Revisiting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> <mml:math xmlns:mml="http://www.w3.org/1998/Mat

The effective potential of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>O</a:mi><a:mo stretchy="false">(</a:mo><a:mi>N</a:mi><a:mo stretchy="false">)</a:mo></a:math> model at large <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>N</e:mi></e:math> limit is reinvestigated with varying pion mass and temperature. For large pion masses and high temperatures, we find the phenomenologically favored vacuum, located on the upper branch of the double-branched effective potential for physical <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>m</g:mi><g:mi>π</g:mi></g:msub></g:math>, moves to the lower branch and becomes no longer a local minimum but a saddle point. The existence and running of the tachyon pole are also discussed. These phenomena indicate that the applicable energy range 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> model is more and more limited as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>m</m:mi><m:mi>π</m:mi></m:msub></m:math> becomes larger and the temperature goes higher. With the effective coupling constant defined from the effective potential, the possible correspondence between the two branches of the effective potential and the two phases of the theory (distinguished by positive or negative coupling) is verified even with nonzero explicit symmetry breaking and at finite temperature. Also, we generalize the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>N</o:mi><o:mo>/</o:mo><o:mi>D</o:mi></o:math> modified <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>O</q:mi><q:mo stretchy="false">(</q:mo><q:mi>N</q:mi><q:mo stretchy="false">)</q:mo></q:math> model to study the thermal trajectory of the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>σ</u:mi></u:math> pole with the cross-channel contributions considered and find the thermal <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>σ</w:mi></w:math> pole trajectory resembles its counterpart with varying pion masses at zero temperature. Published by the American Physical Society 2024