Stanislaw D. Glazek

Nonperturbative QCD: A weak-coupling treatment on the light front.

In this work the determination of low-energy bound states in Quantum Chromodynamics is recast so that it is linked to a weak-coupling problem. This allows one to approach the solution with the same techniques which solve Quantum Electrodynamics: namely, a combination of weak-coupling diagrams and many-body quantum mechanics. The key to eliminating necessarily nonperturbative effects is the use of a bare Hamiltonian in which quarks and gluons have nonzero constituent masses rather than the zero masses of the current picture. The use of constituent masses cuts off the growth of the running coupling constant and makes it possible that the running coupling never leaves the perturbative domain. For stabilization purposes an artificial potential is added to the Hamiltonian, but with a coefficient that vanishes at the physical value of the coupling constant. The weak-coupling approach potentially reconciles the simplicity of the Constituent Quark Model with the complexities of Quantum Chromodynamics. The penalty for achieving this perturbative picture is the necessity of formulating the dynamics of QCD in light-front coordinates and of dealing with the complexities of renormalization which such a formulation entails. We describe the renormalization process first using a qualitative phase space cell analysis, and we then set up a precise similarity renormalization scheme with cutoffs on constituent momenta and exhibit calculations to second order. We outline further computations that remain to be carried out. There is an initial nonperturbative but nonrelativistic calculation of the hadronic masses that determines the artificial potential, with binding energies required to be fourth order in the coupling as in QED. Next there is a calculation of the leading radiative corrections to these masses, which requires our renormalization program. Then the real struggle of finding the right extensions to perturbation theory to study the strong-coupling behavior of bound states can begin.

Limit cycles in quantum theories.

Renormalization group limit cycles and more chaotic behavior may be commonplace for quantum Hamiltonians requiring renormalization, in contrast to experience based on classical models with critical behavior, where fixed points are far more common. We discuss the simplest quantum model Hamiltonian identified so far that exhibits a renormalization group with both limit cycle and chaotic behavior. The model is a discrete Hermitian matrix with two coupling constants, both governed by a nonperturbative renormalization group equation that involves changes in only one of these couplings and is soluble analytically.

Effective confining potentials for QCD

We observe that the linear potential used as a leading approximation for describing color confinement in the instant form of dynamics corresponds to a quadratic confining potential in the front form of dynamics. In particular, the instant-form potentials obtained from lattice gauge theory and string models of hadrons agree with the potentials determined from models using front-form dynamics and light-front holography, not only in their shape, but also in their numerical strength.

Universality, marginal operators, and limit cycles

The universality of renormalization-group limit-cycle behavior is illustrated with a simple discrete Hamiltonian model. A nonperturbative renormalization-group equation for the model is soluble analytically at criticality and exhibits one marginal operator (made necessary by the limit cycle) and an infinite set of irrelevant operators. Relevant operators are absent. The model exhibits an infinite series of bound-state energy eigenvalues. This infinite series approaches an exact geometric series as the eigenvalues approach zero\char22{}also a consequence of the limit cycle. Wegners eigenvalues for irrelevant operators are calculated generically for all choices of parameters in the model. We show that Wegners eigenvalues are independent of location on the limit cycle, in contrast with Wegners operators themselves, which vary depending on their location on the limit cycle. An example is then used to illustrate numerically how one can tune the initial Hamiltonian to eliminate the first two irrelevant operators. After tuning, the Hamiltonians bound-state eigenvalues converge much more quickly than otherwise to an exact geometric series.

Dynamics of effective gluons

Renormalized Hamiltonians for gluons are constructed using a perturbative boost-invariant renormalization group procedure for effective particles in light-front QCD, including terms up to third order. The effective gluons and their Hamiltonians depend on the renormalization group parameter \ensuremath{\lambda}, which defines the width of momentum-space form factors that appear in the renormalized Hamiltonian vertices. Third-order corrections to the three-gluon vertex exhibit asymptotic freedom, but the rate of change of the vertex with \ensuremath{\lambda} depends in a finite way on regularization of small-x singularities. This dependence is shown in some examples, and a class of regularizations with two distinct scales in x is found to lead to the Hamiltonian running coupling constant whose dependence on \ensuremath{\lambda} matches the known perturbative result from Lagrangian calculus for the dependence of gluon three-point Greens function on the running momentum scale at large scales. In the Fock-space basis of effective gluons with small \ensuremath{\lambda}, the vertex form factors suppress interactions with large kinetic energy changes and thus remove direct couplings of low-energy constituents to high-energy components in the effective bound-state dynamics. This structure is reminiscent of parton and constituent models of hadrons.

