Symmetries of spatial correlators of light and heavy mesons in high temperature lattice QCD

The spatial <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>z</a:mi></a:math> correlators of meson operators in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:msub><c:mrow><c:mi>N</c:mi></c:mrow><c:mrow><c:mi>f</c:mi></c:mrow></c:msub><c:mo>=</c:mo><c:mn>2</c:mn><c:mo>+</c:mo><c:mn>1</c:mn><c:mo>+</c:mo><c:mn>1</c:mn></c:mrow></c:math> lattice QCD with optimal domain-wall quarks at the physical point are studied for seven temperatures in the range of 190–1540 MeV. The meson operators include a complete set of Dirac bilinears (scalar, pseudoscalar, vector, axial vector, tensor vector, and axial-tensor vector), and each for six flavor combinations (<e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mover accent="true"><e:mi>u</e:mi><e:mo stretchy="false">¯</e:mo></e:mover><e:mi>d</e:mi></e:math>, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mover accent="true"><i:mi>u</i:mi><i:mo stretchy="false">¯</i:mo></i:mover><i:mi>s</i:mi></i:math>, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mover accent="true"><m:mi>s</m:mi><m:mo stretchy="false">¯</m:mo></m:mover><m:mi>s</m:mi></m:math>, <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mover accent="true"><q:mi>u</q:mi><q:mo stretchy="false">¯</q:mo></q:mover><q:mi>c</q:mi></q:math>, <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mover accent="true"><u:mi>s</u:mi><u:mo stretchy="false">¯</u:mo></u:mover><u:mi>c</u:mi></u:math>, and <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mover accent="true"><y:mi>c</y:mi><y:mo stretchy="false">¯</y:mo></y:mover><y:mi>c</y:mi></y:math>). In a recent paper [], I focused on the meson correlators of <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi>u</cb:mi></cb:math> and <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>d</eb:mi></eb:math> quarks, and discussed their implications for the effective restoration of <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mrow><gb:mi>S</gb:mi><gb:mi>U</gb:mi><gb:mo stretchy="false">(</gb:mo><gb:mn>2</gb:mn><gb:msub><gb:mrow><gb:mo stretchy="false">)</gb:mo></gb:mrow><gb:mrow><gb:mi>L</gb:mi></gb:mrow></gb:msub><gb:mo>×</gb:mo><gb:mi>S</gb:mi><gb:mi>U</gb:mi><gb:mo stretchy="false">(</gb:mo><gb:mn>2</gb:mn><gb:msub><gb:mrow><gb:mo stretchy="false">)</gb:mo></gb:mrow><gb:mrow><gb:mi>R</gb:mi></gb:mrow></gb:msub></gb:mrow></gb:math> and <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:mi>U</mb:mi><mb:mo stretchy="false">(</mb:mo><mb:mn>1</mb:mn><mb:msub><mb:mo stretchy="false">)</mb:mo><mb:mi>A</mb:mi></mb:msub></mb:math> chiral symmetries, as well as the emergence of approximate <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:mi>S</qb:mi><qb:mi>U</qb:mi><qb:mo stretchy="false">(</qb:mo><qb:mn>2</qb:mn><qb:msub><qb:mo stretchy="false">)</qb:mo><qb:mrow><qb:mi>C</qb:mi><qb:mi>S</qb:mi></qb:mrow></qb:msub></qb:math> chiral spin symmetry. In this work, we extend our study to meson correlators of six