First-order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:math> phase transition in two-flavor QCD at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml

We discuss the thermal <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>C</a:mi><a:mi>P</a:mi></a:math> phase transition in QCD at <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>θ</c:mi><c:mo>=</c:mo><c:mi>π</c:mi></c:math> under a weak magnetic field background, where the electromagnetic scale anomaly gets significant. To be explicit, we work on a two-flavor Nambu–Jona-Lasinio model at <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>θ</e:mi><e:mo>=</e:mo><e:mi>π</e:mi></e:math> in the mean field approximation, including the electromagnetic-scale anomaly term. We find that the thermal <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>C</g:mi><g:mi>P</g:mi></g:math> phase transition becomes first order and the strength of the first order gets more prominent as the magnetic field increases. The associated potential barrier is thermally created by the electromagnetic scale anomaly and gives rise to criticality due to the induced potential of a nonperturbative form <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mo>∼</i:mo><i:mfrac><i:mrow><i:mo stretchy="false">|</i:mo><i:mi>e</i:mi><i:mi>B</i:mi><i:msup><i:mo stretchy="false">|</i:mo><i:mn>3</i:mn></i:msup></i:mrow><i:msub><i:mi>f</i:mi><i:mi>π</i:mi></i:msub></i:mfrac><i:mfrac><i:mrow><i:mo stretchy="false">|</i:mo><i:mi>P</i:mi><i:mo stretchy="false">|</i:mo></i:mrow><i:mrow><i:msup><i:mi>P</i:mi><i:mn>2</i:mn></i:msup><i:mo stretchy="false">+</i:mo><i:msubsup><i:mi>m</i:mi><i:mn>0</i:mn><i:mn>2</i:mn></i:msubsup></i:mrow></i:mfrac></i:math>, where <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>e</p:mi><p:mi>B</p:mi></p:math> denotes the magnetic field strength; <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mi>P</r:mi></r:math> is the <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mi>C</t:mi><t:mi>P</t:mi></t:math>-order parameter, and <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:msub><v:mi>m</v:mi><v:mn>0</v:mn></v:msub></v:math> is the isospin-symmetric current-quark mass.