Non-Perturbative Methods Workshop Abstracts
Workshop Abstracts
Convergence of discrete action to continuum action
does not imply convergence of discrete physics to
continuum physics, and vice versa
A basic criterion used when constructing discrete (in particular, lattice)
versions of quantum field theories is that the discrete action should converge
to the continuum action in the continuum limit. In my talk I will show
that this criterion is neither sufficient nor necessary in general to
ensure convergence of discrete physics (amplitudes, expectation values etc)
to continuum physics. This is demonstrated in several simple examples where
the physical quantities of interest can be exactly evaluated in the continuum.
The Model for QCD Running Coupling Constant
with Dynamically Generated Mass and Enhancement
in the Infrared Region
Nonperturbative studies of strong running coupling
constant in the infrared region are discussed.
Starting from the analyses of the Dyson-Schwinger
equations in the gauge sector of QCD the conclusion
is made on incomplete fixing the perturbation theory
summation ambiguity within "(forced) analytization
procedure" (called also as dispersive approach).
The minimal model for $alpha_s(q^2)$ is proposed
so that the perturbative time-like discontinuity
is preserved and nonperturbative terms not only
remove the Landau singularity but also provide the
ultraviolet convergence of the gluon condensate.
Within this model, on the one hand, the gluon zero
modes are enhanced (dual superconductor property
of the QCD vacuum) and, on the other hand, the dynamical
gluon mass generation is realized with $m_g \simeq
0.6 GeV$. Uncertainty connected with division of
perturbative and nonpertutbative contributions is
discussed with gluon condensate taken as an example.
Thermodynamic properties of one-flavour QCD
We have performed a study of the thermodynamic
properties of 1-flavour QCD for heavy to moderate
quark masses, using the multiboson algorithm. The
order of the deconfinement phase transition is
determined by studying the finite-size scaling
of the Polyakov loop susceptibility on lattices
of 3 different spatial sizes. For heavy quarks
the transition is first order which becomes weaker
as the quark mass decreases. We estimate the the
end point of the first order phase transition to
occur at a quark mass of about 1.1 GeV.
Continuum limit of path integrals in euclidean quantum mechanics
There exist quantum mechanical actions, different from the naive one,
which approach more efficiently the continuum limit of the path
integral. Several actions, based on well established approximations, are
proposed and their ability in describing the approach to the continuum
is studied by means of Monte Carlo simulations. Numerical results are
shown for the anharmonic oscillator. In some cases, the
number of points in the path integral is reduced by a factor of 10, as
compared to the naive action. In all studied cases it is verified that
improvement is due to fluctuations around some classical path. There
also seems to be a limit for systematic improvement, since at
some order all expansions produce actions which, if taken literally,
appear to be unbounded from below, hence preventing importance sampling
of the action. The extension to field theory is discussed.
Infrared and Ultraviolet Behaviour of the Running Coupling in QCD
The coupled Dyson-Schwinger equations for the gluon and ghost propagators in QCD are shown to
have solutions that correspond to a unique running coupling that has a finite infrared fixed
point and the expected logarithmic decrease in the ultraviolet. The infrared coupling is
large enough to support chiral symmetry breaking; and quarks are technically not confined,
but they cannot be isolated.
Infrared fixed point of the running coupling in QCD
The coupling of the ghost and gluon Dyson-Schwinger equations yields a
uniquely determined running coupling with a large infrared fixed point. I will
discuss the numerical method used to solve these equations and will show some
new intriguing results concerning the infrared behaviour of the gluon and ghost
propagator.
The analytic structure of heavy quark propagators
We develop a formalism for calculating the
renormalised heavy quark propagator in the
region of the bare fermion mass pole
$p^2 = m_Q^2$ from an approximate Dyson-Schwinger
equation expanded in $1/m_Q$. We are
particularly interested in the effect on the
analytic structure of the heavy quark propagator
if the bare vertex approximation is replaced by
the Ball-Chiu Ansatz, and the effect of ignoring
the correct asymptotic ultraviolet behaviour of
the gluon propagator.
