Implementing Unitarity in Perturbation Theory

Unitarity cannot be perserved order by order in ordinary perturbation theory because the constraint UU^\dagger=\1 is nonlinear. However, the corresponding constraint for $K=\ln U$, being $K=-K^\dagger$, is linear so it can be maintained in every order in a perturbative expansion of $K$. The perturbative expansion of $K$ may be considered as a non-abelian generalization of the linked-cluster expansion in probability theory and in statistical mechanics, and possesses similar advantages resulting from separating the short-range correlations from long-range effects. This point is illustrated in two QCD examples, in which delicate cancellations encountered in summing Feynman diagrams of are avoided when they are calculated via the perturbative expansion of $K$. Applications to other problems are briefly discussed.Comment: to appear in Phys. Rev.

Multiple Reggeon Exchange from Summing QCD Feynman Diagrams

Multiple reggeon exchange supplies subleading logs that may be used to restore unitarity to the Low-Nussinov Pomeron, provided it can be proven that the sum of Feynman diagrams to all orders gives rise to such multiple regge exchanges. This question cannot be easily tackled in the usual way except for very low-order diagrams, on account of delicate cancellations present in the sum which necessitate individual Feynman diagrams to be computed to subleading orders. Moreover, it is not clear that sums of high-order Feynman diagrams with complicated criss-crossing of lines can lead to factorization implied by the multi-regge scenario. Both of these difficulties can be overcome by using the recently developed nonabelian cut diagrams. We are then able to show that the sum of $s$-channel-ladder diagrams to all orders does lead to such multiple reggeon exchanges.Comment: uu-encoded latex file with 11 postscript figures (20 pages

Stability of Filters for the Navier-Stokes Equation

Data assimilation methodologies are designed to incorporate noisy observations of a physical system into an underlying model in order to infer the properties of the state of the system. Filters refer to a class of data assimilation algorithms designed to update the estimation of the state in a on-line fashion, as data is acquired sequentially. For linear problems subject to Gaussian noise filtering can be performed exactly using the Kalman filter. For nonlinear systems it can be approximated in a systematic way by particle filters. However in high dimensions these particle filtering methods can break down. Hence, for the large nonlinear systems arising in applications such as weather forecasting, various ad hoc filters are used, mostly based on making Gaussian approximations. The purpose of this work is to study the properties of these ad hoc filters, working in the context of the 2D incompressible Navier-Stokes equation. By working in this infinite dimensional setting we provide an analysis which is useful for understanding high dimensional filtering, and is robust to mesh-refinement. We describe theoretical results showing that, in the small observational noise limit, the filters can be tuned to accurately track the signal itself (filter stability), provided the system is observed in a sufficiently large low dimensional space; roughly speaking this space should be large enough to contain the unstable modes of the linearized dynamics. Numerical results are given which illustrate the theory. In a simplified scenario we also derive, and study numerically, a stochastic PDE which determines filter stability in the limit of frequent observations, subject to large observational noise. The positive results herein concerning filter stability complement recent numerical studies which demonstrate that the ad hoc filters perform poorly in reproducing statistical variation about the true signal

Large-scale Bias and Efficient Generation of Initial Conditions for Non-Local Primordial Non-Gaussianity

We study the scale-dependence of halo bias in generic (non-local) primordial non-Gaussian (PNG) initial conditions of the type motivated by inflation, parametrized by an arbitrary quadratic kernel. We first show how to generate non-local PNG initial conditions with minimal overhead compared to local PNG models for a general class of primordial bispectra that can be written as linear combinations of separable templates. We run cosmological simulations for the local, and non-local equilateral and orthogonal models and present results on the scale-dependence of halo bias. We also derive a general formula for the Fourier-space bias using the peak-background split (PBS) in the context of the excursion set approach to halos and discuss the difference and similarities with the known corresponding result from local bias models. Our PBS bias formula generalizes previous results in the literature to include non-Markovian effects and non-universality of the mass function and are in better agreement with measurements in numerical simulations than previous results for a variety of halo masses, redshifts and halo definitions. We also derive for the first time quadratic bias results for arbitrary non-local PNG, and show that non-linear bias loops give small corrections at large-scales. The resulting well-behaved perturbation theory paves the way to constrain non-local PNG from measurements of the power spectrum and bispectrum in galaxy redshift surveys.Comment: 43 pages, 10 figures. v2: references added. 2LPT parallel code for generating non-local PNG initial conditions available at http://cosmo.nyu.edu/roman/2LP

Self Consistent Expansion for the Molecular Beam Epitaxy Equation

Motivated by a controversy over the correct results derived from the dynamic renormalization group (DRG) analysis of the non linear molecular beam epitaxy (MBE) equation, a self-consistent expansion (SCE) for the non linear MBE theory is considered. The scaling exponents are obtained for spatially correlated noise of the general form $D({\vec r - \vec r',t - t'}) = 2D_0 | {\vec r - \vec r'} |^{2\rho - d} \delta ({t - t'})$. I find a lower critical dimension $d_c (\rho) = 4 + 2\rho$, above, which the linear MBE solution appears. Below the lower critical dimension a r-dependent strong-coupling solution is found. These results help to resolve the controversy over the correct exponents that describe non linear MBE, using a reliable method that proved itself in the past by predicting reasonable results for the Kardar-Parisi-Zhang (KPZ) system, where DRG failed to do so.Comment: 16 page

