Curvature of the chiral phase transition line from the magnetic equation of state of (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>)-flavor QCD

Abstract

We analyze the dependence of the chiral phase transition temperature on baryon number and strangeness chemical potentials by calculating the leading order curvature coefficients in the light and strange quark flavor basis as well as in the conserved charge (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>B</a:mi></a:mrow></a:math>, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:mi>S</c:mi></c:mrow></c:math>) basis. Making use of scaling properties of the magnetic equation of state (MEoS) and including diagonal as well as off-diagonal contributions in the expansion of the energylike scaling variable that enters the parametrization of the MEoS, allows to explore the variation of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>T</e:mi><e:mi>c</e:mi></e:msub><e:mo stretchy="false">(</e:mo><e:msub><e:mi>μ</e:mi><e:mi>B</e:mi></e:msub><e:mo>,</e:mo><e:msub><e:mi>μ</e:mi><e:mi>S</e:mi></e:msub><e:mo stretchy="false">)</e:mo><e:mo>=</e:mo><e:msub><e:mi>T</e:mi><e:mi>c</e:mi></e:msub><e:mo stretchy="false">(</e:mo><e:mn>1</e:mn><e:mo>−</e:mo><e:mo stretchy="false">(</e:mo><e:msubsup><e:mi>κ</e:mi><e:mn>2</e:mn><e:mi>B</e:mi></e:msubsup><e:msubsup><e:mover accent="true"><e:mi>μ</e:mi><e:mo stretchy="false">^</e:mo></e:mover><e:mi>B</e:mi><e:mn>2</e:mn></e:msubsup><e:mo>+</e:mo><e:msubsup><e:mi>κ</e:mi><e:mn>2</e:mn><e:mi>S</e:mi></e:msubsup><e:msubsup><e:mover accent="true"><e:mi>μ</e:mi><e:mo stretchy="false">^</e:mo></e:mover><e:mi>S</e:mi><e:mn>2</e:mn></e:msubsup><e:mo>+</e:mo><e:mn>2</e:mn><e:msubsup><e:mi>κ</e:mi><e:mn>11</e:mn><e:mrow><e:mi>B</e:mi><e:mi>S</e:mi></e:mrow></e:msubsup><e:msub><e:mover accent="true"><e:mi>μ</e:mi><e:mo stretchy="false">^</e:mo></e:mover><e:mi>B</e:mi></e:msub><e:msub><e:mover accent="true"><e:mi>μ</e:mi><e:mo stretchy="false">^</e:mo></e:mover><e:mi>S</e:mi></e:msub><e:mo stretchy="false">)</e:mo><e:mo stretchy="false">)</e:mo></e:math> along different lines in the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mo stretchy="false">(</u:mo><u:msub><u:mi>μ</u:mi><u:mi>B</u:mi></u:msub><u:mo>,</u:mo><u:msub><u:mi>μ</u:mi><u:mi>S</u:mi></u:msub><u:mo stretchy="false">)</u:mo></u:math> plane. On lattices with fixed cutoff in units of temperature, <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>a</y:mi><y:mi>T</y:mi><y:mo>=</y:mo><y:mn>1</y:mn><y:mo>/</y:mo><y:mn>8</y:mn></y:math>, we find <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msubsup><ab:mi>κ</ab:mi><ab:mn>2</ab:mn><ab:mi>B</ab:mi></ab:msubsup><ab:mo>=</ab:mo><ab:mn>0.015</ab:mn><ab:mo stretchy="false">(</ab:mo><ab:mn>1</ab:mn><ab:mo stretchy="false">)</ab:mo></ab:math>, <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:msubsup><eb:mi>κ</eb:mi><eb:mn>2</eb:mn><eb:mi>S</eb:mi></eb:msubsup><eb:mo>=</eb:mo><eb:mn>0.0124</eb:mn><eb:mo stretchy="false">(</eb:mo><eb:mn>5</eb:mn><eb:mo stretchy="false">)</eb:mo></eb:math> and <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:msubsup><ib:mi>κ</ib:mi><ib:mn>11</ib:mn><ib:mrow><ib:mi>B</ib:mi><ib:mi>S</ib:mi></ib:mrow></ib:msubsup><ib:mo>=</ib:mo><ib:mo>−</ib:mo><ib:mn>0.0050</ib:mn><ib:mo stretchy="false">(</ib:mo><ib:mn>7</ib:mn><ib:mo stretchy="false">)</ib:mo></ib:math>. We show that the chemical potential dependence along the line of vanishing strangeness chemical potential is about 10% larger than along the strangeness neutral line. The latter differs only by about 3% from the curvature on a line of vanishing strange quark chemical potential, <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:msub><mb:mi>μ</mb:mi><mb:mi>s</mb:mi></mb:msub><mb:mo>=</mb:mo><mb:mn>0</mb:mn></mb:math>. We also show that close to the chiral limit the strange quark mass contributes like an energylike variable in scaling relations for pseudocritical temperatures. The chiral phase transition temperature decreases with decreasing strange quark mass, <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:msub><ob:mi>T</ob:mi><ob:mi>c</ob:mi></ob:msub><ob:mo stretchy="false">(</ob:mo><ob:msub><ob:mi>m</ob:mi><ob:mi>s</ob:mi></ob:msub><ob:mo stretchy="false">)</ob:mo><ob:mo>=</ob:mo><ob:msub><ob:mi>T</ob:mi><ob:mi>c</ob:mi></ob:msub><ob:mo stretchy="false">(</ob:mo><ob:msubsup><ob:mi>m</ob:mi><ob:mi>s</ob:mi><ob:mrow><ob:mi>phy</ob:mi></ob:mrow></ob:msubsup><ob:mo stretchy="false">)</ob:mo><ob:mo stretchy="false">(</ob:mo><ob:mn>1</ob:mn><ob:mo>−</ob:mo><ob:mn>0.097</ob:mn><ob:mo stretchy="false">(</ob:mo><ob:mn>2</ob:mn><ob:mo stretchy="false">)</ob:mo><ob:mo stretchy="false">(</ob:mo><ob:msub><ob:mi>m</ob:mi><ob:mi>s</ob:mi></ob:msub><ob:mo>−</ob:mo><ob:msubsup><ob:mi>m</ob:mi><ob:mi>s</ob:mi><ob:mrow><ob:mtext>phys</ob:mtext></ob:mrow></ob:msubsup><ob:mo stretchy="false">)</ob:mo><ob:mo>/</ob:mo><ob:msubsup><ob:mi>m</ob:mi><ob:mi>s</ob:mi><ob:mrow><ob:mi>phy</ob:mi></ob:mrow></ob:msubsup><ob:mo>+</ob:mo><ob:mi mathvariant="script">O</ob:mi><ob:mo stretchy="false">(</ob:mo><ob:mo stretchy="false">(</ob:mo><ob:mi mathvariant="normal">Δ</ob:mi><ob:msub><ob:mi>m</ob:mi><ob:mi>s</ob:mi></ob:msub><ob:msup><ob:mo stretchy="false">)</ob:mo><ob:mn>2</ob:mn></ob:msup><ob:mo stretchy="false">)</ob:mo></ob:math>. Published by the American Physical Society 2024

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