Green function Monte Carlo study of correlation
functions in the (2+1)-dimensional U(1) lattice
gauge model
A ``forward walking'' Quantum Monte Carlo (QMC) algorithm has been
developed to calculate correlation functions for Hamiltonian Lattice
U(1) theory in (2+1) dimensions. Wilson loops, Polyakov loops, spacelike
and timelike correlation functions are studied, the aim being to show
that string tensions and mass gaps can be computed successfully by
QMC methods. Comparisons are made with other approaches.
Recent Results from the Global Colour Model of QCD
We present various new results from the Global Colour Model (GCM) of QCD
including the quark-gluon coupling down to gluon momentum q = 0.5GeV,
nucleon Faddeev computations for the nucleon core, and meson dressing
of the nucleon core to produce the physical nucleon.
The standard model on the lattice
I will discuss the problems with
formulating the standard model
on the lattice, paying particular
attention to the issues of chiral
symmetry.
Amplitude analysis in particle decays
We provide the best possible covariant amplitude
decompositions in weak decays, with which to
compare data, including partial decay rates,
helicity amplitudes and polarizations. The
systematic dependence of the amplitudes on
masses and quantum numbers of participating
particles can then shed light on the underlying
field theoretic dynamics.
A Proposal for "Topologically Unquenched" Lattice-QCD
I propose to split the QCD fermion functional determinant into two
factors, the first referring to a standard configuration in each
topological sector and the second describing the effect of the smooth
deviation of the actual configuration from the reference
configuration. Then "topologically unquenched" QCD is defined as to
take the first (topological) factor fully into account and to set
the second factor only to one. I will try to argue that
"topologically unquenched" QCD is an excellent starting point for
approaching full QCD in lattice simulations as it gets the main
qualitative features right from the beginning.
Improved Chiral Perturbation Theory Approach to
the Nucleon-Nucleon Interaction
Some problems of the existing chiral
perturbation theory approach to the
nucleon-nucleon interaction are considered
(minimal subtraction, cut-off regularization).
An improved treatment of this interaction is
suggested.
Series expansion studies of QED in (1+1)D and (2+1)D
Strong-coupling series expansions are calculated for "positronium"
energy states in Hamiltonian lattice QED in (1+1)D (the Schwinger model)
and (2+1)D. The series are obtained using linked-cluster methods, and
extrapolated towards the continuum limit using Pade or integrated
differential approximants. The results are compared with those from
other methods.
A Solution to Coupled Dyson--Schwinger Equations
for Gluons and Ghosts in Landau Gauge
A truncation scheme for the Dyson--Schwinger equations of QCD in Landau gauge
is presented which implements the Slavnov--Taylor identities for the 3--point
vertex functions. Neglecting contributions from 4--point correlations such as
the 4--gluon vertex function and irreducible scattering kernels, a closed
system of equations for the propagators is obtained. For the pure gauge
theory without quarks this system of equations for the propagators of gluons
and ghosts is solved in an approximation which allows for an analytic
discussion of its solutions in the infrared: The gluon propagator is shown
to vanish for small spacelike momenta whereas the ghost propagator is found
to be infrared enhanced. The running coupling of the non--perturbative
subtraction scheme approaches an infrared stable fixed point at a
critical value of the coupling, $lpha_c \simeq 9.5$. The gluon
propagator is shown to have no Lehmann representation. The results for
the propagators obtained here compare favorably with recent lattice
calculations.
Electromagnetic form factors of light vector mesons
The electromagnetic form factors G_E, G_M, and G_Q,
charge radii, magnetic and quadrupole moments, and decay widths
of the light vector mesons (rho, K*+ and K*0)
are calculated within a Lorentz-covariant,
Dyson-Schwinger equation (DSE) based model,
using algebraic quark propagators that incorporate
confinement and dynamical chiral symmetry breaking effects.
