Schwinger-Dyson Gap Equation Research Articles

The hardest TC self-energy behavior and radiative corrections in a 331-TC model

The solution of the phenomenological problems of technicolor (TC) models may reside in the different dynamical behaviors of the technifermions self-energy appearing in walking (or quasi-conformal) theories. Motivated by recent results, where it is shown how the boundary conditions (BC) of the anharmonic oscillator representation of the Schwinger–Dyson gap equation (SDE) to [Formula: see text] are directly related with the mass anomalous dimensions, and different BC cause a change in the ultraviolet asymptotic behavior of the self-energies, in this paper, we verify that it is possible to have a hard technifermion self-energy in TC models originated through radiative corrections coming from the interactions mediated by the new massive neutral and charged gauge bosons, [Formula: see text] and [Formula: see text] in the context of a 331-TC model.

Dynamical gap generation in a two-dimensional Dirac semimetal with a deformed Dirac cone

According to the extensive theoretical and experimental investigations, it is widely accepted that the long-range Coulomb interaction is too weak to generate a dynamical excitonic gap in graphene with a perfect Dirac cone. We study the impact of the deformation of Dirac cone on dynamical gap generation. When a uniaxial strain is applied to graphene, the Dirac cone is made elliptical in the equal-energy plane and the fermion velocity becomes anisotropic. The applied uniaxial strain has two effects: it decreases the fermion velocity; it increases the velocity anisotropy. After solving the Dyson-Schwinger gap equation, we show that dynamical gap generation is promoted by the former effect, but is suppressed by the latter one. For suspended graphene, we find that the systems undergoes an excitonic insulating transition when the strain is roughly 7.34$\%$. We also solve the gap equation in case the Dirac cone is tiled, which might be realized in the organic material $\alpha$-(BEDT-TTF)$_{2}$I$_{3}$, and find that the tilt of Dirac cone can suppress dynamical gap generation. It turns out that the geometry of the Dirac cone plays an important role in the formation of excitonic pairing.

Walking on the ladder: 125 GeV technidilaton, or Conformal Higgs

The walking technicolor based on the ladder Schwinger-Dyson gap equation is studied, with the scale-invariant coupling being an idealization of the Caswell-Banks-Zaks infrared fixed point in the "anti-Veneziano limit", such that $N_C \rightarrow \infty$ with $N_C \cdot \alpha(\mu^2)=$ fixed and $N_F/N_C=$ fixed ($\gg 1$), of the $SU(N_C)$ gauge theory with massless $N_F$ flavors near criticality. We show that the 125 GeV Higgs can be naturally identified with the technidilaton (TD) predicted in the walking technicolor, a pseudo Nambu-Goldstone (NG) boson of the spontaneous symmetry breaking of the approximate scale symmetry. Ladder calculations yield the TD mass $M_\phi$ from the trace anomaly as $M_\phi^2 F_\phi^2= -4 \langle \theta_\mu^\mu \rangle = - \frac{\beta(\alpha (\mu^2))}{\alpha(\mu^2)}\, \langle G_{\lambda \nu}^2(\mu^2)\rangle \simeq N_C N_F\frac{16}{\pi^4} m_F^4$, independently of the renormalization point $\mu$, where $m_F$ is the dynamical mass of the technifermion, and $F_\phi={\cal O} (\sqrt{N_F N_C}\, m_F)$ the TD decay constant. It reads $M_\phi^2\simeq (\frac{v_{\rm EW}}{2} \cdot \frac{5 v_{\rm EW}}{F_\phi})^2 \cdot [\frac{8}{N_F}\frac{4}{N_C}]$, ($v_{\rm EW}=246$ GeV), which implies $F_\phi\simeq 5 \,v_{\rm EW} $ for $M_\phi \simeq 125\, {\rm GeV}\simeq \frac{1}{2} v_{\rm EW}$ in the one-family model ($N_C=4, N_F=8$), in good agreement with the current LHC Higgs data. The result reflects a generic scaling $ M_\phi^2/v_{\rm EW}^2\sim M_\phi^2/F_\phi^2 \sim m_F^2 /F_\phi^2 \sim 1/(N_F N_C) \rightarrow 0 $ as a vanishing trace anomaly, namely the TD has a mass vanishing in the anti-Veneziano limit, similarly to $\eta^\prime$ meson as a pseudo-NG boson of the ordinary QCD with vanishing $U(1)_A$ anomaly in the Veneziano limit ($N_F/N_C \ll 1$).

