A New Approach for Calculating Spin Susceptibility in QED3

Based on the study of the linear response of the fermion propagator to the presence of an external scalar field, a new method for calculating the staggered spin susceptibility in QED3 is presented, in which the influence of the full vertex function is included. The numerical values of the staggered spin susceptibility are calculated within the framework of the rainbow-ladder approximation of the Dyson-Schwinger approach. A comparison between the result calculated using the full vertex and that using the bare vertex is given.     

Scalar Susceptibility of QCD from Dyson-Schwinger Approach

In quantum chromodynamics （QCD）, the scalar susceptibility represents the modification of the quark condensate, to a small perturbation of the parameter responsible for the explicit breaking of the symmetry, i.e., the current quark mass. By studying the linear response of the dressed quark propagator to the presence of a nonzero quark mass, we derive a model-independent formula for the scalar susceptibility, which contains the dressed quark propagator G（p） and the dressed scalar vertex F（p, 0）. The numerical values of the scalar susceptibility Xs are calculated within the framework of the rainbow-ladder approximation of the Dyson-Schwinger approach by employing two typical forms of model gluon propagator.     

Diquark confinement in an extended NJL model

In a Nambu-Jona-Lasinio model supplemented with an infrared cutoff in addition to the ultraviolet cutoff we study the issue whether diquarks are confined when the model is extended beyond the rainbow-ladder approximation. The gap equation, obtained in a truncation scheme motivated via a non-trivial quark-gluon vertex function, is solved to determine the constituent quark mass if chiral symmetry is spontaneously broken. In a second step, the Bethe-Salpeter equations for mesons and diquarks beyond the ladder approximation are derived, taking care to preserve Goldstone's theorem in the pion channel. While the obtained masses of pseudo-scalar and vector mesons are only moderately shifted compared to the values in the ladder approximation, we observe that scalar diquarks disappear from the physical spectrum and therefore are confined. For axial-vector diquarks we observe indications that the same mechanism may also work, but the NJL model allows no conclusive answer in this channel.     

Nucleon properties in a few mode bag — an effective approximation to QCD

The Hamiltonian of Quantum Chromodynamics is approximated by restricting the spatial dependencies of the quark-antiquark and gluon fields to a small number of modes (few-mode approximation). As a working hypothesis, we assume that the ground states of the dynamical sectors of the theory are automatically gauge invariant so that Gauss's law need not be strictly satisfied by a trial state. We derive and apply a method for explicit calculation of expectation values for wavefunctions in colour space depending on the twoSU (3) invariants acting as dynamical variables. An interacting ground state is constructed by a dynamical dressing transformation of the bare fermionic vacuum which may be regarded as a few-mode approximation to a quark-gluon (or mesonic) condensate. Based on this vacuum a nucleon state is formulated quite similar to the solition bag construction. Some of its effective properties are calculated by a variational calculation and comparison is made with experimental data as well as with other theoretical models of bag-type.     

Bethe-Salpeter equations: mesons beyond the rainbow-ladder truncation

We investigate masses of light mesons from a coupled system of Dyson–Schwinger (DSE) and Bethe– Salpeter equations (BSE),taking into account dominant non-Abelian,sub-leading Abelian,and dominant pion cloud contributions to the dressed quark-gluon vertex.The axial-vector Ward-Takahashi identity preserving Bethe-Salpeter kernel is constructed and the spectrum of light mesons calculated.Our model goes significantly beyond the rainbow-ladder.We find that sub-leading Abelian corrections are further dynamically suppressed,and that our results supersede early qualitative predictions from simple truncation schemes.     

A path integral formula for quark condensate states in?a?modified?PQCD

A modified version of PQCD considered in previous works is investigated here in the case of retaining only the quark condensate. The Green function generating functional is expressed in a form in which Dirac’s delta functions are now absent from the free propagators. The new expansion implements the dimensional transmutation effect through a single interaction vertex in addition to the standard ones in massless QCD. The new vertex suggest a way for constructing an alternative to the SM, in which the mass and CKM matrices could be generated by the instability of massless QCD under the production of the top quark and other fermions condensates, in a kind of generalized Nambu–Jona-Lasinio mechanism. The results of a two loop evaluation of the vacuum energy indicate that the quark condensate is dynamically generated. However, the energy as a function of the condensate parameter is again unbounded from below in this approximation. Assuming the existence of a minimum of the vacuum energy at the experimental value of the top quark mass m _q =173 GeV, we evaluate the two particle propagator in the quark–anti-quark channel in zero order in the coupling and a ladder approximation in the condensate vertex. Adopting the notion from the former top quark models in which the Higgs field corresponds to the quark condensate, the results suggest that the Higgs particle could be represented by a meson which might appear at energies around twice the top quark mass.     

