Analytical insights into the interplay of momentum, multiplicity, and the speed of sound in heavy-ion collisions

We introduce a minimal model of ultracentral heavy-ion collisions to study the relation between the speed of sound of the produced plasma and the final particles' energy and multiplicity. We discuss how the particles' multiplicity <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>N</a:mi><a:mtext>tot</a:mtext></a:msub></a:math> and average energy <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:msub><b:mi>E</b:mi><b:mtext>tot</b:mtext></b:msub><b:mo>/</b:mo><b:msub><b:mi>N</b:mi><b:mtext>tot</b:mtext></b:msub></b:mrow></b:math> is related to the speed of sound <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi>c</c:mi><c:mi>s</c:mi></c:msub></c:math> by <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:msubsup><d:mi>c</d:mi><d:mi>s</d:mi><d:mn>2</d:mn></d:msubsup><d:mo>=</d:mo><d:mi>d</d:mi><d:mo form="prefix">ln</d:mo><d:mrow><d:mo>(</d:mo><d:msub><d:mi>E</d:mi><d:mtext>tot</d:mtext></d:msub><d:mo>/</d:mo><d:msub><d:mi>N</d:mi><d:mtext>tot</d:mtext></d:msub><d:mo>)</d:mo></d:mrow><d:mo>/</d:mo><d:mi>d</d:mi><d:mo form="prefix">ln</d:mo><d:msub><d:mi>N</d:mi><d:mtext>tot</d:mtext></d:msub></d:mrow></d:math> if the fluid is inviscid, its speed of sound is constant and all final particles can be measured. We show that finite rapidity cuts on the particles' multiplicity <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>N</g:mi></g:math> and energy <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mi>E</h:mi></h:math> introduce corrections between <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:msubsup><i:mi>c</i:mi><i:mi>s</i:mi><i:mn>2</i:mn></i:msubsup></i:math> and <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mi>d</j:mi><j:mo form="prefix">ln</j:mo><j:mo>(</j:mo><j:mi>E</j:mi><j:mo>/</j:mo><j:mi>N</j:mi><j:mo>)</j:mo><j:mo>/</j:mo><j:mi>d</j:mi><j:mo form="prefix">ln</j:mo><j:mi>N</j:mi></j:mrow></j:math> that depend on the system's lifetime. We study analytically these deviations with the Gubser hydrodynamic solution, finding that, for ultrarelativistic bosons, they scale as the ratio of the freeze-out temperature <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mi>T</m:mi><m:mi>FO</m:mi></m:msub></m:math> over the maximum initial temperature of the fluid <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:msub><n:mi>T</n:mi><n:mn>0</n:mn></n:msub></n:math>; the nonthermodynamic aspect of these corrections is highlighted through their dependence on the system's initial conditions. Published by the American Physical Society 2024