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Register a new userStudy of the conformal hyperscaling relation through the Schwinger-Dyson equation
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We study corrections to the conformal hyperscaling relation in the conformal window of the large Nf QCD by using the ladder Schwinger-Dyson (SD) equation as a concrete dynamical model. From the analytical expression of the solution of the ladder SD equation, we identify the form of the leading mass correction to the hyperscaling relation. We find that the anomalous dimension, when identified through the hyperscaling relation neglecting these corrections, yields a value substantially lower than the one at the fixed point gamma_m^* for large mass region. We further study finite-volume effects on the hyperscaling relation, based on the ladder SD equation in a finite space-time with the periodic boundary condition. We find that the finite-volume corrections on the hyperscaling relation are negligible compared with the mass correction. The anomalous dimension, when identified through the finite-size hyperscaling relation neglecting the mass corrections as is often done in the lattice analyses, yields almost the same value as that in the case of the infinite space-time neglecting the mass correction, i.e., a substantially lower value than gamma_m^* for large mass. We also apply the finite-volume SD equation to the chiral-symmetry-breaking phase and find that when the theory is close to the critical point such that the dynamically generated mass is much smaller than the explicit breaking mass, the finite-size hyperscaling relation is still operative. We also suggest a concrete form of the modification of the finite-size hyperscaling relation by including the mass correction, which may be useful to analyze the lattice data.
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Motivated by recent progress on many flavor QCD on a lattice, we investigate conformal/walking dynamics by using Schwinger-Dyson (SD) equation within an improved ladder approximation for two-loop running coupling. By numerically solving the SD equation, we obtain a pole mass $m_{p}$, pion decay constant $f_{pi}$, and investigate the chiral symmetry breaking and mass anomalous dimension $gamma_{m}$ in the presence of IR cutoffs $Lambda_{mathrm{IR}}$. We find that the chiral symmetry breaking is suppressed if IR cutoff $Lambda_{mathrm{IR}}$ becomes larger than the critical value near the dynamical mass ($Lambda_{mathrm{IR}}$ $simeq m_{D}$) In the conformal phase the $gamma_{m}$ is strongly suppressed by IR cutoffs for $Lambda _{mathrm{IR}}$ $simeq m_{p}$. We, then, obtain finite size hyperscaling (FSS) relation by adapting a linearized approximation for the SD equation, and examine the $gamma_{m}$ The results offer valuable insight and suggestion for analyses in lattice gauge theories.
Any practical application of the Schwinger-Dyson equations to the study of $n$-point Greens functions of a field theory requires truncations, the best known being finite order perturbation theory. Strong coupling studies require a different approach. In the case of QED, gauge covariance is a powerful constraint. By using a spectral representation for the massive fermion propagator in QED, we are able to show that the constraints imposed by the Landau-Khalatnikov-Fradkin transformations are linear operations on the spectral densities. Here we formally define these group operations and show with a couple of examples how in practice they provide a straightforward way to test the gauge covariance of any viable truncation of the Schwinger-Dyson equation for the fermion 2-point function.
In view of the properties of mesons in hot strongly interacting matter the properties of the solutions of the truncated Dyson-Schwinger equation for the quark propagator at finite temperatures within the rainbow-ladder approximation are analysed in some detail. In Euclidean space within the Matsubara imaginary time formalism the quark propagator is not longer a O(4) symmetric function and possesses a discrete spectra of the fourth component of the momentum. This makes the treatment of the Dyson-Schwinger and Bethe-Salpeter equations conceptually different from the vacuum and technically much more involved. The question whether the interaction kernel known from vacuum calculations can be applied at finite temperatures remains still open. We find that, at low temperatures, the model interaction with vacuum parameters provides a reasonable description of the quark propagator, while at temperatures higher than a certain critical value $T_c$ the interaction requires stringent modifications. The general properties of the quark propagator at finite temperatures can be inferred from lattice QCD calculations. We argue that, to achieve a reasonable agreement of the model calculations with that from lattice QCD, the kernel is to be modified in such a way as to screen the infra-red part of the interaction at temperatures larger than $T_c$. For this, we analyse the solutions of the truncated Dyson-Schwinger equation with existing interaction kernels in a large temperature range with particular attention on high temperatures in order to find hints to an adequate temperature dependence of the interaction kernel to be further implemented in to the Bethe-Salpeter equation for mesons. This will allow to investigate the possible in medium modifications of the meson properties as well as the conditions of quark deconfinement in hot matter.
We derive the Dyson-Schwinger equation of a link variable in SU(n) lattice gauge theory in minimal Landau gauge and confront it with Monte-Carlo data for the different terms. Preliminary results for the lattice analog of the Kugo-Ojima confinement criterion is also shown.
The gluon propagator plays a central role in determining the dynamics of QCD. In this work we demonstrate for BRST quantised QCD that the Dyson-Schwinger equation imposes significant analytic constraints on the structure of this propagator. In particular, we find that these constraints control the appearance of massless components in the gluon spectral density.
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