INFLUENCE OF A UNIFORM MAGNETIC FIELD ON DYNAMICAL CHIRAL SYMMETRY BREAKING IN QED3

We study dynamical chiral symmetry breaking (DCSB) in an effective QED3 theory of d-wave high temperature cuprate superconductors under a uniform magnetic field. At zero temperature, the external magnetic field induces a mixed state by generating vortices in the condensate of charged holons. The growing magnetic field suppresses the superfluid density and thus reduces the gauge field mass which is opened via the Anderson–Higgs mechanism. By numerically solving the Dyson–Schwinger gap equation, we show that the massless fermions acquires a dynamical gap through DCSB mechanism when the magnetic field strength H is above a critical value H c and the fermion flavors N is below a critical value N c . Further, it is found that both N c and the dynamical fermion gap increase as the magnetic field H grows. It is expected that our result can be tested in phenomena in high temperature cuprate superconductors.

Dynamic gap generation in graphene under the long-range Coulomb interaction

Dynamic gap generation in graphene under the long-range Coulomb interaction is studied by the Dyson–Schwinger gap equation beyond the instantaneous approximation. Once the dependence of the dynamic gap on the energy has been considered, the critical interaction strength αc decreases to 0.542. If the renormalization of the fermion velocity is considered, αc will become αc = 1.02. This indicates that the dependence on the energy and the renormalization of the fermion velocity are both important for dynamic gap generation in graphene under long-range Coulomb interaction.

Dynamical fermion mass generation and exciton spectra in graphene

The Coulomb interaction between massless Dirac fermions may induce dynamical chiral symmetry breaking by forming excitonic pairs in clean graphene, leading to semimetal-insulator transition. If the Dirac fermions have zero bare mass, an exact continuous chiral symmetry is dynamically broken and thus there are massless Goldstone excitons. If the Dirac fermions have a small bare mass, an approximate continuous chiral symmetry is dynamically broken and the resultant Goldstone type excitons become massive, which is analogous to what happens in QCD. In this paper, after solving Dyson-Schwinger gap equation in the presence of a small bare fermion mass, we found a remarkable reduction of the critical Coulomb interaction strength for excitonic pair formation and a strong enhancement of dynamical fermion mass. We then calculate the masses of Goldstone type excitons using the SVZ sum rule method and operator product expansion technique developed in QCD and find that the exciton masses are much larger than bare fermion mass but smaller than dynamical fermion mass gap. We also study the spin susceptibilities and estimate the masses of non-Goldstone type excitons using the same tools.

DENSITY DEPENDENT STRONG COUPLING CONSTANT OF QCD DERIVED FROM COMPACT STAR DATA

Since 1996 there is major influx of X-ray and γ-ray data from binary stars, one or both of which are compact objects that are difficult to explain as neutron stars since they contain a mass M in too small a radius R. The suggestion has been put forward that these are strange quark stars (SS) explainable in a simple model with chiral symmetry restoration (CSR) for the quarks and the M, R and other properties like QPOs (quasi-periodic oscillations) in their X-ray power spectrum. It would be nice if this astrophysical data could shed some light on fundamental properties of quarks obeying QCD. One can relate the strong coupling constant of QCD, αs to the quark mass through the Dyson–Schwinger gap equation using the real time formalism of Dolan and Jackiw. This enables us to obtain the density dependence of αs from the simple CSR referred to above. This way fundamental physics, difficult to extract from other models like for example lattice QCD, can be constrained from present-day compact star data and may be put back to modeling the dense quark phase of early universe.

Gauge coupling instability and dynamical mass generation in N=1 three-dimensional supersymmetric QED

Using superfield Dyson-Schwinger equations, we compute the infrared dynamics of the semi-amputated full vertex, corresponding to the effective running gauge coupling, in N-flavour {\mathcal N}=1 supersymmetric QED(3). It is shown that the presence of a supersymmetry-preserving mass for the matter multiplet stabilizes the infrared gauge coupling against oscillations present in the massless case, and we therefore infer that the massive vacuum is thus selected at the level of the (quantum) effective action. We further demonstrate that such a mass can indeed be generated dynamically in a self-consistent way by appealing to the superfield Dyson-Schwinger gap equation for the full matter propagator.

