Limit cycles of effective theories

A simple example is used to show that renormalization group limit cycles of effective quantum theories can be studied in a new way. The method is based on the similarity renormalization group procedure for Hamiltonians. The example contains a logarithmic ultraviolet divergence that is generated by both real and imaginary parts of the Hamiltonian matrix elements. Discussion of the example includes a connection between asymptotic freedom with one scale of bound states and the limit cycle with an entire hierarchy of bound states.Comment: 8 pages, 3 figures, revtex

Renormalized Poincar\'e algebra for effective particles in quantum field theory

Using an expansion in powers of an infinitesimally small coupling constant $g$, all generators of the Poincar\'e group in local scalar quantum field theory with interaction term $g \phi^3$ are expressed in terms of annihilation and creation operators $a_\lambda$ and $a^\dagger_\lambda$ that result from a boost-invariant renormalization group procedure for effective particles. The group parameter $\lambda$ is equal to the momentum-space width of form factors that appear in vertices of the effective-particle Hamiltonians, $H_\lambda$. It is verified for terms order 1, $g$, and $g^2$, that the calculated generators satisfy required commutation relations for arbitrary values of $\lambda$. One-particle eigenstates of $H_\lambda$ are shown to properly transform under all Poincar\'e transformations. The transformations are obtained by exponentiating the calculated algebra. From a phenomenological point of view, this study is a prerequisite to construction of observables such as spin and angular momentum of hadrons in quantum chromodynamics.Comment: 17 pages, 5 figure

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 order 1. The small width effective hamiltonian is projected on a small subset of the effective basis states. The resulting small matrix is diagonalized and the exact bound state energy is obtained with accuracy of the order of 10% using the first three terms in the expansion. We briefly describe options for improving the accuracy.Comment: plain latex file, 15 pages, 6 latex figures 1 page each, 1 tabl

Large-momentum convergence of Hamiltonian bound-state dynamics of effective fermions in quantum field theory

Contributions to the bound-state dynamics of fermions in local quantum field theory from the region of large relative momenta of the constituent particles, are studied and compared in two different approaches. The first approach is conventionally developed in terms of bare fermions, a Tamm-Dancoff truncation on the particle number, and a momentum-space cutoff that requires counterterms in the Fock-space Hamiltonian. The second approach to the same theory deals with bound states of effective fermions, the latter being derived from a suitable renormalization group procedure. An example of two-fermion bound states in Yukawa theory, quantized in the light-front form of dynamics, is discussed in detail. The large-momentum region leads to a buildup of overlapping divergences in the bare Tamm-Dancoff approach, while the effective two-fermion dynamics is little influenced by the large-momentum region. This is illustrated by numerical estimates of the large-momentum contributions for coupling constants on the order of between 0.01 and 1, which is relevant for quarks.Comment: 22 pages, 9 figure

Flow equation solution for the weak to strong-coupling crossover in the sine-Gordon model

A continuous sequence of infinitesimal unitary transformations, combined with an operator product expansion for vertex operators, is used to diagonalize the quantum sine-Gordon model for 2 pi < beta^2 < infinity. The leading order of this approximation already gives very accurate results for the single-particle gap in the strong-coupling phase. This approach can be understood as an extension of perturbative scaling theory since it links weak to strong-coupling behavior in a systematic expansion. The approach should also be useful for other strong-coupling problems that can be formulated in terms of vertex operators.Comment: 4 pages, 1 figure, minor changes (typo in Eq. (3) corrected, references added), published versio

Color van der Waals forces between heavy quarkonia in effective QCD

The perturbative renormalization group for light-front QCD Hamiltonian produces a logarithmically rising interquark potential already in second order, when all gluons are neglected. There is a question if this approach produces also color van der Waals forces between heavy quarkonia and of what kind. This article shows that such forces do exist and estimates their strength, with the result that they are on the border of exclusion in naive approach, while more advanced calculation is possible in QCD.Comment: 7 pages, elsart, bibliography in .bbl file, to be submitted to Physics Letters

