Comparison of chiral limit studies in a curvature mass versus on-shell renormalized quark-meson model using ChPT

Consistent chiral limit has been investigated in the curvature mass parametrized quark-meson (QM) model with the quark one-loop vacuum term (QMVT) employing the infrared regularized chiral perturbation theory predicted scaling of the pion, kaon decay constants <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msub><a:mrow><a:mi>f</a:mi></a:mrow><a:mrow><a:mi>π</a:mi></a:mrow></a:msub></a:mrow></a:math>, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:msub><c:mrow><c:mi>f</c:mi></c:mrow><c:mrow><c:mi>K</c:mi></c:mrow></c:msub></c:mrow></c:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:msubsup><e:mrow><e:mi>M</e:mi></e:mrow><e:mrow><e:mi>η</e:mi></e:mrow><e:mrow><e:mn>2</e:mn></e:mrow></e:msubsup><e:mo>=</e:mo><e:msubsup><e:mrow><e:mi>m</e:mi></e:mrow><e:mrow><e:mi>η</e:mi></e:mrow><e:mrow><e:mn>2</e:mn></e:mrow></e:msubsup><e:mo>+</e:mo><e:msubsup><e:mrow><e:mi>m</e:mi></e:mrow><e:mrow><e:msup><e:mrow><e:mi>η</e:mi></e:mrow><e:mrow><e:mo>′</e:mo></e:mrow></e:msup></e:mrow><e:mrow><e:mn>2</e:mn></e:mrow></e:msubsup></e:mrow></e:math> when the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>π</g:mi></g:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>K</i:mi></i:math> meson masses are reduced as one moves away from the physical point in the Columbia plot. Comparing the QMVT model Columbia plots with the corresponding Columbia plots computed in the very recent work of Tiwari [], using the on-shell renormalized QM (RQM) model and the earlier work of Resch [] using functional renormalization group techniques in the extended mean field approximation of the QM (e-MFA:QM-FRG) model, it has been estimated how the first, second, and crossover chiral transition regions in the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>m</k:mi><k:mi>π</k:mi></k:msub><k:mo>−</k:mo><k:msub><k:mi>m</k:mi><k:mi>K</k:mi></k:msub></k:math>(<m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>m</m:mi><m:mrow><m:mi>u</m:mi><m:mi>d</m:mi></m:mrow></m:msub><m:mo>−</m:mo><m:msub><m:mi>m</m:mi><m:mi>s</m:mi></m:msub></m:math>) and the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>μ</o:mi><o:mo>−</o:mo><o:msub><o:mi>m</o:mi><o:mi>K</o:mi></o:msub></o:math>(<q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>μ</q:mi><q:mo>−</q:mo><q:msub><q:mi>m</q:mi><q:mi>s</q:mi></q:msub></q:math>) planes get modified by different methods of implementing the quark one-loop vacuum fluctuations in the QM model. Since both the e-MFA:QM-FRG and the QMVT model use curvature meson masses to fix the parameters and the dimensional regularization of vacuum divergences are incorporated equivalently, the differences in their results can be attributed to different methods of approaching the chiral limit. The first order regions in the QMVT model while being much smaller than those in the RQM model have similar features but moderately smaller area than those in the e-MFA:QM-FRG study. In going to the chiral limit, the vacuum mass <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>m</s:mi><s:mi>σ</s:mi></s:msub><s:mo>=</s:mo><s:mn>530</s:mn><s:mtext> </s:mtext><s:mtext> </s:mtext><s:mi>MeV</s:mi></s:math> that is taken at the physical point does not change in the QMVT model whereas it decreases significantly in the e-MFA:QM-FRG study. Being different from the pole mass <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>m</u:mi><u:mi>σ</u:mi></u:msub><u:mo>=</u:mo><u:mn>530</u:mn><u:mtext> </u:mtext><u:mtext> </u:mtext><u:mi>MeV</u:mi></u:math>, the RQM model vacuum curvature mass <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:msub><w:mi>m</w:mi><w:mrow><w:mi>σ</w:mi><w:mo>,</w:mo><w:mi>c</w:mi></w:mrow></w:msub></w:math> increases toward the chiral limit from its minimum value at the physical point.