24 research outputs found
Dyson instability for 2D nonlinear O(N) sigma models
For lattice models with compact field integration (nonlinear sigma models
over compact manifolds and gauge theories with compact groups) and satisfying
some discrete symmetry, the change of sign of the bare coupling g_0^2 at zero
results in a mere discontinuity in the average energy rather than the
catastrophic instability occurring in theories with integration over
arbitrarily large fields. This indicates that the large order of perturbative
series and the non-perturbative contributions should have unexpected features.
Using the large-N limit of 2-dimensional nonlinear O(N) sigma model, we discuss
the complex singularities of the average energy for complex 't Hooft coupling
lambda= g_0^2N. A striking difference with the usual situation is the absence
of cut along the negative real axis. We show that the zeros of the partition
function can only be inside a clover shape region of the complex lambda plane.
We calculate the density of states and use the result to verify numerically the
statement about the zeros. We propose dispersive representations of the
derivatives of the average energy for an approximate expression of the
discontinuity. The discontinuity is purely non-perturbative and contributions
at small negative coupling in one dispersive representation are essential to
guarantee that the derivatives become exponentially small when lambda -> 0^+ We
discuss the implications for gauge theories.Comment: 10 pages, 10 figures uses revte
Asymptotically Universal Crossover in Perturbation Theory with a Field Cutoff
We discuss the crossover between the small and large field cutoff (denoted
x_{max}) limits of the perturbative coefficients for a simple integral and the
anharmonic oscillator. We show that in the limit where the order k of the
perturbative coefficient a_k(x_{max}) becomes large and for x_{max} in the
crossover region, a_k(x_{max}) is proportional to the integral from -infinity
to x_{max} of e^{-A(x-x_0(k))^2}dx. The constant A and the function x_0(k) are
determined empirically and compared with exact (for the integral) and
approximate (for the anharmonic oscillator) calculations. We discuss how this
approach could be relevant for the question of interpolation between
renormalization group fixed points.Comment: 15 pages, 11 figs., improved and expanded version of hep-th/050304
The non-perturbative part of the plaquette in quenched QCD
We define the non-perturbative part of a quantity as the difference between
its numerical value and the perturbative series truncated by dropping the order
of minimal contribution and the higher orders. For the anharmonic oscillator,
the double-well potential and the single plaquette gauge theory, the
non-perturbative part can be parametrized as A (lambda)^B exp{-C/lambda} and
the coefficients can be calculated analytically. For lattice QCD in the
quenched approximation, the perturbative series for the average plaquette is
dominated at low order by a singularity in the complex coupling plane and the
asymptotic behavior can only be reached by using extrapolations of the existing
series. We discuss two extrapolations that provide a consistent description of
the series up to order 20-25. These extrapolations favor the idea that the
non-perturbative part scales like (a/r_0)^4 with a/r_0 defined with the force
method. We discuss the large uncertainties associated with this statement. We
propose a parametrization of ln((a/r_0)) as the two-loop universal terms plus a
constant and exponential corrections. These corrections are consistent with
a_{1-loop}^2 and play an important role when beta<6. We briefly discuss the
possibility of calculating them semi-classically at large beta.Comment: 13 pages, 16 figures, uses revtex, contains a new section with the
uncertainties on the extrapolations, refs. adde
Fisher's zeros as boundary of renormalization group flows in complex coupling spaces
We propose new methods to extend the renormalization group transformation to
complex coupling spaces. We argue that the Fisher's zeros are located at the
boundary of the complex basin of attraction of infra-red fixed points. We
support this picture with numerical calculations at finite volume for
two-dimensional O(N) models in the large-N limit and the hierarchical Ising
model. We present numerical evidence that, as the volume increases, the
Fisher's zeros of 4-dimensional pure gauge SU(2) lattice gauge theory with a
Wilson action, stabilize at a distance larger than 0.15 from the real axis in
the complex beta=4/g^2 plane. We discuss the implications for proofs of
confinement and searches for nontrivial infra-red fixed points in models beyond
the standard model.Comment: 4 pages, 3 fig
Critical Exponents, Hyperscaling and Universal Amplitude Ratios for Two- and Three-Dimensional Self-Avoiding Walks
We make a high-precision Monte Carlo study of two- and three-dimensional
self-avoiding walks (SAWs) of length up to 80000 steps, using the pivot
algorithm and the Karp-Luby algorithm. We study the critical exponents
and as well as several universal amplitude ratios; in
particular, we make an extremely sensitive test of the hyperscaling relation
. In two dimensions, we confirm the predicted
exponent and the hyperscaling relation; we estimate the universal
ratios , and (68\% confidence
limits). In three dimensions, we estimate with a
correction-to-scaling exponent (subjective 68\%
confidence limits). This value for agrees excellently with the
field-theoretic renormalization-group prediction, but there is some discrepancy
for . Earlier Monte Carlo estimates of , which were , are now seen to be biased by corrections to scaling. We estimate the
universal ratios and ; since , hyperscaling holds. The approach to
is from above, contrary to the prediction of the two-parameter
renormalization-group theory. We critically reexamine this theory, and explain
where the error lies.Comment: 87 pages including 12 figures, 1029558 bytes Postscript
(NYU-TH-94/09/01
A Nonperturbative Study of Inverse Symmetry Breaking at High Temperatures
The optimized linear -expansion is applied to multi-field scalar theories at high temperatures. Using the imaginary time
formalism the thermal masses are evaluated perturbatively up to order
which considers consistently all two-loop contributions. A
variational procedure associated with the method generates nonperturbative
results which are used to search for parameters values for inverse symmetry
breaking (or symmetry nonrestoration) at high temperatures. Our results are
compared with the ones obtained by the one-loop perturbative approximation, the
gap equation solutions and the renormalization group approach, showing good
agreement with the latter method. Apart from strongly supporting inverse
symmetry breaking (or symmetry nonrestoration), our results reveal the
possibility of other high temperature symmetry breaking patterns for which the
last term in the breaking sequence is .Comment: 28 pages,5 eps figures (uses epsf), RevTeX. Only a small misprint in
Eq. (2.10) and a couple of typos fixe
Wilson loops to 20th order numerical stochastic perturbation theory
We calculate Wilson loops of various sizes up to 20 loops in SU(3) pure
lattice gauge theory at different lattice sizes for Wilson gauge action using
the technique of numerical stochastic perturbation theory. This allows us to
investigate the perturbative series for various Wilson loops at high loop
orders. We observe differences in the behavior of those series as function of
the loop order. Up to we do not find evidence for the factorial growth
of the expansion coefficients often assumed to characterize an asymptotic
series. Based on the actually observed behavior we sum the series in a model
parametrized by hypergeometric functions. Alternatively we estimate the total
series in boosted perturbation theory using information from the first 14
loops. We introduce generalized ratios of Wilson loops of different sizes.
Together with the corresponding Wilson loops from standard Monte Carlo
measurements they enable us to assess their non-perturbative parts.Comment: 29 pages, 21 figures, version accepted for publication in Phys. Rev.
D, some inconsistencies removed, more details added concerning the Langevin
simulation, references added and update
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