18,781 research outputs found
On Perturbation theory improved by Strong coupling expansion
In theoretical physics, we sometimes have two perturbative expansions of
physical quantity around different two points in parameter space. In terms of
the two perturbative expansions, we introduce a new type of smooth
interpolating function consistent with the both expansions, which includes the
standard Pad\'e approximant and fractional power of polynomial method
constructed by Sen as special cases. We point out that we can construct
enormous number of such interpolating functions in principle while the "best"
approximation for the exact answer of the physical quantity should be unique
among the interpolating functions. We propose a criterion to determine the
"best" interpolating function, which is applicable except some situations even
if we do not know the exact answer. It turns out that our criterion works for
various examples including specific heat in two-dimensional Ising model,
average plaquette in four-dimensional SU(3) pure Yang-Mills theory on lattice
and free energy in c=1 string theory at self-dual radius. We also mention
possible applications of the interpolating functions to system with phase
transition.Comment: 31+11 pages, 15 figures, 9 tables, 1 Mathematica file; v3: minor
correction
Ising n-fold integrals as diagonals of rational functions and integrality of series expansions
We show that the n-fold integrals of the magnetic susceptibility
of the Ising model, as well as various other n-fold integrals of the "Ising
class", or n-fold integrals from enumerative combinatorics, like lattice Green
functions, correspond to a distinguished class of function generalising
algebraic functions: they are actually diagonals of rational functions. As a
consequence, the power series expansions of the, analytic at x=0, solutions of
these linear differential equations "Derived From Geometry" are globally
bounded, which means that, after just one rescaling of the expansion variable,
they can be cast into series expansions with integer coefficients. We also give
several results showing that the unique analytical solution of Calabi-Yau ODEs,
and, more generally, Picard-Fuchs linear ODEs, with solutions of maximal
weights, are always diagonal of rational functions. Besides, in a more
enumerative combinatorics context, generating functions whose coefficients are
expressed in terms of nested sums of products of binomial terms can also be
shown to be diagonals of rational functions. We finally address the question of
the relations between the notion of integrality (series with integer
coefficients, or, more generally, globally bounded series) and the modularity
of ODEs.Comment: This paper is the short version of the larger (100 pages) version,
available as arXiv:1211.6031 , where all the detailed proofs are given and
where a much larger set of examples is displaye
Linked cluster expansions beyond nearest neighbour interactions: convergence and graph classes
We generalize the technique of linked cluster expansions on hypercubic
lattices to actions that couple fields at lattice sites which are not nearest
neighbours. We show that in this case the graphical expansion can be arranged
in such a way that the classes of graphs to be considered are identical to
those of the pure nearest neighbour interaction. The only change then concerns
the computation of lattice imbedding numbers. All the complications that arise
can be reduced to a generalization of the notion of free random walks,
including hopping beyond nearest neighbour. Explicit expressions for
combinatorical numbers of the latter are given. We show that under some general
conditions the linked cluster expansion series have a non-vanishing radius of
convergence.Comment: 20 pages, latex2
Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings
We investigate one-loop four-point scattering of non-abelian gauge bosons in
heterotic string theory and identify new connections with the corresponding
open-string amplitude. In the low-energy expansion of the heterotic-string
amplitude, the integrals over torus punctures are systematically evaluated in
terms of modular graph forms, certain non-holomorphic modular forms. For a
specific torus integral, the modular graph forms in the low-energy expansion
are related to the elliptic multiple zeta values from the analogous open-string
integrations over cylinder boundaries. The detailed correspondence between
these modular graph forms and elliptic multiple zeta values supports a recent
proposal for an elliptic generalization of the single-valued map at genus zero.Comment: 57+22 pages, v2: references updated, version published in JHE
Triviality problem and the high-temperature expansions of the higher susceptibilities for the Ising and the scalar field models on four-, five- and six-dimensional lattices
High-temperature expansions are presently the only viable approach to the
numerical calculation of the higher susceptibilities for the spin and the
scalar-field models on high-dimensional lattices. The critical amplitudes of
these quantities enter into a sequence of universal amplitude-ratios which
determine the critical equation of state. We have obtained a substantial
extension through order 24, of the high-temperature expansions of the free
energy (in presence of a magnetic field) for the Ising models with spin s >=
1/2 and for the lattice scalar field theory with quartic self-interaction, on
the simple-cubic and the body-centered-cubic lattices in four, five and six
spatial dimensions. A numerical analysis of the higher susceptibilities
obtained from these expansions, yields results consistent with the widely
accepted ideas, based on the renormalization group and the constructive
approach to Euclidean quantum field theory, concerning the no-interaction
("triviality") property of the continuum (scaling) limit of spin-s Ising and
lattice scalar-field models at and above the upper critical dimensionality.Comment: 17 pages, 10 figure
N-vector spin models on the sc and the bcc lattices: a study of the critical behavior of the susceptibility and of the correlation length by high temperature series extended to order beta^{21}
High temperature expansions for the free energy, the susceptibility and the
second correlation moment of the classical N-vector model [also known as the
O(N) symmetric classical spin Heisenberg model or as the lattice O(N) nonlinear
sigma model] on the sc and the bcc lattices are extended to order beta^{21} for
arbitrary N. The series for the second field derivative of the susceptibility
is extended to order beta^{17}. An analysis of the newly computed series for
the susceptibility and the (second moment) correlation length yields updated
estimates of the critical parameters for various values of the spin
dimensionality N, including N=0 [the self-avoiding walk model], N=1 [the Ising
spin 1/2 model], N=2 [the XY model], N=3 [the Heisenberg model]. For all values
of N, we confirm a good agreement with the present renormalization group
estimates. A study of the series for the other observables will appear in a
forthcoming paper.Comment: Revised version to appear in Phys. Rev. B Sept. 1997. Revisions
include an improved series analysis biased with perturbative values of the
scaling correction exponents computed by A. I. Sokolov. Added a reference to
estimates of exponents for the Ising Model. Abridged text of 19 pages, latex,
no figures, no tables of series coefficient
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