Equations from non-linear chiral transformations

In comparison with theWT chiral identity which is indispensable for renormalization theory, relations deduced from the non-linear chiral transformation have a totally different physical significance. We wish to show that non-linear chiral transformations are powerful tools to deduce useful integral equations for propagators. In contrast to the case of linear chiral transformations, identities derived from non-linear ones contain more involved radiative effects and are rich in physical content. To demonstrate this fact we apply the simplest non-linear chiral transformation to the Nambu-Jona-Lasinio model, and show how our identity is related to the Dyson-Schwinger equation and Bethe-Salpeter amplitudes of the Higgs and π. Unlike equations obtained from the effective potential, our resultant equation is exact and can be used for events beyond the LEP energy.     

Photon-Z mixing in the Weinberg-Salam model: Effective charges and the a = −3 gauge

We study some properties of the Weinberg-Salam model connected with the photon-Z mixing. We solve the linear Dyson-Schwinger equations between full and 1PI boson propagators. The task is made easier by the two-point function Ward identities that we derive to all orders and in any gauge. Some aspects of the renormalization of the model are also discussed. We display the exact mass-dependent one-loop two-point functions involving the photon and Z field in any linear ξ-gauge. The special gauge a = ξ^?1 = ?3 is shown to play a peculiar role. In this gauge, the Z field is multiplicatively renormalizable (at the one-loop level), and one can construct both electric and weak effective charges of the theory from the photon and Z propagators, with a very simple expression similar to that of the QED Petermann, Stueckelberg, Gell-Mann and Low charge.     

Investigation of continuous-time quantum walk via spectral distribution associated with adjacency matrix

Using the spectral distribution associated with the adjacency matrix of graphs, we introduce a new method of calculation of amplitudes of continuous-time quantum walk on some rather important graphs, such as line, cycle graph C_n, complete graph K_n, graph G_n, finite path and some other finite and infinite graphs, where all are connected with orthogonal polynomials such as Hermite, Laguerre, Tchebichef, and other orthogonal polynomials. It is shown that using the spectral distribution, one can obtain the infinite time asymptotic behavior of amplitudes simply by using the method of stationary phase approximation (WKB approximation), where as an example, the method is applied to star, two-dimensional comb lattices, infinite Hermite and Laguerre graphs. Also by using the Gauss quadrature formula one can approximate the infinite graphs with finite ones and vice versa, in order to derive large time asymptotic behavior by WKB method. Likewise, using this method, some new graphs are introduced, where their amplitudes are proportional to the product of amplitudes of some elementary graphs, even though the graphs themselves are not the same as the Cartesian product of their elementary graphs. Finally, by calculating the mean end to end distance of some infinite graphs at large enough times, it is shown that continuous-time quantum walk at different infinite graphs belong to different universality classes which are also different from those of the corresponding classical ones.     

Large volume behaviour of Yang-Mills propagators

We investigate finite volume effects in the propagators of Landau gauge Yang-Mills theory using Dyson-Schwinger equations on a 4-dimensional torus. In particular, we demonstrate explicitly how the solutions for the gluon and the ghost propagator tend towards their respective infinite volume forms in the corresponding limit. This solves an important open problem of previous studies where the infinite volume limit led to an apparent mismatch, especially of the infrared behaviour, between torus extrapolations and the existing infinite volume solutions obtained in 4-dimensional Euclidean space-time. However, the correct infinite volume limit is approached rather slowly. The typical scales necessary to see the onset of the leading infrared behaviour emerging already imply volumes of at least 10-15 fm in lengths. To reliably extract the infrared exponents of the infinite volume solutions requires even much larger ones. While the volumes in the Monte-Carlo simulations available at present are far too small to facilitate that, we obtain a good qualitative agreement of our torus solutions with recent lattice data in comparable volumes.     

Are scattering amplitudes dual to super Wilson loops?

The MHV scattering amplitudes in planar N=4 SYM are dual to bosonic light-like Wilson loops. We explore various proposals for extending this duality to generic non-MHV amplitudes. The corresponding dual object should have the same symmetries as the scattering amplitudes and be invariant to all loops under the chiral half of the N=4 superconformal symmetry. We analyze the recently introduced supersymmetric extensions of the light-like Wilson loop (formulated in Minkowski space-time) and demonstrate that they have the required symmetry properties at the classical level only, up to terms proportional to field equations of motion. At the quantum level, due to the specific light-cone singularities of the Wilson loop, the equations of motion produce a nontrivial finite contribution which breaks some of the classical symmetries. As a result, the quantum corrections violate the chiral supersymmetry already at one loop, thus invalidating the conjectured duality between Wilson loops and non-MHV scattering amplitudes. We compute the corresponding anomaly to one loop and solve the supersymmetric Ward identity to find the complete expression for the rectangular Wilson loop at leading order in the coupling constant. We also demonstrate that this result is consistent with conformal Ward identities by independently evaluating corresponding one-loop conformal anomaly.     