Renormalization approach to many-particle systems

This paper presents a renormalization approach to many-particle systems. By starting from a bare Hamiltonian ${\cal H}= {\cal H}_0 +{\cal H}_1$ with an unperturbed part ${\cal H}_0$ and a perturbation ${\cal H}_1$,we define an effective Hamiltonian which has a band-diagonal shape with respect to the eigenbasis of ${\cal H}_0$. This means that all transition matrix elements are suppressed which have energy differences larger than a given cutoff $\lambda$ that is smaller than the cutoff $\Lambda$ of the original Hamiltonian. This property resembles a recent flow equation approach on the basis of continuous unitary transformations. For demonstration of the method we discuss an exact solvable model, as well as the Anderson-lattice model where the well-known quasiparticle behavior of heavy fermions is derived.Comment: 11 pages, final version, to be published in Phys. Rev.

Boost-Invariant Running Couplings in Effective Hamiltonians

We apply a boost-invariant similarity renormalization group procedure to a light-front Hamiltonian of a scalar field phi of bare mass mu and interaction term g phi^3 in 6 dimensions using 3rd order perturbative expansion in powers of the coupling constant g. The initial Hamiltonian is regulated using momentum dependent factors that approach 1 when a cutoff parameter Delta tends to infinity. The similarity flow of corresponding effective Hamiltonians is integrated analytically and two counterterms depending on Delta are obtained in the initial Hamiltonian: a change in mu and a change of g. In addition, the interaction vertex requires a Delta-independent counterterm that contains a boost invariant function of momenta of particles participating in the interaction. The resulting effective Hamiltonians contain a running coupling constant that exhibits asymptotic freedom. The evolution of the coupling with changing width of effective Hamiltonians agrees with results obtained using Feynman diagrams and dimensional regularization when one identifies the renormalization scale with the width. The effective light-front Schroedinger equation is equally valid in a whole class of moving frames of reference including the infinite momentum frame. Therefore, the calculation described here provides an interesting pattern one can attempt to follow in the case of Hamiltonians applicable in particle physics.Comment: 24 pages, LaTeX, included discussion of finite x-dependent counterterm

Note on restoring manifest rotational symmetry in hyperfine and fine structure in light-front QED

We study the part of the renormalized, cutoff QED light-front Hamiltonian that does not change particle number. The Hamiltonian contains interactions that must be treated in second-order bound state perturbation theory to obtain hyperfine structure. We show that a simple unitary transformation leads directly to the familiar Breit-Fermi spin-spin and tensor interactions, which can be treated in degenerate first-order bound-state perturbation theory, thus simplifying analytic light-front QED calculations. To the order in momenta we need to consider, this transformation is equivalent to a Melosh rotation. We also study how the similarity transformation affects spin-orbit interactions.Comment: 17 pages, latex fil

Similarity Renormalization, Hamiltonian Flow Equations, and Dyson's Intermediate Representation

A general framework is presented for the renormalization of Hamiltonians via a similarity transformation. Divergences in the similarity flow equations may be handled with dimensional regularization in this approach, and the resulting effective Hamiltonian is finite since states well-separated in energy are uncoupled. Specific schemes developed several years ago by Glazek and Wilson and contemporaneously by Wegner correspond to particular choices within this framework, and the relative merits of such choices are discussed from this vantage point. It is shown that a scheme for the transformation of Hamiltonians introduced by Dyson in the early 1950's also corresponds to a particular choice within the similarity renormalization framework, and it is argued that Dyson's scheme is preferable to the others for ease of computation. As an example, it is shown how a logarithmically confining potential arises simply at second order in light-front QCD within Dyson's scheme, a result found previously for other similarity renormalization schemes. Steps toward higher order and nonperturbative calculations are outlined. In particular, a set of equations analogous to Dyson-Schwinger equations is developed.Comment: REVTex, 32 pages, 7 figures (corrected references