Effect of a spurious component of the potential in Faddeev-type equations

In the integro-differential-equation approach (IDEA) and its associated hyperspherical-harmonic expansion method (HHEM) terms appear involving the potential for a single pair of particles. This introduces a spurious component in the potential that is eliminated in the Schrödinger equation by summing over all pairs. We calculate the contribution of this spurious potential in the solution of the HHEM for an even number of bosons in the ground state, using the Afnan-Tang S3 and Malfliet-Tjon MT5 interactions.Unité de Recherche des Universités Paris 11 et Paris 6 Associée au CNRS     

Convergence of an Exact Quantization Scheme

It has been shown by Voros [V1] that the spectrum of the one-dimensional homogeneous anharmonic oscillator (Schrödinger operator with potential q^2M, M=2,3,...) is a fixed point of an explicit non-linear transformation. We show that this fixed point is globally and exponentially attractive in spaces of properly normalized sequences.Partially supported by Faperj and CNPq, BrazilCurrent address:Laboratoire de Probabilités et Modéles aléatoires, Université Pierre et Marie Curie–Boi te courrier 188, 75252–Paris Cedex 05, France. E-mail: artur@ccr.jussieu.fr     

The quantum group structure of 2D gravity and minimal models II: The genus-zero chiral bootstrap

The chiral operator-algebra of the quantum-group-covariant operators (of vertex type) is completely worked out by making use of the operator-approach suggested by the Liouville theory, where the quantum-group symmetry is explicit. This completes earlier articles along the same line. The relationship between the quantum-group-invariant (of IRF type) and quantum-group-covariant (of vertex type) chiral operator-algebras is fully clarified, and connected with the transition to the shadow world for quantum-group symbols. The corresponding 3-j symbol dressing is shown to reduce to the simpler transformation of Babelon and one of the authors (J.-L. G.) in a suitable infinite limit defined by analytic continuation. The above two types of operators are found to coincide when applied to states with Liouville momenta going to in a suitable way. The introduction of quantum-group-covariant operators in the three dimensional picture gives a generalization of the quantum-group version of discrete three-dimensional gravity that includes tetrahedra associated with 3-j symbols and universalR-matrix elements. Altogether the present work and a previous parallel article gives the concrete realization of Moore and Seiberg's scheme that describes the chiral operator-algebra of two-dimensional gravity and minimal models.Unité Propre du Centre National de La Recherche Scientifique, associée à l'École Normale Supérieure et à l'Université de Paris-Sud     

On the algebraic structure ofN=2 string theory

InN=2 string theory the chiral algebra expresses the generations and anti-generations of the theory and the Yukawa couplings among them and is thus crucial to the phenomenological properties of the theory. Also the connection with complex geometry is largely through the algebras. These algebras are systematically investigated in this paper. A solution for the algebras is found in the context of rational conformal field theory based on Lie algebras. A statistical mechanics interpretation for the chiral algebra is given for a large family of theories and is used to derive a rich structure of equivalences among the theories (dihedralities). The Poincaré polynomials are shown to obey a resolution series which cast these in a form which is a sum of complete intersection Poincaré polynomials. It is suggested that the resolution series is the proper tool for studying allN=2 string theories and, in particular, exposing their geometrical nature.     

Chiral transition and deconfinement transition in QCD with the highly improved staggered quark (HISQ) action

We report preliminary results on the chiral and deconfinement aspects of the QCD transition at finite temperature using the Highly Improved Staggered Quark (HISQ) action on lattices with temporal extent of N _τ = 6 and 8. The chiral aspects of the transition are studied in terms of quark condensates and the disconnected chiral susceptibility. We study the deconfinement transition in terms of the strange quark number susceptibility and the renormalized Polyakov loop. We find reasonably good agreement between our results and the recent continuum extrapolated results obtained with the stout staggered quark action.     

Unconventional properties of dislocations in quasicrystals

Dislocations in quasicrystals are defined as the intersections of dislocations in a high dimensional lattice with an irrational cut which figures the physical space. This definition confers to them a number of unusual geometrical properties which can be studied either by suitable extensions of the Volterra process, or by topological approaches, which often offer complementary points of view and are presented in this paper. Amongst these unusual properties, the production of stacking faults under shear at low temperature, reshuffling processes on stacking faults, and properties of non-commutativity which could have some incidences on the interplay between dislocations in deformation processes are mentioned.Dedicated to Dr. Frantiek Kroupa in honour of his 70^th birthday.Unité de Recherche Associée 009 du CNRS, associée aux Universités de Paris VI at Paris VIIWe are very grateful to Dr. Vladimir Dmitrienko for discussions and useful remarks.     