The QCD β-function from global solutions to Dyson-Schwinger equations

We study quantum chromodynamics from the viewpoint of untruncated Dyson-Schwinger equations turned to an ordinary differential equation for the gluon anomalous dimension. This non-linear equation is parameterized by a function P(x) which is unknown beyond perturbation theory. Still, very mild assumptions on P(x) lead to stringent restrictions for possible solutions to Dyson-Schwinger equations.We establish that the theory must have asymptotic freedom beyond perturbation theory and also investigate the low energy regime and the possibility for a mass gap in the asymptotically free theory.     

Null-plane charges and fourier transforms for explicitly and spontaneously broken symmetries

It is shown that the existence of null-plane charges and null-plane Fourier transforms (which enter into the discussion of current algebra at infinite momentum) depends on the high energy asymptotic behaviour of off mass shell scattering amplitudes; s In s behaviour is the maximum growth allowed. Under the assumption that asymptotic states exist, these results also hold in the case of spontaneously broken chiral symmetry with massless pseudoscalar Goldstone bosons. In this case, the resulting charges are the non-conserved pole-free charges suggested by Carlitz et al.     

Local conserved currents for CPN σ-models

We present a new method to derive an infinite series of conserved local charges for the two-dimensional CP^N σ-models. The generating relation for the conservation laws is a couple of first-order nonlinear differential equations. The method displays transparently the connection of the local charges with nonlocal dynamical charges of CP^N models previously found.     

Integral sum rules and inequalities for ππ partial wave amplitudes

Exact integral relations on all ππ partial waves are derived from crossing symmetry. As a corollary we also derive an infinite number of inequalities fors-wave amplitudes.     

Chiral phase transition from the Dyson-Schwinger equations in a finite spherical volume

Within the framework of the Dyson-Schwinger equations and by means of Multiple Reflection Expansion,we study the effect of finite volume on the chiral phase transition in a sphere, and discuss in particular its influence on the possible location of the critical end point(CEP). According to our calculations, when we take a sphere instead of a cube, the influence of finite volume on phase transition is not as significant as previously calculated. For instance,as the radius of the spherical volume decreases from infinite to 2 fm, the critical temperature T c, at zero chemical potential and finite temperature, drops only slightly. At finite chemical potential and finite temperature, the location of CEP shifts towards smaller temperature and higher chemical potential, but the amplitude of the variation does not exceed 20%. As a result, we find that not only the size of the volume but also its shape have a considerable impact on the phase transition.     

How crossing affects the analytic relationship between low energy ππ amplitudes and their asymptotic behaviour

Different physical assumptions about the asymptotic behaviour of ππ amplitudes are realised in the different number of substractions involved in fixed t dispersion relations for the various amplitudes and their inverses. The fact that each new dispersion relation must be consistent with s - t crossing leads to a number of conditions relating low energy ππ amplitudes to their high energy behaviour. These are discussed in detail. Such relationships supplement finite energy sum rules with which they are compared. The dispersive sum rules, crossing conditions, and finite energy sum rules we discuss are applied to recent phenomenological solutions to Roy's equations and shown not to narrow the presently accepted range of threshold parameters. These results are in marked contrast to the conclusions of other recent studies. To complete the study of finite energy sum rules we consider the behaviour of the isospin zero t-channel amplitude and estimate the asymptotic ππ total cross-section. We present evidence to suggest that the pomeron is late-developing in meson-meson scattering.     

Multi-Reggeon processes in QCD

We discuss the multi-Regge form of QCD amplitudes and outline a way how to prove this form. The key to the proof is given by the “bootstrap” requirement. This requirement leads to an infinite set of bootstrap relations for multiparticle production amplitudes. On the other hand, all these amplitudes are expressed in terms of the gluon trajectory and a finite number of Reggeon vertices. Therefore, it is extremely nontrivial to satisfy all these relations. However, it turns out that all of them can be fulfilled if the vertices and trajectory submit to several bootstrap conditions. Fulfillment of all these relations secures the Reggeized form of the radiative corrections order by order in perturbation theory.     