flavor contents, and first observe the hierarchical restoration of chiral symmetries in QCD, from <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:mrow><ub:mi>S</ub:mi><ub:mi>U</ub:mi><ub:mo stretchy="false">(</ub:mo><ub:mn>2</ub:mn><ub:msub><ub:mrow><ub:mo stretchy="false">)</ub:mo></ub:mrow><ub:mrow><ub:mi>L</ub:mi></ub:mrow></ub:msub><ub:mo>×</ub:mo><ub:mi>S</ub:mi><ub:mi>U</ub:mi><ub:mo stretchy="false">(</ub:mo><ub:mn>2</ub:mn><ub:msub><ub:mrow><ub:mo stretchy="false">)</ub:mo></ub:mrow><ub:mrow><ub:mi>R</ub:mi></ub:mrow></ub:msub><ub:mo>×</ub:mo><ub:mi>U</ub:mi><ub:mo stretchy="false">(</ub:mo><ub:mn>1</ub:mn><ub:msub><ub:mrow><ub:mo stretchy="false">)</ub:mo></ub:mrow><ub:mrow><ub:mi>A</ub:mi></ub:mrow></ub:msub></ub:mrow></ub:math> to <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" display="inline"><cc:mrow><cc:mi>S</cc:mi><cc:mi>U</cc:mi><cc:mo stretchy="false">(</cc:mo><cc:mn>3</cc:mn><cc:msub><cc:mrow><cc:mo stretchy="false">)</cc:mo></cc:mrow><cc:mrow><cc:mi>L</cc:mi></cc:mrow></cc:msub><cc:mo>×</cc:mo><cc:mi>S</cc:mi><cc:mi>U</cc:mi><cc:mo stretchy="false">(</cc:mo><cc:mn>3</cc:mn><cc:msub><cc:mrow><cc:mo stretchy="false">)</cc:mo></cc:mrow><cc:mrow><cc:mi>R</cc:mi></cc:mrow></cc:msub><cc:mo>×</cc:mo><cc:mi>U</cc:mi><cc:mo stretchy="false">(</cc:mo><cc:mn>1</cc:mn><cc:msub><cc:mrow><cc:mo stretchy="false">)</cc:mo></cc:mrow><cc:mrow><cc:mi>A</cc:mi></cc:mrow></cc:msub></cc:mrow></cc:math>, and to <kc:math xmlns:kc="http://www.w3.org/1998/Math/MathML" display="inline"><kc:mi>S</kc:mi><kc:mi>U</kc:mi><kc:mo stretchy="false">(</kc:mo><kc:mn>4</kc:mn><kc:msub><kc:mo stretchy="false">)</kc:mo><kc:mi>L</kc:mi></kc:msub><kc:mo>×</kc:mo><kc:mi>S</kc:mi><kc:mi>U</kc:mi><kc:mo stretchy="false">(</kc:mo><kc:mn>4</kc:mn><kc:msub><kc:mo stretchy="false">)</kc:mo><kc:mi>R</kc:mi></kc:msub><kc:mo>×</kc:mo><kc:mi>U</kc:mi><kc:mo stretchy="false">(</kc:mo><kc:mn>1</kc:mn><kc:msub><kc:mo stretchy="false">)</kc:mo><kc:mi>A</kc:mi></kc:msub></kc:math>, as the temperature is increased from 190 to 1540 MeV. Moreover, we compare the temperature windows for the emergence of the approximate <sc:math xmlns:sc="http://www.w3.org/1998/Math/MathML" display="inline"><sc:mi>S</sc:mi><sc:mi>U</sc:mi><sc:mo stretchy="false">(</sc:mo><sc:mn>2</sc:mn><sc:msub><sc:mo stretchy="false">)</sc:mo><sc:mrow><sc:mi>C</sc:mi><sc:mi>S</sc:mi></sc:mrow></sc:msub></sc:math> symmetry in light and heavy vector mesons, and find that the temperature windows are dominated by the <wc:math xmlns:wc="http://www.w3.org/1998/Math/MathML" display="inline"><wc:mo stretchy="false">(</wc:mo><wc:mover accent="true"><wc:mi>u</wc:mi><wc:mo stretchy="false">¯</wc:mo></wc:mover><wc:mi>c</wc:mi><wc:mo>,</wc:mo><wc:mover accent="true"><wc:mi>s</wc:mi><wc:mo stretchy="false">¯</wc:mo></wc:mover><wc:mi>c</wc:mi><wc:mo>,</wc:mo><wc:mover accent="true"><wc:mi>c</wc:mi><wc:mo stretchy="false">¯</wc:mo></wc:mover><wc:mi>c</wc:mi><wc:mo stretchy="false">)</wc:mo></wc:math> sectors. Published by the American Physical Society 2024