Hadron-quark hybrid model, modular transformation and Roberge-Weiss transition

TL;DR

This paper reformulates a hadron-quark hybrid model using modular transformations to analyze the Roberge-Weiss transition, revealing geometric and symmetry changes of a torus structure at different temperatures and chemical potentials.

Contribution

The study introduces a modular transformation approach to the hadron-quark hybrid model, providing new geometric insights into the Roberge-Weiss transition and the behavior of the associated torus.

Findings

Torus vanishes at the Roberge-Weiss transition point.

Discontinuity in the geometrical object appears for T > TRW.

Symmetry of the transformed moduli is broken above TRW.

Abstract

In the framework of modular transformations, we reformulate the recently proposed hadron-quark hybrid model when the imaginary baryonic chemical potential is introduced. As a result, we can consider the torus, which is characterized by the complex number densities of baryons (antibaryons) and quarks (antiquarks). We apply this model to analyze the Roberge-Weiss transition. It is shown that the torus vanishes at the baryonic chemical potential where the Roberge-Weiss transition appears because the number density of baryons (antibaryons) is not linearly independent of the number density of quarks (antiquarks). When the temperature T is lower than the Roberge-Weiss transition temperature TRW, the torus shrinks smoothly to the one-dimensional object at the Roberge-Weiss transition point, but the discontinuity does not appear. On the other hand, the discontinuity of the geometrical object appears when T>TRW. We also calculate the modulus of the torus and transform it into the fundamental region. The transformed moduli are symmetric below TRW, but the symmetry is broken above TRW.