Assessing the reconstruction of the critical line in the QCD phase diagram from imaginary to real chemical potential

We test a technique adopted in the lattice simulations framework, to reconstruct the chiral-phase boundary at real chemical potential, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>μ</a:mi></a:mrow></a:math>, via extrapolation from imaginary <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>μ</c:mi></c:math>. We use a low-energy effective model, the quark-meson model, both in the mean-field approximation and within the functional renormalization group, the latter in the local potential approximation. The model provides results both for real and imaginary values of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>μ</e:mi></e:math>; thus a direct comparison can be performed between the prediction of the model for real values of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>μ</g:mi></g:math> and the ones obtained via extrapolation from the results at imaginary <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>μ</i:mi></i:math>. We compute an effective convergence radius for the reconstruction technique, <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>μ</k:mi><k:mrow><k:mi>conv</k:mi></k:mrow></k:msub></k:math>, and find <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>μ</m:mi><m:mrow><m:mi>conv</m:mi></m:mrow></m:msub><m:mo>≈</m:mo><m:mn>263</m:mn><m:mtext> </m:mtext><m:mtext> </m:mtext><m:mi>MeV</m:mi></m:math>. This value sustains the validity of the reconstruction technique also for finite and moderate values of the chemical potential. On the other hand, within our model, <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>μ</o:mi><o:mrow><o:mi>conv</o:mi></o:mrow></o:msub></o:math> is smaller then the value of <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>μ</q:mi></q:math> where we find the actual critical endpoint. Near this point of the phase diagram, we find a discrepancy between the actual phase boundary and the one obtained via extrapolation of <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mo>≈</s:mo><s:mn>100</s:mn><s:mo>%</s:mo></s:math>. Therefore, our results show that the location of the critical endpoint obtained via reconstruction from imaginary <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>μ</u:mi></u:math> might be considered with due caution.