[Paper Review] Dyson-Schwinger Approach to Color-Superconductivity: Effects of Selfconsistent Gluon Dressing

Daniel Müller, Michael Buballa+1 more|arXiv (Cornell University)|Mar 9, 2016

High-Energy Particle Collisions Research4 citations

TL;DR

This paper extends the Dyson-Schwinger approach to dense QCD by incorporating self-consistent gluon dressing with dynamically generated quark masses and superconducting gaps, significantly enhancing critical temperatures for color-superconducting phases by a factor of ~2 compared to previous approximations using bare quarks. The improved truncation yields consistent Debye and Meissner masses, validating the framework against weak-coupling expectations.

ABSTRACT

The phase diagram of dense QCD at nonvanishing temperatures and large quark chemical potentials is studied with Dyson-Schwinger equations for 2+1 quark flavors, focusing on color-superconducting phases with 2SC and CFL-like pairing. The truncation scheme of our previous investigations is extended to include the dressing of gluons with selfconsistently determined quarks, i.e., taking into account the dynamical masses and superconducting gaps of the quarks in the gluon polarization. As a consequence the gluon screening is reduced, leading to an enhancement of the critical temperatures of the color-superconducting phases by about a factor of 2 as compared to the case where the gluons are dressed with bare quarks. We also calculate the Debye and Meissner masses of the gluons and show that they are consistent with weak-coupling results.

Motivation & Objective

• To improve the Dyson-Schwinger equation (DSE) framework for dense QCD by replacing the use of bare quarks in gluon polarization with dynamically dressed quarks.
• To address the inconsistency in prior DSE studies where vacuum observables like the pion decay constant $f_\pi$ could not be simultaneously matched with chiral critical temperatures.
• To investigate the impact of self-consistent quark dressing—including dynamical masses and superconducting gaps—on the phase structure of color-superconducting phases (2SC and CFL-like) at finite temperature and chemical potential.
• To compute and validate the Debye and Meissner masses of gluons in the weak-coupling limit as a consistency check for the truncation scheme.
• To provide a more reliable description of the QCD phase diagram at high baryon density, especially in the color-superconducting regime, by reducing reliance on the HTL-HDL approximation.

Proposed method

• The study employs a self-consistent truncation scheme in which the gluon polarization function is computed using quark propagators with dynamically generated masses and anomalous self-energies (Cooper pairs), rather than bare quarks.
• The quark loop in the gluon polarization is regularized via subtraction of vacuum contributions at zero temperature and chemical potential, ensuring numerical stability despite divergent cancellations.
• A color-diagonal representation of the gluon propagator is achieved by diagonalizing the polarization tensor using a unitary transformation, accounting for off-diagonal color components in 2SC and CFL-like phases.
• The quark-gluon vertex is treated consistently within the truncation, with renormalization procedures applied to maintain gauge consistency.
• The Debye and Meissner masses of the gluons are calculated both in the weak-coupling limit and at strong coupling, enabling cross-checks with known perturbative results.
• The full system of DSEs is solved numerically for $2+1$ quark flavors, including realistic strange quark mass, with the anomalous self-energy matrices parametrized in a color-flavor basis to describe 2SC and CFL-like pairing.

Experimental results

Research questions

• RQ1 How does the inclusion of self-consistently dressed quarks in the gluon polarization affect the critical temperatures of color-superconducting phases in dense QCD?
• RQ2 Can the improved truncation scheme simultaneously reproduce vacuum observables like $f_\pi$ and finite-density phase transition temperatures?
• RQ3 To what extent do the Debye and Meissner masses of gluons remain consistent with weak-coupling expectations in the self-consistent framework?
• RQ4 How does the self-consistent dressing of gluons alter the phase diagram of dense QCD, particularly the stability and extent of 2SC and CFL-like phases?
• RQ5 What is the quantitative impact of replacing bare quarks with dressed quarks in the gluon polarization loop on the critical temperature of color superconductivity?

Key findings

• The critical temperatures for color-superconducting phases are enhanced by a factor of approximately 2 when gluon polarization is computed with self-consistently dressed quarks instead of bare quarks.
• The inclusion of dynamical quark masses and superconducting gaps in the gluon polarization reduces gluon screening, leading to stronger effective interactions and higher critical temperatures.
• The calculated Debye and Meissner masses of the gluons are consistent with weak-coupling results, validating the truncation scheme and the treatment of the quark-gluon vertex.
• The model successfully reconciles vacuum observables—such as the pion decay constant $f_\pi$—with finite-temperature phase transition temperatures, resolving a key inconsistency present in earlier studies.
• The phase diagram shows that the CFL phase is favored at low temperatures and chemical potentials above 500–600 MeV, while the 2SC phase dominates at lower chemical potentials and extends into a narrow temperature band above the CFL phase.
• The numerical implementation is robust, with regularization via subtraction of vacuum contributions ensuring numerical stability without introducing significant truncation errors.