On the origin of the $$N_c^1$$ scaling in the confined but chirally symmetric phase at high T

There is lattice evidence that the QCD matter above the chiral restoration temperature <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>T</mml:mi> <mml:mrow><mml:mi>ch</mml:mi></mml:mrow> </mml:msub> </mml:math> and below the deconfinement temperature <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>T</mml:mi> <mml:mi>d</mml:mi></mml:msub> </mml:math> , called stringy fluid, is characterized by approximate chiral spin symmetry, which is a symmetry of confinement in QCD with light quarks. The energy density, pressure and entropy density in the stringy fluid scale as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>N</mml:mi> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn></mml:msubsup> </mml:math> , which is in contrast to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>N</mml:mi> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn></mml:msubsup> </mml:math> scaling in the hadron gas and to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>N</mml:mi> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn></mml:msubsup> </mml:math> scaling in the quark-gluon plasma. Here we clarify the origin of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>N</mml:mi> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn></mml:msubsup> </mml:math> scaling. We employ a solvable field-theoretical large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>N</mml:mi> <mml:mi>c</mml:mi></mml:msub> </mml:math> chirally symmetric and confining model. In vacuum the confining potential induces a spontaneous breaking of chiral symmetry. The mesons are spatially localized states of quarks and antiquarks. Still in the confining regime the system undergoes the chiral restoration phase transition at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>T</mml:mi> <mml:mrow><mml:mi>ch</mml:mi></mml:mrow> </mml:msub> </mml:math> because of Paili blocking of the quark levels required for the existence of the quark condensate, by the thermal excitation of quarks and antiquarks. The same Paili blocking leads to a delocalization of the color singlet low-spin meson-like states that become infinitely large in the chiral limit. Consequently the stringy fluid represents a very dense medium of the overlapping huge color-singlet low-spin quark-antiquark systems. The Bethe-Salpeter equation that determines the rest-frame excitation energies of the color-singlet quark-antiquark system is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>N</mml:mi> <mml:mi>c</mml:mi></mml:msub> </mml:math> -independent both in vacuum and in the medium in the confining regime. The excitation energy of the quark-antiquark color-singlet systems scales as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>N</mml:mi> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn></mml:msubsup> </mml:math> , i.e. as meson mass in vacuum. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>N</mml:mi> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn></mml:msubsup> </mml:math> scaling of the energy density in the stringy fluid is provided by the fluctuations of the color-singlet quark-antiquark systems.