55 research outputs found
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
, all generators of the Poincar\'e group in local scalar quantum field
theory with interaction term are expressed in terms of annihilation
and creation operators and that result from a
boost-invariant renormalization group procedure for effective particles. The
group parameter is equal to the momentum-space width of form factors
that appear in vertices of the effective-particle Hamiltonians, . It
is verified for terms order 1, , and , that the calculated generators
satisfy required commutation relations for arbitrary values of .
One-particle eigenstates of 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
Renormalization approach to many-particle systems
This paper presents a renormalization approach to many-particle systems. By
starting from a bare Hamiltonian with an
unperturbed part and a perturbation ,we define an
effective Hamiltonian which has a band-diagonal shape with respect to the
eigenbasis of . This means that all transition matrix elements are
suppressed which have energy differences larger than a given cutoff
that is smaller than the cutoff 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.
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
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
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