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SU($N_c$) confinement and color $N_c$-ality

V. Tomas Mari Surkau, Urko Reinosa

Abstract

In a recent work, we have argued that the net quark number gained by a bath of quarks and gluons upon bringing an external static quark probe, while being equal to $1$ in the high temperature, deconfined phase, is equal to $0$ or $3$ in the low temperature, confined phase, depending on the value of the quark chemical potential. This establishes a clear-cut connection between the confinement of a medium as probed by order parameters such as the Polyakov loop, and the ability of that same medium to screen the quark probe into states with the same net quark content as mesons or baryons. In this work, we extend these findings to the SU($N_c$) color group, consider color probes in various (fundamental and non-fundamental) representations, and conjecture a rationale of how these results could be recovered from a purely thermodynamic argument that shortcuts the use of the Euclidean functional framework, while emphasizing the role of color $N_c$-ality as a proxy for center symmetry within the Minkowskian framework.

SU($N_c$) confinement and color $N_c$-ality

Abstract

In a recent work, we have argued that the net quark number gained by a bath of quarks and gluons upon bringing an external static quark probe, while being equal to in the high temperature, deconfined phase, is equal to or in the low temperature, confined phase, depending on the value of the quark chemical potential. This establishes a clear-cut connection between the confinement of a medium as probed by order parameters such as the Polyakov loop, and the ability of that same medium to screen the quark probe into states with the same net quark content as mesons or baryons. In this work, we extend these findings to the SU() color group, consider color probes in various (fundamental and non-fundamental) representations, and conjecture a rationale of how these results could be recovered from a purely thermodynamic argument that shortcuts the use of the Euclidean functional framework, while emphasizing the role of color -ality as a proxy for center symmetry within the Minkowskian framework.

Paper Structure

This paper contains 32 sections, 199 equations, 10 figures.

Table of Contents

  1. Introduction
  2. General set-up
  3. Fundamental representations
  4. Net quark number
  5. Heavy-quark regime
  6. Low-temperature limit
  7. Asymptotic behavior
  8. Interpretation
  9. Ordering
  10. High-temperature limit
  11. Non-fundamental representations
  12. $N_c$-ality of SU($N_c$) representations
  13. Low-temperature behavior
  14. Tree-level Polyakov loops
  15. Character identities
  16. ...and 17 more sections

Figures (10)

  • Figure 1: Young tableaux of the fundamental representations of SU($N_c$).
  • Figure 2: Net quark number responses $\Delta Q_{1,2}$ of a thermal bath upon bringing one- or two-quark probes in a fundamental representation of SU(3), at low temperature, and as functions of $\mu/M$.
  • Figure 3: Free energies $\Delta F_{1,2}$ gained by a thermal bath upon bringing one- or two-quark probes in a fundamental representation of SU(3) scaled by the quark mass $M$, at low temperature, and as functions of $\mu/M$.
  • Figure 4: Net quark numbers $1+\Delta Q_1$ and $2+\Delta Q_2$ gained by a thermal bath upon bringing one- or two-quark probes in a fundamental representation of SU(3), at low temperature, and as functions of $\mu/M$.
  • Figure 5: Top: Free energy $\Delta F_1$ gained by a thermal bath upon bringing a quark probe into the fundamental representation of SU(2) scaled by the quark mass $M$, at low temperature, and as a function of $\mu/M$. Bottom: Corresponding net quark number responses.
  • ...and 5 more figures