Isospin Symmetry Breaking in Non-Perturbative QCD

At finite isospin chemical potential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>μ</mi><mi>I</mi></msub></semantics></math></inline-formula>, the tension between measured decays and partial branching ratios of neutral and charged bosons as functions of dimuon mass squared and the Standard Model (SM) isospin asymmetry can be analyzed in nonperturbative QCD-effective models, for instance, the Polyakov linear sigma-model. With almost first-principle derivation of the explicit isospin symmetry breaking, namely, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>σ</mi><mo stretchy="false">¯</mo></mover><mn>3</mn></msub><mo>=</mo><msub><mi>f</mi><msup><mi>K</mi><mo>±</mo></msup></msub><mo>−</mo><msub><mi>f</mi><msup><mi>K</mi><mn>0</mn></msup></msub></mrow></semantics></math></inline-formula> the isospin sigma field, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>h</mi><mn>3</mn></msub><mo>=</mo><msubsup><mi>m</mi><msub><mi>a</mi><mn>0</mn></msub><mn>2</mn></msubsup><mfenced separators="" open="(" close=")"><msub><mi>f</mi><msup><mi>K</mi><mo>±</mo></msup></msub><mo>−</mo><msub><mi>f</mi><msup><mi>K</mi><mn>0</mn></msup></msub></mfenced></mrow></semantics></math></inline-formula> the third generator of the matrix of the explicit symmetry breaking <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><msub><mi>T</mi><mi>a</mi></msub><msub><mi>h</mi><mi>a</mi></msub></mrow></semantics></math></inline-formula>. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><msup><mi>K</mi><mo>±</mo></msup></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><msup><mi>K</mi><mn>0</mn></msup></msub></semantics></math></inline-formula> are decay constants of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>K</mi><mo>±</mo></msup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>K</mi><mn>0</mn></msup></semantics></math></inline-formula>, respectively. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><msub><mi>a</mi><mn>0</mn></msub></msub></semantics></math></inline-formula> is the mass of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>a</mi><mn>0</mn></msub></semantics></math></inline-formula> meson. Accordingly, the QCD phase structure could be extended to finite <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>μ</mi><mi>I</mi></msub></semantics></math></inline-formula>. With the thermal and density dependence of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>a</mi><mn>0</mn></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><msup><mi>K</mi><mo>±</mo></msup></msub></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><msup><mi>K</mi><mn>0</mn></msup></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>σ</mi><mo stretchy="false">¯</mo></mover><mn>3</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>h</mi><mn>3</mn></msub></semantics></math></inline-formula> are accordingly expressed in dependence on the temperatures and the chemical potentials. We find that the resulting critical chiral temperatures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>χ</mi></msub></semantics></math></inline-formula> decrease with the increase in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>μ</mi><mi>B</mi></msub></semantics></math></inline-formula> and/or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>μ</mi><mi>I</mi></msub></semantics></math></inline-formula>. We conclude that the (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi>χ</mi></msub><mo>−</mo><msub><mi>μ</mi><mi>I</mi></msub></mrow></semantics></math></inline-formula>) boundary has almost the same structure as that of the (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi>χ</mi></msub><mo>−</mo><msub><mi>μ</mi><mi>B</mi></msub></mrow></semantics></math></inline-formula>) plane.