QCD at finite temperature

Quantum chromodynamics is studied at finite temperatures and densities using the temperature Green functions method. For the Green functions the renormalized Schwinger-Dyson equations are obtained and their qualitative properties are discussed. The equality of the renormalization constants for the equations obtained at T, μ ≠ 0 with those for quantum field theory is pointed out. General properties of the gluon polarization tensor are investigated at T, μ ≠ 0. The temperature Green functions are calculated within the one-loop approximation using both relativistic and axial gauges. The fulfilment of the Slavnov-Taylor identities is verified. The asymptotic behaviour of the polarization tensor at T, μ ≠ 0 is established and the excitation spectrum of quark-gluon plasma is found. Both Fermi and Bose excitations are considered and the gauge invariance of the spectra is demonstrated. The renormalization group extension of the dispersion laws into the regions of high temperatures and densities is presented. The exact representation of the thermodynamical potential in QCD is found in terms of the temperature Green functions. For the quark-gluon plasma the thermodynamical potential is calculated with the g³-term taken into account. The equation of state of the hot quark-gluon plasma is found and its properties are discussed. The complete evolutional diagram of the hadronic matter is outlined. The phase curve asymptotics, which put bounds on the quark-gluon plasma domain, are found for the two limiting cases (μ = 0, T → T₀; T = 0, μ → μ₀). The phase transition of the hot quark-gluon plasma placed in external Abelian field is studied. The instability of such plasma has been found and the program of its stabilization is indicated. The infrared behaviour of the non-Abelian gauge theory is studied for finite temperatures when power divergencies are essential. The propagator of transverse gluons is shown to be singular for momenta |p| ˜ g²T and this cannot be avoided by summing the simplest bubble chains. The infrared asymptotic behaviour of the tree-gluon vertex is found and the results obtained are checked using the Slavnov-Taylor identities. The Green functions asymptotics found indicate either an instability of the quark-gluon plasma in the infrared momentum domain or the inconsistency of the perturbational methods. A non-perturbative approach to the infrared problem in QCD is developed within the axial gauge. The closed equations for the structure functions that determine the gluon polarization tensor are obtained by using the Slavnov-Taylor identities to found approximately the three-gluon vertex. It is shown that the solution of the equations obtained by iterations does not remove the infrared singularity from the temperature Green functions. The nonperturbative solution of such equations is discussed.     

Vacuum Polarisation Effects in the Lorentz Violating Electrodynamics

In this work we reconsider the one loop calculation for the vacuum polarisation tensor in the Lorentz violating quantum electrodynamics. The electron propagator is “dressed” by a Lorentz breaking extra term in the fermion Lagrangian density. We check gauge invariance and use the Schwinger–Dyson equation to discuss the full photon propagator. After a discussion on a possible photon mass shift, we show how a finite quantum correction can be chosen in a unique way in order to ensure—in the spirit of spontaneously broken theories—the standard normalisation conditions for the vacuum polarisation tensor. Then we comment on possible observable physical consequences on the Lamb-shift.     

Four-fermion interaction from torsion as dark energy

The observed small, positive cosmological constant may originate from a four-fermion interaction generated by the spin-torsion coupling in the Einstein–Cartan–Sciama–Kibble gravity if the fermions are condensing. In particular, such a condensation occurs for quark fields during the quark-gluon/hadron phase transition in the early Universe. We study how the torsion-induced four-fermion interaction is affected by adding two terms to the Dirac Lagrangian density: the parity-violating pseudoscalar density dual to the curvature tensor and a spinor-bilinear scalar density which measures the nonminimal coupling of fermions to torsion.     

Birkhoff Type Decompositions and the Baker–Campbell–Hausdorff Recursion

We describe a unification of several apparently unrelated factorizations arising from quantum field theory, vertex operator algebras, combinatorics and numerical methods in differential equations. The unification is given by a Birkhoff type decomposition that was obtained from the Baker–Campbell–Hausdorff formula in our study of the Hopf algebra approach of Connes and Kreimer to renormalization in perturbative quantum field theory. There we showed that the Birkhoff decomposition of Connes and Kreimer can be obtained from a certain Baker–Campbell–Hausdorff recursion formula in the presence of a Rota–Baxter operator. We will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for Lie algebras that arose in vertex operator algebra and conformal field theory, and the even-odd decomposition of combinatorial Hopf algebra characters as well as the Lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations.     