Calculated charge radii are 0.63 fm for the rho,
0.51 fm for the K*+, and r^2 = -0.0014 fm^2 for the K*0;
magnetic moments (in natural magnetons) are 2.41 for the rho,
2.08 for K*+, and -0.020 for K*0.
The calculated static properties of the rho agree with those
found in light-front calculations, but the form factors differ
at larger q^2 due to our use of more realistic quark propagators.
Quark Contributions to Magnetic and Electric
Moments of Hadrons
The results for the calculations of the electric
and magnetic dipole moments of the rho meson,
using the propagators and vertices derived from
QCD Dyson-Schwinger equations, are presented.
Current progress on a similar calculation for
the nucleon will also be discussed.
Vacuum with "polarized" instantons in non-Abelian
gauge theory
The models in non-Abelia gauge theory are found
in which topological excitations (instantons and anti-instantons) are
"polarized", i.e. have a preferred orientation.
The vacuum with polarized instantons is the new
gauge-theory phase. An interest in this phase
is enhanced by a hope is that excitations
above "polarized" vacuum have spin 2. This fact,
if confirmed, would make these excitations be
a new candidate for the role of the graviton.
Electromagnetic corrections to nonperturbative strong interaction
quantities
An expression is derived for the electromagnetic correction to any
model of strong interactions specified by field theoretical
equations. The correction is nonperturbative with respect to the
strong interaction and does not depend on the gauge used to calculate
physical quantities.
Light Hadron Spectroscopy from Mean-Field Improved Lattice Actions
The lattice QCD field is currently undergoing a revolution in the
manner in which improvements of the approach are being implemented.
The discovery of new perturbative techniques has shifted the focus
away from ``brute force'' to the consideration of improved actions
involving next-nearest neighbors. These new lattice actions provide
scaling in the pure gauge sector for lattice spacings as large as 0.4
fm. I will focus on recent results for the masses and dispersions of
light hadrons calculated in lattice QCD using an order a^2
mean-field-improved gluon action and an order a^2 mean-field-improved
next-nearest-neighbor fermion action originally proposed by Hamber and
Wu. Two lattices of constant volume with coarse lattice spacings of
approximately 0.40 fm and 0.24 fm are considered. The results reveal
some scaling violations at the coarser lattice spacing on the order of
5%. At the finer lattice spacing, the calculated mass ratios
reproduce state-of-the-art results using unimproved actions. Good
dispersion and rotational invariance are also found. These actions
hold the promise of finally moving beyond the quenched approximation
in a quantitative sense. The relative merit of alternative choices
for improvement operators is assessed through close comparisons with
other plaquette-based mean-field-improved actions.
Pseudoscalar mesons as QCD bound states
Independent of assumptions about the form of the quark-quark scattering
kernel, $K$, we derive the explicit relation between the
flavour-nonsinglet pseudoscalar meson Bethe--Salpeter amplitude,
$\Gamma_H$, and the dressed-quark propagator in the chiral limit. In
addition to a term proportional to $\gamma_5$, $\Gamma_H$ necessarily
contains qualitatively and quantitatively important terms proportional
to $\gamma_5\,\gamma\cdot P$ and $\gamma_5\,\gamma\cdot k\,k\cdot P$,
where $P$ is the total momentum of the bound state. Using only the term
proportional to $\gamma_5$ it is impossible to satisfy the axial-vector
Ward-Takahashi identity. The rainbow-ladder Ansatz for $K$, with a
simple model for the dressed-quark-quark interaction, is used to
illustrate and elucidate these general results. The model preserves the
one-loop renormalisation group structure of QCD. The ultraviolet
asymptotic behaviour of the scalar functions in the meson BS amplitude
is fully determined by the behaviour of the chiral limit quark mass
function. The high-energy behaviour of the pion electromagnetic form
factor is dominated by the pseudovector part of the BS amplitude,
whereas low-energy pion observables are dominated by the pseudoscalar
part.