Magnetization and dynamically induced finite densities in three-dimensional Chern-Simons QED

In (2 + 1)-dimensional QED with a Chem-Simons term, we show that spontaneous magnetization occurs in the context of finite density vacua, which are the lowest Landau levels fully or half occupied by fermions. Charge condensation is shown to appear so as to complement the fermion anti-fermion condensate, which breaks the flavor U(2 N) symmetry and causes fermion mass generation. The solutions to the Schwinger-Dyson gap equation show that the fermion self-energy contributes to the induction of a finite fermion density and/or fermion mass. The magnetization can be supported by charge condensation for theories with the Chem-Simons coefficient κ = Ne 22 gp, and κ = Ne 2/4 π, under the Gauss law constraint. For κ = Ne 2/4 π, both the magnetic field and the fermion mass are simultaneously generated in the half-filled ground state, which breaks the U(2 N) symmetry as well as the Lorentz symmetry.

CRITICAL BEHAVIOR IN (2+1)-DIMENSIONAL QED WITH A FOUR-FERMION INTERACTION

We analyze QED3 with a scalar-scalar type four-fermion interaction in the framework of 1/N expansion. We have derived the Dyson-Schwinger gap equation up to the next-to-leading order and determined the dynamically generated fermion mass as a function of the number of fermion flavors N and a scaled coupling θ. We observe that the critical behaviors are separated by the point [Formula: see text] and, contrary to the leading behaviors, there are mass generations even when θ ≥ θc with the critical flavor Nc. The critical line on the Nc−θc plane is presented. We also discuss the vacuum stability of our model, and confirm that our nontrivial solutions are energetically preferred to the perturbative solution Σ(p)=0.

FINITE-TEMPERATURE AND FINITE-DENSITY QED: THE SCHWINGER-DYSON EQUATION IN THE REAL-TIME FORMALISM

We derive, based on the real-time formalism (especially thermo-field-dynamics), the Schwinger-Dyson gap equation for the fermion propagator in QED and the four-fermion model at finite temperature and density. We discuss some advantages of the real-time formalism in solving the self-consistent gap equation, in comparison with the ordinary imaginary-time formalism. Once we specify the vertex function, we can write down the SD equation with only continuous variables without performing the discrete sum over Matsubara frequencies which cannot be performed in advance without further approximation in the imaginary-time formalism. By solving the SD equation obtained in this way, we find the chiral-symmetry-restoring transition at finite temperature and present the associated phase diagram of strong-coupling QED. In solving the SD equation, we consider two approximations: instantaneous-exchange and p0-independent ones. The former has a direct correspondence in the imaginary-time formalism; the latter is a new approximation beyond the former, since it is able to incorporate new thermal effects which have been overlooked in the ordinary imaginary-time solution. However, the two approximations are shown to give qualitatively the same results on the finite-temperature phase transition.

Techniodor

A theory with a slowly varying running coupling constant can have several scales of chiral symmetry breaking all separated from the scale of confinement. This is indicated by a numerical analysis of the Schwinger-Dyson gap equation. Such a theory is placed in the context of technicolor and Glashow's odor force.

Goldstone Bosons as Bound States in the Quark-Gluon Model

We examine the implications of a Nambu-Goldstone realization of chiral symmetry in the quark-gluon model. The context of this examination is that of a renormalizable, finite theory, so eigenvalue conditions are assumed to be satisfied. We discuss the solutions to the Schwinger-Dyson gap equation for the fermion self-energy $\ensuremath{\Sigma}({p}^{2})$ that exhibit spontaneous breaking of the vacuum symmetry. In the leading-order Bethe-Salpeter approximation the boundary conditions to the homogeneous, linear integral equations stipulate the vacuum symmetry. It is shown how the Goldstone bosons emerge as bound states, as suggested by Nambu and Jona-Lasinio. We also examine the Goldstone alternative in the Bethe-Salpeter equation for fermion-fermion scattering. Explicit symmetry breaking is introduced by additional Abelian vector gluons coupling to hypercharge and isospin besides baryon number. The eigenvalue condition for the fine-structure constant is consequently model-dependent but takes a simple form. We also consider the influence of explicit symmetry breaking on the ground-state mesons and indicate how the solutions to the eigenvalue problem regulate the structure of symmetry breaking.