Functional renormalization group study of thermodynamic geometry around the phase transition of quantum chromodynamics

We investigate the thermodynamic geometry of the quark-meson model at finite temperature, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>T</a:mi></a:math>, and quark number chemical potential, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>μ</c:mi></c:math>. We extend previous works by the inclusion of fluctuations exploiting the functional renormalization group approach. We use recent developments to recast the flow equation into the form of an advection-diffusion equation. We adopt the local potential approximation for the effective average action. We focus on the thermodynamic curvature, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>R</e:mi></e:math>, in the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mo stretchy="false">(</g:mo><g:mi>μ</g:mi><g:mo>,</g:mo><g:mi>T</g:mi><g:mo stretchy="false">)</g:mo></g:math> plane, in proximity of the chiral crossover, up to the critical point of the phase diagram. We find that the inclusion of fluctuations results in a smoother behavior of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>R</k:mi></k:math> near the chiral crossover. Moreover, for small <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>μ</m:mi></m:math>, <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>R</o:mi></o:math> remains negative, signaling the fact that bosonic fluctuations reduce the capability of the system to completely overcome the fermionic statistical repulsion of the quarks. We investigate in more detail the small <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>μ</q:mi></q:math> region by analyzing a system in which we artificially lower the pion mass, thus approaching the chiral limit in which the crossover is actually a second order phase transition. On the other hand, as <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>μ</s:mi></s:math> is increased and the critical point is approached, we find that <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>R</u:mi></u:math> is enhanced and a sign change occurs, in agreement with mean field studies. Hence, we completely support the picture that <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>R</w:mi></w:math> is sensitive to a crossover and a phase transition, and provides information about the effective behavior of the system at the phase transition. Published by the American Physical Society 2024