XAX: a multi-ton, multi-target detection system for dark matter, double beta decay and pp solar neutrinos

A multi-target detection system XAX, comprising concentric 10 ton targets of 136Xe and 129/131Xe, together with a geometrically similar or larger target of liquid Ar, is described. Each is configured as a two-phase scintillation/ionization TPC detector, enhanced by a full 4pi array of ultra-low radioactivity Quartz Photon Intensifying Detectors (QUPIDs) replacing the conventional photomultipliers for detection of scintillation light. It is shown that background levels in XAX can be reduced to the level required for dark matter particle (WIMP) mass measurement at a 10^-10 pb WIMP-nucleon cross section, with single-event sensitivity below 10^-11 pb. The use of multiple target elements allows for confirmation of the A^2 dependence of a coherent cross section, and the different Xe isotopes provide information on the spin-dependence of the dark matter interaction. The event rates observed by Xe and Ar would modulate annually with opposite phases from each other for WIMP mass >~100 GeV/c^2. The large target mass of 136Xe and high degree of background reduction allow neutrinoless double beta decay to be observed with lifetimes of 10^27-10^28 years, corresponding to the Majorana neutrino mass range 0.01-0.1 eV, the most likely range from observed neutrino mass differences. The use of a 136Xe-depleted 129/131Xe target will also allow measurement of the pp solar neutrino spectrum to a precision of 1-2%.Comment: 16 pages with 17 figure

Unconditional Continuous Variable Dense Coding

We investigate the conditions under which unconditional dense coding can be achieved using continuous variable entanglement. We consider the effect of entanglement impurity and detector efficiency and discuss experimental verification. We conclude that the requirements for a strong demonstration are not as stringent as previously thought and are within the reach of present technology

Simulations of lattice animals and trees

The scaling behaviour of randomly branched polymers in a good solvent is studied in two to nine dimensions, using as microscopic models lattice animals and lattice trees on simple hypercubic lattices. As a stochastic sampling method we use a biased sequential sampling algorithm with re-sampling, similar to the pruned-enriched Rosenbluth method (PERM) used extensively for linear polymers. Essentially we start simulating percolation clusters (either site or bond), re-weigh them according to the animal (tree) ensemble, and prune or branch the further growth according to a heuristic fitness function. In contrast to previous applications of PERM, this fitness function is {\it not} the weight with which the actual configuration would contribute to the partition sum, but is closely related to it. We obtain high statistics of animals with up to several thousand sites in all dimension 2 <= d <= 9. In addition to the partition sum (number of different animals) we estimate gyration radii and numbers of perimeter sites. In all dimensions we verify the Parisi-Sourlas prediction, and we verify all exactly known critical exponents in dimensions 2, 3, 4, and >= 8. In addition, we present the hitherto most precise estimates for growth constants in d >= 3. For clusters with one site attached to an attractive surface, we verify the superuniversality of the cross-over exponent at the adsorption transition predicted by Janssen and Lyssy. Finally, we discuss the collapse of animals and trees, arguing that our present version of the algorithm is also efficient for some of the models studied in this context, but showing that it is {\it not} very efficient for the `classical' model for collapsing animals.Comment: 17 pages RevTeX, 29 figures include

Two-soliton solution for the derivative nonlinear Schr\"odinger equation with nonvanishing boundary conditions

An explicit two-soliton solution for the derivative nonlinear Schr\"odinger equation with nonvanishing boundary conditions is derived, demonstrating details of interactions between two bright solitons, two dark solitons, as well as one bright soliton and one dark soliton. Shifts of soliton positions due to collisions are analytically obtained, which are irrespective of the bright or dark characters of the participating solitons.Comment: 11 pages, 4 figures. Phys. Lett. A 2006 (in press

Numerical evidence for the spin-Peierls state in the frustrated quantum antiferromagnet

We study the spin-$1\over2$ Heisenberg antiferromagnet with an antiferromagnetic $J_3$ (third nearest neighbor) interaction on a square lattice. We numerically diagonalize this ``$J_1$-$J_3$'' model on clusters up to 32-sites and search for novel ground state properties as the frustration parameter $J_3/J_1$ changes. For ``larger'' $J_3/J_1$ we find enhancement of incommensurate spin order, in agreement with spin-wave, large-$N$ expansions, and other predictions. But for intermediate $J_3/J_1$, the low lying excitation energy spectrum suggests that this incommensurate order is short-range. In the same region, the first excited state has the symmetries of the columnar dimer (spin-Peierls) state. The columnar dimer order parameter suggests the presence of long-range columnar dimer order. Hence, this spin-Peierls state is the best candidate for the ground state of the $J_1$-$J_3$ model in an intermediate $J_3/J_1$ region.Comment: RevTeX file with five postscript figures uuencode