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We study a simple class of unitary renormalization group 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 with f near 1 divergent couplings arise as bound state thresholds emerge. To illustrate and diagnose this behavior, we numerically study Hamiltonian flows 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. We analyze the shift mechanism analytically in a 3x3 matrix model, which displays the essense of this RG behavior., Comment: 27 pages, 3 figures

Quantum field theories require a cutoff to regulate divergences that result from local interactions, and yet physical results can not depend on the value of this cutoff. The renormalization group employs a transformation that changes the cutoff to isolate hamiltonians that produce cutoff-independent eigenvalues. The similarity renormalization group is based on similarity transformations that regulate off-diagonal matrix elements, forcing the hamiltonian towards a band-diagonal form as the cutoff is lowered. This avoids pathologies that plagued tradition transformations acting on hamiltonians, making it possible to produce a well-behaved perturbative approximation of renormalized hamiltonians in asymptotically free theories. We employ a simple two-dimensional delta function example to illustrate this new renormalization technique., Comment: 28 pages, 7 postscript figures

It may be possible to approximate the full pion wave function in light-front QCD using only $q \qbar$ and $q \qbar g$ Fock space components. Removing zero modes and using invariant-mass cutoffs that make a constituent approximation possible leads to non-canonical terms in the QCD hamiltonian, and forces us to work in the broken symmetry phase of QCD in which chiral symmetry breaking operators must appear directly in the hamiltonian because the vacuum is trivial. Assuming any candidate interactions are local in the transverse direction, I argue that they are probably relevant operators so that perturbation theory is not modified at high energies. Since light-front chiral symmetry corresponds to quark helicity conservation, we can readily identify candidate chiral symmetry breaking interactions. The only candidate relevant operator is the quark-gluon emission/absorption operator with a quark spin flip. I argue that this operator can only produce the physical $\pi-\rho$ mass splitting if the $q \qbar g$ component of the pion is significant., Comment: LATEX, 6 pages. To appear in Proceedings of HD 2000, From Hadrons to Strings

We calculate a renormalized Hamiltonian for pure-glue QCD and diagonalize it. The renormalization procedure is designed to produce a Hamiltonian that will yield physical states that rapidly converge in an expansion in free-particle Fock-space sectors. To make this possible, we use light-front field theory to isolate vacuum effects, and we place a smooth cutoff on the Hamiltonian to force its free-state matrix elements to quickly decrease as the difference of the free masses of the states increases. The cutoff violates a number of physical principles of light-front pure-glue QCD, including Lorentz covariance and gauge covariance. This means that the operators in the Hamiltonian are not required to respect these physical principles. However, by requiring the Hamiltonian to produce cutoff-independent physical quantities and by requiring it to respect the unviolated physical principles of pure-glue QCD, we are able to derive recursion relations that define the Hamiltonian to all orders in perturbation theory in terms of the running coupling. We approximate all physical states as two-gluon states, and use our recursion relations to calculate to second order the part of the Hamiltonian that is required to compute the spectrum. We diagonalize the Hamiltonian using basis-function expansions for the gluons' color, spin, and momentum degrees of freedom. We examine the sensitivity of our results to the cutoff and use them to analyze the nonperturbative scale dependence of the coupling. We investigate the effect of the dynamical rotational symmetry of light-front field theory on the rotational degeneracies of the spectrum and compare the spectrum to recent lattice results. Finally, we examine our wave functions and analyze the various sources of error in our calculation., Comment: 75 pages, 17 figures, 1 table

The Schrodinger equation with a two-dimensional delta-function potential is a simple example of an asymptotically free theory that undergoes dimensional transmutation. Renormalization requires the introduction of a mass scale, which can be lowered perturbatively until an infrared cutoff produced by non-perturbative effects such as bound state formation is encountered. We outline the effective field theory and similarity renormalization group techniques for producing renormalized cutoff hamiltonians, and illustrate the control of logarithmic and inverse-power-law errors both techniques provide., Comment: 17 pages, 3 figures, to be published in Proceedings of the 1998 YITP-Workshop on QCD and Hadron Physics, Kyoto, Japan