Monte Carlo Simulations with Complex-Valued Measures
A simulation method based on the RG blocking is shown to yield
statistical errors smaller than that of the crude MC using absolute
values of the original measures. The method is applied to the
simulation of 2D Ising model with complex-valued temperature.
Functional Methods for Fermion Propagators
The functional expression for the fermion Propagator in an external field
is expanded in a perturbation series which is resummed to provide a
non-perturbative expansion of the propagator.
It is then shown how to use this result to
1) Discuss possible chaotic behaviour of the quantum system
2) Obtain similar results for a coupled fermion-boson field theory
Dynamical symmetry breaking and other miracles in a magnetic field
In 3+1 and 2+1 dimensions, a constant magnetic field is a strong
catalyst of dynamical chiral symmetry breaking, leading to the
generation of a fermion dynamical mass even at the weakest attractive
interactions between fermions. The effect is illustrated in QED and in the
Nambu-Jona-Lasinio model. Possible applications of this effect in particle
physics, cosmology, and condensed matter physics are discussed. It is also
shown how this effect is connected with a quantum field theoretical analog
of the Aharonov-Bohm effect.
J/Psi production within the NRQCD factorisation formalism
Relativistic corrections to J/Psi photoproduction in the color singlet
channel
are computed within the factorization formalism of non-relativistic QCD.
The formalism provides a consistent framework in which observables are
expressed in terms of a series of non-perturbative matrix elements of NRQCD.
These matrix elements describe the hadronization of a heavy quark
anti-quark
state in a definite spin, angular momentum and color configuration.
The formation of the quark and anti-quark is a short distance process
which is calculated in terms of the strong coupling constant.
Mass production and precision engineering
(1) Calculation of physical quantities, like realistic
dynamically generated masses, requires the precision
construction of non-perturbative interactions,
(2) Tests of the mechanisms of chiral symmetry breaking
become possible with the precision detection of decays.
Blueprints for (1) and finger-prints for (2) will be outlined.
Analytic structure of scalar composites in the
gauged Nambu-Jona-Lasinio model
A method is introduced for solving the
Schwinger-Dyson equations for the Yukawa vertex of
the GNJL model in specific kinematic regimes.
This allows one to derive an analytic expression
for the scalar propagator, which is valid along
the entire critical curve separating a chiral
symmetric and a dynamically chiral
symmetry broken phase. The dynamics of
the scalar composites and the conformal phase
transition will be discussed.
Hadrons at extremes of temperature and density
Dyson-Schwinger equations are applied to the study of the phase
transition to a quark gluon plasma at finite temperature, T, and
chemical potential, mu. The transition is first order for all nonzero
mu but second order at mu=0, and the equation of state in the
deconfined phase approaches the Stefan-Boltzmann limit only for large
values of T and mu, which has consequences for quark matter in neutron
stars. Bound states disappear at the transition boundary, and the
response of meson masses to changes in T and mu is anticorrelated with
that of their decay constants.
The quasilocal background field method applied to
gauge theories
An extension of the background field method of
Brown and Duff, beyond the covariantly constant
limit, has been applied to Yang-Mills theory in
interaction with matter. The form of the counter-
terms has been evaluated up to mass dimension 6,
in arbitrary space-time dimension.
Variational Methods in the Worldline
Representation of Field Theory
Polaron variational methods are applied to the
field theoretic Green functions in the worldline
formalism after the bosonic degrees of freedom
have been integrated out. Recent progress in
dealing with fermionic systems, in particular QED,
by means of this non-pertubative technique is
discussed.
Thermodynamics of Lattice QCD
I will discuss some of my recent work with Lagae and Kogut on the
thermodynamics of Lattice QCD with 2 flavours of quarks, in the chiral limit.