On the phenomenological three-graviton vertex

In General Relativity, the graviton interacts in three-graviton vertex with a tensor that is not the energy-momentum tensor of the gravitational field. We consider the possibility that the graviton interacts with the definite gravitational energy-momentum tensor that we previously found in the G ² approximation. This tensor in a gauge, where nonphysical degrees of freedom do not contribute, is remarkable, because it gives positive gravitational energy density for the Newtonian center in the same manner as the electromagnetic energy-momentum tensor does for the Coulomb center. We show that the assumed three-graviton vertex does not lead to contradiction with the precession of Mercury’s perihelion. In the S-matrix approach used here, the external gravitational field has only a subsidiary role, similar to the external field in quantum electrodynamics. This approach with the assumed vertex leads to the gravitational field that cannot be obtained from a consistent gravity equation.     

Nonlinear dynamics of soft boson collective excitations in hot QCD plasma III: Bremsstrahlung and energy losses

Within the framework of semiclassical approximation a general formalism for deriving an effective current generating bremsstrahlung of arbitrary number of soft gluons (longitudinal or transverse ones) in scattering of higher-energy parton off thermal parton in hot quark-gluon plasma with subsequent extension to two and more scatterers is obtained. For the case of static color centers, an expression for energy loss induced by usual bremsstrahlung of lowest-order with allowance for an effective temperature-induced gluon mass and finite mass of the projectile (heavy quark) is derived. The detailed analysis of contribution to radiation energy loss associated with existence of effective three-gluon vertex induced by hot QCD medium is performed. It is shown that in general, the bremsstrahlung associated with this vertex has no sharp direction (as in the case of usual bremsstrahlung) and therefore here, we can expect an absence of suppression effect due to multiple scattering. For the case of two-color static scattering centers it was shown that the problem of calculation of bremsstrahlung induced by four-gluon hard thermal loop (HTL) vertex correction can be reduced to the problem of the calculation of bremsstrahlung induced by three-gluon HTL correction. It was shown that for limiting value of soft gluon occupation number N_k 1/α_s all higher processes of bremsstrahlung of arbitrary number of soft gluons become of the same order in coupling, and the problem of resummation of all relevant contributions to radiation energy loss of fast parton, arises. An explicit expression for matrix element of two soft gluon bremsstrahlung in small angles approximation is obtained.     

Quark Gluon Condensate,Virtuality and Susceptibility of QCD Vacuum

We study vacuum of QCD in this work. The structure of non-local quark vacuum condensate, values of various local quark and gluon vacuum condensates, quark-gluon mixed vacuum condensate, quark and gluon virtuality in QCD vacuum state, quark dynamical mass and susceptibility of QCD vacuum state to external field are predicted by use of the solutions of Dyson-Schwinger equations in “rainbow” approximation with a modeling gluon propagator and three different sets of quark-quark interaction parameters. Our theoretical predictions are in good agreement with the correspondent empirical values used widely in literature, and many other theoretical calculations. The quark propagator and self-energy functions are also obtained from the numerical solutions of Dyson-Schwinger equations. This work is centrally important for studying non-perturbative QCD, and has many important applications both in particle and nuclear physics.     

Effect of porosity on the permeability tensor of polycrystalline ferrites (independent grain approximation)

The effect of porosity on the form of the permeability tensor is calculated using the independent-grain approximation; this procedure is similar to the Schlömann method. The theoretical curve for the resonant-field distribution is approximated by the Lorentz curve using the method of least squares. It is shown that with this approximation, porosity increases the width of the ferromagnetic resonance line for a non-porous material by the width of the Lorentz distribution curve; thus the resonant field shifts toward lower values. Formulas are obtained for the resonant-field shift due to porosity and the broadening in the ferromagnetic resonance line; these formulas differ somewhat from the Schlömann formulas. In order to check the working formulas and the applicability of the independent-grain approximation, measurements were performed on the tensor for magnesium-chromium-copper ferrites with variable porosity and a magnetization on the order of 1200 gauss at a frequency of 4000 mHz. Specimens having the form of longitudinally magnetized circular cylinders were used so that there was no degeneration in uniform precession of magnetization with long spin waves. The observed effect of porosity on the width of the ferromagnetic resonance line (determined by measuring the tensor) was found to be in good quantitative agreement with calculation. The shift in the resonant field due to porosity was negligibly small, which also agreed with calculation. The experimental results show that when there is no degeneration in uniform precession with spin waves, the independent-grain approximation can be used in experiments even when the magnetization and resonant field are approximately equal.Here, we must allow for the static magnetic susceptibility in the formulas for the resonant-field shift and the broadening in the ferromagnetic resonance line.     

