Columbia plot based on the symmetry-improved Cornwall-Jackiw-Tomboulis formalism in the linear sigma model

We study the Columbia plot for the chiral phase transition in the framework of a three-flavor linear sigma model based on the Cornwall-Jackiw-Tomboulis (CJT) formalism. The conventional CJT approach with the Hartree truncation suffers from artificial chiral breaking, leading to the violation of the Nambu-Goldstone theorem and the (anomalous) chiral Ward-Takahashi identities. We apply the symmetry-improved CJT formalism to resolve this issue. We observe a first-order phase transition and a tricritical point in the light-quark mass regime, which is relatively independent of the size of the sigma meson, in contrast to the conventional CJT approach. The tricritical point, found on the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>m</a:mi> <a:mi>s</a:mi> </a:msub> </a:math> axis, is at <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:msubsup> <c:mrow> <c:mi>m</c:mi> </c:mrow> <c:mrow> <c:mi>s</c:mi> </c:mrow> <c:mrow> <c:mi>tri</c:mi> </c:mrow> </c:msubsup> <c:mo>/</c:mo> <c:msubsup> <c:mrow> <c:mi>m</c:mi> </c:mrow> <c:mrow> <c:mi>s</c:mi> </c:mrow> <c:mrow> <c:mi>phys</c:mi> </c:mrow> </c:msubsup> <c:mo>=</c:mo> <c:mn>0.175</c:mn> </c:mrow> </c:math> with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msubsup> <e:mi>m</e:mi> <e:mi>s</e:mi> <e:mrow> <e:mi>phys</e:mi> </e:mrow> </e:msubsup> </e:math> being the physical strange quark mass in real-life QCD. The critical pion mass in the three-flavor symmetric limit, on the second-order boundary, is measured at <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mrow> <g:msub> <g:mrow> <g:mi>m</g:mi> </g:mrow> <g:mrow> <g:mi>π</g:mi> </g:mrow> </g:msub> <g:mo>∼</g:mo> <g:mn>52.4</g:mn> <g:mtext> </g:mtext> <g:mtext> </g:mtext> <g:mi>MeV</g:mi> </g:mrow> </g:math> , with the critical temperature <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mrow> <i:msub> <i:mrow> <i:mi>T</i:mi> </i:mrow> <i:mrow> <i:mi>c</i:mi> </i:mrow> </i:msub> <i:mo>∼</i:mo> <i:mn>51.7</i:mn> <i:mtext> </i:mtext> <i:mtext> </i:mtext> <i:mi>MeV</i:mi> </i:mrow> </i:math> .