Pseudoscalar meson nonet at zero and finite temperature

Theoretical understanding of experimental results from relativistic heavy-ion collisions requires a microscopic approach to the behavior of QCD n-point functions at finite temperatures, as given by the hierarchy of Dyson-Schwinger equations, properly generalized within the Matsubara formalism. The convergence of sums over Matsubara modes is studied. The technical complexity of finite-temperature calculations mandates modeling. We present a model where the QCD interaction in the infrared, nonperturbative domain is modeled by a separable form. Results for the mass spectrum of light quark flavors (u, d, s) and for the pseudoscalar bound-state amplitudes at finite temperature are presented. Talk presented by D. Klabučar at the “Dense Matter In Heavy Ion Collisions and Astrophysics” Conference, JINR, Dubna, August 21–September 1, 2006. The text was submitted by the authors in English.     

A nonperturbative solution to the Dyson-Schwinger-equations of QCD

This is the first of two papers in which we discuss a nonperturbatively modified solution to the Euclidean Dyson-Schwinger equations for the 7 superficially divergent proper verticesΓ of QCD. It takes the formΣ _n g ²ⁿ Γ( ⁿ ) where eachΓ( ⁿ ) approaches its perturbative form at large momenta. At lower momenta, it differs from that form by an additional non-analyticg ² dependence through a dynamical mass scaleb, proportional toΛ _qcd and associated with a pole dependence on the momentum invariants. In the zeroth-order two-point functions, these nonperturbative modifications amount to a generalized Schwinger mechanism, leading to propagators without particle poles. The termsΓ(⁰), representing the Feynman rules of the modified iterative solution, can become self-consistent in the DS equations through a mechanism of “nonperturbative logarithms” which we explain. The mechanism is tied to the presence of divergent loops, and thus represents a pure quantum effect, similar to quantum anomalies. It restricts formation of nonperturbativeΓ(⁰)'s to the 7 primitively divergent vertices, thus escaping the infinite nature of the DS hierarchy. In a given loop order, the self-consistency problem reduces to a finite set of algebraic equations.     

On the formalism of relativistic many body theory

The thermodynamic potential is constructed as an effective action functional of the various n point amplitudes (n ? 4). One of the functionals is used to obtain the equations of state as simple, convergent expressions involving the conventionally renormalized charges and masses.     

Rigorous construction of planar diagram field theories in four dimensional Euclidean space

Asymptotically free quantum field theories with planar Feynman diagrams [such as SU(∞) gauge theory] are considered in 4 dimensional Euclidean space. It is shown that if all particles involved have non-vanishing masses and if the coupling constant(s) γ (org ²) are small enough (λ≦λ^crit), then an absolutely convergent procedure exists to obtain Green functions that uniquely solve the Dyson-Schwinger equations.     

Solving coupled Dyson-Schwinger equations on a compact manifold

We present results for the gluon and ghost propagators in SU (N) Yang-Mills theory on a four-torus at zero and non-zero temperatures from a truncated set of Dyson-Schwinger equations. When compared to continuum solutions at zero temperature sizeable modifications due to the finite volume of the manifold, especially in the infrared, are found. Effects due to non-vanishing temperatures T, on the other hand, are minute for T < 250 MeV.     

The size of finite size effects in lattice gauge theories

If a quantum field is enclosed in a spatial box of finite volume, its mass spectrum depends on the box size L. For field theories in the continuum Lüscher has shown to all orders in perturbation theory that for large L this dependence is related to certain scattering amplitudes of the infinite volume theory. We derived the corresponding relations for lattice field theories. Assuming their validity for lattice gauge theory outside the perturbative region the magnitude of finite size effects on the spectrum is determined by a glueball coupling constant. This quantity is estimated by strong coupling methods.     

The flexural vibration of a line-stiffened plate with fluid loading,part I: Analysis

This paper considers the vibration of a symmetrical system consisting of an infinite fluid loaded plate bearing a finite number of parallel stiffeners. The system is driven at the central stiffener by a travelling wave line force. Formal solutions for the equations of motion are found in terms of cosine transforms. Manipulation of the equations allows the problem to be reduced to the solution of a set of linear algebraic equations in the vibration amplitudes at the stiffeners. The coefficients in these equations depend in a simple way upon the stiffener parameters, and upon particular values of the cosine transform of a function which depends only on the plate and fluid parameters, and the stiffener positions.     

Finite nuclei in the reggeon “toy model”

Hadron–nucleus amplitudes at high energies are studied in the “toy” Regge model in zero transverse dimension for finite nuclei, when the standard series of fan diagrams is converted into a finite sum and loses physical sense at quite low energies. Taking into account all the loop contributions by numerical methods we find a physically meaningful amplitudes at all energies. They practically coincide with the amplitudes for infinite nuclei. A surprising result is that for finite nuclei and small enough triple pomeron coupling the infinite series of fan diagrams describes the amplitude quite well in spite of the fact that in reality the series should be cut and as such deprived of any physical sense at high energies.