Implications of the electromagnetic scale anomaly to the QCD chiral phase transition in the smaller quark mass regime: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>p</mml:mi><

The decrease of the chiral pseudocritical temperature <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>T</a:mi><a:mrow><a:mi>p</a:mi><a:mi>c</a:mi></a:mrow></a:msub></a:math> with an applied strong magnetic field has been extensively investigated by various QCD low-energy effective models and lattice QCD at the physical point. We find that this decreasing feature may not hold in the case with a weak magnetic field and still depends on quark masses: when the quark masses get smaller, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>T</c:mi><c:mrow><c:mi>p</c:mi><c:mi>c</c:mi></c:mrow></c:msub></c:math> turns to increase with the weak magnetic field. This happens due to the significant electromagnetic scale anomaly contribution in the thermomagnetic medium. We demonstrate this salient feature by employing the Nambu-Jona-Lasinio model with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:mn>2</e:mn><e:mo>+</e:mo><e:mn>1</e:mn></e:mrow></e:math> quark flavors including the electromagnetic scale anomaly contribution. We observe that at <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:mo stretchy="false">(</g:mo><g:msub><g:mrow><g:mi>m</g:mi></g:mrow><g:mrow><g:mn>0</g:mn><g:mi>c</g:mi></g:mrow></g:msub><g:mo>,</g:mo><g:msub><g:mrow><g:mi>m</g:mi></g:mrow><g:mrow><g:mi>s</g:mi><g:mi>c</g:mi></g:mrow></g:msub><g:mo stretchy="false">)</g:mo><g:mo>≃</g:mo><g:mo stretchy="false">(</g:mo><g:mn>2</g:mn><g:mo>,</g:mo><g:mn>20</g:mn><g:mo stretchy="false">)</g:mo><g:mtext> </g:mtext><g:mtext> </g:mtext><g:mi>MeV</g:mi></g:mrow></g:math> for the isospin symmetric mass for up and down quarks, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>m</m:mi><m:mn>0</m:mn></m:msub></m:math>, and the strange quark mass, <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>m</o:mi><o:mi>s</o:mi></o:msub></o:math>, <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>T</q:mi><q:mrow><q:mi>p</q:mi><q:mi>c</q:mi></q:mrow></q:msub></q:math> decreases with the magnetic field if the quark masses exceed the critical values, and increases as the quark masses become smaller. Related cosmological implications, arising when the supercooled electroweak phase transition or dark QCD cosmological phase transition is considered along with a primordial magnetic field, are also briefly addressed.