Temperature derivative divergence of the electric conductivity and thermal photon emission rate at the critical end point from holography

The thermal photon emission rate <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mfrac> <a:mrow> <a:mi>d</a:mi> <a:mi mathvariant="normal">Γ</a:mi> </a:mrow> <a:mrow> <a:mi>d</a:mi> <a:mi>k</a:mi> </a:mrow> </a:mfrac> </a:math> and dc electric conductivity <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:msub> <d:mi>σ</d:mi> <d:mi>Q</d:mi> </d:msub> </d:math> of the strongly coupled quark-gluon plasma are investigated around the critical end point in a <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:msub> <f:mi>N</f:mi> <f:mi>f</f:mi> </f:msub> <f:mo>=</f:mo> <f:mn>2</f:mn> <f:mo>+</f:mo> <f:mn>1</f:mn> </f:math> holographic QCD model with parameters obtained from machine learning. It is found that both thermal photon emission rate and electric conductivity grow most obviously around <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:msub> <h:mi>T</h:mi> <h:mi>c</h:mi> </h:msub> </h:math> , which agrees with the previous studies, and the result of electric conductivity at zero chemical potential resembles the lattice results. Moreover, it is found that both the temperature derivative of the electric conductivity and thermal photon emission rate diverge at the critical end point.