Asymptotic freedom and bound states in Hamiltonian dynamics

We study a model of asymptotically free theories with bound states using the similarity renormalization group for Hamiltonians. We find that the renormalized effective Hamiltonians can be approximated in a large range of widths by introducing similarity factors and the running coupling constant. This approximation loses accuracy for the small widths on the order of the bound state energy and it is improved by using the expansion in powers of the running coupling constant. The coupling constant for small widths is of order 1. The small width effective Hamiltonian is projected on a small subset of the effective basis states. The resulting small matrix is diagonalized exactly and the bound state energy of the original theory is obtained with accuracy of the order of 10% using the first three terms in the perturbative expansion of the effective Hamiltonian. We briefly describe options for improving the accuracy. @S0556-2821~98!03006-9#

Reinterpretation of gluon condensate in dynamics of hadronic constituents

We describe an approximate quantum mechanical picture of hadrons in Minkowski space in the context of a renormalization group procedure for effective particles (RGPEP) in a light-front Hamiltonian formulation of QCD. The picture suggests that harmonic oscillator potentials for constituent quarks in lightest mesons and baryons may result from the gluon condensation inside hadrons, rather than from an omnipresent gluon condensate in vacuum. The resulting boost-invariant constituent dynamics at the renormalization group momentum scales comparable with Lambda_QCD, is identified using gauge symmetry and a crude mean-field approximation for gluons. Besides constituent quark models, the resulting picture also resembles models based on AdS/QCD ideas. However, our hypothetical picture significantly differs from the models by the available option for a systematic analysis in QCD, in which the new picture may be treated as a candidate for a first approximation. This option is outlined by embedding our presentation of the crude and simple hadron picture in the context of RGPEP and a brief outlook on hadron phenomenology. Several appendices describe elements of the formalism required for actual calculations in QCD, including an extension of RGPEP beyond perturbation theory.

Impact of bound states on similarity renormalization group transformations

We study a simple class of unitary renormalization group (RG) transformations governed by a parameter f in the range [0; 1]. For f = 0, the transformation is one introduced by Wegner in condensed matter physics, and for f = 1 it is a simpler transformation that is being used in nuclear theory. The transformation with f = 0 diagonalizes the Hamiltonian but in the transformations withf near 1 divergent couplings arise as bound state thresholds emerge. To illustrate and diagnose this behavior, we numerically study Hamiltonian ows in two simple models with bound states: one with asymptotic freedom and a related one with a limit cycle. The f = 0 transformation places bound-state eigenvalues on the diagonal at their natural scale, after which the bound states decouple from the dynamics at much smaller momentum scales. At the other extreme, the f = 1 transformation tries to move bound-state eigenvalues to the part of the diagonal corresponding to the lowest momentum scales available and inevitably diverges when this scale is taken to zero. Intermediate values of f cause intermediate shifts of bound state eigenvalues down the diagonal and produce increasingly large coupling constants to do this. In discrete models, there is a critical value, fc, below which bound state eigenvalues appear at their natural scale and the entire ow to the diagonal is well-behaved. We analyze the shift mechanism analytically in a 3x3 matrix model,

Asymptotic freedom in the front-form Hamiltonian for quantum chromodynamics of gluons

Asymptotic freedom of gluons in QCD is obtained in the leading terms of their renormalized Hamiltonian in the Fock space, instead of considering virtual Greens functions or scattering amplitudes. Namely, we calculate the three-gluon interaction term in the front-form Hamiltonian for effective gluons in the Minkowski space-time using the renormalization group procedure for effective particles (RGPEP), with a new generator. The resulting three-gluon vertex is a function of the scale parameter,

Model of the AdS/QFT duality

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