Revisiting O(N) σ model at unphysical pion masses and high temperatures

Roy-equation analysis on lattice data of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>π</a:mi><a:mi>π</a:mi></a:math> scattering phase shifts at <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>m</c:mi><c:mi>π</c:mi></c:msub><c:mo>=</c:mo><c:mn>391</c:mn><c:mtext> </c:mtext><c:mtext> </c:mtext><c:mi>MeV</c:mi></c:math> reveals that the lowest <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>f</e:mi><e:mn>0</e:mn></e:msub></e:math> meson becomes a bound state under this condition. In addition, there is a pair of complex poles below threshold generated by crossing symmetry [X.-H. Cao , ]. We use the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>N</g:mi><g:mo>/</g:mo><g:mi>D</g:mi></g:math> method to partially recover crossing symmetry of the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>O</i:mi><i:mo stretchy="false">(</i:mo><i:mi>N</i:mi><i:mo stretchy="false">)</i:mo></i:math> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>σ</m:mi></m:math> model amplitude at leading order of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mn>1</o:mn><o:mo>/</o:mo><o:mi>N</o:mi></o:math> expansion, and qualitatively reproduce the pole structure and pole trajectories with varying pion masses as revealed by Roy-equation analyses. The <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>σ</q:mi></q:math> pole trajectory with varying temperature is also discussed and found to be similar to its properties when varying <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>m</s:mi><s:mi>π</s:mi></s:msub></s:math>. As the temperature increases, the complex <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>σ</u:mi></u:math> poles firstly move from the second Riemann sheet to the real axis becoming two virtual state poles, and then one virtual state pole moves to the first sheet turning into a bound state pole and finally tends to the pion pole position at high temperature which is as expected from the chiral symmetry restoration. Our results provide further evidence that the lowest <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:msub><w:mi>f</w:mi><w:mn>0</w:mn></w:msub></w:math> state extracted from experiments and lattice data plays the role of the <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>σ</y:mi></y:math> meson in the spontaneous breaking of chiral symmetry. Finally, we also briefly discuss the problems of the effective potential in the situation when <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msub><ab:mi>m</ab:mi><ab:mi>π</ab:mi></ab:msub></ab:math> and the temperature become large. Published by the American Physical Society 2024