Chemical Potential Dependence of Vertices

Based on the rainbow-ladder approximation of the Dyson-Schwinger equations and the assumption of the analyticity of the fermion-boson vertex in the neighborhood of zero chemical potential (μ=0) and neglecting the μ-dependence of the dressed gluon propagator, we apply the method in [Phys. Rev. C 71 (2005) 015205] of studying the dressed quark propagator at finite chemical potential to prove that the general fermion-boson vertex at finite μ can also be obtained from the one at μ=0 by a simple shift of variables. Using this result we extend the results of [Phys. Lett. B 420 (1998) 267] to the situation of finite chemical potential and show that under the approximations we have taken, the Gell-Mann-Oakes-Renner relation also holds at finite chemical potential.     

Nonlocal Four-Quark Condensate and Vacuum Susceptibilities

Based on the quark propagator in the instanton dilute liquid approximation, we have determined the nonlocal four-quark condensate at the mean-field level in the framework of global color symmetry model. By making use of this formula, the vacuum susceptibility (pionsusceptibility) was calculated in this paper. The theoretical analysis and numerical calculation show that even at the mean-field level the naive vacuum saturation approximation is not a good approximation when we consider nonlocal four-quark condensate.     

CALCULATION OF KAON ELECTROMAGNETIC FORM FACTOR

The kaon meson electromagnetic form factor is calculated in the framework of coupled Schwinger-Dyson and Bethe-Salpeter formulation in simplified impulse approximation (dressed vertex) with modified flat-bottom potential, which is a combination of the flat-bottom potential taking into consideration the infrared and ultraviolet asymptotic behaviours of the effective quark-gluon coupling. All the numerical results give a good fit to experimental values.     

UA(1) anomaly and $ \eta^{{\prime}}_{}$ -mass from an infrared singular quark-gluon vertex

The U _A(1) problem of QCD is inevitably tied to the infrared behaviour of quarks and gluons with its most visible effect being the -mass. A dimensional argument of Kogut and Susskind showed that the mixing of the pseudoscalar flavour-singlet mesons with gluons can provide a screening of the Goldstone pole in this channel if the full quark-quark interaction is strongly infrared singular as ∼ 1/k ⁴ . We investigate this idea using previously obtained results for the Landau gauge ghost and gluon propagator, together with recent determinations for the singular behaviour of the quark-gluon vertex. We find that, even with an infrared vanishing gluon propagator, the singular structure of the quark-gluon vertex for certain kinematics is apposite for yielding a non-zero screening mass.     

Gauge Symmetry and Transverse Symmetry Transformations in Gauge Theories

The transverse symmetry transformations associated with the normal symmetry transformations are proposed to build the transverse constraints on the basic vertices in gauge theories. I show that, while the BRST symmetry in non-Abelian gauge theory QCD (Quantum Chromodynamics) leads to the Slavnov-Taylor identity for the quark-gluon vertex which constrains the longitudinal part of thevertex, the transverse symmetry transformation associated with the BRST symmetry enables to derive the transverse Slavnov-Taylor identity for the quark-gluon vertex, which constrains the transverse part of the quark-gluon vertex from the gauge symmetry of QCD.     

Bethe-Salpeter Meson Masses Beyond Ladder Approximation

The effect of quark-gluon vertex dressing on the ground-state masses of the u/d-quark pseudoscalar, vector and axialvector mesons is considered with the Dyson-Schwinger equations. This extends the ladder-rainbow Bethe-Salpeter kernel to two-loops. To render the calculations feasible for this exploratory study, we employ a simple infrared dominant model for the gluon exchange that implements the vertex dressing. The resulting model, involving two distinct representations of the effective gluon exchange kernel, preserves both the axial-vector Ward-Takahashi identity and charge conjugation symmetry. Numerical results confirm that the pseudoscalar meson retains its Goldstone boson character. The vector meson mass, already at a very acceptable value at ladder level, receives only 30MeV of attraction from this vertex dressing. For the axial-vector states, which are about 300MeV too low in ladder approximation, the results are mixed: The 1^+– state receives 290MeV of repulsion, but the 1⁺⁺ state is lowered further by 30MeV. The exotic channels 0^–– and 1^–+ are found to have no states below 1.5GeV in this model.