Probing saturonlike limits in QCD systems

High-occupancy quantum chromodynamics (QCD) matter enters a saturated regime when its entropy or occupancy approaches the unitarity bound <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mo>∼</a:mo> <a:mn>1</a:mn> <a:mo>/</a:mo> <a:mi>α</a:mi> </a:math> , the “saturon” criterion. We test this criterion for protons and nuclei at small <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>x</c:mi> </c:math> using analytic and numerical solutions of the BK equation. From these solutions we construct the gluon occupancy <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>N</e:mi> <e:mi>g</e:mi> </e:msub> <e:mo stretchy="false">(</e:mo> <e:mi>x</e:mi> <e:mo stretchy="false">)</e:mo> </e:math> and a thermodynamic entropy <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>S</i:mi> <i:mo stretchy="false">(</i:mo> <i:mi>x</i:mi> <i:mo stretchy="false">)</i:mo> </i:math> via an Unruh-like temperature <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>T</m:mi> <m:mo>=</m:mo> <m:msub> <m:mi>Q</m:mi> <m:mi>s</m:mi> </m:msub> <m:mo>/</m:mo> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mi>π</m:mi> <m:mo stretchy="false">)</m:mo> </m:math> and an emergent gluon mass <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msub> <q:mi>M</q:mi> <q:mi>g</q:mi> </q:msub> <q:mo>∼</q:mo> <q:msub> <q:mi>Q</q:mi> <q:mi>s</q:mi> </q:msub> </q:math> . For protons, both <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msub> <s:mi>N</s:mi> <s:mi>g</s:mi> </s:msub> </s:math> and <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>S</u:mi> </u:math> rise toward small <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:mi>x</w:mi> </w:math> yet stay below <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mn>1</y:mn> <y:mo>/</y:mo> <y:msub> <y:mi>α</y:mi> <y:mi>s</y:mi> </y:msub> </y:math> in our baseline setup. For nuclei, by contrast, the nuclear entropy <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:msub> <ab:mi>S</ab:mi> <ab:mi>A</ab:mi> </ab:msub> </ab:math> attains the <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mn>1</cb:mn> <cb:mo>/</cb:mo> <cb:msub> <cb:mi>α</cb:mi> <cb:mi>s</cb:mi> </cb:msub> </cb:math> benchmark in a small- <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mi>x</eb:mi> </eb:math> window where the proton does not. This singles out nuclei as the natural environment to search for saturon-like behavior and motivates precision small- <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:mi>x</gb:mi> </gb:math> measurements and high-occupancy <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"> <ib:mi>p</ib:mi> <ib:mi>A</ib:mi> </ib:math> and <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"> <kb:mi>A</kb:mi> <kb:mi>A</kb:mi> </kb:math> collisions.