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Higher Representations: Confinement and Large N

Francesco Sannino

TL;DR

This work analyzes confinement and deconfinement transitions in SU($N$) gauge theories with fermions in two-index symmetric or antisymmetric representations. By examining center symmetry and employing a Polyakov-loop effective theory, the author shows that even $N$ can yield a second-order transition in the 3D Ising universality class, while odd $N$ exhibits complete center breaking, with these patterns persisting at large $N$ and not mapping to super Yang–Mills. The paper extends the Svetitsky–Yaffe framework to these representations, studies free-energy-based degrees of freedom, and argues that the Appelquist–Cohen–Schmaltz inequality imposes no new constraints on these theories. It also discusses how adding matter in the same higher-dimensional representation affects the conclusions and highlights the confinement–chiral symmetry interplay as a fertile ground for insights applicable to QCD-like theories.

Abstract

We investigate the confining phase transition as function of temperature for theories with dynamical fermions in the two index symmetric and antisymmetric representation of the gauge group. By studying the properties of the center of the gauge group we predict for an even number of colors a confining phase transition, if second order, to be in the universality class of Ising in three dimensions. This is due to the fact that the center group symmetry does not break completely for an even number of colors. For an odd number of colors the center group symmetry breaks completely. This pattern remains unaltered at large number of colors. We claim that the confining/deconfining phase transition in these theories at large N is not mapped in the one of super Yang-Mills. We extend the Polyakov loop effective theory to describe the confining phase transition of the theories studied here for a generic number of colors. Our results are not modified when adding matter in the same higher dimensional representation of the gauge group. We comment on the interplay between confinement and chiral symmetry in these theories and suggest that they are ideal laboratories to shed light on this issue also for ordinary QCD. We compare the free energy as function of temperature for different theories. We find that the conjectured thermal inequality between the infrared and ultraviolet degrees of freedom computed using the free energy does not lead to new constraints on asymptotically free theories with fermions in higher dimensional representation of the gauge group.

Higher Representations: Confinement and Large N

TL;DR

This work analyzes confinement and deconfinement transitions in SU() gauge theories with fermions in two-index symmetric or antisymmetric representations. By examining center symmetry and employing a Polyakov-loop effective theory, the author shows that even can yield a second-order transition in the 3D Ising universality class, while odd exhibits complete center breaking, with these patterns persisting at large and not mapping to super Yang–Mills. The paper extends the Svetitsky–Yaffe framework to these representations, studies free-energy-based degrees of freedom, and argues that the Appelquist–Cohen–Schmaltz inequality imposes no new constraints on these theories. It also discusses how adding matter in the same higher-dimensional representation affects the conclusions and highlights the confinement–chiral symmetry interplay as a fertile ground for insights applicable to QCD-like theories.

Abstract

We investigate the confining phase transition as function of temperature for theories with dynamical fermions in the two index symmetric and antisymmetric representation of the gauge group. By studying the properties of the center of the gauge group we predict for an even number of colors a confining phase transition, if second order, to be in the universality class of Ising in three dimensions. This is due to the fact that the center group symmetry does not break completely for an even number of colors. For an odd number of colors the center group symmetry breaks completely. This pattern remains unaltered at large number of colors. We claim that the confining/deconfining phase transition in these theories at large N is not mapped in the one of super Yang-Mills. We extend the Polyakov loop effective theory to describe the confining phase transition of the theories studied here for a generic number of colors. Our results are not modified when adding matter in the same higher dimensional representation of the gauge group. We comment on the interplay between confinement and chiral symmetry in these theories and suggest that they are ideal laboratories to shed light on this issue also for ordinary QCD. We compare the free energy as function of temperature for different theories. We find that the conjectured thermal inequality between the infrared and ultraviolet degrees of freedom computed using the free energy does not lead to new constraints on asymptotically free theories with fermions in higher dimensional representation of the gauge group.

Paper Structure

This paper contains 10 sections, 31 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Thermal Wilson Line
  3. The effective theory
  4. Yang-Mills
  5. Fundamental, Adjoint and A/S representation for matter fields
  6. Free Energy and Thermal Degrees of Freedom
  7. Adding Matter
  8. Confinement versus Chiral Symmetry
  9. Thermal D.O.F.
  10. Conclusions and Outlook

Figures (1)

  • Figure 1: Degrees of freedom counted according to the free energy for different theories. The left panel represents the degree of freedom count as function of the temperature for a given number of colors. Note that at finite number of colors there are no light degrees of freedom. The right panel represent the large N count of the thermal degrees of freedom. It is clear that the $S/A$ theories become degenerate with SYM both at infinite $T$ and zero $T$. However the degree of freedom count for the fundamental representation disagrees at zero temperature at large N from the YM count since we have a Goldstone boson here.