2,2) supersymmetries.

The perils of `soft' SUSY breaking

We consider a two dimensional SU(N) gauge theory coupled to an adjoint Majorana fermion, which is known to be supersymmetric for a particular value of fermion mass. We investigate the `soft' supersymmetry breaking of the discrete light cone quantization (DLCQ) of this theory. There are several DLCQ formulations of this theory currently in the literature and they naively appear to behave differently under `soft' supersymmetry breaking at finite resolution. We show that all these formulations nevertheless yield identical bound state masses in the decompactification limit of the light-like circle. Moreover, we are able to show that the supersymmetry-inspired version of DLCQ (so called `SDLCQ') provides the best rate of convergence of DLCQ bound state masses towards the actual continuum values, except possibly near or at the critical fermion mass. In this last case, we discuss improved extrapolation schemes that must supplement the DLCQ algorithm in order to obtain correct continuum bound state masses. Interestingly, when we truncate the Fock space to two particles, the SDLCQ prescription presented here provides a scheme for improving the rate of convergence of the massive 't Hooft model. Thus the supersymmetry-inspired SDLCQ prescription is applicable to theories without supersymmetry.

Pauli interference in the 't Hooft model: heavy quark expansion and quark–hadron duality

Pauli Interference in decays of heavy flavor mesons, a genuinely nonleptonic nonperturbative effect, is considered in the 't Hooft model. Analytic summation of the exclusive decay widths yields the same expression as obtained in the OPE-based approach. Threshold-related violations of local duality in PI are found to be suppressed. A novel case is identified where the OPE effect is completely saturated by a single final state.

Quark-hadron duality in the ’t Hooft model for meson weak decays: Different quark diagram topologies

We compare the effects of different quark diagram topologies on the weak hadronic width of heavy-light mesons in the large N_c limit. We enumerate the various topologies and show that the only one dominant (or even comparable) in powers of N_c to the noninteracting spectator "tree" diagram is the "annihilation" diagram, in which the valence quark-antiquark pair annihilate weakly. We compute the amplitude for this diagram in the 't Hooft model (QCD in 1+1 spacetime dimensions with a large number of colors N_c) at the hadronic level and compare to the Born term partonic level. We find that quark-hadron duality is not well satisfied, even after the application of a smearing procedure to the hadronic result. A number of interesting subtleties absent from the tree diagram case arise in the annihilation diagram case, and are described in detail.

Heavy flavor decays, OPE, and duality in the two-dimensional ’t Hooft model

The 't Hooft model (two-dimensional QCD in the limit of large number of colors) is used as a laboratory for exploring various aspects of the heavy quark expansions in the nonleptonic and semileptonic decays of heavy flavors. We perform a complete operator analysis and construct the operator product expansion (OPE) up to terms O(1/m_Q^4), inclusively. The OPE-based predictions for the inclusive widths are then confronted with the "phenomenological" results, obtained by summation of all open exclusive decay channels, one by one. The summation is carried out analytically, by virtue of the 't Hooft equation. The two alternative expressions for the total widths match. We comment on the recent claim in the literature of a 1/m_Q correction to the total width which would be in clear conflict with the OPE result. The issue of duality violations both in the simplified setting of the 't Hooft model and in actual QCD is discussed. The amplitude of oscillating terms is estimated.

Real time correlators in hot(2+1)-dimensional QCD

We use dimensional reduction techniques to relate real time finite T correlation functions in (2+1) dimensional QCD to bound state parameters in a generalized 't Hooft model with an infinite number of heavy quark and adjoint scalar fields. While static susceptibilities and correlation functions of the DeTar type can be calculated using only the light (static) gluonic modes, the dynamical correlators require the inclusion of the heavy modes. In particular we demonstrate that the leading T perturbative result can be understood in terms of the bound states of the 2d model and that consistency requires bound state trajectories composed of both quarks and adjoint scalars. We also propose a non-perturbative expression for the dynamical DeTar correlators at small spatial momenta.

How quark-hadron duality works in QCD

We discuss the issue of the local quark-hadron duality at high energies in two- and four-dimensional QCD. A mechanism of the dynamical realization of the quark-hadron duality in two-dimensional QCD in the limit of large number of colors, N c → ∞, (the 't Hooft model) is considered. A similar mechanism of dynamical smearing may be relevant in four-dimensional QCD. Although particular details of our results are model-dependent, the general features of the duality implementation conjectured previously get further support.

Illustrative example of how quark-hadron duality might work

We discuss the issue of the local quark-hadron duality at high energies in two- and four-dimensional QCD. A mechanism of the dynamical realization of the quark-hadron duality in two- dimensional QCD in the limit of large number of colors, N_c -> \infty, (the 't Hooft model) is considered. A similar mechanism of dynamical smearing may be relevant in four-dimensional QCD. Although particular details of our results are model-dependent, the general features of the duality implementation conjectured previously get further support.