We have studied the breaking of the U(1) axial symmetry as hadronic matter
is heated through the phase transition to a quark-gluon plasma. I will discuss
the mesonic screening lengths and the role played by instantons in the plasma
phase. If time permits I might also discuss some of our work using a lattice
QCD action which includes chiral 4-fermion interactions that permit us to
simulate at zero quark masses. This promises to allow us to measure the
critical exponents at this chiral/deconfinement transition.
The infrared behaviour of the gluon propagator from lattice QCD
The gluon propagator in the Landau gauge is calculated in quenched QCD
on a large lattice (32^3x64) at beta=6.0. In order to assess finite
volume and finite lattice spacing artefacts, we also calculate the
propagator on a smaller volume for two different values of the lattice
spacing. New structure seen in the infrared region survives
conservative cuts to the lattice data, and serves to exclude a number
of models that have appeared in the literature.
Lattice calculations of the massive Schwinger model
The Schwinger model (1+1D QED) is the simplest of all gauge field
theories, and can be solved analytically in the two limits of strong coupling,
e>>m, and weak coupling, e<<m. The model also exhibits many of the same
phenomena as QCD, such as confinement and chiral symmetry breaking
with a U(1) axial current anomaly. As a result this makes the massive
Schwinger model an ideal test bed for various lattice techniques which
can later be applied to QCD. We use Hamiltonian lattice techniques and
exactly diagonalize the resulting matrix to calculate eigenvalues
for the massive Schwinger model. Our numerical results are more accurate
than the previous results obtained using this method.
Variational calculation of the effective action
An indication of spontaneous symmetry breaking
is found in the two-dimensional $\lambda\phi^4$
model, where an attention is payed to a
functional form of an effective action.
An effective energy, which is an effective action
for a static field, is obtained as a functional
of the classical field from the ground state of
hamiltonian $H[J]$ interacting with a constant
external field. The energy and wavefunction of
the ground state are calculated in terms of
DLCQ (Discretized Light-Cone Quantization)
under antiperiodic boundary condition. A field
configuration which is physically meaningful
is found as a solution of the quantum mechanical
Euler-Lagrange equation in the $J o 0$ limit.
It is shown that there exists a nontrivial
field configuration in the broken phase of
$Z_2$ symmetry because of a boundary effect.
Test of gauge covariance
of fermion-photon vertex in three dimensional quantum electrodynamics
We study the gauge covariance of fermion-photon vertex in quenched,
massless 3 dimensional quantum electrodynamics. A previous result by Dong
et. al. is tested by using the invariance of photon polarization scalar
under gauge transformation. We also examine whether the restriction in the
earlier work is sufficient to predict the photon polarization scalar
accurately.
Spontaneous Mass Generation in Supersymmetric QED3
Supersymmetry (SUSY) is an exciting mathematical symmetry which, though
yet to be verified expermentally, gives a real and possibly the only hope
of unifying gravity with the other forces. QED3 is an important example of
a confining theory with the desirable property of being reasonably
accessable to analysis. It is therefore an interesting exercise to study
the behaviour of SUSY QED3. In this talk I shall describe SUSY QED3 and
the investigation of spontaneous mass-generation using the Schwinger-Dyson
equation and SUSY Ward identities.
Renormalized Strong-Coupling Quenched QED in Four Dimensions
We study renormalized quenched strong-coupling QED in four dimensions
in arbitrary covariant gauge using an ultraviolet cut-off regularization.
Above the critical coupling leading to dynamical chiral symmetry breaking,
we show that there is no finite chiral limit. This behaviour is found to be
independent of the detailed choice of photon-fermion proper vertex
in the Dyson-Schwinger equation formalism, provided that the vertex
is consistent with the Ward-Takahashi identity and multiplicative
renormalizability. The first results from a study using dimensional
regularization in place of the cut-off approach will also be
presented.