Light-front quantum chromodynamics may lead to an accurate constituent approximation for the low-energy properties of hadrons. This requires a cutoff that violates explicit gauge invariance and Lorentz covariance, leading to the calculation of a renormalized QCD hamiltonian using a similarity renormalization group. Renormalization repairs broken symmetries, and in light-front field theory it moves dynamical effects that usually require large numbers of partons to few-body effective interactions. This has been shown to work in QED through dominant contributions to the Lamb shift. In QCD logarithmic confinement arises at second order, and initial bound state calculations produce reasonable results., Comment: 26 pages, 5 figures, to be published in Proceedings of the APCTP-RCNP Joint International School on Physics of Hadrons and QCD, Osaka, Japan

Hamiltonian light-front field theory can be used to solve for hadron states in QCD. To this end, a method has been developed for systematic renormalization of Hamiltonian light-front field theories, with the hope of applying the method to QCD. It assumed massless particles, so its immediate application to QCD is limited to gluon states or states where quark masses can be neglected. This paper builds on the previous work by including particle masses non-perturbatively, which is necessary for a full treatment of QCD. We show that several subtle new issues are encountered when including masses non-perturbatively. The method with masses is algebraically and conceptually more difficult; however, we focus on how the methods differ. We demonstrate the method using massive phi^3 theory in 5+1 dimensions, which has important similarities to QCD., Comment: 7 pages, 2 figures. Corrected error in Eq. (11), v3: Added extra disclaimer after Eq. (2), and some clarification at end of Sec. 3.3. Final published version

We develop a systematic method for computing a renormalized light-front field theory Hamiltonian that can lead to bound states that rapidly converge in an expansion in free-particle Fock-space sectors. To accomplish this without dropping any Fock sectors from the theory, and to regulate the Hamiltonian, we suppress the matrix elements of the Hamiltonian between free-particle Fock-space states that differ in free mass by more than a cutoff. The cutoff violates a number of physical principles of the theory, and thus the Hamiltonian is not just the canonical Hamiltonian with masses and couplings redefined by renormalization. Instead, the Hamiltonian must be allowed to contain all operators that are consistent with the unviolated physical principles of the theory. We show that if we require the Hamiltonian to produce cutoff-independent physical quantities and we require it to respect the unviolated physical principles of the theory, then its matrix elements are uniquely determined in terms of the fundamental parameters of the theory. This method is designed to be applied to QCD, but for simplicity, we illustrate our method by computing and analyzing second- and third-order matrix elements of the Hamiltonian in massless phi-cubed theory in six dimensions., Comment: 47 pages, 6 figures; improved referencing, minor presentation changes

Light-front coordinates offer a scenario in which a constituent approximation of hadron structure can emerge from QCD. This requires cutoffs that violate Lorentz covariance and gauge invariance, and a new renormalization group formalism based on a similarity transformation is used with coupling coherence to fix counterterms that restore these symmetries. The counterterms contain functions of longitudinal momentum fractions which severely complicate renormalization, but they also offer possible resolutions of apparent contradictions between the constituent picture and QCD. The similarity transformation and coupling coherence are applied to QED; and it is shown that the resultant Hamiltonian leads to the correct bound state results. The same techniques are applied to QCD and it is shown that a simple confinement mechanism and a reasonable description of heavy quark bound states emerge naturally., Comment: 44 pages, LATEX (including 4 ps-figures), uses epsfig.sty. Based on lectures given at the NATO Advanced Study Institute: Confinement, Duality, and Non-Perturbative Aspects of QCD, Newton Institute, Cambridge, 23 June - 4 July, 1997

Light-front Hamiltonian methods are being developed to attack bound-state problems in QCD. In this paper we advance the state of the art for these methods by computing the well-known Lamb shift in hydrogen starting from first principles of QED. There are obvious but significant qualitative differences between QED and QCD. In this paper, we discuss the similarities that may survive in a non-perturbative QCD calculation in the context of a precision non-perturbative QED calculation. Central to the discussion are how a constituent picture arises in a gauge field theory, how bound-state energy scales emerge to guide the renormalization procedure, and how rotational invariance emerges in a light-front calculation., Comment: 35 pages, latex, 5 postscript figures included

A constituent parton picture of hadrons with logarithmic confinement naturally arises in weak coupling light-front QCD. Confinement provides a mass gap that allows the constituent picture to emerge. The effective renormalized Hamiltonian is computed to ${\cal O}(g^2)$, and used to study charmonium and bottomonium. Radial and angular excitations can be used to fix the coupling $\alpha$, the quark mass $M$, and the cutoff $\Lambda$. The resultant hyperfine structure is very close to experiment., Comment: 9 pages, 1 latex figure included in the text. Published version (much more reader-friendly); corrected error in self-energy

