Energy hierarchies governing quarkonium dynamics in heavy ion collisions

In this paper, we critically examine hierarchies between the energy scales that determine quarkonium dynamics in the quark gluon plasma. A particularly important role is played by the ratio of the binding energy of species (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>E</a:mi><a:mi>b</a:mi></a:msub></a:math>) and the medium scales; temperature (<b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>T</b:mi></b:math>) and Debye mass (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi>m</c:mi><c:mi>D</c:mi></c:msub></c:math>). It is well known that if these ratios are much larger than 1 then the dominant process governing quarkonium evolution is dissociation by thermal gluons (gluodissociation). On the other hand, if the ratio <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:msub><d:mi>E</d:mi><d:mi>b</d:mi></d:msub><d:mo>/</d:mo><d:mi>T</d:mi></d:mrow></d:math> is much smaller than one then quarkonium dynamics is dominated by screening and Landau damping of the exchanged gluons. Here we show that over most of the evolution, the scale hierarchies do not fall in either limit and one needs to use the full structure of the gluonic spectral function to follow the dynamics of the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>Q</e:mi><e:mover accent="true"><e:mi>Q</e:mi><e:mo>¯</e:mo></e:mover></e:mrow></e:math> pair. This has a significant bearing when we follow the quantum dynamics of quarkonia in the medium. The inverse medium relaxation time is also <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mo>≈</g:mo><g:mi>T</g:mi></g:mrow></g:math> and if <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:msub><h:mi>E</h:mi><h:mi>b</h:mi></h:msub></h:math> is comparable (or larger) in magnitude to <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mi>T</i:mi></i:math>, the quantum evolution of <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mi>Q</j:mi><j:mover accent="true"><j:mi>Q</j:mi><j:mo>¯</j:mo></j:mover></j:mrow></j:math> is nonlocal in time within the Brownian approximation. Published by the American Physical Society 2024