On Segal-Wilson's construction for the τ-functions of the constrained KP hierarchies

In this Letter, we study the constrained KP hierarchies by employing Segal-Wilson's theory on the -functions of the KP hierarchy. We first describe the elements of the Grassmannian which correspond to solutions of the constrained KP hierarchy, and then we show how to construct its rational and soliton solutions from these elements of the Grassmannian.     

Lie algebraic approach to τ-functions and its equations

A general construction of equations satisfied by the components of -functions is given by considering the tensor products of modules. As an example, the homogeneous basic realization of is given, leading to NLS equations.     

Three-loop β-functions for the superstring and heterotic string

The β-function for the (1,1) superstring is calculated explicitly up to three loops, using two different choices of the fermion quantum field. In both cases the bosonic contribution is exactly cancelled by the fermionic contribution. The calculation is repeated for the heterotic string and the non-vanishing β-function is shown to be compatible with an O(α′²) effective action in which there are no terms cubic in the Riemann tensor- or gauge-field strength. A renormalization group equation for the gauge field is constructed and its β-function is shown to vanish at the two-loop level.     

A new “dual” symmetry structure of the KP hierarchy

A new infinite set of commuting additional (“ghost”) symmetries is proposed for the KP-type integrable hierarchy. These symmetries allow for a Lax representation in which they are realized as standard isospectral flows. This gives rise to a new double-KP hierarchy embedding “ghost” and original KP-type Lax hierarchies connected to each other via a “duality” mapping exchanging the isospectral and “ghost” “times”. A new representation of the 2D Toda lattice hierarchy as a special Darboux-Bäcklund orbit of the double-KP hierarchy is found and parametrized entirely in terms of (adjoint) eigenfunctions of the original KP subsystem.     

ß-functions and the exact renormalization group

We relate ß-functions to the flow of relevant couplings in the exact renormalization group. The specific case of a cutoff γφ⁴ theory in four dimensions is discussed in detail. The underlying idea of convergence of the flow of effective lagrangians is developed to identify the ß-functions. A perturbative calculation of the ß-functions using the exact flow equations is sketched. The analysis may be extended to any system with a cutoff.     

On the linearization of the coupled Harry－Dym soliton hierarchy

This paper is devoted to the study of the underlying linearities of the coupled Harry--Dym (cHD) soliton hierarchy, including the well-known cHD equation. Resorting to the nonlinearization of Lax pairs, a family of finite-dimensional Hamiltonian systems associated with soliton equations are presented, constituting the decomposition of the cHD soliton hierarchy. After suitably introducing the Abel--Jacobi coordinates on a Riemann surface, the cHD soliton hierarchy can be ultimately reduced to linear superpositions, expressed by the Abel--Jacobi variables.     

On the nonexistence of Liouvillian first integrals for generalized Liénard polynomial differential systems

We consider generalized Liénard polynomial differential systems of the form ? = y, ? = -g(x) - f (x) y, with f (x) and g(x) two polynomials satisfying deg(g) ≤ deg(f). In their work, Llibre and Valls have shown that, except in some particular cases, such systems have no Liouvillian first integral. In this letter, we give a direct and shorter proof of this result.     

Bäcklund transformation for the first flows of the relativistic Toda hierarchy and associated properties

In this communication we study a class of one parameter dependent auto-Bäcklund transformations for the first flow of the relativistic Toda lattice and also a variant of the usual Toda lattice equation. It is shown that starting from the Hamiltonian formalism such transformations are canonical in nature with a well defined generating function. The notion of spectrality is also analyzed and the separation variables are explicitly constructed.     

On an isomonodromy deformation equation without the Painlevé property

We show that the fourth-order nonlinear ODE which controls the pole dynamics in the general solution of equation P _I ² compatible with the KdV equation exhibits two remarkable properties: (1) it governs the isomonodromy deformations of a 2 × 2 matrix linear ODE with polynomial coefficients, and (2) it does not possess the Painlevé property. We also study the properties of the Riemann-Hilbert problem associated to this ODE and find its large-t asymptotic solution for physically interesting initial data.     

On the Grüneisen parameter for crystals and glasses

Additional data confirming the close relation between the Grüneisen parameter and the Poisson ratio are obtained. The estimate of the Grüneisen parameter based on the Leont’ev formula using the data on density, bulk compression modulus, and velocity of sound is in conformity with the results of calculations based on the Grüneisen and Belomestnykh-Tesleva equations.     

Search for the decay τ− → vτηπ−

We have searched for the decay τ⁻ → v_τηπ⁻ using data accumulated by the ARGUS detector at the e⁺e⁻ storage ring DORIS II at DESY. No η signal was found in the π⁺π⁻π⁰ subsystems of the decay τ⁻ →v_τπ⁻π⁻π⁺ π⁰. We obtain an upper limit for the branching ratio of the decay τ⁻ → v_τηπ⁻ of 1.3% at the 95% confidence level.     

Where to look for solving the gauge hierarchy problem?

A mass of the Higgs boson close to 126 GeV may give a hint that the standard model of particle physics is valid up to the Planck scale. We discuss perspectives for the solution of the gauge hierarchy problem at high scales. Scenarios with an ultraviolet fixed point have predicted a Higgs boson mass very close to 126 GeV if the fixed point value of the quartic scalar coupling is small. In this case the top quark pole mass should be close to 172 GeV.     

On the relationship between string low-energy effective actions and O(α′3) σ-model β-functions

We examine in detail the connection between low-energy string effective actions and the σ-model β-functions up to O(α^′3) in the torsion-free case. In particular we show that the relationship between, on the one hand, the equations of motion derived from the O(α^′3) string effective action, and, on the other hand, the O(α^′3) σ-model β-functions necessarily involves derivative operators acting on the β-functions.     

Polynomial τ-Functions of the NLS-Toda Hierarchy and the Virasoro Singular Vectors

A family of polynomial -functions for the NLS-Toda hierarchy is constructed. The hierarchy is associated with the homogeneous vertex operator representation of the affine algebra of type A ₁ ⁽¹⁾. These -functions are given explicitly in terms of Schur functions that correspond to rectangular Young diagrams. It is shown that an arbitrary polynomial -function which is an eigenvector of d, the degree operator of , is contained in the family. By the construction, any -function in the family becomes a Virasoro singular vector. This consideration gives rise to a simple proof of known results on the Fock representation of the Virasoro algebra with c = 1.     

On a geometrical property of the Bäcklund transformation of the sine-Gordon equation

We show how a solution of the Bäcklund transformation equations for the sine-Gordon equation determines an explicit isometry from the underlying surface to the upper half plane. We then use the action on the upper half to find explicitly the general solution of the equations.     

Determination of an upper limit for the mass of the τ-neutrino at LEP

An upper limit for the mass is determined through the kinematic reconstruction of the decay5 ^± in the OPAL detector at LEP. The limit is obtained using a new method based on the comparison of the two-dimensional distribution of energy and invariant mass of the five-pion system with expectations from different neutrino mass hypotheses. From a sample of five events surviving the selection criteria we obtain an upper limit of 74 MeV at 95% confidence level. It is the first measurement at LEP energies, where the larger average multiplicity of events makes the suppression of this background more robust compared to lower energies.     

On the effective action and field renormalization for variationalλø4

We perform a study of the effective action within the gaussian variational approach for λλ₃₊₁⁴ with the aim of understanding better field renormalization. The difficulties in obtaining a finite effective action in this approach are made clear.