Variational mass perturbation theory for light-front bound-state equations

We investigate the mesonic light-front bound-state equations of the 't Hooft and Schwinger model in the two-particle, i.e. valence sector, for small fermion mass. We perform a high precision determination of the mass and light-cone wave function of the lowest lying meson by combining fermion mass perturbation theory with a variational approach. All calculations are done entirely in the fermionic representation without using any bosonization scheme. In a step-by-step procedure we enlarge the space of variational parameters. For the first two steps, the results are obtained analytically. Beyond that we use computer algebraic and numerical methods. We achieve good convergence so that the calculation of the meson mass squared can be extended to third order in the fermion mass. Within the numerical treatment we include higher Fock states up to six particles. Our results are consistent with all previous numerical investigations, in particular lattice calculations. For the massive Schwinger model, we find a small discrepancy (less than 2 percent) in comparison with known bosonization results. Possible resolutions of this discrepancy are discussed.

Explicit quark-hadron duality in heavy-light meson weak decays in the ’t Hooft model

We compute the nonleptonic weak decay width of a heavy-light meson in 1+1 spacetime dimensions with a large number of QCD colors (the 't Hooft model) as a function of the heavy quark mass. In this limit, QCD is exactly soluble, and decay modes are dominated by two-particle final states. We compare the results to the tree-level partonic decay width of the heavy quark in order to test quark-hadron duality in this universe. We find that this duality is surprisingly well satisfied in the heavy quark limit, in that the difference between the sum of exclusive partial widths and the tree-level partonic width approaches a constant as M --> infinity, and the deviation is well-fit by a small 1/M correction. We comment on the meaning of this conclusion and its implications for the use of quark-hadron duality in hadronic physics.

Fermion condensates and the trivial vacuum of light-cone quantum field theory

We discuss the definition of condensates within light-cone quantum field theory. As the vacuum state in this formulation is trivial, we suggest to abstract vacuum properties from the particle spectrum. The latter can in principle be calculated by solving the eigenvalue problem of the light-cone Hamiltonian. We focus on fermionic condensates which are order parameters of chiral symmetry breaking. As a paradigm identity we use the Gell-Mann-Oakes-Renner relation between the quark condensate and the observable pion mass. We examine the analogues of this relation in the 't Hooft and Schwinger model, respectively. A brief discussion of the Nambu-Jona-Lasinio model is added.

Two-dimensional QCD with matter in the adjoint representation: What does it teach us?

We analyse the highly excited states in QCD 2( N c → ∞) with adjoint matter by using such general methods as dispersion relations, duality and unitarity. We find the Hagedorn-like spectrum ϱ( m) ∼ m − a exp( β H m) where the parameters β H and a can be expressed in terms of the asymptotics of the matrix elements f n k ∼ 〈0| Tr( Ψ − Ψ) k|n k〉 . We argue that the asymptotical values f n k do not depend on k (after appropriate normalization). Thus, we obtain β H = ( 2 π )√ π g 2N c and a = −3 2 in the case of Majorana fermions in the adjoint representation. The Hagedorn temperature is the limiting temperature in this case. We also argue that the chiral condensate 〈0| Tr( Ψ − Ψ)|0〉 is not zero in the model. Contrary to the 't Hooft model, this condensate does not break down any continuous symmetries and can not be considered as an order parameter. Thus, no Goldstone boson appears as a consequence of the condensation. We also discuss a few apparently different but actually tightly related problems: master field, condensate, wee partons and constituent quark model in the light-cone framework.

Chiral condensate, master field and all that in QCD 2 ( N → ∞)

We discuss the various aspects of two-dimensional QCD 2 ( N → ∞) (the 't Hooft model). Our main interest (motivated by the corresponding analysis in the four-dimensional QCD) is the vacuum structure of the theory. We use very general methods in the analysis, such as dispersion relations and duality, in order to relate the known spectrum of QCD 2 to the different vacuum characteristics. We explicitly calculate (in terms of physical parameters like masses and matrix elements) the chiral condensate as well as the mixed vacuum condensates: (0| q(gϵ μνG μν) nq|0) ∼ M eff 2n(0| qq|0) . We interpret the factorization property for the mixed vacuum condensates as a reminiscent of the master field at large N.

Light cone formalism for QCD string with quarks

The light-cone Hamiltonian is derived from the general gauge- and Lorentz-invariant expression for the q q Green's function, containing confinement via the area law for the Wilson loop. The resulting Hamiltonian contains in a nonadditive way contributions from quark and string degrees of freedom, exemplifying in a nontrivial way nonperturbative dynamics on the light cone. Different limiting regimes are considered, e.g. in two dimensions the Hamiltonian reduces to that of the 't Hooft model for 1 + 1 QCD.