Renormalisation of composite operators in lattice perturbation theory with clover fermions: non-forward matrix elements

We consider the renormalisation of lattice QCD operators with one and two covariant derivatives related to the first and second moments of generalised parton distributions and meson distribution amplitudes. Employing the clover fermion action we calculate their non-forward quark matrix elements in one-loop lattice perturbation theory. For some representations of the hypercubic group commonly used in simulations we determine the sets of all possible mixing operators and compute the matrices of the renormalisation factors in one-loop approximation. We describe how tadpole improvement is applied to the results. PACS 11.15.Ha; 12.38.Bx; 12.38.Gc An erratum to this article can be found at     

Berezin-Toeplitz Operators, a Semi-Classical Approach

This article is devoted to the Toeplitz Operators [4] in the context of the geometric quantization [11, 15]. We propose an ansatz for their Schwartz kernel. From this, we deduce the main known properties of the principal symbol of these operators and obtain new results : we define their covariant and contravariant symbols, which are full symbols, and compute the product of these symbols in terms of the Kähler metric. This gives canonical star products on the Kählerian manifolds. This ansatz is also useful to introduce the notion of microsupport.     

Semiclassical States Associated with Isotropic Submanifolds of Phase Space

We define classes of quantum states associated with isotropic submanifolds of cotangent bundles. The classes are stable under the action of semiclassical pseudo-differential operators and covariant under the action of semiclassical Fourier integral operators. We develop a symbol calculus for them; the symbols are symplectic spinors. We outline various applications.     

Covariant Anomalies and Functional Determinants

We analize the algebraic structure of consistent and covariant anomalies in gauge and gravitational theories: using a complex extension of the Lie algebra it is possible to describe them in a unified way. Then we study their representations by means of functional determinants, showing how the algebraic solution determines the relevant operators for the definition of the effective action. Particular attention is devoted to the Lorentz anomaly: we obtain by functional methods the covariant anomaly for the spin-current and for the energy-momentum tensor in presence of a curved background. With regard to the consistent sector we are able to give a general functional solution only for d = 2: using the characterization derived from the extended algebra, we find a continuous family of operators whose determinant describes the effective action of chiral spinors in curved space. We compute this action and we generalize the result in presence of a U(1) gauge connection.     

Integrable Fredholm Operators and Dual Isomonodromic Deformations

The Fredholm determinants of a special class of integrable integral operators K supported on the union of m curve segments in the complex λ-plane are shown to be the τ-functions of an isomonodromic family of meromorphic covariant derivative operators , having regular singular points at the 2m endpoints of the curve segments, and a singular point of Poincaré index 1 at infinity. The rank r of the corresponding vector bundle over the Riemann sphere equals the number of distinct terms in the exponential sum defining the numerator of the integral kernel. The matrix Riemann–Hilbert problem method is used to deduce an identification of the Fredholm determinant as a τ-function in the sense of Segal–Wilson and Sato, i.e., in terms of abelian group actions on the determinant line bundle over a loop space Grassmannian. An associated dual isomonodromic family of covariant derivative operators , having rank n= 2m, and r finite regular singular points located at the values of the exponents defining the kernel of K is derived. The deformation equations for this family are shown to follow from an associated dual set of Riemann–Hilbert data, in which the r?les of the r exponential factors in the kernel and the 2m endpoints of its support are interchanged. The operators are analogously associated to an integral operator whose Fredholm determinant is equal to that of K. Received: 10 June 1997 / Received revised: 16 February 2001 / Accepted: 27 November 2001     

Einstein-aether theory with a Maxwell field: General formalism

We extend the Einstein-aether theory to include the Maxwell field in a nontrivial manner by taking into account its interaction with the time-like unit vector field characterizing the aether. We also include a generic matter term. We present a model with a Lagrangian that includes cross-terms linear and quadratic in the Maxwell tensor, linear and quadratic in the covariant derivative of the aether velocity four-vector, linear in its second covariant derivative and in the Riemann tensor. We decompose these terms with respect to the irreducible parts of the covariant derivative of the aether velocity, namely, the acceleration four-vector, the shear and vorticity tensors, and the expansion scalar. Furthermore, we discuss the influence of an aether non-uniform motion on the polarization and magnetization of the matter in such an aether environment, as well as on its dielectric and magnetic properties. The total self-consistent system of equations for the electromagnetic and the gravitational fields, and the dynamic equations for the unit vector aether field are obtained. Possible applications of this system are discussed. Based on the principles of effective field theories, we display in an appendix all the terms up to fourth order in derivative operators that can be considered in a Lagrangian that includes the metric, the electromagnetic and the aether fields.     

