29,858 research outputs found
Manifestly Finite Perturbation Theory for the Short-Distance Expansion of Correlation Functions in the Two Dimensional Ising Model
In the spirit of classic works of Wilson on the renormalization group and
operator product expansion, a new framework for the study of the theory space
of euclidean quantum field theories has been introduced. This formalism is
particularly useful for elucidating the structure of the short-distance
expansions of the -point functions of a renormalizable quantum field theory
near a non-trivial fixed point. We review and apply this formalism in the study
of the scaling limit of the two dimensional massive Ising model.
Renormalization group analysis and operator product expansions determine all
the non-analytic mass dependence of the short-distance expansion of the
correlation functions. An extension of the first order variational formula to
higher orders provides a manifestly finite scheme for the perturbative
calculation of the operator product coefficients to any order in parameters. A
perturbative expansion of the correlation functions follows. We implement this
scheme for a systematic study of correlation functions involving two spin
operators. We show how the necessary non-trivial integrals can be calculated.
As two concrete examples we explicitly calculate the short-distance expansion
of the spin-spin correlation function to third order and the spin-spin-energy
density correlation function to first order in the mass. We also discuss the
applicability of our results to perturbations near other non-trivial fixed
points corresponding to other unitary minimal models.Comment: 38 pages with 1 figure, UCLA/93/TEP/4
Boundary integral equation based numerical solutions of helmholtz transmission problems for composite scatters
In this dissertation, an in-depth comparison between boundary integral equation solvers and Domain Decomposition Methods (DDM) for frequency domain Helmholtz transmission problems in composite two-dimensional media is presented. Composite media are characterized by piece-wise constant material properties (i.e., index of refraction) and thus, they exhibit interfaces of material discontinuity and multiple junctions. Whenever possible to use, boundary integral methods for solution of Helmholtz boundary value problems are computationally advantageous. Indeed, in addition to the dimensional reduction and straightforward enforcement of the radiation conditions that these methods enjoy, they do not suffer from the pollution effect present in volumetric discretization. The reformulation of Helmholtz transmission problems in composite media in terms of boundary integral equations via multi-traces constitutes one of the recent success stories in the boundary integral equation community. Multi-trace formulations (MTF) incorporate local Dirichlet and Neumann traces on subdomains within Green’s identities and use restriction and extension by zero operators to enforce the intradomain continuity of the fields and fluxes. Through usage of subdomain Calderon projectors, the transmission problem is cast into a linear system form whose unknowns are local Dirichlet and Neumann traces (two such traces per interface of material discontinuity) and whose operator matrix consists of diagonal block boundary integral operators associated with the subdomains and extension/projections off diagonal blocks. This particular form of the matrix operator associated with MTF is amenable to operator preconditioning via Calderon projectors.
DDM rely on subdomain solutions that are matched via transmission conditions on the subdomain interfaces that are equivalent to the physical continuity of fields and traces. By choosing the appropriate transmission conditions, the convergence of DDM for frequency domain scattering problems can be accelerated. Traditionally, the intradomain transmission conditions were chosen to be the classical outgoing Robin/impedance boundary conditions. When the ensuing DDM linear system is solved via Krylov subspace methods, the convergence of DDM with classical Robin transmission conditions is slow and adversely affected by the number of subdomains. Heuristically, this behavior is explained by the fact that Robin boundary conditions are first order approximations of transparent boundary conditions, and thus there is significant information that is reflected back into a given subdomain from adjacent subdomains. Clearly, using more sophisticated transparent boundary conditions facilitates the information exchange between subdomains. For instance, Dirichlet-to-Neumann (DtN) operators of adjacent domains or suitable approximations of these can be used in the form of generalized Robin boundary conditions to increase the rate of the convergence of iterative solvers of DDM linear systems. The approximations of DtN operators that are expressed in terms of Helmholtz hypersingular operators (e.g., the normal derivative of the double layer operator) are used in this dissertation. The incorporation of these in a DDM framework is subtle, and an effective method is proposed to blend these transmission operators in the presence of multiple junctions. Conceptually, the information exchange between subdomains is realized through certain Robin-to-Robin (RtR) operators, which how to compute robustly via integral equation formulations is shown.
All of the Helmholtz boundary integral operators that feature in Calderon’s calculus are discretized via Nystr¨om methods that rely on sigmoid transforms, trigonometric interpolation, and singular kernel splitting. Sigmoid transforms are means to polynomially accumulate discretization points toward corners without compromising the discretization density in smooth boundary portions. A wide variety of numerical results is presented in this dissertation that illustrate the merits of each of the two approaches (MTF and DDM) for the solution of transmission problems in composite domains
The Operator Product Expansion of N=4 SYM and the 4-point Functions of Supergravity
We give a detailed Operator Product Expansion interpretation of the results
for conformal 4-point functions computed from supergravity through the AdS/CFT
duality. We show that for an arbitrary scalar exchange in AdS(d+1) all the
power-singular terms in the direct channel limit (and only these terms) exactly
match the corresponding contributions to the OPE of the operator dual to the
exchanged bulk field and of its conformal descendents. The leading logarithmic
singularities in the 4-point functions of protected N=4 super-Yang Mills
operators (computed from IIB supergravity on AdS(5) X S(5) are interpreted as
O(1/N^2) renormalization effects of the double-trace products appearing in the
OPE. Applied to the 4-point functions of the operators Ophi ~ tr F^2 + ... and
Oc ~ tr FF~ + ..., this analysis leads to the prediction that the double-trace
composites [Ophi Oc] and [Ophi Ophi - Oc Oc] have anomalous dimension -16/N^2
in the large N, large g_{YM}^2 N limit. We describe a geometric picture of the
OPE in the dual gravitational theory, for both the power-singular terms and the
leading logarithms. We comment on several possible extensions of our results.Comment: 42 page
Composite Operators in QCD
We give a formula for the derivatives of a correlation function of composite
operators with respect to the parameters (i.e., the strong fine structure
constant and the quark mass) of QCD in four-dimensional euclidean space. The
formula is given as spatial integration of the operator conjugate to a
parameter. The operator product of a composite operator and a conjugate
operator has an unintegrable part, and the formula requires divergent
subtractions. By imposing consistency conditions we derive a relation between
the anomalous dimensions of the composite operators and the unintegrable part
of the operator product coefficients.Comment: 26 page
A higher-order singularity subtraction technique for the discretization of singular integral operators on curved surfaces
This note is about promoting singularity subtraction as a helpful tool in the
discretization of singular integral operators on curved surfaces. Singular and
nearly singular kernels are expanded in series whose terms are integrated on
parametrically rectangular regions using high-order product integration,
thereby reducing the need for spatial adaptivity and precomputed weights. A
simple scheme is presented and an application to the interior Dirichlet Laplace
problem on some tori gives around ten digit accurate results using only two
expansion terms and a modest programming- and computational effort.Comment: 7 pages, 2 figure
Multiloop Superstring Amplitudes from Non-Minimal Pure Spinor Formalism
Using the non-minimal version of the pure spinor formalism, manifestly
super-Poincare covariant superstring scattering amplitudes can be computed as
in topological string theory without the need of picture-changing operators.
The only subtlety comes from regularizing the functional integral over the pure
spinor ghosts. In this paper, it is shown how to regularize this functional
integral in a BRST-invariant manner, allowing the computation of arbitrary
multiloop amplitudes. The regularization method simplifies for scattering
amplitudes which contribute to ten-dimensional F-terms, i.e. terms in the
ten-dimensional superspace action which do not involve integration over the
maximum number of 's.Comment: 23 pages harvmac, added acknowledgemen
- …