Spectral Action for Torsion with and without Boundaries

We derive a commutative spectral triple and study the spectral action for a rather general geometric setting which includes the (skew-symmetric) torsion and the chiral bag conditions on the boundary. The spectral action splits into bulk and boundary parts. In the bulk, we clarify certain issues of the previous calculations, show that many terms in fact cancel out, and demonstrate that this cancellation is a result of the chiral symmetry of spectral action. On the boundary, we calculate several leading terms in the expansion of spectral action in four dimensions for vanishing chiral parameter θ of the boundary conditions, and show that θ = 0 is a critical point of the action in any dimension and at all orders of the expansion.     

Observables in nuclear fragmentation

We study the method of moments correlations as a tool to discriminate between different fragmentation mechanisms. We show, using simple models, that the variance of the fragment distribution, the size of the heaviest residue and the scaling of these quantities with the size of the fragmenting system give significant insight into the nature of the fragmentation mechanism.Unité de Recherche des Universités Paris XI et Paris VI associée au CNRS     

Relativistic interactions without fields or potentials

Employing Poincaré degrees of freedomM ^jk=(¯K,¯J) andP ^k=(E,¯p) transforming linearly (but inhomogeneously) under the action of the Poincaré group we define a number of quantities which we later identify with physical observables. The identifications are consistent with the nonrelativistic limit and with other requirements following from the Poincaré covariance. Next, we treat a free relativistic particle as composed of two interacting parts. Relativistic quantum commutation relations for their Poincaré algebras and a kind of (inverse) relativistic correspondence principle are used to generate (quasi-) classical equations of their relative motion. A simple example based on these ideas is explicitly solved.I am indebted to Prof. B. Laurent, Dr. S. Flodmark, and Prof. I. Fischer-Hjalmars for pointing this out to me.     

Scaling behaviour of the effective chiral action and stability of the chiral soliton

The effective chiral action is evaluated within a novel improved heat-kernel expansion, which includes gradients of the chiral field in a non-perturbative way. This expansion is valid for both small and large momenta. The exact scaling behaviour of the effective action of a localized chiral field with respect to changing its spatial size is found. From this it is proved that the radiatively induced derivative terms cannot absolutely stabilize the chiral soliton against collapsing. The collapsing of the soliton is, however, accompanied by a vanishing of the baryon charge. It is argued that the effective chiral action constrained to a fixed baryon number may still admit stable soliton configurations.     

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.     

Sharpenings of Li's Criterion for the Riemann Hypothesis

Exact and asymptotic formulae are displayed for the coefficients λ _n used in Li's criterion for the Riemann Hypothesis. For n → ∞ we obtain that if (and only if) the Hypothesis is true, λ _n ∼ n(A log n + B) (with A > 0 and B explicitly given, also for the case of more general zeta or L-functions); whereas in the opposite case, λ _n has a non-tempered oscillatory form. ^★Institut de Mathématiques de Jussieu-Chevaleret (CNRS UMR 7586), Université Paris 7, F-75251 Paris CEDEX 05, France.     

Nonlinear chiral models: Soliton solutions and spatio-temporal chaos

Nonlinear effective Lagrangian models with a chiral symmetry have been used to describe strong interactions at low energy, for a long time. The Skyrme model and the chiral quark-meson model are two such models, which have soliton solutions which can be identified with the baryons. We describe the various kinds of soliton states in these nonlinear models and discuss their physical significance and uses in this review. We also study these models from the view point of classical nonlinar dynamical systems. We consider fluctuations around theB=1 soliton solutions of these models (B, being the baryon number) and solve the spherically symmetric, time-dependent systems. Numerical studies indicate that the phase space around the Skyrme soliton solution exhibits spatio-temporal chaos. It is remarkable that topological solitons signifying stability/order and spatio-temporal chaos coexist in this model. In contrast with this, the soliton of the quark-meson model is stable even for large perturbations.     

An explicit construction of the quantum group in chiral WZW-models

It is shown how a chiral Wess-Zumino-Witten theory with globally defined vertex operators and a one-to-one correspondence between fields and states can be constructed. The Hilbert space of this theory is the direct sum of tensor products of representations of the chiral algebra and finite dimensional internal parameter spaces. On this enlarged space there exists a natural action of Drinfeld's quasi-quantum groupA _{g, t} which commutes with the action of the chiral algebra and plays the rôle of an internal symmetry algebra. TheR matrix describes the braiding of the chiral vertex operators and the coassociator gives rise to a modification of the duality property.For genericq the quasi-quantum group is isomorphic to the coassociative quantum groupU _q (g) and thus the duality property of the chiral theory can be restored. This construction has to be modified for the physically relevant case of integer level. The quantum group has to be replaced by the corresponding truncated quasiquantum group, which is not coassociative because of the truncation. This exhibits the truncated quantum group as the internal symmetry algebra of the chiral WZW model, which therefore has only a modified duality property. The case ofg=su(2) is worked out in detail.     