Alternative approaches to the construction of the Poincare-invariant spin operators of the Dirac particles

At present a number of methods of constructing the Poincare-invariant spin operators for relativistic particles with half-integer spin in the one-particle theory are well known. The method of odd operator constructing, the Lorentz method of bilinear covariant form transformation, and the method with the Foldy–Wouthuysen representation belong to them. New approaches to the construction of spin operators are developed in the present work, namely, a method of separating space-like component directly from the spin matrices of bilinear covariant forms, including the method of multiplication of the covariant Hamiltonian of the Dirac equation by these matrices. By this means we succeeded in constructing the Poincare-invariant spin operators by simpler and mathematically faultless methods. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 10–15, November, 2008.     

Geometric significance of the spinor Lie derivative. I

In a previous article, the writer explored the geometric foundation of the generally covariant spinor calculus. This geometric reasoning can be extended quite naturally to include the Lie covariant differentiation of spinors. The formulas for the Lie covariant derivatives of spinors, adjoint spinors, and operators in spin space are deduced, and it is observed that the Lie covariant derivative of an operator in spin space must vanish when taken with respect to a Killing vector. The commutator of two Lie covariant derivatives is calculated; it is noted that the result is consistent with the geometric interpretation of the Jacobi identity for vectors. Lie current conservation is seen to spring from the result that the operator of spinor affine covariant differentiation commutes with the operator of spinor Lie covariant differentiation with respect to a Killing vector. It is shown that differentiations of the spinor field defined geometrically are Lorentz-covariant.     

Superprojectors

We present a simple algorithm for constructing the N-extended superfield projection operators for irreducible representations of supersymmetry, and explicitly perform all simplifications due to spinor derivative algebra. The method is based on covariant expansion of a general superfield in terms of chiral superfields, and requires no knowledge of Casimir operators. We list these superprojectors for various N = 1, 2, and 4 superfields, and apply our results to quantized the linearized N = 2 vector multiplet in a supersymmetric gauge.     

Covariant derivatives on null submanifolds

The degenerate nature of the metric on null hypersurfaces makes it difficult to define a covariant derivative on null submanifolds. Recent approaches using decomposition to define a covariant derivative on null hypersurfaces are investigated, with examples demonstrating the limitations of the methods. Motivated by Geroch’s work on asymptotically flat spacetimes, conformal transformations are used to construct a covariant derivative on null hypersurfaces, and a condition on the Ricci tensor is given to determine when this construction can be used. Several examples are given, including the construction of a covariant derivative operator for the class of spherically symmetric hypersurfaces.     

Testing the electroweak phase transition in scalar extension models at lepton colliders

We study the electroweak phase transition in three scalar extension models beyond the Standard Model.Assuming new scalars are decoupled at some heavy scale, we use the covariant derivative expansion method to derive all of the dimension-6 effective operators, whose coefficients are highly correlated in a specific model. We provide bounds to the complete set of dimension-6 operators by including the electroweak precision test and recent Higgs measurements. We find that the parameter space of strong first-order phase transitions(induced by the |H|~6 operator)can be probed extensively in Zh production at future electron-positron colliders.     

Separation of variables in the diffusion equation. II

The covariant and symmetry properties of the linear diffusion equation having a scalar matrix of variable diffusion coefficients are studied. By means of differential symmetry operators of order no higher than two, a complete separation of variables is effected for the stationary and nonstationary cases.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 40–45, December, 1976.     

