Nucleon and singly heavy baryons from the QCD instanton vacuum

We construct an effective chiral theory for the nucleon, based on the low-energy effect quantum chromodynamics (QCD) partition function from the QCD instanton vacuum. We fully consider the momentum-dependent dynamical quark mass whose value at the zero virtuality of the quark is determined by the gap equation from the instanton vacuum, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>M</a:mi><a:mn>0</a:mn></a:msub><a:mo>=</a:mo><a:mn>359</a:mn><a:mtext> </a:mtext><a:mtext> </a:mtext><a:mi>MeV</a:mi></a:math>. The nucleon emerges as a state of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>N</c:mi><c:mi>c</c:mi></c:msub></c:math> valence quarks bound by the pion mean field, which was created self-consistently by the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>N</e:mi><e:mi>c</e:mi></e:msub></e:math> valence quarks. In the large Euclidean time, the classical nucleon mass is evaluated by minimizing the sum of the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>N</g:mi><g:mi>c</g:mi></g:msub></g:math> discrete-level energies and the Dirac-continuum energy: <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>M</i:mi><i:mi>cl</i:mi></i:msub><i:mo>=</i:mo><i:mn>1.2680</i:mn><i:mtext> </i:mtext><i:mtext> </i:mtext><i:mi>GeV</i:mi></i:math>. The pion mean-field solution turns out broader than the local chiral quark-soliton model. The zero-mode quantization furnishes the nucleon with proper quantum numbers such as the spin and isospin. We compute the moment of inertia <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>I</k:mi><k:mo>=</k:mo><k:mn>1.3853</k:mn><k:mtext> </k:mtext><k:mtext> </k:mtext><k:mi>fm</k:mi></k:math> by using the self-consistent mean-field solution, which yields the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi mathvariant="normal">Δ</m:mi><m:mo>−</m:mo><m:mi>N</m:mi></m:math> mass splitting <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:msub><p:mi>M</p:mi><p:mrow><p:mi mathvariant="normal">Δ</p:mi><p:mo>−</p:mo><p:mi>N</p:mi></p:mrow></p:msub><p:mo>=</p:mo><p:mn>213.67</p:mn><p:mtext> </p:mtext><p:mtext> </p:mtext><p:mi>MeV</p:mi></p:math>. In the same manner, singly heavy baryons can be described as a bound state of the <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>N</s:mi><s:mi>c</s:mi></s:msub><s:mo>−</s:mo><s:mn>1</s:mn></s:math> valence quarks with the corresponding pion mean field, with the heavy quark regarded as a static color source. The mass splitting of the singly heavy baryons is obtained to be <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>M</u:mi><u:mrow><u:msub><u:mi mathvariant="normal">Σ</u:mi><u:mi>Q</u:mi></u:msub><u:mo>−</u:mo><u:msub><u:mi mathvariant="normal">Λ</u:mi><u:mi>Q</u:mi></u:msub></u:mrow></u:msub><u:mo>=</u:mo><u:mn>206.20</u:mn><u:mtext> </u:mtext><u:mtext> </u:mtext><u:mi>MeV</u:mi></u:math>, which are in good agreement with the experimental data. The effective chiral theory developed in the present work will provide a solid theoretical framework to investigate gluonic observables of both the light and singly heavy baryons.