We study the QED bound-state problem in a light-front hamiltonian approach. Starting with a bare cutoff QED Hamiltonian, $H_{_{B}}$, with matrix elements between free states of drastically different energies removed, we perform a similarity transformation that removes the matrix elements between free states with energy differences between the bare cutoff, $\Lambda$, and effective cutoff, $\lam$ ($\lam < \Lam$). This generates effective interactions in the renormalized Hamiltonian, $H_{_{R}}$. These effective interactions are derived to order $\alpha$ in this work, with $\alpha \ll 1$. $H_{_{R}}$ is renormalized by requiring it to satisfy coupling coherence. A nonrelativistic limit of the theory is taken, and the resulting Hamiltonian is studied using bound-state perturbation theory (BSPT). The effective cutoff, $\lam^2$, is fixed, and the limit, $0 \longleftarrow m^2 \alpha^2\ll \lam^2 \ll m^2 \alpha \longrightarrow \infty$, is taken. This upper bound on $\lam^2$ places the effects of low-energy (energy transfer below $\lam$) emission in the effective interactions in the $| e {\overline e} > $ sector. This lower bound on $\lam^2$ insures that the nonperturbative scale of interest is not removed by the similarity transformation. As an explicit example of the general formalism introduced, we show that the Hamiltonian renormalized to $O(\alpha)$ reproduces the exact spectrum of spin splittings, with degeneracies dictated by rotational symmetry, for the ground state through $O(\alpha^4)$. The entire calculation is performed analytically, and gives the well known singlet-triplet ground state spin splitting of positronium, $7/6 \alpha^2 Ryd$. We discuss remaining corrections other than the spin splittings and how they can be treated in calculating the spectrum with higher precision., Comment: 46 pages, latex, 3 Postscript figures included, section on remaining corrections added, title changed, error in older version corrected, cutoff placed in a window

Light-front field theory offers a scenario in which a constituent picture of hadrons may arise, but only if cutoffs that violate explicit covariance and gauge invariance are used. The perturbative renormalization group can be used to approximate the cutoff QCD hamiltonian, and even at lowest orders the resultant hamiltonian displays interesting phenomenological features. A general scheme for computing and using these hamiltonians is discussed and it is explicitly shown that a confining interaction appears when the hamiltonian is computed to second order., Comment: 15 pages, latex, no figures. To appear in Proceedings of the workshop `Theory of Hadrons and Light-Front QCD,' Poland, August 1994

Light-front coordinates offer a scenario in which a constituent picture of hadron structure can emerge from QCD, after several difficulties are addressed. Field theoretic difficulties force us to introduce cutoffs that violate Lorentz covariance and gauge invariance, and a new renormalization group formalism based on a similarity transformation is used with coupling coherence to fix cuonterterms that restore these symmetries. The counterterms contain functions of longitudinal momentum fractions that severely complicate renormalization, but they also offer possible resolutions of apparent contradictions between the constituent picture and QCD. The similarity transformation and coupling coherence are applied to QED; and it is shown that the resultant Hamiltonian leads to standard lowest order bound state results, with the Coulomb interaction emerging naturally. The same techniques are applied to QCD and with physically motivated assumptions it is shown that a simple confinement mechanism appears., Comment: corrected version of 1994 lectures, 84 pages, LATEX. Published in "Hadron Physics 94," Proceedings, Gramado, Brazil (World Scientific, 1995)

A perturbative renormalization group is formulated for the study of Hamiltonian light-front field theory near a critical Gaussian fixed point. The only light-front renormalization group transformations found that can be approximated by dropping irrelevant operators and using perturbation theory near Gaussian fixed volumes, employ invariant-mass cutoffs. These cutoffs violate covariance and cluster decomposition, and allow functions of longitudinal momenta to appear in all relevant, marginal, and irrelevant operators. These functions can be determined by insisting that the Hamiltonian display a coupling constant coherence, with the number of couplings that explicitly run with the cutoff scale being limited and all other couplings depending on this scale only through perturbative dependence on the running couplings. Examples are given that show how coupling coherence restores Lorentz covariance and cluster decomposition, as recently speculated by Wilson and the author. The ultimate goal of this work is a practical Lorentz metric version of the renormalization group, and practical renormalization techniques for light-front quantum chromodynamics., Comment: TEX with macros included. 11 figures available upon request. 120 pages. Ohio State preprint OSU-NT-93-117