On the Bilinear Equations for Fredholm Determinants Appearing in Random Matrices

Abstract

It is shown how the bilinear differential equations satisfied by Fredholm determinants of integral operators appearing as spectral distribution functions for random matrices may be deduced from the associated systems of nonautonomous Hamiltonian equations satisfied by auxiliary canonical phase space variables introduced by Tracy and Widom. The essential step is to recast the latter as isomonodromic deformation equations for families of rational covariant derivative operators on the Riemann sphere and interpret the Fredholm determinants as isomonodromic τ -functions.     

Gravity and the frame field

The covariant derivative of a single massive fermion field on a Riemannian manifold is defined. The standard method of defining free bosonic Lagrangians from the fermion covariant derivative does not give the usual Lagrangian density for the free gravitational field. We express the fermion Lagrangian mass term as a frame field term added to the covariant derivative; this extended covariant derivative defines a gravitational Lagrangian density proportional to the usual scalar curvatureR, plus a term quadratic in the curvature components. The quadratic term is expected to be negligible at distances much greater than the fermion Compton wavelength, and is of a general form widely studied in recent years. The frame field term used to derive this gravitational Lagrangian is essentially the same as that used previously to derive the electroweak interaction boson mass matrix without using the Higgs-Kibble mechanism.     

Basic Properties of Symplectic Dirac Operators

Symplectic Dirac operators, acting on symplectic spinor fields introduced by B.~Kostant in geometric quantization, are canonically defined in a similar way as the Dirac operator on Riemannian manifolds. These operators depend on a choice of a metaplectic structure as well as on a choice of a symplectic covariant derivative on the tangent bundle of the underlying manifold. This paper performs a complete study of these relations and shows further basic properties of the symplectic Dirac operators. Various examples are given for illustration. Received: 1 July 1996 / Accepted: 24 September 1996     

Direct construction of the effective action of chiral gauge fermions in the anomalous sector

The anomaly implies an obstruction to a fully chiral covariant calculation of the effective action in the abnormal-parity sector of chiral theories. The standard approach then is to reconstruct the anomalous effective action from its covariant current. In this work, we use a recently introduced formulation which allows one to directly construct the non-trivial chiral invariant part of the effective action within a fully covariant formalism. To this end we develop an appropriate version of Chan’s approach to carry out the calculation within the derivative expansion. The result to four derivatives, i.e., to leading order in two and four dimensions and next-to-leading order in two dimensions, is explicitly worked out. Fairly compact expressions are found for these terms.     

Semiclassical Orthogonal Polynomials, Matrix Models and Isomonodromic Tau Functions

The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such measures. These are shown to preserve the generalized monodromy of the associated rank-2 rational covariant derivative operators. The corresponding matrix models, consisting of unitarily diagonalizable matrices with spectra supported on these contours are analyzed, and it is shown that all coefficients of the associated spectral curves are given by logarithmic derivatives of the partition function or, more generally, the gap probabilities. The associated isomonodromic tau functions are shown to coincide, within an explicitly computed factor, with these partition functions. Research supported in part by the Natural Sciences and Engineering Research Council of Canada, the Fonds FCAR du Québec and EC ITH Network HPRN-CT-1999-000161.     

Covariant technique for nonlocal terms of one-loop radiative currents

Within the bounds of the recently proposed covariant perturbation theory for nonlocal terms of an effective action, we develop a technique for the coincidence limits of the heat kernel, Green functions, polarization operators, and their covariant derivatives. We briefly discuss applications of the obtained results for radiative currents to expectation values and matrix elements of the various field operators.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 14–21, October, 1991.     

Spacelike energy of timelike unit vector fields on a Lorentzian manifold

On a Lorentzian manifold, we define a new functional on the space of unit timelike vector fields given by the L₂ norm of the restriction of the covariant derivative of the vector field to its orthogonal complement. This spacelike energy is related with the energy of the vector field as a map on the tangent bundle endowed with the Kaluza–Klein metric, but it is more adapted to the situation. We compute the first and second variation of the functional and we exhibit several examples of critical points on cosmological models as generalized Robertson–Walker spaces and Gödel universe, on Einstein and contact manifolds and on Lorentzian Berger’s spheres. For these critical points we have also studied to what extent they are stable or even absolute minimizers.