Strangeness-correlations on the pseudocritical line in ( 2+1 )-flavor QCD

We present some lattice QCD results on first (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msubsup><a:mrow><a:mi>χ</a:mi></a:mrow><a:mrow><a:mn>1</a:mn></a:mrow><a:mrow><a:mi>i</a:mi></a:mrow></a:msubsup></a:mrow></a:math>) and second (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msubsup><c:mi>χ</c:mi><c:mn>2</c:mn><c:mi>i</c:mi></c:msubsup></c:math>) cumulants of and correlations (<e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:msubsup><e:mrow><e:mi>χ</e:mi></e:mrow><e:mrow><e:mn>11</e:mn></e:mrow><e:mrow><e:mi>i</e:mi><e:mi>j</e:mi></e:mrow></e:msubsup></e:mrow></e:math>) among net baryon-number (<g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>B</g:mi></g:math>), strangeness (<i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>S</i:mi></i:math>) and electric charge (<k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>Q</k:mi></k:math>) along the pseudocritical line [<m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>T</m:mi><m:mrow><m:mi>p</m:mi><m:mi>c</m:mi></m:mrow></m:msub><m:mo stretchy="false">(</m:mo><m:msub><m:mi>μ</m:mi><m:mi>B</m:mi></m:msub><m:mo stretchy="false">)</m:mo></m:math>] in the temperature (<q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>T</q:mi></q:math>)–baryon chemical potential (<s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>μ</s:mi><s:mi>B</s:mi></s:msub></s:math>) phase diagram of (<u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mrow><u:mn>2</u:mn><u:mo>+</u:mo><u:mn>1</u:mn></u:mrow></u:math>)-flavor QCD. We point out that violations of sum rules among second order cumulants, which hold in the isospin symmetric limit of vanishing electric charge chemical potential, are small along the <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:msub><w:mi>T</w:mi><w:mrow><w:mi>p</w:mi><w:mi>c</w:mi></w:mrow></w:msub><w:mo stretchy="false">(</w:mo><w:msub><w:mi>μ</w:mi><w:mi>B</w:mi></w:msub><w:mo stretchy="false">)</w:mo></w:math> for the entire range of <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msub><ab:mi>μ</ab:mi><ab:mi>B</ab:mi></ab:msub></ab:math> covered in the RHIC beam energy scan. For the strangeness neutral matter produced in heavy-ion collisions this leads to a close relation between <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msubsup><cb:mi>χ</cb:mi><cb:mn>11</cb:mn><cb:mrow><cb:mi>B</cb:mi><cb:mi>S</cb:mi></cb:mrow></cb:msubsup></cb:math> and <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:msubsup><eb:mi>χ</eb:mi><eb:mn>11</eb:mn><eb:mrow><eb:mi>Q</eb:mi><eb:mi>S</eb:mi></eb:mrow></eb:msubsup></eb:math>. We compare lattice QCD results for <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msubsup><gb:mi>χ</gb:mi><gb:mn>11</gb:mn><gb:mrow><gb:mi>B</gb:mi><gb:mi>S</gb:mi></gb:mrow></gb:msubsup><gb:mo>/</gb:mo><gb:msubsup><gb:mi>χ</gb:mi><gb:mn>2</gb:mn><gb:mi>S</gb:mi></gb:msubsup></gb:math> along the <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:msub><ib:mi>T</ib:mi><ib:mrow><ib:mi>p</ib:mi><ib:mi>c</ib:mi></ib:mrow></ib:msub><ib:mo stretchy="false">(</ib:mo><ib:msub><ib:mi>μ</ib:mi><ib:mi>B</ib:mi></ib:msub><ib:mo stretchy="false">)</ib:mo></ib:math> line with preliminary experimental measurements of <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:msubsup><mb:mi>χ</mb:mi><mb:mn>11</mb:mn><mb:mrow><mb:mi>B</mb:mi><mb:mi>S</mb:mi></mb:mrow></mb:msubsup><mb:mo>/</mb:mo><mb:msubsup><mb:mi>χ</mb:mi><mb:mn>2</mb:mn><mb:mi>S</mb:mi></mb:msubsup></mb:math> for collision energies <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:mrow><ob:mn>7.7</ob:mn><ob:mtext> </ob:mtext><ob:mtext> </ob:mtext><ob:mi>GeV</ob:mi><ob:mo>≤</ob:mo><ob:msqrt><ob:mrow><ob:msub><ob:mrow><ob:mi>s</ob:mi></ob:mrow><ob:mrow><ob:mi>N</ob:mi><ob:mi>N</ob:mi></ob:mrow></ob:msub></ob:mrow></ob:msqrt><ob:mo>≤</ob:mo><ob:mn>62.4</ob:mn><ob:mtext> </ob:mtext><ob:mtext> </ob:mtext><ob:mi>GeV</ob:mi></ob:mrow></ob:math>. While we find good agreements for <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:msqrt><qb:msub><qb:mi>s</qb:mi><qb:mrow><qb:mi>N</qb:mi><qb:mi>N</qb:mi></qb:mrow></qb:msub></qb:msqrt><qb:mo>≥</qb:mo><qb:mn>39</qb:mn><qb:mtext> </qb:mtext><qb:mtext> </qb:mtext><qb:mi>GeV</qb:mi></qb:math>, differences are sizeable at smaller values of <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:msqrt><sb:msub><sb:mi>s</sb:mi><sb:mrow><sb:mi>N</sb:mi><sb:mi>N</sb:mi></sb:mrow></sb:msub></sb:msqrt></sb:math>. Moreover, we compare lattice QCD results for the ratio of the strangeness (<ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:msub><ub:mi>μ</ub:mi><ub:mi>S</ub:mi></ub:msub></ub:math>) to baryon (<wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:msub><wb:mi>μ</wb:mi><wb:mi>B</wb:mi></wb:msub></wb:math>) chemical potentials, which define a strangeness neutral system with fixed electric charge to baryon number density, with experimental results obtained by the STAR collaboration for <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:msub><yb:mi>μ</yb:mi><yb:mi>S</yb:mi></yb:msub><yb:mo>/</yb:mo><yb:msub><yb:mi>μ</yb:mi><yb:mi>B</yb:mi></yb:msub></yb:math> using strange baryon yields on the freeze-out line. Finally, we determine the baryon chemical potential at the freeze-out (<ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"><ac:msubsup><ac:mi>μ</ac:mi><ac:mi>B</ac:mi><ac:mi>f</ac:mi></ac:msubsup></ac:math>) by comparing <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" display="inline"><cc:msubsup><cc:mi>χ</cc:mi><cc:mn>1</cc:mn><cc:mi>B</cc:mi></cc:msubsup><cc:mo>/</cc:mo><cc:msubsup><cc:mi>χ</cc:mi><cc:mn>2</cc:mn><cc:mi>B</cc:mi></cc:msubsup></cc:math> along the <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" display="inline"><ec:msub><ec:mi>T</ec:mi><ec:mrow><ec:mi>p</ec:mi><ec:mi>c</ec:mi></ec:mrow></ec:msub><ec:mo stretchy="false">(</ec:mo><ec:msub><ec:mi>μ</ec:mi><ec:mi>B</ec:mi></ec:msub><ec:mo stretchy="false">)</ec:mo></ec:math> with the experimentally measured net-proton cumulants <ic:math xmlns:ic="http://www.w3.org/1998/Math/MathML" display="inline"><ic:msubsup><ic:mi>χ</ic:mi><ic:mn>1</ic:mn><ic:mi>p</ic:mi></ic:msubsup><ic:mo>/</ic:mo><ic:msubsup><ic:mi>χ</ic:mi><ic:mn>2</ic:mn><ic:mi>p</ic:mi></ic:msubsup></ic:math>. We find that <kc:math xmlns:kc="http://www.w3.org/1998/Math/MathML" display="inline"><kc:mrow><kc:mo stretchy="false">{</kc:mo><kc:msubsup><kc:mrow><kc:mi>μ</kc:mi></kc:mrow><kc:mrow><kc:mi>B</kc:mi></kc:mrow><kc:mrow><kc:mi>f</kc:mi></kc:mrow></kc:msubsup><kc:mo>,</kc:mo><kc:msub><kc:mrow><kc:mi>T</kc:mi></kc:mrow><kc:mrow><kc:mi>p</kc:mi><kc:mi>c</kc:mi></kc:mrow></kc:msub><kc:mo stretchy="false">(</kc:mo><kc:msubsup><kc:mrow><kc:mi>μ</kc:mi></kc:mrow><kc:mrow><kc:mi>B</kc:mi></kc:mrow><kc:mrow><kc:mi>f</kc:mi></kc:mrow></kc:msubsup><kc:mo stretchy="false">)</kc:mo><kc:mo stretchy="false">}</kc:mo></kc:mrow></kc:math> are consistent with the freeze-out parameters of the statistical-model fits to experimentally measured hadron yields for <qc:math xmlns:qc="http://www.w3.org/1998/Math/MathML" display="inline"><qc:msqrt><qc:msub><qc:mi>s</qc:mi><qc:mrow><qc:mi>N</qc:mi><qc:mi>N</qc:mi></qc:mrow></qc:msub></qc:msqrt><qc:mo>≥</qc:mo><qc:mn>11.5</qc:mn><qc:mtext> </qc:mtext><qc:mtext> </qc:mtext><qc:mi>GeV</qc:mi></qc:math>.