Wavelet Analysis of Fractal Boundaries. Part 2: Multifractal Analysis

This second part deals with the global analysis of the boundary of domains . We develop methods for determining the dimensions of the sets where the local behaviors introduced in Part 1 occur. These methods are based on analogies with the thermodynamic formalism in statistical physics and lead to new classification tools for fractal domains.The first author is supported by the Institut Universitaire de France.This work was performed while the second author was at the Laboratoire d’Analyse et de Mathématiques Appliquées (University Paris XII) and at the Istituto di Matematica Applicata e Tecnologie Informatiche (Pavia, Italy) and partially supported by the Société de Secours des amis des Sciences and the TMR Research Network “Breaking Complexity”.     

The role of strange sea quarks in chiral extrapolations on the lattice

Since the strange quark has a light mass of order , fluctuations of sea pairs may play a special role in the low-energy dynamics of QCD by inducing significantly different patterns of chiral symmetry breaking in the chiral limits N _f = 2 (m _u = m _d = 0, m _s physical) and N _f = 3 (m _u = m _d = m _s = 0). This effect of vacuum fluctuations of pairs is related to the violation of the Zweig rule in the scalar sector, described through the two O(p ⁴) low-energy constants L ₄ and L ₆ of the three-flavour strong chiral lagrangian. In the case of significant vacuum fluctuations, three-flavour chiral expansions might exhibit numerical competition between leading- and next-to-leading-order terms according to the chiral counting, and chiral extrapolations should be handled with special care. We investigate the impact of the fluctuations of pairs on chiral extrapolations in the case of lattice simulations with three dynamical flavours in the isospin limit. Information on the size of the vacuum fluctuations can be obtained from the dependence of the masses and decay constants of pions and kaons on the light quark masses. Even in the case of large fluctuations, corrections due to the finite size of spatial dimensions can be kept under control for large enough boxes ( fm).Received: 20 October 2004, Revised: 28 December 2004, Published online: 9 February 2005Laboratoire de Physique Théorique: LPT is an Unité Mixte de Recherche du CNRS et de lUniversité Paris-Sud 11 (UMR 8627).     

The quantum group structure associated with non-linearly extended Virasoro algebras

Recently, an infinite family of chiral Virasoro vertex operators, whose exchange algebra is given by the universalR-matrix ofSL(2) _q , has been constructed. In the present paper, the case of non-linearly (W-) extended Virasoro symmetries, related to the algebrasA _N,N>1, is considered along the same line. Contrary to the previous case (A ₁) the standardR-matrix ofSL(N+1)q does not come out, and a different solution of Yang and Baxter's equations is derived. The associated quantum group structure is displayed.Unité Propre du Centre National de la Recherche Scientifique, associée à l'École Normale Supérieure et à l'Université de Paris-Sud     

Two-dimensional Lorentz-Weyl anomaly and gravitational Chern-Simons theory

Two-dimensional chiral fermions and bosons, more generally conformal blocks of two-dimensional conformal field theories, exhibit Weyl-, Lorentz- and mixed Lorentz-Weyl anomalies. A novel way of computing these anomalies for a system of chiral bosons of arbitrary conformal spinj is sketched. It is shown that the Lorentz- and mixed Lorentz-Weyl anomalies of these theories can be cancelled by the anomalies of a three-dimensional classical Chern-Simons action for the spin connection, expressed in terms of the dreibein field. Some tentative applications of this result to string theory are indicated.     

Fluctuations as probe of the QCD phase transition and freeze-out in heavy ion collisions at LHC and RHIC

We discuss the relevance of higher order cumulants of net baryon number fluctuations for the analysis of freeze-out and critical conditions in heavy ion collisions at LHC and RHIC. Using properties of O(4) scaling functions, we discuss the generic structure of these higher cumulants at vanishing baryon chemical potential and apply chiral model calculations to explore their properties at non-zero baryon chemical potential. We show that the ratios of the sixth to second and eighth to second order cumulants of the net baryon number fluctuations change rapidly in the transition region of the QCD phase diagram. Already at vanishing baryon chemical potential they deviate considerably from the predictions of the hadron resonance gas model which reproduce the second and fourth order cumulants of the net proton number fluctuations at RHIC. We point out that the sixth order cumulants of baryon number and electric charge fluctuations remain negative at the chiral transition temperature. Thus, they offer the possibility to probe the proximity of the chemical freeze-out to the crossover line.